Robust Qualitative and Quantitative Methods
for
Disturbance Rejection
and
Fault Accommodation
A Thesis presented
by
Stephen Paul Linder
to The Department of Mechanical, Industrial, and
Manufacturing Engineering
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the field of
Computer System Engineering
Northeastern University
Boston, Massachusetts
December 1997
Dedicated
to
Mom and Dad
The tide rises, the tide falls,
The twilight darkens, the curlew calls;
Along the sea-sands damp and brown
The traveler hastens toward the town,
And the tide rises, the tide falls.
Darkness settles on roofs and walls,
But the sea, the sea in the darkness calls;
The little waves, with their soft, white hands,
Efface the footprints in the sands,
And the tide rises, the tide falls.
The moving breaks; the steeds in their stalls
Stamp and neigh, as the hostler calls;
The day returns, but nevermore
Returns the traveler to the shore,
And the tide rises, the tide falls.
Henry W. Longfellow (1807-1882)
The Tide Rises, The Tide Falls
© 1997
Stephen Paul Linder
ii
Abstract
Disturbances caused by unmeasured inputs, plant perturbations or faulty actuators degrade
the robustness and performance of both control and diagnostic systems. When these disturbances are unknown, robust accommodation of disturbances requires, to date, elaborate
architectures. We, however, propose two new simple accommodation methodologies: one
quantitative and the other qualitative. The quantitative methodology uses the integral action
of the Proportional Integral (PI) Observer to estimate and accommodate disturbances.
This methodology has been further generalized to the PI Adaptive Observer and the
Proportional Fading-Integral (PFI) Observer. The PI Adaptive Observer expands the
applicability of integral action to systems with unknown parameters, while the PFI Observer
can also accommodate transitory disturbances of unknown origin. Our second methodology
is based on qualitatively robust models, where the models subsume all possible plant
perturbations and disturbances. Robust control objectives, even for simple plants, can be
achieved without the precise models of the standard quantitative robust control techniques,
but with qualitative stability and tracking behaviors. Validation of both methodologies has
been achieved through extensive simulation studies. Our simulations shows that integral
action is effective in estimating disturbances caused by unmeasured inputs, plant
perturbations or faulty actuators, and a PI Observer-based controller will even outperform a
Linear Quadratic Regulator in the presence of nonlinear actuator faults. The validation of
the qualitative approach was done with the 1992 ACC Robust Control Benchmark. The
QRC controller is the first published controller for the Benchmark that uses intelligent
control techniques; it achieves stability robustness and tracking robustness comparable to
the best published linear compensators for the Benchmark.
Keywords: disturbance rejection, qualitative modeling , hybrid system, fuzzy observers
PH.D. THESIS FOR STEPHEN PAUL LINDER
iii
It is easier to have some vague notion about any subject, no
matter what, than to arrive at the real truth about a single
question, however simple that may be.
René Descartes (1596 – 1650),
from Rules for the Direction of the Mind
Never tell people how to do things. Tell them what to do, and
they will surprise you with their ingenuity.
George S. Patton
Education is hanging around until you've caught on.
Robert Frost
iv
Table of Contents
Acknowledgements ...................................................................... xv
Chapter 1 Introduction ................................................................ 1
1.
Background ..........................................................................................................6
1.1 Quantitative Approaches.....................................................................................6
1.2 Qualitative Approaches..................................................................................... 11
2.
Motivation .......................................................................................................... 19
2.1 Disturbance Rejection....................................................................................... 19
2.2 Quantitative versus Qualitative Models ............................................................. 20
2.3 Improving Robust Fuzzy Control Paradigms ..................................................... 20
3.
Goal Statement ................................................................................................... 21
4.
Solution Statement ............................................................................................. 22
4.1 Step 1 - PI Observer and PI Kalman Filter ........................................................ 22
4.2 Step 2 - PI Adaptive Observer .......................................................................... 23
4.3 Step 3 - Robust PFI Adaptive Observer ............................................................ 23
4.4 Step 4 - Robust State Feedback-based fuzzy controller ..................................... 23
4.5 Step 5 - Robust PFI observer-based QRC controller ........................................ 23
5.
Validating Results .............................................................................................. 24
6.
Applications........................................................................................................ 25
7.
Thesis Outline..................................................................................................... 26
Chapter 2 Background ............................................................... 27
1.
Quantitative Observers ...................................................................................... 28
1.1 Deterministic .................................................................................................... 28
1.2 Stochastic......................................................................................................... 32
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2.
Application of Quantitative Observers..............................................................34
2.1 Robust Observer-Based State Feedback Control ...............................................34
2.2 Fault Estimation and Accommodation...............................................................37
Chapter 3 Validation Benchmarks ............................................41
1.
Robust Control Benchmarks..............................................................................42
1.1 ACC ’92 Robust Control Benchmark ................................................................43
2.
Benchmark Extensions .......................................................................................53
2.1 Failure Scenarios for the ACC Benchmark ........................................................53
2.2 Adaptive Observer Benchmark..........................................................................54
Chapter 4 Integral Action ..........................................................59
1.
PI Adaptive Observer.........................................................................................60
2.
Robust PI Kalman Filter....................................................................................63
3.
Kalman Filter with a Fading Integral Term......................................................66
4.
Tuning Integral Action.......................................................................................68
4.1 Tuning the Integral Gain ...................................................................................68
4.2 Tuning the Fading Rate.....................................................................................68
Chapter 5 Robust Fuzzy Control...............................................71
1.
Robust Fuzzy Control.........................................................................................72
1.1 Selecting a Proper Qualitative Plant Model .......................................................74
1.2 Designing a Fuzzy Controller ............................................................................74
2.
Robust Fuzzy Controller for Benchmark...........................................................76
2.1 Fuzzy Spring Process Model.............................................................................77
2.2 Estimating Plant State .......................................................................................79
2.3 Fuzzy Compensator ..........................................................................................81
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Chapter 6 Simulation Results for Integral Action ................... 85
1.
Simple Integral Action ....................................................................................... 86
1.1 PI Observer for Kudva Plant ............................................................................. 88
1.2 PI Adaptive Observer for Kudva Plant .............................................................. 93
1.3 PI Kalman Filter for ACC Benchmark............................................................. 108
2.
Fading Integral Action ..................................................................................... 113
2.1 PFI Kalman filter for the ACC Benchmark ...................................................... 113
Chapter 7 Simulation Results for Robust Fuzzy Control...... 119
1.
Qualitatively Robust Fuzzy Controller............................................................ 120
2.
Simulation Results............................................................................................ 122
2.1 Fuzzy Stability Behavior ................................................................................. 123
2.2 Effect of Tracking Behavior on Stability ......................................................... 124
2.3 Fuzzy Control with Full State Feedback.......................................................... 126
2.4 Fuzzy Control with Output Feedback.............................................................. 127
2.5 Stability Robustness Comparison .................................................................... 130
Chapter 8 Conclusion............................................................... 133
1.
Quantitative Methods - Integral Action .......................................................... 134
1.1 PI Adaptive Observer ..................................................................................... 134
1.2 PFI Observer .................................................................................................. 135
2.
Qualitative Methods - QRC............................................................................. 136
3.
Future Research ............................................................................................... 138
3.1 Integral Action................................................................................................ 138
3.2 Qualitative Approach...................................................................................... 139
Appendix I QRC Controller – No Noise .................................. 141
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1.
Fuzzy Controller for Benchmark .....................................................................142
1.1 Qualitative Modeling of Spring .......................................................................142
1.2 Implementing Stability and Performance Objectives.........................................145
2.
Qualitatively Robust Fuzzy Controller ............................................................150
2.1 Scenario 1 results - impulse on m1 and m2 .......................................................152
2.2 Scenario 4 results - Step Response..................................................................154
Appendix II FIS Definitions......................................................157
1.
Original FIS Files from ACC 1997 Paper........................................................158
1.1 FIS Definition for Process Model....................................................................158
1.2 FIS Definition for Controller ...........................................................................160
2.
FIS Files for Final Version of QRC Compensator...........................................164
2.1 FIS File for Process Model..............................................................................164
2.2 FIS Definition for Controller ...........................................................................166
Bibliography...............................................................................171
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Table of Figures
Figure 1. Plant and observer in an open loop configuration. .............................................. 28
Figure 2. Adaptive Observer schematic taken from (Narendra and Annaswamy 1989). ..... 31
Figure 3. State feedback regulation utilizing an observer................................................... 34
Figure 4. The effects of actuator faults (solid line) in distorting desired actuator output
(gray dashed lines) can be seen for the four actuator faults: (a) gain mismatch, (b)
deadzone, (c) backlash and (d) saturation. ................................................................ 39
Figure 5. The dual mass, single spring plant used in the ACC benchmark.......................... 42
Figure 6. Series compensator, C, and plant, P, in a negative feedback loop. ...................... 44
Figure 7. Plant output y after unit impulse disturbance (with duration of 0.5 seconds and
amplitude of 2.0) to m2 for compensated systems from (Ly 1990; Marrison and Stengel
1995). ...................................................................................................................... 49
Figure 8. Compensator output u after unit impulse disturbance to m2 for compensated
systems from (Ly 1990; Marrison and Stengel 1995). ............................................... 49
Figure 9. The response of a controller from (Collins 1990) to mismatch in frequency. The
plant output y is shown in (a) for an input disturbance of 0.500 (solid line) and 0.495
rad/sec (dotted line). (b) shows graphically the small mismatch between the two input
disturbances. ............................................................................................................ 52
Figure 10. Simulation Scenario 1: plant and observer in an open loop configuration with
input measurement disturbance, din. .......................................................................... 54
Figure 11. Simulation Scenario 2: state feedback regulation utilizing an adaptive observer
with input measurement disturbance, din. .................................................................. 54
Figure 12. Fuzzy Controller consisting of a linear state estimator, fuzzy process model and
fuzzy compensator which superimposes stability and tracking behaviors. .................. 72
Figure 13. Input and output membership function of the Spring Observer. ........................ 77
Figure 14. Output surface of the Spring Fuzzy Process Model ......................................... 77
Figure 15. Simulink simulation for comparing the effect of integral action on the disturbance
rejection properties of open-loop (a) and closed-loop observers (b). ........................ 87
Figure 16. Details of the Simulink simulation of Observer-based regulator block. ............ 89
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Figure 17. Details of the Simulink model of the PI Observer block....................................89
Figure 18. Response of open-loop P and PI observer to a step input disturbance, din = 0.2,
use in Adaptive Observer Benchmark plant. Integral offset, v, from the PI observer is
shown in (a). (b) shows the cumulative error estimating the hidden state x2 for both
the P and PI observer................................................................................................90
Figure 19. Response of closed-loop P and PI observer to a step input disturbance, din = 0.2,
using the Adaptive Observer Benchmark plant. Integral offset, v, from the PI observer
is shown in (a). (b) shows the cumulative error estimating the hidden state x2 for both
the P and PI observer................................................................................................91
Figure 20. Estimates of the hidden state x2 generated by a closed-loop PI (a) and the P (b)
observer. ..................................................................................................................92
Figure 21. Details of a Simulink simulation of a PI Adaptive Observer. ............................93
Figure 22. Response of open-loop P and PI adaptive observer to a step input disturbance,
din = 0.2, use in Adaptive Observer Benchmark plant. Integral offset, v, from the PI
adaptive observer is shown in (a). (b) shows the cumulative error estimating the
hidden state x2 for both the P and PI adaptive observer. ............................................95
Figure 23. Parameter estimates generated by the PI (solid line) and P adaptive observer
(dotted line) for Scenario 1 of the Adaptive Observer Benchmark. ..........................96
Figure 24. Cumulative error for estimating a (a) and b (b) system parameters....................97
Figure 25. Response of closed-loop P and PI adaptive observer to a step input disturbance,
din = 0.2, use in Adaptive Observer Benchmark plant. Integral offset, v, from the PI
adaptive observer is shown in (a). (b) shows the cumulative error estimating the
hidden state x2 for both the P and PI adaptive observer. ............................................98
Figure 26. Estimates of the hidden state x2 generated by the closed-loop PI (a) and the P (b)
adaptive observer used in observer based state feedback controller for the Adaptive
Observer Benchmark, Scenario 2. .............................................................................99
Figure 27. Parameter estimates generated by the PI (solid line) and P adaptive observer
(dotted line) based state feedback controller) for Scenario 2..................................100
Figure 28. Response of closed-loop P and PI adaptive observer to a triangle input
disturbance, din = 0.2, use in Adaptive Observer Benchmark plant. Integral offset, v,
x
from the PI adaptive observer is shown in (a). (b) shows the cumulative error
estimating the hidden state x2 for both the P and PI adaptive observer..................... 101
Figure 29. Estimates of the hidden state x2 (solid line) and actual state (dashed line) from
the closed-loop PI (a) and the P (b) regulator while disturbed by a triangle wave. ... 102
Figure 30. Estimates of a and b parameters from the closed-loop PI (solid line) and the P
(dashed line) Adaptive Observer while disturbed by a triangle wave. ....................... 103
Figure 31. The performance of the LQR (dashed line), the P observer-based regulator (gray
dashed line) and the PI observer-based regulator in the presence of four common
actuator faults is compared in the first two columns; Column (a) gives the plant output,
and Column (b) the cumulative output error. Column (c) show the disturbance created
by the actuator fault (gray line) and its estimate by the integral action (black line). .. 106
Figure 32. The performance of the LQR (dashed line), the P adaptive observer-based
regulator (gray dashed line) and the PI adaptive observer-based regulator in the
presence of four common actuator faults is compared in the first two columns; Column
(a) gives the plant output, and Column (b) the cumulative output error. Column (c)
show the disturbance created by the actuator fault (gray line) and its estimate by the
integral action (black line)....................................................................................... 107
Figure 33. The response to a step disturbance (a) for Full State Feedback (LQR), P Kalman
filter based controller and the PI Kalman filter controller. (b) gives the integral action
for a integral gain KI = -10. .................................................................................... 109
Figure 34. Plant output, y, and integral action, v, for three frequencies of sine wave
disturbance: (a) and(b) 0.5 rad/sec; (c) and (d) 0.4 rad/sec; and (e) and (f) 0.2 rad/sec.110
Figure 35. The response of the plant output y to a nonlinear disturbance is given in (a). (b)
shows the disturbance (gray line) and the resulting integral action v (solid line). (c)
compares the cumulative control error for the three regulators................................ 112
Figure 36. The effects of process noise and integral fading on the rejection of a unit impulse
to m2 for a perturbed plant with spring constant k =0.8. (a) shows the output for a PI
Kalman Filter-based controller over a range of p while (b) shows the output of the
correspond PFI Kalman Filter-based controller (KI = -20) for a range of KF. (c) and (d)
give the respective error in estimating the state x1. .................................................. 114
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Figure 37. A comparison of the P Kalman Filter and PFI Kalman Filter-based regulators and
a LQR with full state feedback for a perturbed plant with spring constant k =0.8. (a)
shows the output y, (b) the regulator output u, (c) the respective estimating error, ε,
for state x1 and (d) the estimate BIv of the perturbation term ∆x from the PFI Kalman
Filter.......................................................................................................................116
Figure 38. The peak regulator output versus plant spring constant for a P Kalman Filter based controller with several levels of process noise and for a PFI Kalman Filter-based
regulator with p = 0.1 for two levels of integral action. ...........................................117
Figure 39. Simulink model of plant used for Robust Control Benchmark.........................120
Figure 40. Simulink Simulation of the Robust Control Benchmark system with series
compensation..........................................................................................................121
Figure 41. Performance of stability behavior using FSFB (black) and state estimates derived
from a PFI Kalman filter (gray) is compared. The spring length L (a) and Actuator
output u (b) are shown after a unit impulse disturbance to m2 for k = 0.5 to 3.0 in steps
of 0.5......................................................................................................................123
Figure 42. Comparison of the two tracking behaviors using only output feedback, where the
black lines are from the more robust Tracking Behavior A and the gray lines are from
the better performing Tracking Behavior B . The plant output y (a) and spring length L
(b) are shown after a unit impulse disturbance to m2 for k = 0.6 to 2.0 in steps of 0.4.124
Figure 43. Performance comparison of two full state feedback controllers: the fuzzy
controller (black and the LQR (gray) and Fuzzy Controller (black) after unit impulse
disturbance to m2 for k = 1, 2, 3, 4. Figure (a) shows plant output y, (b) the spring
length L, (c) the compensator output u, and (d) the cumulative compensator output Σu
(for k = 0.5 to 4.5 in steps of 0.5). ..........................................................................126
Figure 44. Performance comparison of fuzzy controller using state estimates from a PFI
Kalman filter (black lines) and Comp1, a 5th order H2 compensator from Marrison and
Stengel (gray lines). Figure (a) shows the output response to a unit impulse
disturbance to m2 for k = 0.6 to k = 2.0 in steps of 0.2, (b) for m1 = 0.6 to m1 = 2.0 in
steps of 0.2, and (c) for m2 = 0.8 to m2 = 2.0 in steps of 0.2. Figure (d) shows the
tracking of a unit step command for k = 0.6 to k = 2.0 in steps of 0.2. .....................128
xii
Figure 45. Responses to an impulse at m2 for the nominal plant (k = 1.0) for Fuzzy
controllers QRC A and QRC B and the linear controllers Comp1 and Comp3. Figure
(a) shows that the fuzzy compensators have a lower overshoot in the output while
Figure (b) shows that the fuzzy controllers also dampening vibrations faster. .......... 130
Figure 46. A comparison of stability robustness and vibration suppression robustness for
various Benchmark compensators is made by showing the range of spring constants for
which the compensated system in response to a unit impulse to m2 is stable and has
L < 0.05 after 15 seconds. Numeric labels give extent of k. ................................ 131
Figure 47. Input and output membership function of the Spring Observer. ...................... 143
Figure 48. Structure of Fuzzy Controller ........................................................................ 145
Figure 49. Simulink model of plant used for Robust Control Benchmark. ....................... 150
Figure 50. Simulink Simulation of the Robust Control Benchmark system with series
compensation. ........................................................................................................ 151
Figure 51. (a) Position and velocity of m1 and m2 after a unit impulse disturbance on m2 with
k= 4. for Fuzzy Design A. (b) The corresponding output for Design 3. ................... 152
Figure 52. (a) Response of plant output y to a unit impulse disturbance on m2 with k = 1.
(b) The corresponding clipped view of compensator output u with the minimum values
for Design A and Design B being -9.9 and -5.1, respectively. .................................. 153
Figure 53. Tracking response to unit step, with plant output y, Figure (a), and compensator
output u, Figure (b). ............................................................................................... 154
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Table of Tables
Table 1. The number of positive responses to 9 survey questions for 311 polled companies
that implemented fuzzy controllers............................................................................17
Table 2. Rules for determining spring state. ......................................................................78
Table 3. Coefficients for output equations used by vibration suppression behavior. ...........82
Table 4. Fuzzy Rules for Controlling Plant........................................................................83
Table 5. Coefficients for output equations used by Tracking Behavior A and B.................84
Table 6. Rules for determining spring state. ....................................................................144
Table 7. Fuzzy Rules for Controlling Plant......................................................................148
Table 8. Output Polynomial Coefficients.........................................................................149
xiv
Acknowledgements
In everyone's life, at some time, our inner fire goes out. It is
then burst into flame by an encounter with another human
being. We should all be thankful for those people who
rekindle the inner spirit.
Albert Schweitzer
I would like to thank my advisor, Prof. Bahram Shafai, for his guidance and endless
patience with me in the last two years. Without his help I would have never even begun this
dissertation. The countless hours spent reading my papers by Dr. Zbigniew Korona were
also indispensable in the completion of this thesis; he also showed the way by finishing two
years before me. Prof. Thomas Cullinane, Prof. Gerald Voland, and Prof. Sagar Kamarthi
complete my thesis committee and provided much encouragement and guidance.
I would also like to acknowledge the help of several others: David Custer and Prof. James
Paradis of MIT for their guidance with regards to the proper and effective use of the
English language; Catherine Lavelle for editing my thesis; and my parents for their
encouragement in this quest. Several members of the staff at Northeastern were
indispensable in my work. Laura McGann, the departmental secretary, was always there to
answer questions and a friendly word. The staff at the College of Engineering computer
center, Kyle McDonald, James Jones and Dan Calandriello, time, and time again, proved to
be indispensable in keeping my computer up running, and provided valuable insight into the
workings of the network and computer systems.
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CHAPTER
INTRODUCTION
Chapter 1
Introduction
Everything should be made as simple as possible, but not
simpler.
Albert Einstein (1879-1955)
Disturbances caused by unmeasured inputs, plant perturbations or faulty actuators degrade
the robustness and performance of both control and diagnostic systems. These disturbances
can be known or unknown. When the disturbances are known, extensive techniques exist
for accommodating them, while only weaker techniques exists for estimating and
accommodating faults of unknown origin and unknown dynamics. This thesis will
investigate two original techniques for the accommodation of unknown disturbances. The
first technique uses a variation of the linear state observer, the Proportional Integral (PI)
observer, to estimate disturbances. The estimate is then used to reject the effect of the
disturbances. The second technique does not rely on the quantitative linear plant model
needed by the linear observer, but rather relies on qualitative models of the plant and
control behaviors. This technique, Qualitative Robust Control (QRC), uses qualitative
models, based on linguistic terms, which capture the structure of the plant and subsume
perturbations and faults. The controller is then developed using a hierarchy of qualitative
control behaviors, with separate behaviors achieving stability and tracking objectives.
Validation of both methodologies has been achieved through extensive simulation studies.
The PI observer has been shown to be the only known methodology to both estimate and
accommodate unknown disturbances caused by either unmeasured inputs, plant
perturbations or faulty actuators. The QRC methodology has been validated with the
popular 1992 American Control Conference (ACC) Robust Control Benchmark. The QRC
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BACKGROUND
controller is the first published controller for the Benchmark based on intelligent control
techniques. It performs as well as, if not better than, any previous solution.
The state of a system must often be inferred from observations of the system’s input and
output because the state is otherwise inaccessible. These inferences can be made by state
observers, which use measurements of the system’s input and output, in conjunction with a
dynamic model of the system, to estimate the system’s hidden states and predict future
system output. Complications arise when system inputs are corrupted by disturbances,
when inputs to the system are not accessible for measurement, and when precise models of
the system dynamics are unavailable. Ideally the observer is robust, rejecting disturbances
caused by unknown inputs, faulty actuators and plant perturbations, where plant
perturbation are caused by mismatch between the plant and the model of the plant used by
the observer.
Observers designed by engineers to implement controllers and fault detectors have
historically been quantitative, utilizing precise numerical equations to model the system
and characterize inputs and disturbances (Antsaklis 1994). With these quantitative methods
an observer can be designed to reject any single disturbance that can be described
quantitatively (Bar-Shalom and Fortmann 1988; Widrow and Walach 1996). However, if
the disturbance disappears or changes, the observer can not dynamically compensate for the
change.
The PI observer was developed by Shafai (Beale and Shafai 1989; Kim, Shafai et al. 1989)
to extend the robustness of observers by including an integral action in the observer
equation. This thesis extends this work by characterizing integral action for the purpose of
disturbance rejection. First, integral action was extended to the adaptive case, where the
parameters of the plant are unknown, with the PI Adaptive Observer (PIAO). Next, the
integral action of a PI observer was shown to effectively estimate and compensate for an
arbitrary set of disturbances with n distinct injection points, if n independent state
measurements are available. When used with a single output system, integral action allows
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CHAPTER 1
INTRODUCTION
the PI observer to reject any single unknown input, any single faulty actuator, or any single
rank one perturbation, as long as the distribution matrix of the disturbance is known and the
effect of the disturbance is slower than the time constant of the integral action. Increasing
the integral gain allows for the rejection of faster perturbations, but with the negative side
effect of decreasing the filter's stability margin.
The robustness of the PI observer can, however, be adversely effected by transitory
disturbances with unmodelled distribution matrices. This can in fact severely limit the
applicability of the PI observer; simple integral action cannot alone provide a robust
solution to the 1992 ACC Benchmark. This thesis presents a generalization of the PI
observer, the Proportional Fading-Integral (PFI) observer, which discounts the integral
term over time, enabling the rejection of these transitory events with unmodelled
distribution matrices. Additionally, the fading term can improve the stability margin of the
observer, allowing an unstable PI observer to become a stable PFI observer, yet with still
sufficient integral action to reject disturbances.
The second approach to disturbance rejection, Qualitative Robust Control, is based on
qualitative modeling. Qualitative models of the plant, the system to be controlled, and the
control strategies are used to subsume any perturbations in the plant or actuators, and
compensate for unmeasured inputs.
In contrast to the quantitative techniques used with the PI Observer, people in daily life
often utilize symbolic reasoning and a qualitative abstraction of a system to ascertain a
system’s hidden states (e.g. shaking a box to ascertain its contents). These qualitative
abstractions are frequently created using fuzzy logic and are represented as symbolic
linguistic models. These models provide a convenient mechanism for tuning quantitative
observers (Daiß and Kiencke 1995; Roberts, Mills et al. 1995) and improving their
disturbance rejection properties. In addition to providing a mechanism for tuning
quantitative observers fuzzy logic can be used to build fully fuzzy observers (Moore, Harris
et al. 1993; Moore and Harris 1994; Chao and Teng 1996). However, these existing fuzzy
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BACKGROUND
observers rely on machine learning to model a system, rather than using symbolic linguistic
models of the system. An observer based on quantitative abstractions of the system and its
disturbances will function over the entire range of system and disturbance configurations
described by the qualitative model instead of the narrow range of systems described by the
more precise quantitative models used by classical quantitative observers (Moore and Harris
1994). The symbolic linguistic model effectively improves the robustness of the derived
observer.
The use of symbolic linguistic models allows symbolic reasoning to complement quantitative
reasoning. Symbolic reasoning is preferred for many tasks because, in general, people relate
better to reasoning based on alphanumeric symbols (Albus 1991). Since planning and goals
can easily expressed in a expressed in high-level symbolic language, symbolic reasoning
lends itself to the design of controllers with conflicting robustness requirements. With
symbolic reasoning the various robustness requirements can be implemented in a
hierarchical structure and tradeoffs between requirements can be made at a symbolic level.
The QRC methodology for designing fuzzy controllers transcends the quantitative methods
that are utilized by the majority of fuzzy controllers described in the literature (Jang and Sun
1995; Driankov, Hellendoorn et al. 1996). Instead of using these fuzzified versions of
quantitative controllers, our methodology relies on qualitative models of the plant behavior.
As in conventional control where the designer must make tradeoffs between different
performance metrics with the fuzzy controller, the designer must weigh the effects of
different localized qualitative behaviors on the overall performance of the controller.
Our work shows that fuzzy setpoint control can be expanded to provide robust setpoint
control if a qualitative plant model that subsumes all specified plant perturbations is utilized
in designing the controller. A rule-based fuzzy controller is then incrementally designed
based on the “qualitatively robust” plant model. First, stability behaviors are developed and
characterized. Then tracking behaviors are developed to augment the controller. Because
these behaviors are based on a qualitatively robust model of the plant, the stability and
4
CHAPTER 1
INTRODUCTION
tracking behaviors are robust over the extent of plant configurations that are subsumed by
the qualitative plant model. The resulting fuzzy controller supports both stability and
performance robustness and allows for simple compensator tuning by changing linguistically
interpretable rule parameters.
The following sections provide a brief overview of the background material for this
research. The reader is then shown the motivation for this research and a concise goal
statement is given. The introduction continues with a solution statement that briefly
describes the new techniques for improving disturbance rejection and a validation procedure
for these new techniques.
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BACKGROUND
1. Background
This section gives a brief overview and provides a context for our results. Detailed
background material, complete with equations, is given in Chapter 2 of this thesis. We begin
with quantitative techniques and conclude with qualitative approaches to rejecting
disturbances.
1.1 Quantitative Approaches
Quantitative approaches to rejecting disturbances for state observes have been under
development for the last half century. This section begins with a brief overview of the three
major types of observers: Luenberger observer, adaptive observer and Kalman filter. Then
the existing disturbance rejection techniques, which involve the modeling of the disturbance
or the use of multiple models are described. The section concludes with a description of two
applications of state observers: state-based feedback control and fault detection.
1.1.1 State Observers
A device that estimates or observes state variables of a system is called a state observer. A
state observer utilizes measurements of the system inputs and outputs and a model of the
system based on differential or difference equations. Three main quantitative state observers
are: Luenberger observer, adaptive observer and Kalman filter.
In the deterministic case, when no random noise is present, the Luenberger observer and its
extensions (Luenberger 1971; Bhattacharyya 1976; Fairman and Gupta 1980; O'Reilley
1983) are used for time-invariant systems with known parameters. The equation for the
Luenberger observer contains a term that corrects the current state estimates by an amount
proportional to the prediction error: the estimation of the current output minus the actual
measurement. Inclusion of this correction ensures stability and convergence of the observer
even when the system being observed is unstable.
When the parameters of the system are unknown or time varying, an adaptive observer
(Carroll and Lindorff 1973; Kudva and Narendra 1973) must be used. The adaptive
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INTRODUCTION
observer, in addition to estimating the system states, must now also estimate the system
parameters. Achieving this added requirement, while maintaining stability, has resulted in
the development of significantly complex observer structure. Because prediction error can
no longer be unambiguously associated with errors in estimating state, a persistently
exciting signal must be generate to insure the stability of the adaptive observer. Even with
this persistently exciting plant signal, the adaptive observer has significant difficulty
distinguishing between the effects of inaccurate parameter estimates and measurement
disturbances.
The corresponding observer for a stochastic system containing additive noise processes,
with known parameters, is a stochastic observer with a structure attributed to Kalman
(Kalman 1960). This Kalman filter is a recursive solution to Gauss’s original least squares
estimation problem (Sorenson 1970) and builds on the work of Norbert Wiener (Wiener
1949) in estimating the underlying signal from a noisy time series. The Kalman filter is the
optimum estimator when the corrupting noise has a Gaussian probability distribution. Like
the Luenberger observer, the Kalman filter also includes a correction factor to insure
stability and convergence, but for the Kalman filter it is based on the variances of the noise
processes. If accurate estimates of the variance are not available, optimal observer
performance is not obtained (Lin 1994).
All three of these observers have degraded performance in the presence of input
disturbances. The following section discuss the two major techniques for rejecting known
input disturbances.
1.1.2 Disturbance Rejection
Disturbances can be either stochastic or deterministic. While stationary stochastic input
processes with a zero-mean Gaussian distribution can be effectively rejected by a Kalman
filter when accurate noise statistics are available, fixed non-stochastic disturbances can only
be rejected when the observer is augmented with a dynamic model of the disturbance (Lin
1994). Time varying disturbances of either type that can not be modeled as a linear system
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BACKGROUND
are difficult if not impossible to reject. Any ability to compensate for either stochastic or
deterministic disturbances is called disturbance rejection. However, this dissertation will
use the term disturbance rejection only for the rejection of deterministic disturbances, while
the term noise rejection will used for the rejection of stochastic disturbances.
Disturbances severely degrade observer performance. In the tracking domain, where
disturbances are often referred to as maneuvers, a typical radar tracker tracks an aircraft
using a parsimonious constant-velocity model and switches to a constant-acceleration model
only after detecting an acceleration disturbance (Bar-Shalom and Fortmann 1988). The
tracker cannot measure acceleration directly, but must instead infer that an acceleration
disturbance has occurred when the predicted position, calculated by the observer, begins to
diverge from the measured position. This difference, the measurement residual, increases
rapidly as the maneuver progresses and indicates poor state estimation performance. Only if
the disturbance is detected and rejected promptly can the divergence of state estimation be
prevented.
Target trackers often use multiple observers to ensure prompt detection and rejection of
disturbances (Blackman 1986; Bolger 1987; Bar-Shalom and Fortmann 1988) and faults
(Frank and Wünnenberg 1989). Individual observers are tuned to include the dynamics of a
known disturbance. When the set of disturbances is finite, complete coverage of the
disturbance set can be achieved by the tracker with a finite set of observers. Rule-based or
statistical reasoning is then used to analyze the measurement residuals generated by the
observer set with the results of the analysis used to accommodate the maneuver or fault
(Lin 1994). This technique fails if any unmodelled disturbance is present.
Rejection of a single know disturbance can be achieved in a straight forward manner with a
single observer when the system has known parameters. However, no known disturbance
rejection methodologies exist for adaptive observers. Even with a persistently exciting plant
input the adaptive observer has difficult distinguishing between the effect of inaccurate
parameter estimates and disturbances. Additional difficulties exist when the adaptive
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INTRODUCTION
observer is utilized in a closed loop regulator because the controller suppresses the
persistently exciting input signal needed for the convergence of parameter estimates.
1.1.3 Application of Observers
The two main applications of state observers are observer-based state feedback control and
fault detection. Both applications rely on accurate state estimation and suffer from
performance degradation when input disturbances corrupt the observer’s state estimates.
1.1.3.1 Observer-based control
State feedback controllers designed as Linear Quadratic Regulators (LQR) have
impressive robustness properties (Anderson and Moore 1989), including the rejection of
disturbances from unknown inputs, actuator faults and plant perturbations. A LQR however
requires access to system state variables. When state variables are unavailable a state
observer must be included in the feedback loop, but this drastically reduces the robustness
of state feedback control. The Linear Quadratic Gaussian/ Loop Transfer Recovery
(LQG/LTR) techniques were introduced to recover some of the robustness properties of
LQR, but these techniques often do not achieve full loop transfer recovery, and do not have
disturbance rejection properties of the original LQR.
The PI observer, however, can be used to enhance the LTR procedure; the integral action
of the PI observer allows additional freedom in adjusting the controller. A PI observerbased controller can be designed with frequency and time recovery properties approaching
that of full state feedback control (Niemann, Stoustrup et al. 1995), and therefore recover
the robustness property of a full state feedback controller.
An equivalent loop transfer recovery procedure, however, does not exist for the adaptive
observer-based controller. Therefore, LTR techniques can not be used to recover the
disturbance rejection properties of the original LQR. Instead, the adaptive control
community has developed Model Reference Adaptive Control (MRAC) for linear systems
with unknown parameters (Narendra and Lin 1980; Kaufmann, Bar-Kana et al. 1994).
MRAC does not use state information in the controller implementation, but rather relies
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BACKGROUND
only on measurement of the input and output of the system and a separate reference model
to adapt the control gain. While MRAC can adapt to known disturbances (Kaufmann, BarKana et al. 1994) it does not provide estimate of the disturbance or states.
1.1.3.2 Fault Detection.
Fault detection is the second major application of state observers. Faults are often detected
by monitoring the measurement residuals of state observers. Excessive measurement
residuals are interpreted as being indicative of a fault. A tradeoff, however, must be made
between detecting all faults and creating an excessive number of false alarms since
measurement residuals can also be generated by unmodelled plant dynamics, parameter
mismatch or plant input disturbances. Disturbance decoupling is required in order to
distinguish between true faults and the effects of disturbances (Frank and Wünnenberg
1989).
Accurate estimation and robust accommodation of actuator faults can greatly increase the
reliability and flexibility of control systems. Estimation allows for the characterization and
classification of the fault, while actuator fault accommodation increases the robustness of
the control system and allows time for diagnostic evaluation of the fault mechanism. With
sufficiently accurate fault estimates and sufficiently robust accommodation, the dynamics of
a fault can be closely monitored and used for the preemptive scheduling of repairs, without
interrupting normal plant operation. Robust accommodation also allows for the utilization
of less expensive actuators (Tao and Kokotovic 1996). High accuracy and performance can
thus be achieved with components that previously were not precise enough and did not have
sufficiently stable performance characteristics.
Currently however, no simple scheme exists for the design of a controller that both
estimates and accommodates for unknown actuator faults. If the type of fault is known, and
has a priori been characterized by a piecewise linear model, adaptive techniques exist for
estimating the parameters of the fault model (Tao and Kokotovic 1996). Other techniques
can accommodate, but not estimate, a class of faults with a known H∞ bound (Seo and
Byung 1996) and much work has been done in accommodating actuator faults in systems
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INTRODUCTION
with redundant actuators (Cho and Bien 1989; Medanic 1994). More complex
methodologies have also been developed that use computer-automated reconfiguration of
control laws to accommodate for a set of known actuator faults (Kwong, Passino et al.
1995; Menke and Maybeck 1995). Results have also been obtained for a single known
actuator fault; as an example, accommodation of saturation failures caused by faulty aircraft
control surfaces are covered in (Keating, Pachter et al. 1997).
1.2 Qualitative Approaches
Qualitative approaches to observer design, fault detection and control requires the creation
of qualitative models of the underlying quantitative system. These qualitative models must
be designed so they support reasoning that is consistent with the quantitative system. This
coupling of consistent qualitative models with a continuous, quantitative plant is called a
hybrid system (Nerode and Kohn 1993). In contrast, many intelligent systems do not use
qualitative models of the underlying quantitative system. As an example, intelligent systems
based on neural networks often learn to recognize patterns, and reason about the
recognized patterns, but never develop a qualitative model of the patterns.
This section explains how qualitative models are abstracted from quantitative systems and
then introduces the implementation of fuzzy systems from qualitative models. It concludes
with an overview of fuzzy tuning systems for linear observers, linear observer-based fault
detectors and Proportional, Integral and Differential (PID) controllers, and a discussion of
full fuzzy controllers.
1.2.1 Qualitative Abstractions
When a qualitative model is created of a continuous plant, perceptual chunks of
information are abstracted (Albus 1991). Multiple points in the continuous domain are
condensed into perceptual chunks that represent single symbolic concepts. However, an
arbitrary mapping from the continuous quantitative plant to a symbolic qualitative model
may create a model that looses many important details about the plant. Indeed, so many
details may be lost that the model can not support consistent reasoning about the plant. The
qualitative reasoning community therefore has focused on developing chunks that promote
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BACKGROUND
consistent reasoning. Consistency must be preserved in order to have confidence in any
performed symbolic reasoning (Genesereth and Nilsson 1987; Kokar 1992; Linder, Kokar et
al. 1996).
A typical approach to translating quantitative variables into qualitative symbols is to
partition the quantitative variables with some critical values of the quantitative variables.
When the value of the quantitative variable crosses a critical value a qualitative event is
generated (Antsaklis, Passino et al. 1989; Lemmon, Stiver et al. 1993). These events can
then be used as input to a finite state automaton; an automaton that qualitatively models the
state transitions of the continuous plant.
The continuous plant of a hybrid system has traditionally been partitioned with orthogonal,
hyperplanes (Guckenheimer and Johnson 1994) to form hyperboxes (Kuipers 1986;
Kuipers 1994). However, hyperboxes, with their orthogonal boundaries, do not take into
account the interactions between plant variables produced by the dynamics of the plant and
therefore do not produce consistent finite qualitative abstraction of the underlying
continuous system (Strauss 1990). Hyperbox based hybrid systems can be forced to behave
consistently by using smaller hyperboxes in areas of the system space that contain complex
behavior (Kuipers 1994), an approach similar to that utilized in finite-element analysis. This
approach has the drawback of creating many additional qualitative states.
Kokar, in his earlier papers on dimensional analysis (Kokar 1987), suggested a different
approach for developing consistent qualitative abstractions that limits the number of
partitions. Instead of utilizing a grid of hyperboxes, it would be better to utilize hypersurfaces suggested by the physical model of the plant. This work resulted in the Q2
methodology for constructing consistent symbolic models of continuous noise-free dynamic
plants (Kokar 1992; Kokar 1995). Q2 focuses on how to partition a system’s space into a
finite number of qualitative chunks that preserve consistency. Q2 's partitions allow the
construction of a symbolic representation of the underlying plant that is provably consistent
for noise-free general dynamic systems and provides superior qualitative models for noisy
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INTRODUCTION
plants (Linder, Kokar et al. 1996). However, the Q2 methodology is difficult to extend to
complex systems because it requires a complete quantitative model for generating
partitions.
1.2.2 Fuzzy Linguistic Models
Fuzzy linguistic models hold the promise of providing a finite qualitative partition of a
quantitative dynamic system while being applicable to any system that can be described in
linguistic terms. Fuzzy models provide a succinct and robust representation of systems that
lack a complete quantitative model or have uncertain system perturbations. Consistency in
reasoning, however, has not yet been proven for a fuzzy linguistic representation of a
quantitative system.
Fuzzy linguistic models use fuzzy sets (Zadeh 1965) to create a finite number of partitions,
membership functions, of the inputs, outputs and states of a quantitative system. Currently
most fuzzy models are implemented as a set of if-then rules, where the system input is used
to evaluate the rules’ antecedents and the model’s output is the combined output of all the
rules evaluated in parallel (Jang and Sun 1995). This simple logical system, a Fuzzy
Inference System (FIS) (Gulley and Jang 1995), does not implement inference chaining and
can only evaluate a simplified qualitative model of a plant. Recent work has expanded the
usefulness of this structure by providing machine learning methodologies to adapt and tune
fuzzy linguistic models and to automatically generate new models through selforganization.
1.2.3 Learning Fuzzy Models
Learning or tuning allows the initial linguistic fuzzy model developed from heuristic domain
knowledge to be optimized. Learning is achieved by using a neuro-fuzzy structure and
exploiting the supervised learning strategies originally developed for neural networks. These
strategies include gradient descent back-propagation (Rumelhart, Hinton et al. 1986), leastmean-squares, and a hybrid methodology that combines least-squares to optimize linear
parameters and back-propagation to optimize the nonlinear parameters (Jang and Sun
1995).
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BACKGROUND
These same supervised learning methodologies can automatically learn any arbitrary
nonlinear mapping between input and output without an initial linguistic fuzzy model
(Moore and Harris 1994). The resulting self-organized fuzzy models do not necessarily have
a linguistic interpretation that would be recognized by a human expert. Often systems
developed through self-organization are never interpreted linguistically, but are utilized
effectively for pattern matching and curve fitting. Fuzzy networks are often preferred for
curve fitting because the fuzzy rules used by the network have only a local effect, in effect
providing an adaptive mechanism for implementing B-splines (Wu and Harris 1996). As an
example application, neuro-fuzzy curve fitting is used by Roberts, Mills, Charnley and
Harris (Roberts, Mills et al. 1995) to extract estimates from a noisy time series. These
estimates then initialize a Kalman filter, resulting in improved overall performance of the
filter. Self-organized fuzzy models have also found application in fuzzy implementations of
state estimators.
Moore, Harris and Rogers (Moore, Harris et al. 1993) have used least-mean-squares
learning to train a set of fuzzy networks, where each individual fuzzy network is trained
with noisy data to track a target when perturbed by a single unique acceleration
disturbance/maneuver. These fuzzy networks are then used in a hybrid scheme to detect and
identify maneuvers and estimate target position.
A similar approach to solving the least-mean-squares estimation problem has been
developed by Chao and Teng (Chao and Teng 1996). A fuzzy network is trained off-line to
estimate the state of a non-linear process from a sequence of noisy measurements. An
estimation correction term similar to that utilized by a Kalman filter is included in the
observer to ensure stability and convergence. As in the work by Moore, Harris and Rogers
no linguistic qualitative model is employed in the design of the observer.
In contrast to these last two examples of fuzzy estimators where the fuzzy models do not
have a qualitative interpretation, the next section will give examples of how qualitative
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INTRODUCTION
linguistic models are used to automate the tuning of classical linear observers and
controllers. In addition, the application of linguistic models to fuzzy control will be
discussed.
1.2.4 Qualitative Tuning of Quantitative Observers
Qualitative models and machine learning techniques are used extensively in industry to tune
linear systems. Undergraduate control courses introduce tuning by teaching the ZieglerNichols tuning rules for PID controllers (Ziegler and Nichols 1942). These heuristics adapt
the three PID controller parameters based on the step response of the compensated system.
When first developed in 1942, these heuristics were manually employed by an engineer to
tune a PID controller. Recently, most of the tuning systems developed for industrial
controllers rely on qualitative reasoning or machine learning techniques to automate the
pattern recognition needed for tuning (Åström 1989; Hind Jr. and Cooper 1993; Copeland
and Rattan 1994).
Qualitative tuning is also used to adapt many other types of linear and non-linear dynamic
systems. Most relevant to this dissertation is the work on tuning fault detectors and Kalman
filters. The fault detection domain focuses on detecting frictional faults where even small
amounts of unwanted friction severely compromises the positional accuracy of robots and
actuators (Isermann, Keller et al. 1996).
A fault detection system has been developed by Schneider and Frank (Schneider and Frank
1996) for an industrial robot that increases fault detection accuracy by adapting the
detection threshold. A fuzzy rule base adapts the detection threshold with heuristic
qualitative rules that correlate nominal disturbance levels with the robot’s joint velocities
and acceleration. In general, the higher the expected disturbance level for a particular robot
configuration, the higher the fault detection threshold.
Rule-based adaptive compliance to frictional faults in positional actuators has been
investigated by Isermann, Keller and Raab (Isermann, Keller et al. 1996). A complete
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BACKGROUND
autonomous controller was implemented with an 8-bit microcontroller that with 15 heuristic
rules adaptively determined the optimal control parameters from the step response of the
actuator.
Improved estimation accuracy for automobile velocity estimation has been shown by Daiß
and Kiencke (Daiß and Kiencke 1995) using a fuzzy inference system based on a qualitative
model of sensor performance. The qualitative model guides the fusion of measurements
from several sensors, and results in overall improvement in velocity estimates utilized in
antilock braking and traction control systems. Another approach to improving velocity
estimates is shown by Kobayashi, Cheok and Watanabe (Kobayashi, Cheok et al. 1995). A
fuzzy inference system is used to tune the covariance matrices of the Kalman filter. When a
skid or wheel slip is detected, the noise variance for the velocity sensor is increased, biasing
the filter to rely less on the velocity sensor and more heavily on the acceleration sensor.
Fuzzy tuning of quantitative systems is a large field, but by far the largest application of
fuzzy systems is control. The next section describes fuzzy derivatives of quantitative
controllers and controllers based on qualitative models.
1.2.5 Fuzzy Control
A survey of 311 companies that recently implemented fuzzy controllers was made by von
Altrock (Altrock 1995). Table 1 tallies the number of positive response to 9 survey
questions. The survey shows that fuzzy control did not displace many conventional control
designs. Rather fuzzy control was used in new products where the faster implementation
time of fuzzy control and the ability to add new features to a product line convinced
engineers and managers to try fuzzy control. 97% of the companies had a positive
experience with fuzzy control and would use fuzzy control again.
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INTRODUCTION
Table 1. The number of positive responses to 9 survey questions for 311 polled
companies that implemented fuzzy controllers.
Agree
Did the fuzzy controller …
#
2
replace a PID controller?
7
replace a conventional multivariable controller?
35
enable the use of cheaper controller?
56
allow for cheaper sensors?
89
solve a problem that conventional techniques could not previously solve?
130
enable new features?
255
decrease production cost or increase product value?
271
implement faster?
303
perform successfully and would the company use fuzzy control again?
The vast majority of the controllers covered by this survey were of medium complexity.
These controllers were implemented with between 20 and 80 rules, with each rule having up
to 7 terms and usually one output. The survey did not further elaborate on the types of
implementations used. However the majority of fuzzy controllers described in the literature
fall into three main categories (Jang and Sun 1995; Driankov, Hellendoorn et al. 1996):
•
fuzzy PID controllers,
•
fuzzy sliding mode controllers, and
•
fuzzy gain scheduling.
All three compensators realize close-loop control action and are based on quantitative
control techniques. The fuzzy PID controllers and fuzzy slide mode controllers are fuzzy
implementations of the linear quantitative PID controller and a nonlinear quantitative sliding
mode controller. Both controllers use the error term and its derivatives and integrals as
input into a fuzzy rule base. Coleman and Godbode compare the robustness of a fuzzy PID
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BACKGROUND
controller with that of a conventional PID and sliding mode controller and conclude that the
fuzzy controller has equivalent robustness characteristics (Coleman and Godbole 1994).
The fuzzy gain scheduler uses Sugeno type fuzzy rules to interpolate between several
control strategies (Jang and Sun 1995). This methodology is useful for controlling nonlinear
plants that are piece wise linear or for linear plants that have a time varying parameter. An
example of this is a controller built for an inverted pendulum with a variable length
pendulum. Measurements of the pendulum length are used as inputs into a fuzzy rule base
that interpolates the output of a small number of controllers that are optimized for
controlling short, medium length and long pendulums (Gulley and Jang 1995).
Additionally, fuzzy controllers have been developed using qualitative models of target
behavior. These controllers are often developed using the following steps:
1. rules are developed to realize a localized qualitative behavior,
2. global behavior caused by the interpolated localized rules is tested, and
3. behavior is refined by tuning localized behavior and superimposing additional localized
and global behaviors.
Ruspini, Saffiotti and Konolige (Ruspini, Saffiotti et al. 1995) utilize these steps and show
that only 15 rules, representing 6 elemental behaviors, are needed to navigate a simple
maze. These rules reasoned about three sonar sensor outputs and direction, and implement a
"reactive navigation" behavior. Fuzzy logic achieves a consistent global behavior by
interpolating and superimposing the elemental behaviors.
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INTRODUCTION
2. Motivation
This dissertation is motivated by a desire to develop new disturbance rejection
methodologies that are simpler and more robust than existing methodologies. Existing
methodologies for rejecting unknown disturbances use complex paradigms. A quantitative
model of each possible disturbance is required and a complex reasoning method most
coordinate the accommodation of the disturbance. Even when using fuzzy control
techniques researchers rely on fuzzy version of quantitative controllers to achieve
robustness. Qualitative models and behaviors are rarely used.
2.1 Disturbance Rejection
Current techniques for disturbance rejection require either a precise quantitative model of a
disturbance or extensive supervised learning. This requirement limits the disturbance
rejection capabilities of a single observer to a single narrow class of quantifiable
disturbance. If a set of several disturbances need to be rejected a corresponding set of
observers is required, with each individual observer designed to reject only one of the know
possible disturbances. Optimal rejection of the entire set of disturbance then relies on an
autonomous supervisor which selects the appropriate observer outputs to use in
construction the state estimation.
Both the quantitative and learning approach suffer from some serious limitations, including:
•
disturbance must be known a priori,
•
multiple observers must perform in parallel to reject multiple classes of disturbances,
and
•
estimates of disturbances are not available.
New techniques for disturbances reject must be developed that allow the rejection of
unknown set of disturbances with only one observer.
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M OTIVATION
2.2 Quantitative versus Qualitative Models
Robust setpoint control is implemented almost exclusively using conventional techniques
relying on precise quantitative models of plant and disturbances, while qualitative and
intelligent techniques have focused on task oriented control of complex plants. The
conventional robust control techniques first define a set of controllers that can stabilize the
plant, including any disturbances that might be present, and then search for a controller that
minimizes a robustness cost function. After a search process finds the optimal design, direct
incremental tuning of the initial design is difficult because compensator parameters do not
necessarily have a linguistic interpretation to guide the refinement, nor does the designer
know the precision that must be preserved in the individual parameter in order to preserve
the specified performance. These are severe shorting comings for the existing qualitative
approaches to robust control.
2.3 Improving Robust Fuzzy Control Paradigms
Current approaches to fuzzy control are limited by their almost exclusive use of output
feedback. It is imperative that fuzzy control be expanded beyond the prevalent PID
paradigm. Iterative, incremental, design of qualitative models for use in the design of both
the observer and the controller will help in developing robust fuzzy controllers. Rule-based
fuzzy systems can be augmented directly by the addition of new rules and tuned by adjusting
parameters with clear linguistic interpretations. As shown by Ruspini, Saffiotti and Konolige
(Ruspini, Saffiotti et al. 1995) fuzzy logic can achieve a consistent robust global behavior by
interpolating and superimposing elemental behaviors. We need to extend this technique to
include the development of fuzzy controllers which are robust to disturbance.
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INTRODUCTION
3. Goal Statement
The goal of this dissertation is to provide the techniques and methodologies to
•
allow a single state observer or a single observer-based controller to estimate and
accommodate unknown disturbances
•
achieve robustness to disturbances by using qualitative models and behaviors
The set of unknown disturbance to be estimated and accommodated include
•
steps,
•
pulses,
•
sinusoids,
•
perturbation to the plant, and
•
linear and nonlinear actuator faults.
In addition to rejecting these disturbances the observers and controllers should perform
robustly with respect to
•
stability, and
•
measurements corrupted with a zero-mean Gaussian noise process.
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SOLUTION STATEMENT
4. Solution Statement
Two solutions are presented. The first uses the PI Observer and its new variants to estimate
and accommodate unknown disturbances. The second uses fuzzy control based on
qualitative models to accommodate for disturbance. The solution steps involving the
integral action are:
1. characterize the quantitative PI techniques for disturbance rejection for PI observer and
the PI Kalman filter,
2. increase parameter robustness of the PI observer by introducing an adaptive PI
observer, and
3. increase the disturbance rejection robustness of the PI observer by developing the
Proportional Fading-Integral (PFI) observer.
The solution steps for the approach using qualitative models and behaviors are:
4. create design methodology for a robust state-feedback fuzzy controller that use
qualitative behaviors that incorporates robust stability and tracking behaviors, and
5. create a robust hybrid output-feedback controller that combines the fuzzy controller
with a robust PFI observer.
The following section will give a brief overview of the results achieved for each of the six
solution steps.
4.1 Step 1 - PI Observer and PI Kalman Filter
The PI observer and PI Kalman filter has been shown to estimate and reject disturbance for
systems with known parameters and when the injection point of all disturbances are known.
In the presence of faulty actuators, a observer-based regulator with integral action can out
perform even a LQR using full state feedback.
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INTRODUCTION
4.2 Step 2 - PI Adaptive Observer
The PI Adaptive Observer allows for the estimation and rejection of disturbances when
plant parameters are unknown. Convergence of disturbance estimates are compromised only
slightly by the concurrent estimation of plant parameters. This dissertation shows that only
with integral action can the parameters estimates of an adaptive observe converge in the
presence of disturbances.
4.3 Step 3 - Robust PFI Adaptive Observer
The PFI observer, which discounts the integral term over time, enables the rejection of
transitory events with unmodelled distribution matrices. Additionally, the fading term can
improve the stability margin of an observer, allowing an unstable PI observer to become a
stable PFI observer, yet with still sufficient integral action to reject disturbances. In fact,
only with the addition of fading, does integral action provide a robust solution to the 1992
ACC Benchmark.
4.4 Step 4 - Robust State Feedback-based fuzzy controller
Robust rejection of disturbances has been achieved using qualitative behaviors with the
Qualitative Robust Control (QRC) technique. Initially behaviors are developed and
evaluated that stabilize the plant and then the fuzzy system is incrementally augmented with
tracking behaviors that achieve the final performance objectives. Behaviors can be tuned
with the addition of new rules and by the adjustment of parameters that have clear linguistic
interpretations. This methodology produce an extremely robust controller for the ACC
Benchmark, but with the use of full sate feedback.
4.5 Step 5 - Robust PFI observer-based QRC controller
Instead of directly designing a QRC controller using only output feedback, the QRC is first
developed using full state feedback, and then the design is migrated to the output feedback
configuration. A robust PFI observer is used to estimate the state of the plant. This hybrid
solution performed as well if not better than existing ACC Benchmark solutions.
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VALIDATING RESULTS
5. Validating Results
The theory and methodology developed for this dissertation is benchmarked using the very
popular 1992 American Control Conference (ACC) Robust Control Benchmark (Wie and
Bernstein 1990; Wie and Bernstein 1992). The ACC Benchmark has been utilized in over
40 papers and has been the theme of several journal issues. Stengel and Marrison provide an
excellent comparison of solutions from numerous authors in their 1992 review (Stengel and
Marrison 1992). Additional validation was accomplished using our proposed benchmark
based on the plant from Kudva and Narendra’s seminal 1973 paper on adaptive observers
(Kudva and Narendra 1973). Comparisons are made between the PI Observer, the P
Observer and the Linear Quadratic Regulator (LQR). The LQR was selected because it
shows the optimal robustness to plant perturbations of any linear (Anderson and Moore
1989).
The theoretical results obtained for PI observer and its variants were validated with both the
Kudva plant and the ACC Benchmark. Both systems are used to demonstrate the efficacy of
integral action in estimating the disturbance of unknown inputs, plant perturbations and
fault actuators. In all cases the integral action allowed the PI observer-based controller to
out perform a P-based controller, and in some cases even outperform a LQR using full state
feedback.
The universal appeal of the ACC Benchmark is it’s simple mass-spring-plant. It provides an
ideal vehicle for benchmarking the QRC methodology because all plant parameters have
qualitative, linguistic interpretation. Extensive comparisons were made with existing
solution for the Benchmark. These simulations validates the superior performance of our
QRC controller with regards to disturbances caused by unknown inputs and plant
perturbations.
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6. Applications
The new technologies described in this thesis can be useful over a wide range of
applications because disturbances are prevalent in real world systems. One application
shown by this thesis is the use of integral action to estimating and actively accommodating
for the disturbances caused by actuator nonlinearities (Tao and Kokotovic 1996). This
capability has a two fold benefit. First, less precise actuators with known non-linearities can
be incorporated into a control system. This results in a great cost reduction and allows for
greater flexibility in selecting appropriate components. Also, the less precise actuator is
often of sturdier design and can require significantly less maintenance. Second, integral
action can be used to monitor the gradual degradation of an actuator. Preventative
maintenance can then be scheduled before the accommodating capabilities of the integral
action is overwhelmed, and a catastrophic failure occurs.
Another application demonstrated by this thesis is the rejection of vibrations in flexible
structures, which include large mechanical structures, such as cranes (Singhose, Porter et al.
1997), and space structures (Pao and Singhose 1997; Singhose, Singh et al. 1997), such as
the new space station Endeavor. The QRC fuzzy controller developed in this thesis for the
flexible structure of the 1992 ACC Robust Control Benchmark is especially well suited for
controlling space structures. Space structures use thrusters for maneuvering and attitude
control. These thrusters are limited in their effectiveness because they are essentially binary
actuators; the thrusters are either on or off. The QRC controller also uses discrete pulses to
dampen vibrations of the structure.
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THESIS OUTLINE
7. Thesis Outline
This dissertation continues with a more detailed description of the material covered in this
introduction. First, the pertinent mathematical background for my research is described in
Chapter 2. Next, the validations benchmarks are described in Chapter 4. The theoretical
results are given in Chapter 5 and 6, while Chapter 7 and 8 give detailed simulation results
that validate the theoretical results. Chapter 9 provides concluding remarks.
26
CHAPTER 2
BACKGROUND
Chapter 2
Background
Do not worry about your difficulties in Mathematics. I can
assure you mine are still greater.
Albert Einstein (1879-1955)
This chapter expands on the background section include in the introduction and presents the
background material most relevant to the proposed research The chapter begins with a brief
summary of the three major quantitative observer implementation: deterministic state
observer, deterministic adaptive state observer, and stochastic Kalman state estimator.
Additionally the Proportional-Integral (PI) variants of the deterministic state is presented.
Next, two application of observers are described: observer-based feedback control, and
fault detection.
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QUANTITATIVE OBSERVERS
din
u
plant
y
observer
x
Figure 1. Plant and observer in an open loop configuration.
1. Quantitative Observers
Quantitative observers are partitioned into deterministic and stochastic variants.
Deterministic observers require relatively noise free measurements of a system’s input and
output while stochastic observers can model additive measurement noise and process noise
and provide the linear least means square estimates of state. This section begins with a
description of the deterministic Luenberger observer and adaptive observer, and then
continues with the stochastic Kalman filter.
1.1 Deterministic
Deterministic observers for time invariant plants with known parameters are commonly
referred to as Luenberger Observers (Luenberger 1971) while observers for plants with
unknown parameters are known simply as adaptive observers.
1.1.1 Luenberger Observer
The Luenberger observer estimates the state variables of linear time-invariant systems with
known parameters. Consider a dynamic system of order n described by the following
equations
x = Ax + Bu
y = Cx
28
( 1)
CHAPTER 2
BACKGROUND
where u(t) are the p inputs and y(t) the m outputs. A, B and C are (n × n), (n × p) and (m ×
n) matrices respectively. A Luenberger observer is described by
x = Ax + L( y − Cx ) + Bu
( 2)
where x is the estimates of the state x. If the system (A, C) is observable then the constant
L can be selected so that (A - LC) is asymptotically stable and x ( t ) asymptotically
approaches x(t). The L( y − Cx ) term provides a provides a proportional correction factor
that insures stability of the observer even when the system (A, B) is unstable. A schematic
of the observer in this open-loop configuration is shown in Figure 1.
1.1.2 Robust PI Observer
The PI observer includes an additional term, the integral of the estimation error, v, in the
observer equation
x = Ax + L( y − y ) + B( u + v )
( 3)
v = K I (Cx − y )
where KI is a (n × n) matrix and is selected so the matrix R,
R = A − LC B
K IC
0
( 4)
is asymptotically stable. Refer to (Niemann, Stoustrup et al. 1995) for a complete discussion
on how to optimally achieve stability.
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QUANTITATIVE OBSERVERS
1.1.3 Adaptive Observer
When the plant has unknown parameters, an observer must be adaptive and capable of
estimating the system parameters A, B and C. Parameters estimates are only required for
those parameters that are known. Estimation of these system parameters is aided by
associating the parameters with the measurements y(t) and u(t) (Carroll and Lindorff 1973;
Kudva and Narendra 1973). This is accomplished by first transforming the system's
unknown and known parameters to left companion observable canonical form
x = [ − a A ]x + bu
( 5)
y = cx = x1
where aT = [a1, a2, ... an], bT = [b1, b2, ... bn], cT = [1, 0, ... 0] and A = I n −1
0
with In-1
representing an ((n -1) × (n -1)) identity matrix. Now only the n parameters in a(t) are
needed to specify A(t) and the n parameters in b(t) are needed to specify (B(t), C(t)).
Introduction of the constant k, a (n × 1) vector, allows further algebraic manipulation of
equations so that the system parameters a(t) and b(t) are associated with the measurements
y(t) and u(t) respectively:
x = Kx + [k − a( t )]y( t ) + b( t )u(t)
y = cx = x1
( 6)
k is selected so the matrix K = [−k | A ] is asymptotically stable. A P adaptive observer can
now be described by
( t ) u(t) + w (t) + w (t)
x = Kx + [k − a ( t )]x1(t) + b
1
2
30
( 7)
CHAPTER 2
BACKGROUND
y =x 1
x
Observer
x1
cT
w1 + w2
e
a
u
Aux Signal
Generation
w1
b
Identifier
w2
Figure 2. Adaptive Observer schematic taken from (Narendra and Annaswamy
1989).
where a (t ) and b(t ) are estimates of the parameters a(t) and b(t). w1(t) and w2(t) are n-
dimensional signal that are required to insure stability of the observer.1 The adaptation laws
for the parameters a(t) and b(t) are
a = Γ 1 (cx − y)w 1
( 8)
b = Γ 2 (cx − y)w 2
The constant matrices Γ1 and Γ2 are proportionality constants that affect the rate of
convergence of the parameter estimates. Figure 2 shows a schematic of the adaptive
observer implementation. Equation ( 7) is implemented by the Observer block, and Equation
( 8) is implemented by the Identifier block.
1
The description of these auxiliary signals is beyond the scope of the proposal; refer to the book by
Narendra and Annaswamy (Narendra and Annaswamy 1989) for further details.
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QUANTITATIVE OBSERVERS
1.2 Stochastic
Stochastic observers are commonly referred to as Kalman filters (Kalman 1960). They
structure is the same as simple linear observe with a gain optimized to solving the on-line
the Linear Least Mean Square (LLMS) estimation problem. The filter dynamically
calculates the prediction and estimation error covariance and at each filter iteration and
calculates the optimal observer gain. However, when the plant is time invariant and the
random noise has a fixed variance the gain of the observer converges to a fixed value; the
observer can then be calculated off-line. Kalman filters are usually implemented digitally,
therefore the discrete version of the filter is presented in the following section.
1.2.1 Kalman Filter
Consider a dynamic system of order n described by the following discrete linear system
model
x(k+1) = Ax(k) + Bu(k) + v(k)
y(k) = Cx(k) + w(k)
( 9)
where v(k) and w(k) are the process and measurement zero-mean Gaussian noise sequence
with Q(k) and R(k) covariance matrices, respectively. The observer equation for the
corresponding Kalman filter (Santina, Stubberud et al. 1994) is
[
]
x k k = Ax k k −1 + K k y k − Cx k k −1 + Bu(k)
( 10)
where the optimal Kalman Gain Kk is given by
(
K k = Pk |k −1C k C k Pk |k −1C′k + R k
)
−1
( 11)
The Kalman gain is defined in terms of the state prediction covariance matrix Pk|k-1. Pk|k-1 is
32
CHAPTER 2
BACKGROUND
Pk |k −1 = A( k − 1)Pk −1|k −1A( k − 1) + Q( k − 1)
( 12)
which is given in terms of the state estimation error covariance at the previous filter
iteration, Pk-1|k-1
Pk −1|k −1 = [ I − K ( k − 1)C( k − 1)]Pk −1|k −2
( 13)
The estimation error covariance definition is recursive because of the Pk-1|k-2 term. If
however the noise covariances and the system do not vary with time a non-recursive
solution for K can be obtained by solving the following algebraic Riccatti equation for the
state prediction covariance matrix Pk|k-1
[
Pk |k −1 = APk |k −1 F'− APk |k −1 C' CPk |k −1 C'+ R
]
−1
CPk |k −1 C'+Q
( 14)
The Matlab m-file destim obtains the Kalman gain by solving this equation.
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APPLICATION OF QUANTITATIVE OBSERVERS
din
u
plant
Kl
y
observer
x
Figure 3. State feedback regulation utilizing an observer.
2. Application of Quantitative Observers
The two main application of state observers are state feedback control and fault detection.
In the control domain, researchers attempt to recover the robust properties of full state
feedback designed with Linear Quadratic Regulator (LQR) (Anderson and Moore 1989)
when using observer-based feedback. In the fault detection domain, observers are used to
overcome the lack of sensors to detect faults and disturbances through the use of analytical
redundancy (Frank and Wünnenberg 1989).
2.1 Robust Observer-Based State Feedback Control
If all states of a system are available a gain matrix K can be designed so that the closed loop
control law u = Kx minimizes a quadratic performance index. This Linear Quadratic
Regulator (LQR) has impressive robustness properties; robustness can be demonstrated for
a large range of disturbances. When states are unavailable an observer must be used to
estimate the system states. The Separation Principle allows the design of the observerbased state feedback controller to be separated into the design of a state feed back
controller and a observer. The separation principles ensures that the controller and observer
are both stable the combined observer-based controller will also be stable. However, a
Loop Transfer Recovery (LTR) procedure must be used to design the combined
observer/controller pair since the use of a state estimates instead of direct measurements of
34
CHAPTER 2
BACKGROUND
state can degrade the performance of the observer-based controller (Jamshidi, Vadiee et al.
1993). When the system to be controlled is deterministic the design procedure is called
LQR/LTR while the equivalent design technique for stochastic systems is called the Liner
Quadratic Gaussian (LQG) / LTR procedure.
The infinite horizon, time-invariant linear quadratic optimization for the design of LQR use
a quadratic performance cost function to optimize the control input u(t) = Kx of the
following form (Lin 1994)
∞
J =
0
[ z T Qz + u T Ru]dt
( 15)
subject to the constraints that
x = Ax + Bu and z = Cx
( 16)
where Q and R are symmetric matrices. Often it is also assumed that Q is positive
semidefinite and R is positive definite.
Q and R are design parameters that give the relative weighting of the state trajectory and
control input. Using the calculus of variations the optimal control action can be derived
from the following Hamiltonian equation
0.5( z T Qz + u T Ru) + λT ( Ax + Bu)
=
( 17)
where λ is the Lagrange multiplier. The major objective of the optimized controller is to
regulate the plant output to zero while stabilizing the closed-loop system. The optimization
procedure must balance the conflicting goals of stabilization and minimization of controller
effort. The Matlab m-file lqr is often used to calculate the optimal gain matrix K given A,
B, Q and R.
The cost function for the stochastic LQG regulator uses the estimate of Equation ( 11) to
give
J =E
∞
0
[z T Qz + u T Ru]dt y
( 18)
subject to the constraints that
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APPLICATION OF QUANTITATIVE OBSERVERS
x = Ax + B[u + w( t )] and y (t) = Cx + v ( t )
( 19)
where v(k) and w(k) are the process and measurement zero-mean Gaussian noise sequence,
respectively. The optimization process relies on the Separation Principle to achieve
optimality while separately optimizing the feedback gain and the Kalman Filter.
36
CHAPTER 2
BACKGROUND
2.2 Fault Estimation and Accommodation
Faults can take several forms, but the literature usually focuses on faulty actuators and
sensors. Additional faults can be caused by excessive friction, component failure, or age
degradation. This thesis will focus on actuator faults and will propose a benchmark for the
accommodation of frictional faults to be used in future research.
Actuator faults can be linear or nonlinear in nature. Gain mismatch and gain offsets are
common examples of idealized linear faults. Backlash and deadzone are often described as
typical idealized “actuator nonlinearities” (Tao and Kokotovic 1996), and actuator
saturation is often indicative of a nonlinear actuator power limit (e.g. all actuators have
fixed output ranges). These faults can occur in combination, or vary in severity with
changes in operating conditions, maintenance schedule and age. Figure 4 shows the effects
of gain mismatch, deadzone, backlash and saturation on actuator output2. The following
sections will describe these four idealized common faults.
2.2.1 Gain Mismatch Faults
Faults caused by gain mismatch are often caused by component drift or miscalibration. The
output of an actuator with this simple linear fault is
u = f ( w) = k atten w
( 20)
Figure 4 shows a gain mismatch with katten = 0.6
2
The actuator inputs are taken from the simulations done with the PI Observer-based regulator.
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APPLICATION OF QUANTITATIVE OBSERVERS
2.2.2 Deadzone Faults
Deadzone faults suppress actuator output for a range of Input values. This fault is often
caused by friction in the actuator. The output of the failed actuator is
u = f ( w) =
w − k dz
if w ≥ k dz
0
if - k dz ≤ w ≤ k dz
w − k dz
if w ≤ − k dz
( 21)
Figure 4 shows a deadzone fault with kdz = 0.3.
2.2.3 Backlash Faults
Backlash is a simple form of hysteresis and differs from the other three faults described here
in that it has memory; the current state of the fault mechanism is a function of it's previous
state. A compact representation of the fault mechanism is
w
if w > 0 and u(t) = w − k bl
0
if w = 0
u ( t ) = f ( w) =
( 22)
Figure 4 shows a backlash fault with kbl = 1.
2.2.4 Saturation Faults
Almost all actuators saturate at some extreme range of inputs, beyond the nominal
operating range of the actuator. However, over time the performance of an actuator can
deteriorate, and saturation begins to overlap the nominal operating range of the actuator.
The output of the failed actuator is
38
CHAPTER 2
BACKGROUND
u
u
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0
5
10
Time (second)
15
-2
0
20
5
(a)
10
Time (second)
15
20
15
20
(b)
u
u
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
0
5
10
Time (second)
15
20
-1.5
0
5
(c)
10
Time (second)
(d)
Figure 4. The effects of actuator faults (solid line) in distorting desired actuator
output (gray dashed lines) can be seen for the four actuator faults: (a) gain
mismatch, (b) deadzone, (c) backlash and (d) saturation.
k sat
u = f ( w) =
w
− k sat
if w ≥ k sat
if - k sat ≤ w ≤ k sat
( 23)
if w ≤ − k sat
Figure 4 shows a saturation fault with ksat = 1.
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CHAPTER 3
VALIDATION BENCHMARKS
Chapter 3
Validation
Benchmarks
This chapter describe the validation benchmarks that we will use to verify the theoretical
results described in the previous chapters. We begin by describing the 1992 ACC Robust
Control Benchmark, a common benchmark scenario for evaluating the disturbance rejection
robustness of controllers. The chapter continues with the additional benchmarks develop to
validate the results of the dissertation and future research. First, an extension to the Robust
Control Benchmark is given to validate the frictional fault rejection properties of observers
and observer based controllers and then a separate benchmark is proposed to validate the
disturbance rejection properties of adaptive observers.
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ROBUST CONTROL BENCHMARKS
input
output
disturbance
u
y = x2
w
k
u
m1
x1,
m2
w
x3 = dx1/dt
x2, x4 = dx2/dt
Figure 5. The dual mass, single spring plant used in the ACC benchmark.
1. Robust Control Benchmarks
The robust control community has two common benchmarks: the ACC ’92 and IFAC ’93
Benchmarks. Named after the two conferences that they were introduce each has a different
focus. The ACC ’92 benchmark was originally introduced at the 1990 (Wie and Bernstein
1990) American Control Conference (ACC) and then augmented at the 1992 ACC (Wie
and Bernstein 1992). The benchmark plant consists of a simple mechanical and the
benchmark scenarios consist of rejecting input disturbances and tracking a unit step. The
IFAC ’93 (Graebe 1994) was introduced at the 12th IFAC World Congress in Sydney
Australia. The benchmark plant is seventh order with a nominal third order realization and
has parameters with no physical interpretation. The exact plant is unknown to the control
engineer and a tracking controller must be designed based only on noisy measurements from
a black box simulation.
The ACC ’92 benchmark, subsequently referred to as the Benchmark, was selected to
validate the theoretical work in qualitative fuzzy control and observer design because the
simple mechanical plant of the Benchmark lends itself to qualitative modeling. The
following sections describe the Benchmark plant and design scenarios.
42
1.1 ACC ’92 Robust Control Benchmark
At the 1990 American Control Conference (ACC) a set of three benchmark scenarios for
robust control was suggested by Wie and Bernstein (Wie and Bernstein 1990) and then
augmented by a fourth scenario at the 1992 ACC (Wie and Bernstein 1992). These
scenarios require the designer to make tradeoffs between maximizing stability and settlingtime robustness while minimizing actuator effort. Scenario 1 and Scenario 4 are used in this
dissertation and will be described after a description of the Benchmark plant.
The Benchmark plant shown in Figure 5 is a simple flexible structure consisting of two
masses connected with a single spring. This dynamic system has a noncollocated sensor
and actuator; the sensor senses the position of m2 while the actuator accelerates m1.
The state space model for the plant is
!
x1
x2
x3
=
0
0
1 0 x1
0
0
0 1 x2
− k m1
k m1
0 0 x3
k m2
− k m2
0 0 x4
x4
%
+
#$
'(
)*
/0
1 m1
&
7
34
0
-.
0
+
1
0
( 24)
9:
0
u+
56
;<
0
w
,
"
2
0
1 m2
8
where
x1 is the position of m1,
x2 is the position of m2,
x3 is the velocity of m1,
x4 is the velocity of m2,
y is the plant output x2,
w is the acceleration disturbance on m2 and
u is the control acceleration on m1.
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ROBUST CONTROL BENCHMARKS
w
e
r
y = x2
u
P
C
-
v
Figure 6. Series compensator, C, and plant, P, in a negative feedback loop.
The following additional variables are needed to complete a description of the closed loop,
series compensated system shown in Figure 6:
v is the sensor noise,
e is the compensator input and
r is the reference input.
The corresponding transfer function between the plant input (actuator output) and plant
output is
T uy =
(k / m1m2 )
[
s s + k (m1 + m2 ) / m1m2
2
2
( 25)
]
The corresponding transfer function between the disturbance and plant output is
T wy =
(1 / m2 )(s + k / m1 )
2
[
s s + k (m1 + m2 ) / m1m2
2
2
]
( 26)
1.1.1 Design Scenario 1
The following design requirements for the compensated system are required:
i) The closed-loop system is stable for m1 = m2 = 1 and 0.5 < k < 2.0.
44
ii) For the disturbance w(t) = unit impulse at t = 0 y has a settling time of 15
seconds for the nominal plant parameters m1 = m2 = k = 1.
The following guidelines were also given:
i) use realistic measurement noise v(t),
ii) achieve reasonable performance and stability robustness,
iii) minimize control effort and
iv) minimize controller complexity.
Settling is achieved when y is bounded by ±0.1 units.
1.1.2 Design Scenario 2
Replace requirement ii) in Design #1
ii) For the disturbance w(t) = A sin (ω
ωt + φ), a frequency ω = 0.5 rad/sec, with
only the disturbance’s frequency known to the designer, achieve asymptotic
rejection of w(t) at y with a settling time of 15 seconds for the plant parameters
m1 = m2 and 10.5 < k < 2.0.
1.1.3 Design Scenario 3
Replace requirement ii) in Design #1
Settling is achieved when y is bounded by ±0.1 units.
ii) Maximize a stability performance measure with respect to the parameters m1, m2
and k with nominal values of m1 = m2 = k = 1.
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ROBUST CONTROL BENCHMARKS
1.1.4 Design Scenario 4
Scenario 4 requires the compensated plant to track a unit-step with the following design
requirements:
i) The compensator output is limited to u ≤ 1.
ii) Settling time and overshoot are both minimized.
iii) Robustness to perturbation in m1, m2 and k with the nominal plant parameters
being m1 = m2 = k = 1.
1.1.5 Existing Solutions for Scenario 1 and 4
The robust setpoint control necessary for implementing the Benchmark is almost exclusively
done using conventional techniques. No intelligent control techniques have been utilized in
the over 30 papers, including the special issues of the Journal of Guidance, Control and
Dynamics and the International Journal of Robust and Nonlinear Control, that utilize the
1990 and 1992 American Control Conference Robust Control Benchmark (Wie and
Bernstein 1990; Wie and Bernstein 1992).
Conventional robust control uses different techniques to minimize a robustness cost
function. The cost function weights several metrics that specify the final compensator
performance, including settling time, actuator saturation, and stability. Published
methodologies include H∞ (Sznaier and Rotstein 1995), H2 (How, Haddad et al. 1994),
H∞/H2 hybrids (Davis, Jr. et al. 1994), generalized LQG (Collins 1990), nonlinear constraint
optimization (Ly 1990), LMI (Folcher and Ghaoui 1994), loop shaping (Schmidt and D.
Benson 1995), and a hybrid design approach in which genetic algorithms optimize a
stochastic robustness function (Marrison and Stengel 1994). All these conventional control
techniques are based on a differential/difference equation model of the plant (Antsaklis
1994). A set of controllers that stabilize the plant is defined and then the set is searched for
a controller that minimizes the robustness cost function. After search finds the optimal
design, direct incremental tuning of the initial design is difficult because compensator
parameters do not necessary have a linguistic interpretation to guide the refinement, nor
46
does the designer know the precision that must be preserved in the individual parameter in
order to preserve the specified performance.
The simple structure of the benchmark initially suggests that it can be controlled by a
conventional Proportional-Integral-Derivative (PID) compensator. The PID control law is
appealing because the parameters of the transfer function
u( s ) = K P +
KI
+ KDs
s
( 27)
have simple linguistic interpretations. Indeed a PID controller can stabilize a mass-springmass plant with a collocated sensor and actuator. However, a PID compensator is of too
low an order to stabilize the Benchmark so a higher order controller is required. The
interpretation of parameters used in the slightly more complex second-order compensator of
Uy-Loi Li’s(Ly 1990)
T =
zu
5.3697s 2 − 1.3702 s − 0.17375
s 2 + 3.6767s + 4.8840
( 28)
are no longer as straight forward. While interpretation can be given in terms of the zeros
and poles of the system (e.g. the compensator models the plant spring as a simple oscillator
with a frequency of 0.34813 rad/sec), superior performance still requires a more complex
fifth-order compensator (Stengel and Marrison 1992). This paper will benchmark two of
these higher order compensators developed by Marrison and Stengel using the linear
quadratic Gaussian regulator methodology (Marrison and Stengel 1995). Design 1 stresses
stability robustness:
−79.3( s − 0.8 )( s + 5.7)( s + 0.11)
T = 2
zu ( s + 3.84 s + 10.24)( s 2 + 6.882s + 13.69)( s + 0.46)
( 29)
Design 3 stresses settling time robustness:
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ROBUST CONTROL BENCHMARKS
−8.2( s − 4.7)( s + 3.9)( s + 0.24)
T = 2
zu ( s + 4.662 s + 13.69)( s 2 + 3132
. s + 7.29)( s + 1.6)
( 30)
Both compensators contain numerous numerical constants that do not have linguistic
interpretations. Tuning of the compensators is difficult since no clear coupling between
individual parameters and individual design constraints exists, and the parameter precision
required to achieve the specified performance is unknown. The next section of the paper
describes a rule based qualitative robust control where every constant has low precision and
a clear linguistic interpretation. Tuning can be performed by either augmenting the design
with additional rules or by adjusting the appropriate compensator parameters.
Results for the Benchmark Design Scenario 1 is given in Figure 7 and Figure 8. Figure 7
shows the response of the compensated system to a unit impulse to mass m2 and Figure 8
gives the corresponding compensator output.
48
3.5
#1 Marrison & Stengel (1995)
#3 Marrison & Stengel (1995)
Uy-Loi Ly (1990)
3
2.5
2
1.5
1
0.5
0
-0.5
0
5
10
15
20
25
30
Figure 7. Plant output y after unit impulse disturbance (with duration of 0.5
seconds and amplitude of 2.0) to m2 for compensated systems from (Ly 1990;
Marrison and Stengel 1995).
1
#1 Marrison & Stengel (1995)
#3 Marrison & Stengel (1995)
Uy-Loi Ly (1990)
0.5
0
-0.5
-1
0
5
10
15
20
25
30
Figure 8. Compensator output u after unit impulse disturbance to m2 for
compensated systems from (Ly 1990; Marrison and Stengel 1995).
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ROBUST CONTROL BENCHMARKS
1.1.6 Existing Solution for Design Scenario 2
Design Scenario 2 for the Robust Control Benchmark requires the design of a controller
design that can reject a sine wave with a frequency of 0.5 rad/sec. Collins, King and
Bernstein (Collins 1990) developed a controller that works robustly for plants that have
spring constants in a range from 0.5 and 2.0. This sixth order controller is given by
T =
86816( s + 0.12322)( s 2 − 0.42562 s + 1.353)( s 2 + 0.04783s + 0.22783)
( s + 253.19)( s + 38.684)( s 2 + 5.0136 s + 7.8279)( s 2 + 0.002242 s + 0.25362)
( 31)
This controller however has limited robustness to uncertainty in the frequency of the
disturbance. The following tabulates the amplitude of steady state oscillations of the plant
output y versus the frequency of the sine wave disturbance for frequencies ranging from 0.5
to 0.25 rad/sec.:
upper
disturbance
bound on
frequency
output y
0.5
0.23
0.48
1.13
0.46
1.77
0.44
2.25
0.42
2.7
0.40
3.1
0.35
3.3
0.30
4.6
0.25
5.5
50
The table distinctly shows that the controller performance is adversely affected by even a
small mismatch between the disturbance frequency and the modeled disturbance frequency.
Figure 9 shows that a one percent deviation from the modeled disturbance frequency causes
the peak regulated output amplitude to more than double.
y
4
0.500 rad/sec
0.495 rad/sec
3
2
1
0
-1
0
50
100
Time (second)
150
(a)
w
1
0.5
0
-0.5
-1
0
50
100
Time (second)
150
(b)
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ROBUST CONTROL BENCHMARKS
Figure 9. The response of a controller from (Collins 1990) to mismatch in frequency.
The plant output y is shown in (a) for an input disturbance of 0.500 (solid line) and
0.495 rad/sec (dotted line). (b) shows graphically the small mismatch between the two
input disturbances.
52
2. Benchmark Extensions
The ACC Benchmark presented in the Background Chapter is a robust control benchmark
and as such is insufficient for benchmarking and validating failure detection and disturbance
rejection properties of adaptive observers. However, the ACC Benchmark can easily be
augmented to include frictional failure scenarios and a simple disturbance rejection
benchmark can be created from the plant used by Kudva and Narendra (Kudva and
Narendra 1973) in their adaptive observer paper. The benchmark presented at the end of
this section based on the Kudva and Narendra plant is called the Adaptive Observer
Benchmark.
2.1 Failure Scenarios for the ACC Benchmark
The ACC Benchmark can easily be augmented to test the robustness of the compensated
system to system failure. The failures consist of Coulomb friction acting on the individual
masses (Schneider and Frank 1996). Three failure modes can exist, with friction on either
m1 or m2, or friction on both masses.
The Failure Scenario #1 has the following design requirements in addition to those for
Scenario #1:
iv) For the set of disturbance w(t) = {δ, 0.1δ, 0.5δ, 2δ, 10δ}, where δ is the unit
impulse at t = 0, y has a settling time of 15 seconds for the nominal plant
parameters m1 = m2 = k = 1 and Coulomb Friction of 0.01 acting on m1.
The Failure Scenario #2 has the following design requirements in addition to those for
Scenario #4:
iv) For the set of disturbance w(t) = {δ, 0.1δ, 0.5δ, 2δ, 10δ}, where δ is the unit
impulse at t = 0, y has a settling time of 15 seconds for the nominal plant
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BENCHMARK EXTENSIONS
din
din
u
u
y = x1
plant
observer
plant
Klq
y = x1
observer
x
x
Figure 10. Simulation Scenario 1: plant
Figure 11. Simulation Scenario 2: state
and observer in an open loop
feedback regulation utilizing an adaptive
configuration with input measurement
observer with input measurement
disturbance, din.
disturbance, din.
parameters m1 = m2 = k = 1 and Coulomb Friction of 0.01 acting on either m1
or m2, or on both m1 or m.
With both scenarios the compensator has no a prior knowledge of whether a failure has
occurred.
2.2 Adaptive Observer Benchmark
The Adaptive Observer Benchmark uses the plant from Kudva and Narendra (Kudva and
Narendra 1973), described by the equations
−5 1
1
x = x + u
− 10 0
2
( 32)
y = [1 0]x = x1
All four of the design scenarios utilize the input
u = 5 sin (t) +5 sin ( 2.5t) ,
( 33)
54
a combination of two distinct sine waves, to excite the two states of the plant and ensure
convergence of the parameter estimates. The initial state and state estimates are
x( 0 ) = x( 0 ) = 0 , and initial parameters estimates, a (t ) or b(t ) , are 102% of the actual
plant parameters:
a 0 = 102
. a = 51
.
10.2
and b 0 = 102
. b=
102
.
2.04
.
( 34)
The first design scenarios benchmarks the open loop performance of an observer shown in
Figure 10, while the second design scenarios shown in Figure 11 test the performance of an
observer-based state feedback regulator.
2.2.1 Design Scenario 1
Given the following set of input disturbances
•
din = 0,
•
din is a step of magnitude 0.2, and
•
din = 0.5 sin (ω
ωt + φ), a frequency ω = 0.5 rad/sec, with only the disturbance’s
frequency known to the designer
design an observer for the plant which demonstrates the following for all disturbances:
•
state estimates converges asymptotically and
•
parameter estimates settle within 40 seconds.
Settling of parameter estimates is achieved when they are is bounded by ±1 % of their
actual value.
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BENCHMARK EXTENSIONS
2.2.2 Design Scenario 2
Given the following set of input disturbances
•
din = 0,
•
din is a step of magnitude 0.2, and
•
din = 0.5 sin (ω
ωt + φ), a frequency ω = 0.5 rad/sec, with only the disturbance’s
frequency known to the designer
design an observer for the plant which when used in an observer-based state feedback
regulator demonstrates the following for all disturbances:
•
state estimates converges asymptotically,
•
parameter estimates settle within 40 seconds and
•
achieves full Loop Transfer Recovery (LTR) after parameter estimates converge.
2.2.3 Design Scenario 3
Fault accommodation is evaluated by using four simple actuator faults described in the
Background Chapter: gain mismatch, deadzone, backlash and saturation on actuator output.
56
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Chapter 4
Integral
Action
Everything that can be invented has been invented.
Charles H. Duell, Commissioner, U.S. Office of Patents, 1899
The chapter begins with the theoretical development of the PI adaptive observer, including
a proof of stability and convergence. Next the Proportional Fading-Integral Observer is
developed. A methodology for selecting the appropriate integral gain and the fading rate is
described and a theoretical proof is given for determining the minimum number of
independent state observations required to reject a set of concurrent disturbances with
distinct injection points.
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PI ADAPTIVE OBSERVER
1. PI Adaptive Observer
The development of the PI adaptive observer parallels the development of the PI observer
from the conventional P observer; a term proportional to the integral of the estimation
error, v, is added to the conventional P observer equation. However, instead of the term
Bv(t) being added to the observer equation, the term b(t)v ( t ) must be added for the
adaptive case. Because the magnitude of the integral offset is now dependent on the
estimate b(t) , and not b, the integral action is no longer unambiguously associated with the
errors in estimating state. With the adaptive version of the PI observer the integral action
can now result from either state estimation error or parameter estimation error. Therefore,
care must be taken in assuring that the parameter adaptation is not completely corrupted by
the observer’s integral action.
The PI adaptive observer is described by the equation
x = Kx + [k − a( t )]x 1(t) + b( t )( u + v ) + w 1(t) + w 2(t)
( 35)
v = K I ( cx − y )
with the corresponding error equations
e = Ke + α(t)x1 + β(t)u + b(t)v + w1(t) + w 2(t),
e1 = ce
( 36)
v = K I e1
where e = x − x , α = a(t) − a and β = b( t ) − b. These error equations differs from the error
equation for the P adaptive observer by the term b(t)v .
Stability and convergence of this new error equation is shown by using an extension of the
Gronwall-Bellman lemma (Narendra and Annaswamy 1989) for almost time-invariant
systems. First the error equations are converted to the following partition form:
)
%
-
3
:
e
8
7
v
;
9
=
&'(
456
K
KI c
0 + 0
./0
1
bˆ (t )
!
+
*+
"#
2
0
,
$
v
α(t)
e
0
β (t)
w ( t ) + w 2 (t )
x1 +
u+ 1
0
0
( 37)
60
~
The error equation has the structure z = ( A + B(t ))z + f (t ) , where
~
A=
K
0 , B (t ) = 0
bˆ (t )
KIc
0
( 38)
and
w (t ) + w 2 (t )
α( t )
β( t )
f(t) =
x1 +
u+ 1
0
0
0
.
We can now use the following theorem to prove stability and convergence of the PIAO:
~
Theorem The PIAO error dynamics z = ( A + B(t ))z + f (t ) converges to zero for t
> t0 if
i)
~
A is
asymptotically stable,
ii) there exists a b0 such that B(t ) ≤ b0 for t > t0 and
iii) there exists a b1 such that f (t ) ≤ b1 for t > t0.
~
Proof: The stability of an error equation in the form z = Az + f (t ) is shown through the
stability proof for the conventional P adaptive observer using the Kalman Yakubovich
Lemma (Kudva and Narendra 1973). Therefore when B(t) = 0, the proof of stability and
convergence for the PI case is the same as the conventional case.
When B(t) ≠ 0, if all the solutions of z = Az are bounded then all solutions of
z = (A + B(t ) )z
!
z ( t ) = e At z ( 0 ) +
t
0
e A ( t − τ ) B ( τ ) z ( τ ) dτ
( 39)
are bounded by the Gronwall-Bellman (Narendra and Annaswamy 1989) lemma for almost
time-invariant systems. The lemma states that
z(t ) ≤ c1 + c1
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B ( τ ) z ( τ ) dτ
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( 40)
61
PI ADAPTIVE OBSERVER
where
c1 = max sup e x (0) , sup e
At
At
( 41)
t ≥0
t ≥0
Because of the bounded assumption on B(τ) and equivalently on
t
0
B( τ) dτ Equation
( 41) reduces to
c 0 B( τ ) dτ
z ( t ) ≤ c1 e 1
t
( 42)
and z(t ) ≤ c 2 . Since the stability of z = Az + f (t ) is insured by the stability of the P
observer, the stability of z = (A + B(t ) )z + f ) is therefore also guaranteed.
62
2. Robust PI Kalman Filter
The PI version of the Kalman filter was first developed by Kim and Shafai (Kim, Shafai et
al. 1989; Kim and Shafai 1990). The development was awkward and disturbance rejection
was not explored. This section begins with a simplified development of integral action for
the Kalman filter and then develops a necessary condition for the rejection of n
simultaneous disturbances injected at n distinct injection points by integral action for either
the PI observer or the PI Kalman filter.
The PI Kalman Filter contains the integral term v, which is added to the original estimator
equation by using the pair of coupled differential equations
xˆ k k = Axˆ k k −1 + K k [ y k − Cxˆ k k −1 ] + Bu(k ) + B I v( k )
( 43)
v (k ) = v(k − 1) + K I [ y ( k ) − Cxˆ k |k −1 ]
where the constant KI determines the time constant of the integral action while BI reflects
the form of the plant perturbations or disturbance injection points. These equations can be
manipulated to give one augmented equation that has the form of the estimator equation for
the standard Kalman filter
zˆ k k = Azˆ k k −1 + K[ y k − C zˆk k −1 ] + B u(k )
( 44)
where
A=
A BI
0
I
, B=
B
0
, C = [C 0], K =
KK
KI
and z =
x
( 45)
v
The optimal gain K for a given process and measurement noise covariance can be derived
using technique developed by Kim and Shafai (Kim, Shafai et al. 1989). However, when
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ROBUST PI KALMAN FILTER
optimizing K for perturbation and disturbance rejection, KK becomes the Kalman gain and
the gain associated with the integral action KI should be selected such that
R=
A − K kC BI
− K IC
( 46)
I
has eigenvalues within the unit disc.
The integral action of the PI Kalman filter is effective in estimating both plant perturbations
and input disturbances that can be modeled by the term BIv. When rejecting plant
perturbations, the perturbation must be in the form D∆
∆E, and D becomes BI. As an
example, the PI Kalman filter can be used to reject the perturbation to the Benchmark plant
caused by changes in the spring constant provided that BI = D = [ 0 0 -1 1]'.
Correspondingly when rejecting input disturbances, the distribution matrix of the
disturbances must be used as BI. As an example, when rejecting a disturbance to m1 the
injection point of the disturbance is x3 and BI = [ 0 0 1 0]'.
However for single output systems, a PI Kalman Filter can not simultaneously reject both a
perturbation and slowly varying disturbance. With only one state measurement the PI
Kalman Filter can reject either (1) a single rank one perturbation or (2) an input disturbance
with a single known injection point. This can be generalized to the following:
Theorem If the plant [A, B, C] of order n is completely observable, and if
r≥ρ
A BI
C
I
−n
where r is the number of independent outputs, then all perturbations and
disturbances modeled by BI can be rejected.
64
The proof is a generalization of the development given by Saif (Saif 1997). The theorem
requires that for the simultaneous rejection of a rank a perturbation and b disturbances with
unique injection points requires (a + b) independent state measurements.
The robustness of PI Kalman filter to plant perturbations and disturbances can be adversely
effected by transitory disturbances. These disturbances which are not modeled by BI create
an offset in the integral term and adversely impact the performance of the PI Kalman filter.
Rejection of these transitory features can be achieved by discounting the integral term over
time.
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KALMAN FILTER WITH A FADING INTEGRAL TERM
3. Kalman Filter with a Fading Integral Term
The fading of the integral term allows integral action to estimate and accommodate
disturbances with known injection points in the presence of transitory disturbances with
unknown injection points. Without fading any small transitory disturbance, who's injection
point is not incorporated in BI, will add a permanent offset to the integral action, precluding
the convergence of the disturbance estimate and the disturbance. The fading-integral is
developed in this section for the Kalman filter; an equivalent development is also valid for
the Luenberger observer.
The Proportional Fading-Integral (PFI) Kalman filter decays the integral term of the PI
Kalman filter, allowing the effects of transient disturbances to decrease with time. The PFI
Kalman filter is described by
xˆ k − k = Axˆ k k −1 + K k [ y k − Cxˆ k k −1 ] + Bu(k ) + B I v (k )
( 47)
v (k ) = ( I − K F ) v (k − 1) + K I [ y (k ) − Cxˆ k |k −1 ]
where the constant KF determines the amount of fading. As for the PI Kalman Filter these
equations can be manipulated to give one augmented equation with the following
parameters
A=
A 0
0
I
, B=
B
0
, C=
C 0
0
I
, K=
K K BI
KI − KF
and z =
xˆ
( 48)
v
Using the same technique as for the PI Kalman filter, the optimal gain K for a given
process and measurement covariance can also be derived for this augmented system. When
designing the filter for perturbation and disturbance rejection it is necessary to insure
stability for the PFI Kalman filter by having the eigenvalues of
66
R=
A − K kC
BI
− K IC
I − KF
( 49)
inside the unit disk. As shown by R, the additional freedom afforded by the KF term allows
the design of PFI Kalman Filter with a larger stability margins than the corresponding PI
Kalman filter. Indeed, an unstable PI Kalman Filter can be made stable with the addition of
fading.
Simple integral action alone can achieve time recovery, but can not reject transitory events
that contribute to the integral action. However, with the PFI Kalman Filter the fading term
KF can be tuned so that the effects of transients on the integral action decay over time.
Faster suppression of transients is achieved by increasing KF , but often it is also necessary
to increase the integral action gain KI so that fading term does not completely suppress the
integral action.
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TUNING INTEGRAL ACTION
4. Tuning Integral Action
Proper rejection of perturbations or disturbances requires that K for the PFI Kalman filter,
or equivalently the PFI observer, be designed by separation. The following four steps are
required:
1. solve for the optimal Kalman gain for the system [A, B, C],
2. determine BI from the perturbation structure and/or the disturbance injection matrices,
3. tune KI so that the transient response of the integral action can follows the fastest
perturbation and/or disturbance, and
4. tune KF so that effects of perturbations and/or disturbances with unknown distribution
matrices decay satisfactorily and the PFI Kalman filter is stable.
4.1 Tuning the Integral Gain
A large integral gain KI shortens the transient response of the integral action to
disturbances, but also leads to overshoot and ringing. The critical integral gain can be
determined by achieving a critically damped transient response to a unit step disturbance.
The speed of the critically damped transient response limits the ability of the PI Kalman
filter to reject fast repetitive disturbances.
An optimal integral gain can often not be found for a given disturbance because the fastest
possible transient response of the integral action is too slow to match a quickly varying
input disturbance. It is however possible to decrease the transient response by
1. increasing the modeled process noise or
2. reducing the measurement noise which allows for larger integral gains.
4.2 Tuning the Fading Rate
The fading of the integral term discounts the integral term over time. Decaying the integral
term allows the rejection of transients that would otherwise cause large offsets in the
estimates of plant perturbation and disturbance. This is analogous to the discounting of old
68
measurements in a P Kalman Filter by increasing the magnitude of the process noise
covariance used by the filter. In fact, increasing the model process noise will also fade the
integral term of a PFI Kalman Filter; however large levels of process noise will increase the
variance of state estimates.
With small values of KF the PFI Kalman filter performance approaches that of the PI
Kalman filter, while for large values of KF the integral action of the filter is suppressed and
the PFI Kalman filter performance approaches that of the standard P Kalman Filter.
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CHAPTER 5
ROBUST FUZZY CONTROL
Chapter 5
Robust
Fuzzy
Control
The chapter describes the Qualitative Robust Control (QRC) methodology. QRC is then
used to construct a fuzzy controller for the 1992 ACC Robust Control Benchmark using
qualitative behaviors that achieve stability and tracking.
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ROBUST FUZZY CONTROL
y
State
Estimator
x̂
u
stability
behavior
fusion
Fuzzy State
Process Model
tracking
behavior
Fuzzy
Compensator
Figure 12. Fuzzy Controller consisting of a linear state estimator, fuzzy
process model and fuzzy compensator which superimposes stability
and tracking behaviors.
1. Robust Fuzzy Control
Fuzzy control has gained a wide practical acceptance, providing a simple, intuitive and
qualitative methodology for control (Yen, Langari et al. 1992; Jamshidi, Vadiee et al.
1993). Currently, the typical implementation of a fuzzy controller consists of a set of ifthen rules, where the controller input e is used to evaluate the rules’ antecedents and the
controller output u is the combined output of all the rules evaluated in parallel. This simple
logical system, a Fuzzy Inference System (FIS) (Gulley and Jang 1995), does not
implement inference chaining and can only evaluate a simplified qualitative model of a plant.
Fuzzy controllers described in the literature fall into three main categories (Jang and Sun
1995; Driankov, Hellendoorn et al. 1996):
•
fuzzy PID controllers,
•
fuzzy sliding mode controllers, and
•
fuzzy gain scheduling.
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CHAPTER 5
ROBUST FUZZY CONTROL
All three categories realize closed-loop control action and are based on quantitative control
techniques. The fuzzy PID controllers and fuzzy slide mode controllers are fuzzy
implementations of the linear quantitative PID controller and a nonlinear quantitative sliding
mode controller. Both controllers use the error term, and its derivatives and integrals, as
input into a FIS (Coleman and Godbole 1994). The fuzzy gain scheduler utilizes Sugeno
fuzzy rules to interpolate between several control strategies (Jang and Sun 1995). This
methodology is useful for controlling nonlinear plants that are piecewise linear or for linear
plants that have a time varying parameter. An example of this is a controller built for an
inverted pendulum with variable length. Measurements of the pendulum length are used as
inputs into a FIS that interpolates the output of a small number of controllers that are
optimized for controlling short, medium and long pendulums (Gulley and Jang 1995).
In addition to these compensators which realize closed-loop control action, a FIS can be
used as a supervisor of a conventional closed-loop controller, including systems where a
FIS adaptively tunes a closed-loop PID controller (Copeland and Rattan 1994).
Our Qualitative Robust Control (QRC) methodology uses an incremental design approach,
based on the work of (Ruspini, Saffiotti et al. 1995), to implement robust fuzzy controllers.
First rules are developed to implement a global stabilizing behavior and then augmented
with behaviors that implement performance goals. Additionally, the compensator achieves
robustness by being designed for a sound, or consistent qualitative plant model that
encompasses all the specified plant perturbations. If the rules developed for the
compensator are consistent with the plant model, then the resulting compensator will
provide robust control. The QRC methodology condenses to a four step procedure:
1. creation of a qualitative plant model
2. creation of a stabilizing compensator
3. augmentation of the initial compensator design with behaviors that achieve robust
performance, and
4. tuning of the linguistically interpretable compensator parameters.
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ROBUST FUZZY CONTROL
At the moment the QRC methodology is only a guide to developing a controller. The
authors are working on making the methodology more rigorous and supporting automatic
generation of rules. The following sections provide additional details.
1.1 Selecting a Proper Qualitative Plant Model
One of the strengths of intelligent control is that it is not based on a quantitative model of
the plant, but rather a simplified qualitative model. Model resolution is sacrificed so that the
qualitative model contains only a finite number of interpretable states and state transitions.
Each qualitative state is based on a finite partition of the plant parameters; exact numerical
values of plant parameters are abstracted to interpretable qualitative terms.
This qualitative partition of plant states and parameters has the benefit of making the
qualitative model more robust to changes in plant parameters. Qualitative abstraction
enhances stability robustness because the qualitative model of the plant does not rely on the
exact values of the plant parameters, but on a finite qualitative partitioning of the plant’s
states and parameters. Abstracting away details of the plant reduces the complexity of the
resulting fuzzy controller, but if too much detail is abstracted away the qualitative model
will no longer be complete and consistent with respect to the control requirements; no
stabilizing controller can be developed.
1.2 Designing a Fuzzy Controller
The fuzzy controller design begins with the qualitative plant model. First state information
must be extracted from the plant output by using a combination of linear methods and fuzzy
process models. The crisp outputs of linear operators and fuzzy process models are used as
input to a Sugeno FIS controller (see Figure 12). The design of the rules for the FIS are
suggested by the plant’s qualitative state transition diagram. Given the connection between
qualitative input events and changes in qualitative states, as shown by the plant’s state
74
CHAPTER 5
ROBUST FUZZY CONTROL
transition diagram, the FIS controller is constructed to generate the qualitative events that
will result in the qualitative state transitions required to realize the desired control actions.
The first control objective is the stabilization of the plant. Stability for the Benchmark
entails the dampening of vibrations after an external disturbance is applied. After
stabilization, the FIS is augmented with rules to achieve performance objectives. Fuzzy
logic lends itself to this methodology, because fuzzy logic deals with possibilities and
reasoning remains consistent even when conflicting requirements generate conflicting rules.
Further tuning of the composite compensator can be made adjusting the relative weighting
of the outputs of the different behaviors and by gating behaviors, allowing them to be
performed only during specified states.
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ROBUST FUZZY CONTROLLER FOR BENCHMARK
2. Robust Fuzzy Controller for Benchmark
The Benchmark problem requires both the achievement of stability and tracking
performance. The physical intuition is that the spring oscillations caused by disturbances
must first be dampened to achieve stability, and then after stability is achieved the goal of
tracking can become paramount. This nonlinear dichotomy is easily supported by fuzzy
logic.
QRC allows the separate development of stability and tracking behaviors; the
superimposition of these behaviors then achieves the final control objective. An analysis of
the transfer function of the mass-spring-mass plant indicates that these behaviors should
exploit the rigid-body-mode of the plant, where the plant behaves as if the masses are rigidly
connected. If the stability behavior can be made to achieve this rigid-body-mode, then the
tracking behavior can treats the mass-spring-mass system as a simple single mass. This
allows a simpler tracking behavior to be effective.
The stability behavior is derived from the heuristic that a control action is most effective in
suppressing plant vibration if it is applied when the spring is neutral, and the control action
opposes the motion of the spring. As an extreme example of this effect, an impulse to a
stationary plant can be rejected with only one complementary impulse of equal magnitude, if
the impulse occurs exactly as the spring relaxes.
This section continues with the development of a fuzzy process model that provides a
qualitative partition of the spring state necessary to suppress vibration. Next, the
ramification of this spring process state on the estimation of the plant is discussed. The
Stability Behavior that suppresses vibration is then described. Finally, we present the
Tracking Behaviors and discuss the superimposition of the stability and tracking behaviors.
76
CHAPTER 5
ROBUST FUZZY CONTROL
zero
1
uf
small_neg
negative
small_pos
positive
spring_state
0.5
0
-2
uf
1
0.6
-1
0
1
spring_length_estimate
2
0.4
zero
negative
0.2
positive
0.5
-2
-1
0
sm_pos
sm_neg
0
0
1
2
delta_spring_length_estimate
-0.2
-0.4
1
uf
-0.6
-2
0.5
-2
0
-1
-0.5
0
0.5
-1
1
spring_state
0
0
1
spring_length_estimate
2 2
delta_spring_length_estimate
Figure 13. Input and output membership Figure 14. Output surface of the Spring
function of the Spring Observer.
Fuzzy Process Model .
2.1 Fuzzy Spring Process Model
A fuzzy process model of the spring needs to provide the qualitative state information
necessary to dampen the vibrations of the plant and achieve stability. This can be achieved
by abstracting the quantitative state of the Benchmark plant to just one qualitative state that
indicates whether the spring is at its neutral length and whether the spring is in the process
of compressing or elongating. This fuzzy process model requires that any estimate provided
by a linear filter of the plant's quantitative state effectively captures
1. the timing of the spring relaxation and
2. the direction of motion of the spring (e.g. compressing and stretching).
Because the fuzzy process model only requires accurate estimates of state with respect to
these two metrics, the linear filter does not need be optimal (e.g. Kalman filter), but can be a
sub-optimal filter derived with robust filter methodologies.
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ROBUST FUZZY CONTROLLER FOR BENCHMARK
Table 2. Rules for determining spring state.
1. If (spring_length_estimate is negative) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
2. If (spring_length_estimate is small_negative) and
(delta_spring_length_estimate is negative) then
(spring_state is compressing_fast_with_neutal_spring)
If (spring_length_estimate is small_negative) and
(delta_spring_length_estimate is sm_neg) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
4. If (spring_length_estimate is small_negative) and
(delta_spring_length_estimate is zero) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
5. If (spring_length_estimate is small_negative) and
(delta_spring_length_estimate is sm_pos) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
6. If (spring_length_estimate is small_negative) and
(delta_spring_length_estimate is positive) then
(spring_state is stretching_fast_with_neutal_spring)
7. If (spring_length_estimate is zero) and
(delta_spring_length_estimate is negative) then
(spring_state is compressing_fast_with_neutal_spring)
8. If (spring_length_estimate is zero) and
(delta_spring_length_estimate is sm_neg) then
(spring_state is compressing_fast_with_neutal_spring)
9. If (spring_length_estimate is zero) and
(delta_spring_length_estimate is zero) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
10. If (spring_length_estimate is zero) and
(delta_spring_length_estimate is sm_pos) then
(spring_state is stretching_fast_with_neutal_spring)
11. If (spring_length_estimate is zero) and
(delta_spring_length_estimate is positive) then
(spring_state is stretching_fast_with_neutal_spring)
12. If (spring_length_estimate is small_positive) and
(delta_spring_length_estimate is negative) then
(spring_state is compressing_fast_with_neutal_spring)
13. If (spring_length_estimate is small_positive) and
(delta_spring_length_estimate is sm_neg) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
14. If (spring_length_estimate is small_positive) and
(delta_spring_length_estimate is zero) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
15. If (spring_length_estimate is small_positive) and
(delta_spring_length_estimate is sm_pos) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
16. If (spring_length_estimate is small_positive) and
(delta_spring_length_estimate is positive) then
(spring_state is stretching_fast_with_neutal_spring)
17. If (spring_length_estimate is positive) then
(spring_state is not_stretching_or_compressing_with_neutal_spring)
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The fuzzy process model utilizes a qualitative spring state that is specified by a qualitative
partition of the spring length, L = x2 - x1, and the spring length velocity, L = x2 − x1 . A
Mamdani Fuzzy Inference System (FIS) applies seventeen rules, shown in Table 2, to infer
the qualitative spring state, QL from the inputs L and L . The output QL is partitioned into
the following qualitative states mapped with triangular and trapezoidal membership
functions to the interval [-1, 1]:
1. spring is compressing rapidly and is relaxed,
2. spring is compressing and is relaxed,
3. spring is not in State 1, 2, 4 or 5.
4. spring is stretching and is relaxed,
5. spring is stretching rapidly and is relaxed,
The input and output membership functions for the process model are shown graphically in
Figure 13, where L, L and QL are partitioned by five membership functions. The stability
behavior is enabled during output states 1, 2, 4 and 5; these states indicate the spring is
vibrating and transitioning through zero. Noise immunity was improved by requiring that
when L is not zero, but rather small_positive or small_negative, that the level of L be
large, either negative or positive, before QL is set to a state indicate a vibration. Only for
zero
values of L will small_positive or small_negative of L result in an output QL which
indicates that a vibration needs to be suppressed. This is shown clearly in the output
decision surface shown in Figure 14.
2.2 Estimating Plant State
Previous versions of this fuzzy controller used the acceleration and jerk of m2 as estimates
of L and QL (Linder and Shafai 1997). This method works extremely well, but fails when
measurement noise is present. The differentiation which is required to obtain the
acceleration and jerk amplifies any high frequency noise associated with the position
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measurement. In order to promote noise immunity the current controller implementation
utilizes a model based filter.
Ideally, in an attempt to maintain technological consistency, a filter based on a fuzzy model
of the plant should be used. However, existing fuzzy estimators are hybrid systems, with
Tanaka observer using fuzzy reasoning about quantitative linear plant models to infer a
quantitative state (Tanaka and Sano 1994). Until a full fuzzy filter is developed the state
must be estimated by a linear estimator. Consider a linear filter for the dynamic system of
order n described by the following equations
x = Ax + Bu
y = Cx
( 50)
where u(t) are the p inputs and y(t) the m outputs, A, B and C are (n × n), (n × p) and (m ×
n) matrices respectively. The filter has the form
x = Ax + L( y − Cx ) + Bu
( 51)
where x is the estimate of the state x. If the system (A, C) is observable then the constant
L can be selected so that (A - LC) is asymptotically stable and x ( t ) asymptotically
approaches x(t). A Kalman filter uses L that minimizes the mean square error of the filter
whereas H∞ filters use L which minimizes the supremum of the estimation error. However,
we use a form of the common Proportional-Integral (PI) Kalman filter (Niemann, Stoustrup
et al. 1995; Linder and Shafai 1997; Saif 1997), the Proportional Fading-Integral (PFI)
Kalman, where the integral action allows the filter to estimate and robustly reject plant
perturbation. The PFI Kalman filter is described by
xˆ = Axˆ + L( y − Cxˆ ) + Bu + B I v
v = K I ( y − Cxˆ ) + K F v
( 52)
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ROBUST FUZZY CONTROL
where the constant KI determines the time constant of the integral action while BI reflects
the form of the plant perturbations or disturbance injection points.
The structure of the spring process model allows the use of linear estimator that is nonoptimal with respect to measurement noise, so long as the estimator can predict accurately
the times when the spring is relaxed and the direction of vibration at these periods of spring
relaxation. Because of this relaxation in performance, it is easier to find a combination L,
BI, KF and KI which performs robustly in conjunction with the spring process model over a
wide range of spring constants. Simulations show that the use of the robust PFI Kalman
filter, rather than the standard P Kalman filter, doubles the stability range of the QRC fuzzy
compensator.
2.3 Fuzzy Compensator
The fuzzy compensator integrates both the stability and tracking behavior. First, a single
Stability Behavior was developed and qualified, and then two tracking behaviors were
developed. As when designing linear controllers, a tradeoff is required between maintaining
the stability robustness of the Stability Behavior and increased tracking performance.
Tracking Behavior A limits its control action to when the spring length is small, minimizing
the interference with the Stability behavior, while Tracking Behavior B adds additional rules
to improve the settling time performance, but at the expense of reducing stability
robustness.
The Stability Behavior requires five rules while Tracking Behavior A required an additional
three rules and Tracking Behavior B requires five rules. These rules used a combination of
six (seven when Tracking Behavior B) Sugeno output functions where the output functions
are a linear combination of the position and velocity inputs to the FIS.
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Table 3. Coefficients for output equations used by vibration suppression behavior.
big
b pos
big
b vel
big
baccel
small
baccel
-1
-3
8
-0.6
2.3.1 Stability Behavior
Stability behavior is realized with 5 fuzzy rules that map the 5 fuzzy partitions of QL to five
output membership functions. These rules are shown at the beginning of Table 4. Vibrations
are suppressed using the following five output functions:
big_stop_spring_stretching
big
big
big
y = b pos
x + bvel
x + baccel
( 53)
small_stop_spring_stretching
small
y = baccel
( 54)
zero
y =0
( 55)
small_stop_spring_compressing
small
y sssc = baccel
( 56)
big_stop_spring_compressing
big
big
big
ybssc = b pos
x + bvel
x − baccel
( 57)
These output equations use four constants whose values are given in Table 3. Actual
big
small
vibration suppression is achieved only by the bias terms baccel
and baccel
. These terms
generate a pulse which opposes the stretching or compression of the spring. The remaining
terms bias the amplitude of the pulse, so that the pulse aids in zeroing both the position and
velocity of the masses.
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Table 4. Fuzzy Rules for Controlling Plant
Rules to supress vibrations
1. If (spring state is compressing_fast_with_neutal_spring) then
(control_output is big_stop_spring_stretching)
2. If (spring state is compressing_slowly_with_neutal_spring) then
(control_output is small_stop_spring_stretching)
3. If (spring state is not_stretching_or_compressing_with_neutal_spring)
then
(control_output is zero)
4. If (spring state is stretching_slowly_at_zero_accel) then
(control_output is small_stop_ spring_compressing)
5. If (spring state is stretching_fast_with_neutal_spring) then
(control_output is big_stop_ spring_compressing)
Additional rules to achieve tracking of m2
6. If (position_error is BigNegative) then
(control_output is zero_large_position)
7. If (position_error is negative) and (spring_length is tinyBell) then
(control_output is zero_small_position)
8. If (position_error is zero) and (velocity is zero) then
(control_output is zero)
9. If (position_error is positive) and (spring_length is tinyBell) then
(control_output is zero_small_position)
10. If (position_error is BigPositive) then
(control_output is zero_large_position)
2.3.2 Tracking Behavior
Tracking behavior has the conflicting goal of settling the plant output so that y< 0.1 after
15 seconds and interfering as little as possible with the stability robustness provided by the
stability behavior. Two versions of the tracking behavior were developed: Tracking
Behavior A which attempts to minimizes any detrimental interaction with the Stability
Behavior and Version B which sacrifices stability, but attempts to minimize the peak
overshoot and settling time for y for the nominal plant with k = 1.
Tracking Behavior A was implemented with Rule 7 to 9, from Table 4. Behavior A uses
these three rules to implement a Sugeno Fuzzy PD controller which becomes active only
when the spring length becomes small. This gating of the tracking behavior minimizes any
detrimental affect the tracking behavior may have on the stability behavior. A smooth gating
is obtained by using a smooth bell curve, the tinyBell membership function, instead of a
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Table 5. Coefficients for output equations used by Tracking Behavior A and B.
psmall
dsmall
Behavior A
-0.4
-2.1
Behavior B
-0.25
-2.5
plarge
dlarge
-0.75
-1.2
simpler triangle membership function. The width of tinyBell at half-height is smaller than
the equivalent zero membership function used by the spring process model, but unlike a
triangle membership function decays smoothly to zero.
Tracking Behavior B was designed so that when there are large position errors the tracking
behavior is superimposed directly on the stability behavior, with the assumption that
stability and vibration suppression are less important for large errors. This modification of
Behavior A requires an additional two rules that are activated when y is large. These
additional two rules are used to increase the PD controller gains when the output error is
large, irrespective of whether the spring length is small, but with the detrimental effect of
reducing the efficacy of the stability behavior.
A Sugeno PD controller was used because it has the advantage of providing a more
compact representation than an equivalent Mamdani fuzzy PD controller. The controller
was implemented with output membership function which consist of a linear combination of
the position and velocity of m2. Tracking Behavior A uses the single output membership
function
zero_small_position
y sssc = psmall x + d small x
( 58)
while Tracking Behavior B uses the additional output membership function
y sssc = pl arg e x + d l arg e x
zero_large_position
( 59)
Table 5 gives the values of the constants used in the output equation for both Tracking
Behavior A and B.
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I
ACTION
Chapter 6
Simulation
Results for
Integral Action
The simulation results presented in this chapter validated the theoretical developments
presented in Chapter 4 for integral action. The chapter presents simulations that validate the
disturbance estimation and accommodation properties of the integral action for the PI
observer and the PI adaptive observer and PFI observer..
Simulations have been performed using Matlab Version 4.2 and Simulink Version 1.3.
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SIMPLE INTEGRAL ACTION
1. Simple Integral Action
The disturbance rejection properties of the integral action used by the PI versions of both
the Luenberger and adaptive observer are characterized using the Adaptive Observer
Benchmark described in the Preliminary Theoretical Results Chapter. Simulink Simulations
were developed to validate the disturbance rejection of properties of the two observer types
in both the open loop and closed-loop regulator configuration.
Design Scenario 1 was utilized to compare the performance of open loop observers with
and without integral action, while Design Scenario 2 was utilized to compare the
performance of closed loop observer-based regulator with and without integral action. The
two Simulink simulations for the adaptive observers are shown in Figure 15. The
equivalent simulation for the non-adaptive observers does not contain the “plot a and b
estimates” block.
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SIMULATION RESULTS FOR INTEGRAL ACTION
P and PI Open Loop
Disturbance Rejection
Output Number 1
u(t) = 5cos(t) +
5cos(2.5t)
+
x' = Ax+Bu
+
Mux
y = Cx+Du
input m3
Plant to Identify
Output Number 2
u(t) = squarewave
0.159 Hz 5 to -5 pp
K
input
disturbance
step
+
+
output
K
C
Plot x's
PI Adaptive
Observer
Sine Wave
P Adaptive
Observer
plot a and b
estimates
(a)
P and PI Regulator
Disturbance Rejection
Mux
Output Number 1
u(t) = 5cos(t) +
5cos(2.5t)
m2
Integral
error
Output Number 2
u(t) = squarewave
0.159 Hz 5 to -5 pp
PI Regulator
plot a and b
estimates
K
common
input
disturbance
for x1
step
P Regulator
Sine Wave
control
results
+
+
+
Mux
input
m5
input
plus disturbance2
control2
x' = Ax+Bu
y = Cx+Du
Plant to Identify
K
K_lqr
(b)
Figure 15. Simulink simulation for comparing the effect of integral action on the
disturbance rejection properties of open-loop (a) and closed-loop observers (b).
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SIMPLE INTEGRAL ACTION
1.1 PI Observer for Kudva Plant
The simulation for non-adaptive observers used the Simulink model of a Luenberger
observer shown in Figure 16. The figure show the configuration of a Luenberger observer
with integral action. The Integral Action block implements the integral action using an
integrator and constant block. Figure 17 shows a detailed Simulink model of the observerbased regulator. The observer output is multiplied by the constant block K_lqr to obtain the
control output u.
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SIMULATION RESULTS FOR INTEGRAL ACTION
1
input
disturbance
2
unmeasured
x1 input
disturbance
+
+
+
-
1
x plant
x' = Ax+Bu
y = Cx+Du
Mux
Plant to Identify
3
unmeasured
x2 input
disturbance
K
C
K
K_lqr
plot state
2
x estimates
+
+
4
output
Disturbance
PI Observer
Figure 16. Details of the Simulink simulation of Observer-based regulator block.
1
plant input
u
2
plant ouput
y = x1
K
B System
Constant
K
L- Lunenburger
Constant
+
+
+
+
1
X estimate
1/s
Integrator
K
A- LC
K
C
Integral
action
Figure 17. Details of the Simulink model of the PI Observer block.
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SIMPLE INTEGRAL ACTION
0.4
v2
0.4
P
0.3
0.3
v1
0.2
0.2
0.1
0.1
0
0
1
2
3
Time (second)
4
(a)
0
0
PI
1
2
3
Time (second)
4
(b)
Figure 18. Response of open-loop P and PI observer to a step input disturbance, din
= 0.2, use in Adaptive Observer Benchmark plant. Integral offset, v, from the PI
observer is shown in (a). (b) shows the cumulative error estimating the hidden
state x2 for both the P and PI observer.
1.1.1 Open Loop Observer
In the open loop configuration the PI observer was able to reject the step plant input
disturbance by utilizing the integral offset v. Figure 18 (a) reveals that after 3 seconds the
values for v begin to settle to their final values. At this point the estimates of by for the PI
observer converge, As shown in Figure 18 (b) the cumulative estimation error for the
hidden state x2 reach a constant value for the PI observer while for the P adaptive observer
the cumulative error is unbounded.
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SIMULATION RESULTS FOR INTEGRAL ACTION
0.5
0.3
v2
0.25
0.4
0.3
0.2
v1
0.15
0.2
PI
0.1
0.1
0
P
Full State
Feedback
0.05
0
1
2
3
Time (second)
4
(a)
0
0
1
2
3
Time (second)
4
(b)
Figure 19. Response of closed-loop P and PI observer to a step input disturbance, din =
0.2, using the Adaptive Observer Benchmark plant. Integral offset, v, from the PI
observer is shown in (a). (b) shows the cumulative error estimating the hidden state
x2 for both the P and PI observer.
1.1.2 Closed Loop Regulator
In the closed loop regulator configuration the PI observer was able to reject the step plant
input disturbance faster than in the open loop case; after only several seconds the integral
offset v shown in Figure 19(a) converges. During this initial settling time for v the PI
regulator fails to track the performance of the full state feedback regulator. After settling
the graph of cumulative error in Figure 19 (b) shows that the PI regulator fully recovers the
performance of the optimal full state feedback regulator, while the P regulator maintains a
proportionally larger error and never achieves full loop recovery. As revealed in Figure 26
this is in part because the P observer has large errors in estimating the hidden state x2 while
the estimates of the PI observer converge quickly. In this regulator configuration the P
adaptive observer has additional difficulties in estimating plant parameters. Figure 20 shows
that the parameter estimates for a using the P observer are unstable and diverge quickly
from their nominal values. In contrast, the estimates based on the PI observer converge
slowly to their nominal values.
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SIMPLE INTEGRAL ACTION
x2
0.1
x2
0.05
0
0.05
0
-0.05
-0.05
-0.1
-0.15
x2
x2
-0.1
-0.2
0
1
2
3
Time (second)
(a)
4
0
1
2
3
Time (second)
4
(b)
Figure 20. Estimates of the hidden state x2 generated by a closed-loop PI (a) and the P
(b) observer.
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K
C
+
e1
1
X estimate
PI Observer
1
plant input
u
3
b estimate
w1 + w2
2
a estimate
Identifier:
calculate
estimate of a and b
Aux Signal
Generation
2
plant ouput
y = x1
q
Aux Signal
Generation
v
Figure 21. Details of a Simulink simulation of a PI Adaptive Observer.
1.2 PI Adaptive Observer for Kudva Plant
The simulation for non-adaptive observers used the Simulink model of an adaptive
observer shown in Figure 21. Integral action is added to the adaptive observer by adding
integral and constant blocks to the PI Observer block of the model. A Simulink model of an
adaptive observer-based regulator is obtained by substituting and adaptive observer for the
Luenberger observer in Figure 16.
The disturbance rejection properties of the PI adaptive observer is evaluated with a step
measurement input disturbance, din of magnitude 0.2, an integral proportionality constant,
KI, of -4. The initial simulations used these adaptive gains from Kudva and Narendra
(Kudva and Narendra 1973) for both the P and PI adaptive observers,
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SIMPLE INTEGRAL ACTION
Γ1 =
140
0
0
75
, and Γ 2 =
5
0
0 7.8
.
( 60)
but the estimates of a2 and b2 failed to converge for the PI adaptive observers. Convergence
however was achieved for both benchmark scenarios after increasing the gains for the PI
adaptive observer to
Γ
PI
1
= 140
0
0
150
, and = 5
0
( 61)
0 15
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SIMULATION RESULTS FOR INTEGRAL ACTION
12
v2
0.4
10
0.3
v1
6
0.2
PI
4
0.1
0
P
8
2
0
0
10
20
30
Time (second)
(a)
40
0
10
20
30
Time (second)
40
(b)
Figure 22. Response of open-loop P and PI adaptive observer to a step input
disturbance, din = 0.2, use in Adaptive Observer Benchmark plant. Integral offset, v,
from the PI adaptive observer is shown in (a). (b) shows the cumulative error
estimating the hidden state x2 for both the P and PI adaptive observer.
1.2.1 Open Loop Adaptive Observer
In the open loop configuration the PI adaptive observer was able to reject the step plant
input disturbance by utilizing the integral offset v. Figure 22 (a) reveals that after 20
seconds the values for v begin to settle to their final values. At this point the estimates of
both state and plant parameters for the PI adaptive observer begin to converge, as shown in
Figure 22 (b) and
Figure 23 respectively. The estimation error for the plants parameters shown in Figure 24
and the estimation error for the hidden state x2 decays asymptotically to zero for the PI
adaptive observer while for the P adaptive observer the estimation error remains constant
throughout the simulation
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SIMPLE INTEGRAL ACTION
P
1
a
5 .6
5 .4
5 .2
5
4 .8
4 .6
a 1P I
0
10
20
30
T im e ( s e c o n d )
40
(a)
a
1 0 .4
P
2
1 0 .2
10
9 .8
9 .6
PI
2
a
0
10
20
30
T im e ( s e c o n d )
40
(b)
1 .1 5
b 1P
1 .1
1 .0 5
1
0 .9 5
0 .9
b 1P I
0
10
20
30
T im e ( s e c o n d )
40
(c)
2 .2
b 2P
2 .1
2
1 .9
1 .8
b 2P I
0
10
20
30
T im e ( s e c o n d )
40
(d)
Figure 23. Parameter estimates generated by the PI (solid line) and P adaptive
observer (dotted line) for Scenario 1 of the Adaptive Observer Benchmark.
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P o bse rve r:
b1
b2
P I o bse rve r:
b1
b2
8
6
4
2
0
0
10
20
30
T im e (second)
40
(a)
P o bs erver:
b1
b2
P I o bs erver:
b1
b2
3
2.5
2
1.5
1
0.5
0
0
10
20
30
Time (second)
40
(b)
Figure 24. Cumulative error for estimating a (a) and b (b) system parameters.
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SIMPLE INTEGRAL ACTION
0.5
1.2
0.4
1
v2
0.3
P
PI
0.8
0.6
0.2
0.4
0.1
0
v1
0
5
10
15
Tim e (second)
Full State
Feedb ack
0.2
20
0
0
(a)
5
10
15
Time (second)
20
(b)
Figure 25. Response of closed-loop P and PI adaptive observer to a step input
disturbance, din = 0.2, use in Adaptive Observer Benchmark plant. Integral offset, v,
from the PI adaptive observer is shown in (a). (b) shows the cumulative error
estimating the hidden state x2 for both the P and PI adaptive observer.
1.2.2 Closed Loop Regulator - Step Disturbance
In the closed loop regulator configuration the PI adaptive observer was able to reject the
step plant input disturbance faster than in the open loop case; after only several seconds the
integral offset v shown in Figure 25(a) converges. During this initial settling time for v the
PI regulator fails to track the performance of the full state feedback regulator. After settling
the graph of cumulative error in Figure 25(b) shows that the PI regulator fully recovers the
performance of the optimal full state feedback regulator, while the P regulator maintains a
proportionally larger error and never achieves full loop recovery. As revealed in Figure 26
this is in part because the P adaptive observer has large errors in estimating the hidden state
x2 while the estimates of the PI adaptive observer converge quickly. In this regulator
configuration the P adaptive observer has additional difficulties in estimating plant
parameters. Figure 27 shows that the parameter estimates for a using the P observer are
unstable and diverge quickly from their nominal values. In contrast, the estimates based on
the PI observer converge slowly to their nominal values.
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0.15
x2
0.1
0.05
0
-0.05
x
-0.1
-0.15
0
5
2
10
15
Time (second)
20
(a)
x2
0.1
0
-0.1
-0.2
x2
0
5
10
15
Tim e (second)
20
(b)
Figure 26. Estimates of the hidden state x2 generated by the closed-loop PI (a) and
the P (b) adaptive observer used in observer based state feedback controller for the
Adaptive Observer Benchmark, Scenario 2.
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a 1P
5 .2
5 .1
5
4 .9
a
4 .8
0
5
PI
1
10
15
T im e ( s e c o n d )
20
(a)
1 0 .5
a
PI
2
10
9 .5
a
9
8 .5
0
5
P
2
10
15
T im e ( s e c o n d )
20
(b)
b 1P
1 .0 2
1
0 .9 8
b 1P I
0 .9 6
0
5
10
15
T im e ( s e c o n d )
20
(c)
2 .2 5
b 2P
2 .2
2 .1 5
2 .1
2 .0 5
b 2P I
2
1 .9 5
0
5
10
15
T im e ( s e c o n d )
20
(d)
Figure 27. Parameter estimates generated by the PI (solid line) and P adaptive
observer (dotted line) based state feedback controller) for Scenario 2.
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0 .8
v
2
0 .6
v
1
0 .4
0 .2
0
0
5
1 0
1 5
T im e ( s e c o n d )
2 0
(a)
2
1 .5
P
P I
1
0 .5
0
F u ll S ta te
F eedback
0
5
1 0
1 5
T im e ( s e c o n d )
2 0
(b)
Figure 28. Response of closed-loop P and PI adaptive observer to a triangle input
disturbance, din = 0.2, use in Adaptive Observer Benchmark plant. Integral offset, v,
from the PI adaptive observer is shown in (a). (b) shows the cumulative error
estimating the hidden state x2 for both the P and PI adaptive observer.
1.2.3 Closed-Loop Regulator - Triangle Disturbance
This section uses a triangle wave to compares the disturbance rejection properties of the P
and PI adaptive observer-based state feedback controller to a LQR controller using full
state feedback. The input to the compensated plant is u = 5sin(t) + 5sin(2.5t), the
disturbance input is a triangle wave and the PI adaptive observer uses an integral gain of KI
= -8 for these simulations. Figure 30 shows that the estimates of the hidden state x2 by the
PI observer adaptive is significantly more accurate than for the P adaptive observer. This
results in superior performance of the PI adaptive observer-based state feedback controller
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as shown in Figure 28. Figure 28 also shows that the PI adaptive observer is able to cancel
the effects of the disturbance by accurately estimating the input disturbance. Figure 30
shows that the cancellation of the input disturbance allows the PI adaptive observer to
accurately estimate the parameters of the system.
0 .1 5
0 .1
0 .0 5
0
-0 .0 5
-0 .1
-0 .1 5
0
5
1 0
1 5
T im e ( s e c o n d )
2 0
(a)
0 .1
0
-0 .1
-0 .2
-0 .3
0
5
1 0
1 5
T im e ( s e c o n d )
2 0
(b)
Figure 29. Estimates of the hidden state x2 (solid line) and actual state (dashed line)
from the closed-loop PI (a) and the P (b) regulator while disturbed by a triangle wave.
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SIMULATION RESULTS FOR INTEGRAL ACTION
5 .5
5
4 .5
a 1 e s tim a te s
0
5
10
15
T im e ( s e c o n d )
20
(a)
1 .1
1 .0 5
1
b
0 .9 5
0
5
1
e s tim a te s
10
15
T im e ( s e c o n d )
20
(b)
11
10
9
8
a 2 e s tim a te s
7
0
5
10
15
T im e ( s e c o n d )
20
(c)
2 .3
b
2 .2 5
2
e s tim a te s
2 .2
2 .1 5
2 .1
2 .0 5
2
0
5
10
15
T im e ( s e c o n d )
20
(d)
Figure 30. Estimates of a and b parameters from the closed-loop PI (solid line) and
the P (dashed line) Adaptive Observer while disturbed by a triangle wave.
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SIMPLE INTEGRAL ACTION
1.2.4 Fault Estimation and Accommodation
Simulations were performed for the Kudva plant using the four actuator faults described in
the Theoretical Results chapter: of gain mismatch, deadzone, backlash and saturation. The
parameters for each fault are:
Fault Type
gain mismatch
deadzone
Parameter Value
katten = 0.6
kdz = 0.3
backlash
kbl = 1
saturation
ksat = 1
The same observer parameters, except where noted, were used in this section as in the
previous sections in this chapter.
1.2.4.1 Fault Estimation and Accommodation with the PI Observer
We selected an integral gain KI = -50 for the PI observer and use an input u = sin(0.5t) .
Figure 31 (c) shows that the integral action of the PIO effectively estimates the disturbances
caused by the actuator fault, leading to superior regulator performance, shown in Figure 31
(a). The graph of the cumulative output error, Figure 31 (b), shows that the PIO-based
regulator out performs both the PO-based regulator and LQR for all four faults. Only for
the backlash fault does the performance of the LQR approach that of the PIO-based
regulator.
1.2.4.2 Estimation and Accommodation with the PI Adaptive Observer
These simulations used an integral action gain for the PI adaptive observer of KI = -20, an
increase from the gain used by the previous simulations (KI = -4). Figure 32 (b) shows that
the integral action of the PIAO precisely estimates the disturbances caused by the actuator
fault, but with a slight degradation in estimation accuracy over the non-adaptive case. This
degradation in performance can be attributed to inaccuracies in estimating plant parameters.
Figure 32 (a) and Figure 32 (c) shows that even with this reduced estimation accuracy, the
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SIMULATION RESULTS FOR INTEGRAL ACTION
PIAO-based regulator outperforms the PAO-based regulator. The graph of the cumulative
output error, Figure 32 (b) shows, however, that the PIAO-based regulator no longer out
performs, in every case, the LQR using full state feedback. The PIAO is superior for gain
mismatch and saturation faults, slightly better for deadzone and inferior to LQR for the
backlash fault. Noted that nonetheless even with this performance degradation with respect
to LQR, only the PIAO-based regulator has the capability of estimating and characterizing
the actuator fault.
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SIMPLE INTEGRAL ACTION
Σy
y
v
0.06
0.8
0.04
Gain Fault
0.4
0.2
0.02
0.6
0
0
0.4
-0.02
-0.2
-0.04
0.2
-0.06
-0.08
0
5
10
Time (second)
15
20
-0.4
0
0
5
10
Time (second)
15
20
Σy
y
Deadzone Fault
10
Time (second)
15
20
5
10
Time (second)
15
20
5
10
Time (second)
15
20
10
Time (second)
15
20
0.3
0.4
0.02
0.01
0.2
0.3
0.1
0.2
-0.1
0
0
-0.01
-0.02
-0.2
0.1
-0.03
-0.3
-0.04
0
5
10
Time (second)
15
20
0
0
5
10
Time (second)
15
20
Σy
y
0.4
0.8
0.04
0.02
-0.4
0
v
0.06
Backlash Fault
5
v
0.03
0.2
0.6
0
0
0.4
-0.02
-0.2
-0.04
0.2
-0.4
-0.06
-0.08
0
5
10
Time (second)
15
20
0
0
5
10
Time (second)
15
20
Σy
y
Saturation Fault
-0.6
0
-0.6
0
y
0.2
2.5
0.1
2
0
1.5
-0.1
1
-0.2
0.5
-0.3
0
0.6
0.4
0.2
0
-0.2
-0.4
0
5
10
Time (second)
(a)
15
20
-0.6
-0.8
0
5
10
Time (second)
(b)
15
20
0
5
(c)
Figure 31. The performance of the LQR (dashed line), the P observer-based regulator
(gray dashed line) and the PI observer-based regulator in the presence of four common
actuator faults is compared in the first two columns; Column (a) gives the plant output,
and Column (b) the cumulative output error. Column (c) show the disturbance created
by the actuator fault (gray line) and its estimate by the integral action (black line).
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SIMULATION RESULTS FOR INTEGRAL ACTION
Σy
Gain Fault
y
0.1
1.4
0.05
1.2
0.2
0.8
-0.05
0
0.6
-0.1
-0.2
0.4
-0.15
-0.4
0.2
5
10
Time (second)
15
20
0
0
5
10
Time (second)
15
20
Σy
y
0.06
Deadzone Fault
0.4
1
0
-0.2
0
-0.6
0
5
10
Time (second)
15
20
5
10
Time (second)
15
20
5
10
Time (second)
15
20
5
10
Time (second)
15
20
v
0.3
1
0.04
0.2
0.02
0.8
0
0.6
-0.02
0.1
0
-0.1
0.4
-0.04
-0.2
0.2
-0.06
-0.08
0
5
10
Time (second)
15
20
0
0
-0.3
5
10
Time (second)
15
20
Σy
y
-0.4
0
v
1.4
0.1
Backlash Fault
v
0.4
1.2
0.05
0.2
1
0
0.8
-0.05
0.6
0
-0.2
0.4
-0.1
-0.4
0.2
-0.15
0
5
10
Time (second)
15
20
0
0
5
10
Time (second)
15
20
Σy
y
-0.6
0
v
Saturation Fault
0.2
1.5
0.1
0
1
0
-0.1
0.5
-0.2
-0.3
0
5
10
Time (second)
15
(a)
20
0
0
5
10
Time (second)
(c)
15
20
-0.5
0
(b)
Figure 32. The performance of the LQR (dashed line), the P adaptive observer-based
regulator (gray dashed line) and the PI adaptive observer-based regulator in the
presence of four common actuator faults is compared in the first two columns; Column
(a) gives the plant output, and Column (b) the cumulative output error. Column (c)
show the disturbance created by the actuator fault (gray line) and its estimate by the
integral action (black line).
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SIMPLE INTEGRAL ACTION
1.3 PI Kalman Filter for ACC Benchmark
The estimation and accommodation properties of integral action for unknown inputs were
also evaluated with the ACC Benchmark plant. A Kalman filter-based regulator was
perturbed by three separate unknown input forces.
1.3.1 Rejecting Unknown Inputs
The simulation of the Benchmark plant was performed with Simulink ®. Three sets of
simulations were performed with an input disturbance perturbed m2 changing from a unit
step, a sine wave with frequency in the range of 0.5 and 0.2 rad/sec and a nonlinear
disturbance formed from the product of two sine waves. Because the disturbance perturbs
only m2, the PI Kalman filter uses BI = [0 0 0 1/m2]' and has an integral term v and an
integral gain KI that are scalars v and KI..
Measurements for all simulations were corrupted with zero mean Gaussian noise with a
period of 0.01 seconds and a standard deviation of 0.1. The performance of the PI Kalman
filter-based state feedback controller was compare to that of full state feedback and a P
Kalman filter-based state feedback controller. The two Kalman filters use process and
measurement noise covariance matrices
0 0
Q=
and V = 0.01 .
0 0.5
( 62)
This gave a Kalman gain matrix for both the P and PI Kalman filters of K = [0.8997 3.6077
0.5634 6.5076]. The PI Kalman filter was constructed simply by augmenting the P Kalman
filter with the integral term, using a gain KI = -10. Additionally, all three controllers use the
same LQR gain matrix KLQR = [0.8997 3.6077 0.5634 6.5076]'. Both matrices were
calculated with the standard Matlab functions.
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SIMULATION RESULTS FOR INTEGRAL ACTION
y
v
P
v
0.8
1
0.6
PI
0.4
0.2
0
0
disturbance
0.5
LQR
10
20
30
40
Time (second)
(a)
0
0
10
20
30
40
Time (second)
(b)
Figure 33. The response to a step disturbance (a) for Full State Feedback (LQR), P
Kalman filter based controller and the PI Kalman filter controller. (b) gives the
integral action for a integral gain KI = -10.
1.3.1.1 Step Disturbance
The PI Kalman filter improves the rejection of a step input disturbance by canceling the step
with an integral action that approximates the step. Figure 33 shows that this disturbance
cancellation allows the PI Kalman filter-based feedback controller performance to approach
that of the LQR regulator, which has noise free access to the plant state.
1.3.1.2 Sine Disturbance
The performance of these three controllers in rejecting sine wave input disturbance can be
partially deduced from their rejection of step disturbances. Figure 34 shows that maximum
amplitude of the system output when perturbed to sine wave disturbances of frequency of
0.2, 0.4 and 0.5 rad/sec. Figure 34 also shows that for each disturbance frequency the
integral action of the PI Kalman filter is able to accurately estimate the input disturbance.
The PI controller’s performance is reduced to the level of the P controller when the
frequency is raised to 0.8 rad/sec and is unstable for a frequency of 0.9 rad/sec.
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SIMPLE INTEGRAL ACTION
Plant Output
PKF-based
PIKF-based
LQR
Disturbance and
estimate v
y
disturbance
integral, v
v
1
0.5
0.5
0
0
-0.5
-0.5
-1
0
10
20
(a)
30
40
y
v
0.5
0
10
20
(b)
30
40
0
10
20
(d)
30
40
1
0.5
0
0
-0.5
-0.5
-1
0
10
20
(c)
30
40
y
v
1
0.5
0.5
0
0
-0.5
-0.5
-1
0
10
20
30
40
Time (second)
(e)
0
10
20
30
40
Time (second)
(f)
Figure 34. Plant output, y, and integral action, v, for three frequencies of sine wave
disturbance: (a) and(b) 0.5 rad/sec; (c) and (d) 0.4 rad/sec; and (e) and (f) 0.2 rad/sec.
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SIMULATION RESULTS FOR INTEGRAL ACTION
1.3.1.3 Nonlinear Disturbance
A nonlinear disturbance was created from the product of two sine waves, The disturbance
w = 2.0 sin ( 0.02t) sin ( 0.05t) has a maximum amplitude of 1.5 and therefore generates a
larger compensated output, as shown in Figure 35. Figure 35 also shows that the PI Kalman
filter-based state feedback controller achieves almost complete loop transfer recovery, while
the conventional P controller has more than twice the error in plant output than full state
feedback.
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SIMPLE INTEGRAL ACTION
y
P
1
0 .5
0
F u ll S ta te
F e ed b a c k
-0 .5
-1
PI
0
50
T im e (s e c o n d )
100
(a)
v
1 .5
1
0 .5
0
- 0 .5
-1
- 1 .5
0
50
T im e ( s e c o n d )
100
Total Control Error
(b)
60
P
40
PI
20
0
F u ll S tate
F ee d b a ck
0
50
T im e (s e c o n d )
100
(c)
Figure 35. The response of the plant output y to a nonlinear disturbance is given
in (a). (b) shows the disturbance (gray line) and the resulting integral action v
(solid line). (c) compares the cumulative control error for the three regulators.
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SIMULATION RESULTS FOR INTEGRAL ACTION
2. Fading Integral Action
The previous section evaluated the properties of integral action with the Benchmark plant.
Nonetheless, the PI observer-based regulator is not able to solve the impulse rejection
Benchmark problem. Whereas the PI observer can compensate for the disturbance caused
by a spring parameter mismatch, it can not also accommodate the unmeasured impulse
input. The following sections show that the PFI variant of the PI observer can discount the
effects of the impulses, and provides a solution for the Benchmark problem that is superior
to that given by the P observer-based controller.
2.1 PFI Kalman filter for the ACC Benchmark
Measurements for all simulations were corrupted with zero mean Gaussian noise with a
period of 0.01 seconds and a standard deviation of 0.1. The performance of the PFI Kalman
filter-based state feedback controller was compare to that of full state feedback, and both a
P and PI Kalman filter-based state feedback controller. The three versions of the Kalman
filter all used a model based on the nominal plant, with m1 = m2 = k = 1. The Kalman filters
use process and measurement noise covariance matrices
p 0
Q=
and V = 0.01 .
0 0
( 63)
where p is varied from 0.1 and 2000. The PI Kalman filter was constructed simply by
augmenting the P Kalman filter with the integral term, using a gain KI = -20. Additionally,
all controllers use the same LQR gain matrix KLQR = [0.8997 3.6077 0.5634 6.5076]'.
Both matrices were calculated with the standard Matlab functions.
Integral action was used to compensate for spring constant perturbations with a BI = [ 0 0 1
1] '. The suppression of offsets to the integral actions by transients was tested with a unit
pulse disturbance at time t = 0 to m2. These simulation were performed with Simulink .
PH
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FADING INTEGRAL ACTION
y
y
KF = 0.01
p = 0.1
0.5
0.5
0
0
p = 1000
-0.5
0
5
10
15
-0.5
0
KF = 100
5
Time (sec)
(a)
ε
ε
0.5
0.5
0
0
-0.5
-1
-1
-1.5
0
-1.5
0
10
Time (sec)
KF = 100
-0.5
p = 0.1
(c)
15
(b)
p = 1000
5
10
Time (sec)
15
KF = 0.01
5
10
15
Time (sec)
(d)
Figure 36. The effects of process noise and integral fading on the rejection of a unit
impulse to m2 for a perturbed plant with spring constant k =0.8. (a) shows the
output for a PI Kalman Filter-based controller over a range of p while (b) shows the
output of the correspond PFI Kalman Filter-based controller (KI = -20) for a range
of KF. (c) and (d) give the respective error in estimating the state x1.
2.1.1 Rejecting transients
A PI Kalman Filter-based controller was designed for the nominal plant and tested on a
perturbed plant with a spring constant, k = 0.8. The efficacy of the fading term and high
modeled process noise in rejecting transients was compared for a range of KF and p. Figure
36 (a) and (b) show that a P Kalman Filter-based regulator with high process noise and a
PFI Kalman Filter-based regulator can achieve comparable output performance. However
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CHAPTER 6
SIMULATION RESULTS FOR INTEGRAL ACTION
Figure 36 (c) and (d) clearly shows that the estimation error for the PFI Kalman filter is
much smaller than for the P Kalman Filter. Increasing the process noise enhances robustness
of the filter, but has the detrimental effect of creating large estimation errors.
2.1.2 Estimating Plant Perturbations with PFI Kalman Filter
The Benchmark problem requires that the integral term of the PFI Kalman filter both (1)
estimate the perturbation ∆x caused by changes in spring constants and (2) simultaneously
reject the effects of an impulse disturbance to m2. Figure 37 (d) shows that for a perturbed
plant, with k = 0.8, the integral action BIv after seven seconds discounts the effects of the
impulse disturbance and begins to effectively estimates the perturbation term ∆x. The fading
integral action allows the PFI Kalman filter-based regulator to achieve superior regulator
output over the standard P Kalman filter-based regulator, shown in Figure 37 (a), and
improves the estimation accuracy of the filter, shown in Figure 37 (c), while requiring only a
nominal increase in compensator effort, Figure 37 (b).
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FADING INTEGRAL ACTION
u
y
1
PFIKF
6
PKF
LQR
PKF
4
2
0.5
0
0
-2
-4
-0.5
PFIKF
-1
0
LQR
-6
5
10
15
0
5
10
Time (sec)
Time (sec)
(a)
(b)
ε
15
0.8
PFIKF
0.6
1
BIv
0.4
0.2
0.5
0
0
-0.2
-0.6
PKF
-1
0
∆x
-0.4
-0.5
5
10
Time (sec)
(c)
15
-0.8
0
5
10
15
Time (sec)
(d)
Figure 37. A comparison of the P Kalman Filter and PFI Kalman Filter-based
regulators and a LQR with full state feedback for a perturbed plant with spring
constant k =0.8. (a) shows the output y, (b) the regulator output u, (c) the respective
estimating error, ε, for state x1 and (d) the estimate BIv of the perturbation term ∆x
from the PFI Kalman Filter.
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CHAPTER 6
SIMULATION RESULTS FOR INTEGRAL ACTION
80
p = 2000
60
p = 100
p = 10
p=1
40
p = 0.1
KI = -90
20
KI = -20
0
0.5
1
1.5
Spring Constant (k)
2
Figure 38. The peak regulator output versus plant spring constant for a P Kalman
Filter -based controller with several levels of process noise and for a PFI Kalman
Filter-based regulator with p = 0.1 for two levels of integral action.
2.1.3 Increasing Robustness
The robustness to plant perturbation of a Kalman filter-based regulator can be increased by
using large modeled process noise. The tradeoff between robustness to plant perturbation
and the level of actuator effort required to achieve this robustness is shown in Figure 38. As
an example, a ten fold increase in process noise, from 10 to 100, can increase the robustness
to perturbation by 50%, but at the expense of doubled compensator effort. However, a PFI
Kalman Filter-based regulator with KI = -90, KF = 2.5 and p = 0.1, shows a five-fold
increase in robustness over the PKF using the same p = 0.1, without the increase in
compensator effort. Compensator effort can be further reduced by using a PFI Kalman
Filter-based regulator with smaller integral gain (KI = -20, KF = 2.6 and p = 0.1.
PH
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CHAPTER 7
SIMULATION RESULTS FOR ROBUST FUZZY CONTROL
Chapter 7
Simulation
Results for
Robust
Fuzzy
Control
O, woe is me,
To have seen what I have seen, see what I see!
Hamlet, William Shakespeare
The simulation results presented in this chapter validated the theoretical results for
qualitative disturbance rejection presented in Chapter 3. We validate the robustness of the
Qualitative Robust Control (QRC) developed for the ACC Robust Control Benchmark
described in Chapter 2.
Simulations have been performed using Matlab Version 4.2 and Simulink Version 1.3.
PH
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QUALITATIVELY ROBUST FUZZY CONTROLLER
3
Control Input
u
4
Disturbance
om m1
w2
+
+
f(u)
k/m1
Coulombic
Friction1
*
f(u)
1/m1
*
-
+
+
+
1/s
f(u)
k/m2
*
-
+
+
1/s
+
f(u)
1/m2
1
x1
3
dx1/dt
+
1
Plant
parameters
2
Disturbance
on m2
w1
1/s
*
1/s
2
x2
4
dx2/dt
Figure 39. Simulink model of plant used for Robust Control Benchmark.
1. Qualitatively Robust Fuzzy Controller
The performance of the QRC designed fuzzy controllers were evaluated using the Robust
Control Benchmark. Two QRC fuzzy controller designs were compared to the performance
of controller Design 1 and 3 of Marrison and Stengel (Marrison and Stengel 1995)
described in the Background Chapter. Design 1 stress stability robustness and Design 3
stresses settling time robustness. The four controller designs were tested and results
tabulated for six spring constants: 0.2, 0.5, 1.0, 2.0, 4.0 and 8.0. ts0.1 is the settling time of
plant output x2 to within ±0.1 units of the final value, ts0.05 is the settling time of x2 to
within ±0.05 units of the final value, x2max is the maximum value of the plant output, umax is
the maximum actuator output and Σu15.0 is the total actuator output for the first 15 seconds
of the simulation. ts0.05 is used as a measure of how fast the vibrations of the flexible
structure are dampened. A controller that is proficient at stabilizing the spring will dampen
x2 quickly.
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CHAPTER 7
SIMULATION RESULTS FOR ROBUST FUZZY CONTROL
1
Spring Constant
k
1
Mass One
m1
Clock
Mux
Mux
yout
To Workspace
Parameters
1
Mass Two
m2
Mux2
disturbance
Mux
+
-
reference
Sum
robust
bench mark
Compensator 1
Marrison and Stengel
Mux1
Plant
Output
+
+
Band-Limited
White Noise
Compensator
Input
Compensator
Output
Figure 40. Simulink Simulation of the Robust Control Benchmark system with series
compensation.
The Simulink model of the of the robust control benchmark plant is shown in Figure 39.
The model was designed so that the three plant parameters, m1, m2 and k, can be adjusted
dynamically. This model is used the Simulink Simulation shown in Figure 40 that
implements the design scenarios. Scenario 1 test the disturbance rejection properties of the
controller while Scenario 3 test the step tracking properties of the controller. A unit impulse
was simulated by a pulse of amplitude of 4 units and duration of 0.25 seconds.
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SIMULATION RESULTS
2. Simulation Results
The performance of our QRC fuzzy controller was investigated using computer simulations
for two scenarios: when complete state information was available and when a state observer
was required to estimate the plant state. The QRC controller using Full State Feedback
(FSFB) was benchmarked against the Linear Quadratic Regulator (LQR). The LQR was
selected because it shows the optimal robustness to plant perturbations of any linear
controller (Anderson and Moore 1989). Equivalently, the QRC controller using output
feedback and a robust Kalman Filter to estimate state was compared to the H2 compensators
Comp1 and Comp3. Comp1 design stresses stability robustness while Comp3 stresses
performance robustness.
The compensators were evaluated for their ability to reject an impulse disturbance to m2
using several metrics. tL≤ 0.05, the time it takes L to settle within ±0.05 units of the final
value, is used to measure the stability of a compensated plant. The effect of vibration
suppression on stability robustness was evaluated by comparing the range of spring
constants k for which L settles within ±0.05 units of the final value in less than 15 seconds
to the stability radius of the compensated plant. Tracking performance was measured by
comparing the metric ymax, the maximum value of the plant output, and ty ≤ 0.1, the settling
time of plant output y to within ±0.1 units of the final value. Since stability robustness and
performance is enhanced by increased compensator output, Σu15.0, the total actuator output
for the first 15 seconds of the simulation was measured to insure that comparable levels of
effort were used by all compensators.
All measurements, of both plant outputs and states, were corrupted with zero mean
Gaussian noise with a period of 0.01 seconds and a standard deviation of 0.02. The LQR
and LQG3 controllers use the same LQR gain matrix KLQR = [0.8997 3.6077 0.5634
3
Note that because the Plant's first Markov parameter, C⋅B = 0, full loop transfer recovery can not be
achieved when designing a LQG for the Benchmark (Saberi, Chen et al. 1993).
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SIMULATION RESULTS FOR ROBUST FUZZY CONTROL
u
L
1.5
1
0.5
0.5
0
0
-0.5
-0.5
-1
0
10
20
30
40
0
10
20
Time (seconds)
Time (seconds)
(a)
(b)
30
40
Figure 41. Performance of stability behavior using FSFB (black) and state estimates
derived from a PFI Kalman filter (gray) is compared. The spring length L (a) and
Actuator output u (b) are shown after a unit impulse disturbance to m2 for k = 0.5 to
3.0 in steps of 0.5.
6.5076]'. The PFI Kalman filters used a model based on the nominal plant, with m1 = m2 = k
= 1 and process and measurement noise covariance matrices
1 0
Q=
and V = 0.01 ,
0 0
( 64)
which gives a Kalman gain K = [41.3051 6.4508 29.1938 20.8062]. Integral action, which
compensates for spring constant perturbations, used a distribution matrix BI = [ 0 0 1 1] '
and an integral gain KI = -20.
Both the Kalman gain and the LQR gain were calculated with the standard Matlab
functions. All the fuzzy compensator configurations modeled the actuator as first order
system with T = 40/(s+40). These simulation were performed with Simulink .
2.1 Fuzzy Stability Behavior
The robustness of the stability behavior used in the full fuzzy controller was characterized
for both FSFB and output feedback. Figure 41 shows the response to a unit impulse
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SIMULATION RESULTS
y
L
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
0
10
20
30
40
-1
0
(a)
10
20
30
40
(b)
Figure 42. Comparison of the two tracking behaviors using only output feedback,
where the black lines are from the more robust Tracking Behavior A and the gray
lines are from the better performing Tracking Behavior B . The plant output y (a)
and spring length L (b) are shown after a unit impulse disturbance to m2 for k = 0.6
to 2.0 in steps of 0.4.
disturbance to m2 for a range of spring constants: k = 0.5 to 3.0 in steps of 0.5. Both
compensator configurations show excellent vibration suppression properties. The range of
spring constants for which L< 0.05 after 15 seconds is 0.1 ≤ k ≤ 1000 for FSFB and 0.4
≤ k ≤ 2.3 for the output feedback configuration. As expected, FSFB suppresses vibrations
faster, over a wider range of plant perturbations and with less compensator effort, than
output feedback. The fuzzy stability behavior is so effective in suppressing vibrations, that
even the output feedback configuration suppresses vibrations over a wider range of spring
constants than a LQR using FSFB with equivalent actuator effort; the LQR regulator settles
L only in the range 0.5 ≤ k ≤ 1.6.
2.2 Effect of Tracking Behavior on Stability
The addition of tracking behavior reduces the stability robustness of the compensator.
However, if designed correctly a compensator with the less evasive Tracking Behavior A
should be more robust than the compensator incorporating Tracking Behavior B. The effect
of these two tracking behaviors is compared for the output feedback case in Figure 42.
While Behavior A shows higher peak responses and longer settling times, it is faster and
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CHAPTER 7
SIMULATION RESULTS FOR ROBUST FUZZY CONTROL
more robust in settling L. This corresponds to a 50% larger stability radius for Behavior A
than Behavior B.
The Stability Behavior settles L within 15 seconds to L≤ 0.05 for 0.4 ≤ k ≤ 2.3. The
addition of Tracking Behavior A only marginally reduces the range of k to 0.7 ≤ k ≤ 2.3,
where as the performance oriented Tracking Behavior B reduces the range even further to
0.8 ≤ k ≤ 2.0.
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SIMULATION RESULTS
L
y
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
0
10
20
30
40
0
5
10
Time (seconds)
15
20
Time (seconds)
(a)
(b)
Σu
u
6
1.5
k = 4.5
k = 0.5
5
1
4
0.5
3
0
2
-0.5
-1
k = 0.5
k = 1.0
1
0
10
20
30
40
0
0
10
20
Time (seconds)
Time (seconds)
(c)
(d)
30
40
Figure 43. Performance comparison of two full state feedback controllers: the fuzzy
controller (black and the LQR (gray) and Fuzzy Controller (black) after unit
impulse disturbance to m2 for k = 1, 2, 3, 4. Figure (a) shows plant output y, (b) the
spring length L, (c) the compensator output u, and (d) the cumulative compensator
output Σu (for k = 0.5 to 4.5 in steps of 0.5).
2.3 Fuzzy Control with Full State Feedback
The stability robustness and tracking performance of the full QRC controller with Tracking
Behavior B (QRC B) using FSFB was compared to LQR. Figure 43 superimposes the
output of these two controllers for a range of spring constants after a unit impulse
disturbance to m2. In order to insure a fair comparison, the LQR gain KLQR was selected so
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CHAPTER 7
SIMULATION RESULTS FOR ROBUST FUZZY CONTROL
that the peak LQR output was about the same as for the fuzzy controller; in fact, Figure 43
(c) shows that in the range 1 ≤ k ≤ 4 the LQR produces larger peak compensator output.
While the LQR over this range of k suppresses the effect of disturbance on the plant output
y faster, shown in Figure 43 (a), Figure 43 (b) shows that the LQR compensated Plant has
significantly larger oscillations in L. The superior vibration suppression properties of the
QRC compensator contributes to the significantly larger stability margin of the QRC
compensator: 0.2 ≤ k ≤ 1000 for the QRC A, 0.4 ≤ k ≤ 1000 for the QRC B, versus 1 ≤ k ≤
4 for the LQR. Figure 43 (d) shows that the superior performance of the QRC
compensators is achieved while having a total compensator effort that is significantly
smaller for the QRC controller, except when k = 0.5.
2.4 Fuzzy Control with Output Feedback
The stability robustness and tracking performance of the full QRC controller with output
feedback was evaluated using a PFI Kalman filter to estimate plant states. The QRC
controller performance was compared to LQG using an identical PFI Kalman filter and to
the two Marrison and Stengel Compensators: Comp1 given by
T
=
− 79.3( s − 0.8)( s + 5.7)( s + 0.11)
( s + 3.84 s + 10.24)( s 2 + 6.882 s + 13.69)(s + 0.46)
2
( 65)
and Comp3, given by
T
PH
=
− 8.2( s − 4.7)( s + 3.9)( s + 0.24)
( s + 4.662 s + 13.69)( s 2 + 3.132 s + 7.29)(s + 1.6 )
2
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( 66)
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SIMULATION RESULTS
y
y
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
0
10
20
30
40
-0.5
0
10
20
Time (seconds)
Time (seconds)
(a)
(b)
30
40
30
40
y
y
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
0
10
20
Time (seconds)
(c)
30
40
-0.5
0
10
20
Time (seconds)
(d)
Figure 44. Performance comparison of fuzzy controller using state estimates from a
PFI Kalman filter (black lines) and Comp1, a 5th order H2 compensator from
Marrison and Stengel (gray lines). Figure (a) shows the output response to a unit
impulse disturbance to m2 for k = 0.6 to k = 2.0 in steps of 0.2, (b) for m1 = 0.6 to m1
= 2.0 in steps of 0.2, and (c) for m2 = 0.8 to m2 = 2.0 in steps of 0.2. Figure (d) shows
the tracking of a unit step command for k = 0.6 to k = 2.0 in steps of 0.2.
The stability robustness of the two QRC controllers proves to be far superior to that of the
PFI Kalman filter-based LQG, but less robust than Comp1 and Comp3. The QRC A is
stable for 0.1 ≤ k ≤ 3.0 and QRC B is stable for 0.5 ≤ k ≤ 2.1, while the LQG is for 0.8 ≤ k
≤ 1.4 and Comp1 is stable for 0.5 ≤ k ≤ 5.5. However QRC B has superior tracking
performance than Comp1 in the range of spring constants specified by the Benchmark
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CHAPTER 7
SIMULATION RESULTS FOR ROBUST FUZZY CONTROL
problem, 0.5 ≤ k ≤ 2.0. As shown in Figure 44 (a) QRC B has smaller peak output when
perturbed by an impulse to m2, and as shown Figure 44 (d) has smaller overshoot when
tracking a unit step. Additional, Figure 44 (b) and (c) show that QRC B has superior
tracking performance when m1 and m2 are perturbed.
When comparing metrics for the QRC A, QRC B, Comp1 and Comp3 compensators at k
= 0.5, 1.0 and 2.0, the QRC compensators show better vibration suppression behavior and
comparable compensator output. The following tabulates simulation results for an impulse
disturbance on m2:
Comp1
k
ty≤ 0.1
tL≤0.05
ymax
Σu15.0
0.5
1.0
Fuzzy Controller A
2.0
0.5
1.0
2.0
26. 14.4 14.3 25.3 15.0 24.2
>40 14.5 7.5 26.1 6.7 8.8
1.8 2.1 2.0 1.45 1.05 1.58
2.6 2.8 2.7 3.1 2.2 3.9
Comp3
Fuzzy Controller B
k
0.5
1.0
ty≤ 0.1
tL≤ 0.05
ymax
∞4
10.3 10.2 >40 8.0 19.4
33.4 9.5 >40 8.8 14.9
0.87 1.17 1.38 1.02 1.33
Σu15.0
2.3
2.0
2.8
0.5
5.71
1.0
2.2
2.0
4.96
The table cells with the best results for the nominal plant, k = 1, are highlighted. QRC B has
the best settling time for x2, but takes longer to dampen vibrations in the flexible structure
than QRC A. Comp3 has the smallest peak, but takes longer to settle than QRC B and is
not stable in the range 0.5 ≤ k ≤ 2.0 specified in the Benchmark problem. Figure 45
4
k = 0.95 gives a settling time of 19.4 seconds.
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SIMULATION RESULTS
y
L
Comp1
0.8
1.5
0.6
Comp1
QRC A
1
0.4
QRC B
QRC A
0.2
0.5
0
0
-0.2
Comp1
Comp3
-0.5
0
10
20
Time (seconds)
QRC B
30
40
-0.4
(a)
0
10
20
Time (seconds)
30
40
(b)
Figure 45. Responses to an impulse at m2 for the nominal plant (k = 1.0) for Fuzzy
controllers QRC A and QRC B and the linear controllers Comp1 and Comp3.
Figure (a) shows that the fuzzy compensators have a lower overshoot in the output
while Figure (b) shows that the fuzzy controllers also dampening vibrations faster.
compares the nominal plant(m1 = m2= k = 1) output y and the spring length L response to a
unit impulse disturbance to m2.
2.5 Stability Robustness Comparison
Stability robustness varies widely for the compensators evaluated in this paper as shown
graphically in Figure 46. The stability margins are given by the range of spring constants for
which the compensated plants are stable. The best stability margins are offered by the fuzzy
controllers with FSFB while the worse stability margin is for the PFI Kalman filter-based
LQG. Figure 46 also shows that stability margins correspond closely to the range of spring
constants for which L settles quickly.
The limited robustness of the PFI Kalman filter-based compensator shows that the
robustness of the QRC compensators using output feedback is limited by the accuracy of
the state estimation; the limited robustness of the Kalman filter decreases the stability of the
fuzzy compensators by several orders of magnitude. Note, however, that the combination of
the QRC controller and PFI Kalman Filter is more robust than the LQG.
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CHAPTER 7
SIMULATION RESULTS FOR ROBUST FUZZY CONTROL
Range of Spring Constants which give Stable System
1000
0.1
Stability Behavior with FSFB
1000
0.2
Fuzzy Comp A with FSFB
1000
0.4
Fuzzy Comp B with FSFB
7.3
0
LQR
5
0.5
Comp1
5.5
1
Comp3
2.9
0.1
Stablity Behavior with PFI
2.4
0.3
Fuzzy Comp A with PFI
1.6
0.5
Fuzzy Comp B with PFI
0.6
0.8
PFI KF-based LQG
0
2
4
6
8
10
Spring Constant (k)
Range of spring constants for which L< 0.05 after 15 seconds
1000
0.1
Stability Behavior with FSFB
1000
0.9
Fuzzy Comp A with FSFB
1000
0.7
Fuzzy Comp B with FSFB
1.1
0.5
LQR
2.2
1
Comp1
1.8
1.2
Comp3
1.9
0.4
Stablity Behavior with PFI
1.6
0.7
Fuzzy Comp A with PFI
1.2
0.8
Fuzzy Comp B with PFI
1
PFI KF-based LQG
0
0.1
2
4
6
8
10
Spring Constant (k)
Figure 46. A comparison of stability robustness and vibration suppression robustness
for various Benchmark compensators is made by showing the range of spring
constants for which the compensated system in response to a unit impulse to m2 is
stable and has L
 < 0.05 after 15 seconds. Numeric labels give extent of k.
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CHAPTER 8
CONCLUSION
Chapter 8
Conclusion
That it should come to this!
Hamlet, William Shakespeare
The innovative techniques developed in this thesis contribute powerful new tools to the field
of disturbances rejection. Using these new techniques disturbances caused either by
unknown inputs, plant perturbations or actuator faults can be estimated and accommodated.
The integral action approach, using variants of the PI observer, provide accurate
disturbance estimate that allow for the accommodation and identification of disturbances
given (1) the injection points of the disturbances, (2) at least one independent state
measurement for each disturbance and (3) disturbances with time constants longer than the
time constant of the plant. The qualitative approach, using the original techniques of
Qualitative Robust Control (QRC), shows that controllers built using qualitative models and
behaviors are as effective at accommodating disturbances as controllers built using H2 and
H∞ techniques. The QRC controller requires only a small set of simple linguistic rule, with
all parameters having simple interpretations, to achieve the same level performance as the
H2 and H∞ controllers built with complex mathematical constructs, and using obscure
parameters. The remainder of the chapter will address these conclusion in detail and will
close with a section describing on going and future research efforts in these fields.
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QUANTITATIVE METHODS - INTEGRAL ACTION
1. Quantitative Methods - Integral Action
Our simulations show that integral action accurately estimates disturbances caused by
unknown actuator faults, unmeasured inputs and plant perturbation. Estimation accuracy is
compromised only by a small amount for the adaptive case, when plant parameters are
unknown, but estimates for both adaptive and non-adaptive cases are still accurate enough
for the identification and characterization of the disturbance. This allows the magnitude of
the disturbance to be tracked precisely over time, giving a characterization of the
disturbance dynamics, and allows for the preemptive scheduling of repairs.
In addition, when rejection actuator faults, the accurate disturbance estimates allows a PI
observer-based controller to achieve superior performance as compared to an equivalent
LQR; while the performance of the PI adaptive observer-based controller is degraded
somewhat when compared to their non-adaptive counterparts. Our simulations show,
however, that this degradation is not due to inaccurate estimates of the disturbance, but
rather inaccurate plant parameter estimates. The next two section, provide more details
pertinent to the two major variants of the PI observer, the PI adaptive observer and the PFI
observer, developed for this thesis.
1.1 PI Adaptive Observer
The robustness of adaptive observer-based controller has been enhanced by adding an
integral path to the adaptive observer. As with the PI observer, the PI adaptive observer
uses its integral action to successfully reject input disturbances. Cancellation of the
disturbance by the integral action allows accurate estimation of the plant state and in turn
allows the accurate estimation of plant parameters. The benefits of the integral action are
even more apparent for the closed loop configuration, where the PI , and not the P,
adaptive observer-based regulator achieves full loop recovery for the Kudva plant when the
system is disturbed. Additionally, for the closed loop configuration, the PI adaptive
observer generates accurate parameter estimates, while the conventional P adaptive
observer does not.
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CHAPTER 8
CONCLUSION
Our experiments show that PI adaptive observer-based regulator rejects any disturbance
that has a time constant shorter than the time constant of the integral action; the integral
action can effectively cancel the input disturbance. However, a complex disturbance pattern
in a conjunction with a change in plant parameters will confuse the PI observer and
slowdown the rate of convergence. Indeed, integral action, by itself, reduces the
convergence rate of plant parameter estimates, and requires that the adaptive gains for the
PI adaptive observer to be higher than those for the corresponding P adaptive observer.
1.2 PFI Observer
The robustness of the PI observer-based controller has been enhanced by discounting the
integral action over time. This fading of the integral action allows the PFI Kalman filter to
reject biases to the integral action caused by transitory disturbances with unmodelled
distribution matrices. Using the ACC Benchmark problem, it was shown that the PFI
Kalman filter effectively rejects a rank one plant perturbation with a known distribution
matrix, while also rejecting unmeasured impulse inputs.. The integral action v effectively
estimated the known perturbation term while suppressing the effects of the unmodelled
impulse disturbances. The cancellation of the perturbation by the integral action improves
the filter’s state estimates and results in a PFI Kalman filter-based regulator with
dramatically improve robustness as compared to a conventional P Kalman filter using the
same level of actuator effort.
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QUALITATIVE METHODS - QRC
2. Qualitative Methods - QRC
This thesis shows that fuzzy control, based on qualitative behaviors, can achieve
performance comparable or superior to that achieved by linear control when used for the set
point control of simple plants. This result contradicts the assumption held by many that
intelligent control techniques are only useful for the control of complex plants with complex
control objectives. Our QRC methodology uses a superimposition of qualitative stability
and tracking behaviors, instantiated with fuzzy rules that have clear linguistic
interpretations. Using the ACC Robust Control Benchmark, we successfully demonstrate
that QRC compensators are able to perform comparable to, if not better than, compensators
designed with H2 and H∞ robust control techniques. The QRC methodology allows the
fuzzy compensator to be designed incrementally, with stability behaviors being developed
and evaluated initially and then supplemented with tracking behaviors. As with linear
controllers, the additional tracking requirements degrade the stability robustness of the
initial stability behavior. However, by properly gating the tracking behavior, we are able to
develop a tracking behavior that has minimal adverse impact on stability robustness. A
second more aggressive tracking behavior, while improving tracking performance, degrades
the stability robustness of the full QRC compensator by over 50%.
The limited robustness of the QRC compensator is due in part to the limited robustness of
the PI Kalman filter which is incorporated into our compensator. In fact, we show that
fuzzy Stability Behavior, with its relaxed requirements for state information, extends the
robustness of the Kalman filter-based compensator. The Kalman filter-based QRC
compensator more than quadruples the robustness of a conventional Kalman filter-based
LQG. The limitations induced by the Kalman filter are shown with additional simulations
using a full state feedback version of the Benchmark, where a PI Kalman filter is no longer
needed to estimate hidden states. Direct state access increased the robustness of the QRC
136
CHAPTER 8
CONCLUSION
compensator dramatically. The fuzzy controller has substantially greater stability robustness
than even an optimally robust LQR, when using the same level of actuator effort.
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FUTURE RESEARCH
3. Future Research
No man really becomes a fool until he stops asking
questions.
Charles Steinmetz
As with all dissertations, as many questions remain unanswered as answered. There exists
an abundance of topics for current and future research. Both the integral action and the
QRC approach to disturbance rejection can be broadened and improved.
3.1 Integral Action
Our on going research is extending the use of integral action to reject friction faults and
multiple simultaneous disturbances. Our studies of frictional faults use the friction extension
to ACC Benchmark that were developed for this thesis. Study of the rejection of multiple
simultaneous disturbances requires multiple plant outputs, but our published results for
integral action have been confined to single input, single output systems. We have
developed two extensions to the ACC Benchmark, with multiple outputs, in order to
validate our findings. The first extension used the velocity of m2 along with the original
position measurement of m2. These two measurements allow the rejection two simultaneous
disturbances: an actuator fault and a perturbed spring constant. The second extension adds
a third mass to the plant and uses the position of m2 and m3 as plant outputs.
Further works needs to be performed in characterizing the stability of the PI observers with
respect to a given set of disturbances. Augmentation of the estimation equation with both
the model of the disturbance and the integral action will allow the optimal integral gain for a
set of disturbances to be derived using the solutions to the resulting algebraic Riccatti
equation.
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CHAPTER 8
CONCLUSION
3.2 Qualitative Approach
The use of a linear observer to estimate state severely limits the robustness of our QRC
solution for the ACC benchmark. Several possible solutions to this problem need to be
investigated. The first solution improves the robustness of the linear observer by
dynamically adapting, with fuzzy rules, the plant model used by the observer. A more
elegant solution would be to replace the linear observer and the fuzzy spring process model
with a fuzzy version of a state observer.
The complete validation of QRC requires that it be used for the design of controllers for
other plants. Two plants suggest themselves: a two-mass plant with friction and a threemass plant.
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FUTURE RESEARCH
Ignorant men don't know what good they hold in their hands
until they've flung it away.
Sophocles
One cool judgment is worth a thousand hasty councils. The
thing is to supply light and not heat.
Woodrow Wilson
How far you go in life depends on your being tender with the
young, compassionate with the aged, sympathetic with the
striving and tolerant of the weak and strong. Because
someday in your life you will have been all of these.
George Washington Carver
It matters if you just don't give up.
Stephen Hawking
140
APPENDIX I
QRC C
–
NOISE
Appendix I
QRC Controller –
No Noise
This Appendix presents original results for the fuzzy control of the ACC Benchmark. These
results are weaker than presented in the main text because they assume no sensor noise, do
not model the actuator as a physically realizable element and require more rules to
implement the controller.
Simulations have been performed using Matlab Version 4.2 and Simulink Version 1.3.
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FUZZY CONTROLLER FOR BENCHMARK
1. Fuzzy Controller for Benchmark
QRC will now be used to design a fuzzy controller for the Benchmark, beginning with a
qualitative model of the plant that captures whether the spring is compressing or stretching.
Using the heuristic that a control action is most effective when the spring is in its neutral
position, stability is achieved using a FIS controller that accelerates m1, in the direction that
oppose the motion of the spring when the spring is near its neutral position. Position control
is then achieved with additional rules that are fired after the vibration of the structure have
been dampened. The sections which following expand upon the design of controller.
1.1 Qualitative Modeling of Spring
The qualitative model of the Benchmark must include knowledge of whether the spring is
compressing or stretching. The model utilizes a quantitative spring state that is specified by
a qualitative partition of the spring length, x2 - x1, and the spring length velocity, x 2 − x1 .
The number of qualitative spring states necessary to achieve stability is small. Stabilizing the
plant requires us to know whether the spring is compressing or stretching, since control
actions are most effective when the spring is relaxed and near its neutral position. Therefore
to efficiently stabilize the oscillation of the spring we need to differentiate between the
following qualitative states:
1. spring is stretching while it is relaxed,
2. spring is compressing while it is relaxed, and
3. spring is not in State 1 or 2.
However this partition conflicts with the implementation of tracking, since the rules for
controlling stability will try to zero any small perturbation in spring length that might be
needed to perform tracking. The following additional states are therefore needed to
complete the list of states for the process model:
142
APPENDIX I
QRC CONTROLLER – NO NOISE
1
uf
0.5
0
-0.2
-0.1
0
0.1
0.2
0.3
acceleration
1
uf
0.5
0
-5
0
5
jerk
uf
1
0.5
0
-1
-0.5
0
spring state
0.5
1
Figure 47. Input and output membership function of the Spring
Observer.
4. spring is stretching while spring is slightly less than its neutral length, and
5. spring compressing while spring slightly more than its neutral length.
Indirectly the qualitative spring state can be inferred from the acceleration and jerk of m2.
The fuzzy process model of the spring is implemented by a single Mamdani FIS (Driankov,
Hellendoorn et al. 1996) using the acceleration and jerk of m2 as inputs, qualitative state as
output and the 17 rules shown in Table 6. The input and output membership functions are
shown in Figure 47. Acceleration and jerk are partition by five membership functions, and
the output by seven functions. This implementation has further partitioned spring state 4
and 5 in order to fine tune the fuzzy process model.
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FUZZY CONTROLLER FOR BENCHMARK
Table 6. Rules for determining spring state.
1. If (acceleration is negative)
then (spring state is neutral)
2. If (acceleration is sm-neg) and (jerk is negative)
then (spring state is stretching_fast_near_zero_accel)
3. If (acceleration is sm-neg) and (jerk is sm-neg)
then (spring state is stretching_near_zero_accel)
4. If (acceleration is sm-neg) and (jerk is zero)
then (spring state is neutral)
5. If (acceleration is sm-neg) and (jerk is sm-pos)
then (spring state is compressing_near_zero_accel)
6. If (acceleration is sm-neg) and (jerk is positive)
then (spring state is compressing_fast_near_zero_accel)
7. If (acceleration is zero) and (jerk is negative)
then (spring state is stretching_near_zero_accel)
8. If (acceleration is zero) and (jerk is sm-neg)
then (spring state is stretching_at_zero_accel)
9. If (acceleration is zero) and (jerk is zero)
then (spring state is neutral)
10. If (acceleration is zero) and (jerk is sm-pos)
then (spring state is compressing_at_zero_accel)
11. If (acceleration is zero) and (jerk is positive)
then (spring state is compressing_near_zero_accel)
12. If (acceleration is sm-pos) and (jerk is negative)
then (spring state is stretching_fast_near_zero_accel)
13. If (acceleration is sm-pos) and (jerk is sm-neg)
then (spring state is stretching_near_zero_accel)
14. If (acceleration is sm-pos) and (jerk is zero)
then (spring state is neutral)
15. If (acceleration is sm-pos) and (jerk is sm-pos)
then (spring state is compressing_near_zero_accel)
16. If (acceleration is sm-pos) and (jerk is positive)
then (spring state is compressing_fast_near_zero_accel)
17. If (acceleration is positive)
then (spring state is neutral)
This process model differs from the Tanaka observers, which are hybrid fuzzy observers
that use fuzzy reasoning about linear models of the plant to infer a quantitative state
(Tanaka and Sano 1994). Prior state information and the qualitative state transition diagram
is not used in our implementation and only current measurements are used to determine
state. Future versions will implement a complete process model and in effect incorporate a
fuzzy version of a Kalman filter to improve measurement noise immunity.
144
APPENDIX I
QRC CONTROLLER – NO NOISE
z
s
x, x , x
u
x, x
Fuzzy Spring
State Process
Model
Fuzzy
Compensator
Figure 48. Structure of Fuzzy Controller
1.2 Implementing Stability and Performance Objectives
The control goal consists of stabilizing the plant, rejecting disturbances and tracking a
reference input. Stability can be achieved if the system is internally stable (i.e. all state
variables are bounded). The physical intuition is that the spring oscillations must first be
dampened to achieve stability, and then after stability is achieved the goals of disturbance
rejection and tracking can become paramount. The nonlinear nature of the FIS controller
allows this dichotomy.
Figure 48. Structure of Fuzzy Controller shows a schematic of the Fuzzy controller. It
contains a box that calculates the derivatives of the plant output, a FIS implement the spring
process model and a FIS that implements the controller. The FIS controller was
implemented iteratively, Step 1 stabilizes spring oscillations, Step 2 is augment with
additional rules to control velocity, and Step 3 implements the final control that both
stabilizes the plant and tracks a reference.
Step 1 utilizes the first seven rules shown in Table 7. These rules use the following output
member functions:
big_stop_spring_stretching
PH
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big
y bsss = baccel
( 67)
145
FUZZY CONTROLLER FOR BENCHMARK
big_stop_spring_compressing
big
ybssc = − baccel
( 68)
small_stop_spring_stretching
small
yssss = baccel
( 69)
small_stop_spring_compressing
small
ysssc = − baccel
( 70)
big controls the rate of dampening for large oscillations, while small
The coefficient baccel
baccel
controls the rate of dampening for small oscillations when the acceleration is close to zero.
big
dampening
In general you want the later constant to be larger than the former so that baccel
term does not interfere too much with the tracking behavior of the controller after the plant
is stabilized.
Step 2 controls the velocity of m2 with the addition of the next three rules shown Table 7.
These new rules are fired only after the acceleration of m2 becomes small and the plant is
stabilized. The 10 rules use the following output membership functions:
big_stop_spring_stretching
big
big
ybsss = bvel
x + baccel
( 71)
big_stop_spring_compressing
big
big
ybssc = b vel
x − baccel
( 72)
small_stop_spring_stretching
small
small
yssss = b vel
x + baccel
( 73)
small_stop_spring_compressing
small
small
ysssc = b vel
x − baccel
( 74)
y zv = b vel
x
vel
( 75)
zero_velocity
The original four output membership functions now have an additional velocity term. This
term biases the stability rules, 1 to 7, so that not only will the plant be stabilized, but
stabilized with a smaller velocity.
146
APPENDIX I
QRC CONTROLLER – NO NOISE
Finally, Step 3 controls the position of m2 with the addition of the three final rules shown in
Table 7. The combination of 13 rules now stabilize the plant, control velocity and control
position. They use the following output membership functions:
big_stop_spring_stretching
big
big
big
y bsss = bpos
x + bvel
x + baccel
( 76)
big_stop_spring_compressing
big
big
ybssc = b big
x + b vel
x − baccel
pos
( 77)
small_stop_spring_stretching
small
small
yssss = b small
x + b vel
x + baccel
pos
( 78)
small_stop_spring_compressing
small
small
ysssc = b small
x + b vel
x − baccel
pos
( 79)
zero_velocity
y zv = b vel
x
vel
( 80)
zero_position
yzp = b pos x + b pos x
pos
vel
( 81)
The original four output membership functions now have an additional position term that
bias the stabilization to a state with smaller position.
PH
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147
FUZZY CONTROLLER FOR BENCHMARK
Table 7. Fuzzy Rules for Controlling Plant
Rules to zero oscillation of the spring.
1. If (spring state is stretching_fast_near_zero_accel)
then (control_output is big_stop_spring_stretching)
2. If (spring state is stretching_near_zero_accel)
then (control_output is small_stop_spring_stretching)
3. If (spring state is stretching_at_zero_accel)
then (control_output is small_stop_spring_stretching)
4. If (spring state is neutral)
then (control_output is zero)
5. If (spring state is compressing_at_zero_accel)
then (control_output is small_stop_ spring_compressing)
6. If (spring state is compressing_near_zero_accel)
then (control_output is small_stop_ spring_compressing)
7. If (spring state is compressing_fast_near_zero_accel)
then (control_output is big_stop_spring_compressing)
Additional rules to control velocity of m2.
If then ! "
If ! then ! #
If then ! Additional rules to control position of m2.
If then ! $
If ! then ! %
If then ! 148
APPENDIX I
QRC CONTROLLER – NO NOISE
Table 8. Output Polynomial Coefficients
Coefficients for Design A
b big
pos
big
bvel
big
baccel
small
bpos
small
bvel
small
baccel
vel
bvel
pos
b pos
pos
bvel
Controlling
7
x
x
x
-2
-4
7
-4
7
0.6
-0.5
-1
0.6
-5.5
-1
0.6
-5.5
-2.5
-6.5
-5.5
-2.2
-6.5
Coefficients for Design B
x
-2
-4
2.9
-0.5
-1
0.6
The output coefficients were selected and then tuned iteratively. The coefficient utilized by
the previous steps were carried over to the next control design. Table 8 gives the
coefficients values for two complete controller designs; Design A includes the coefficients
for all three steps. The two coefficients that differ between Design A and Design B are
highlighted. Design B has a smaller stability coefficient than Design A. If our qualitative
model is correct Design B will show reduced stability performance but smaller compensator
effort.
PH
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149
QUALITATIVELY ROBUST FUZZY CONTROLLER
3
Control Input
u
4
Disturbance
om m1
w2
+
+
f(u)
k/m1
Coulombic
Friction1
*
f(u)
1/m1
*
-
+
+
+
1/s
f(u)
k/m2
*
-
+
+
1/s
+
f(u)
1/m2
1
x1
3
dx1/dt
+
1
Plant
parameters
2
Disturbance
on m2
w1
1/s
*
1/s
2
x2
4
dx2/dt
Figure 49. Simulink model of plant used for Robust Control Benchmark.
2. Qualitatively Robust Fuzzy Controller
The performance of the QRC designed fuzzy controllers were evaluated using the Robust
Control Benchmark. Two QRC fuzzy controller designs were compared to the performance
of controller Design 1 and 3 of Marrison and Stengel (Marrison and Stengel 1995)
described in the Background Chapter. Design 1 stress stability robustness and Design 3
stresses settling time robustness. The four controller designs were tested and results
tabulated for six spring constants: 0.2, 0.5, 1.0, 2.0, 4.0 and 8.0. ts0.1 is the settling time of
plant output x2 to within ±0.1 units of the final value, ts0.05 is the settling time of x2 to
within ±0.05 units of the final value, x2max is the maximum value of the plant output, umax is
the maximum actuator output and Σu15.0 is the total actuator output for the first 15 seconds
of the simulation. ts0.05 is used as a measure of how fast the vibrations of the flexible
structure are dampened. A controller that is proficient at stabilizing the spring will dampen
x2 quickly.
150
APPENDIX I
QRC CONTROLLER – NO NOISE
1
Spring Constant
k
1
Mass One
m1
Clock
Mux
Mux
yout
To Workspace
Parameters
1
Mass Two
m2
Mux2
disturbance
Mux
+
-
reference
Sum
robust
bench mark
Compensator 1
Marrison and Stengel
Mux1
Plant
Output
+
+
Band-Limited
White Noise
Compensator
Input
Compensator
Output
Figure 50. Simulink Simulation of the Robust Control Benchmark system with series
compensation.
The Simulink model of the of the robust control benchmark plant is shown in Figure 49.
The model was designed so that the three plant parameters, m1, m2 and k, can be adjusted
dynamically. This model is used by the Simulink Simulation shown in Figure 50 that
implements the design scenarios. Scenario 1 test the disturbance rejection properties of the
controller while Scenario 3 test the step tracking properties of the controller. A unit impulse
was simulated by a pulse of amplitude of 4 units and duration of 0.25 seconds.
PH
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151
QUALITATIVELY ROBUST FUZZY CONTROLLER
1.5
1.5
x1
x2
x1 velocity
x2 velocity
1
1
0.5
0.5
0
0
-0.5
0
5
10
Time(sec)
x1
x2
x1 velocity
x2 velocity
15
20
-0.5
0
5
10
Time(sec)
(a)
15
20
(b)
Figure 51. (a) Position and velocity of m1 and m2 after a unit impulse disturbance on
m2 with k= 4. for Fuzzy Design A. (b) The corresponding output for Design 3.
2.1 Scenario 1 results - impulse on m1 and m2
Scenario 1 of the Benchmark tests the disturbance rejection of an impulse on m2 and plant
noise on m1. Figure 51 shows how the fuzzy controller stabilizes the plant after an impulse
on m2. Design A completely dampens the oscillation of the spring, before finishing
tracking, while the conventional compensator does not zero spring oscillations before the
plant output is bounded by ±0.1. The following tabulates the simulation results for an
impulse disturbance on m2:
Marrison and Stengel 1
Spring Constant (k)
x 2 settling time
0.5
26.9
22.4
1.0
14.9
14.6
2.0
14.3
14.6
4.0
13.9
36.8
Peak (x2max)
Total Effort (Σu15.0)
1.8
2.6
2.1
2.8
2.0
2.7
2.1
3.2
x2 settling time (ts0.1)
0.2
∞
Fuzzy Controller Design A
8.0
∞
0.2
33.1
25.8
0.5
23.8
6.6
1.0
11.0
4.0
2.0
11.8
3.2
4.0
24.7
16.2
8.0
25.8
17.9
3.04
4.1
1.94
2.6
1.09
1.7
0.82
2.8
1.36
11.9
1.55
20.5
152
APPENDIX I
QRC CONTROLLER – NO NOISE
y 2
u 1
Design A
Design B
Design #1
Design #3
1.5
Design A
Design B
Design #1
Design #3
0.5
1
0
0.5
-0.5
0
-0.5
0
5
10
Time(sec)
15
20
-1
0
5
10
Time(sec)
(a)
15
20
(b)
Figure 52. (a) Response of plant output y to a unit impulse disturbance on m2 with k
= 1. (b) The corresponding clipped view of compensator output u with the minimum
values for Design A and Design B being -9.9 and -5.1, respectively.
Spring Constant (k)
0.2
∞
x2 settling time
Marrison and Stengel 3
0.5
1.0
2.0
4.0
∞5
x 2 settling time
7.7
33.5
10.2
13.8
11.1
> 40
Peak (x2max)
Total Effort (Σu15.0)
0.87
2.3
1.17
2.8
1.24
2.8
8.0
0.2
∞
∞
Fuzzy Controller Design B
0.5
1.0
2.0
4.0
8.0
19.4
5.8
11.0
4.5
15.5
7.1
25.1
15.6
31.2
26.8
1.46
2.6
1.13
2.1
1.43
2.1
1.78
2.1
2.20
1.6
The table cells with the best results for the nominal plant, k = 1, are highlighted. Design 3
has the best settling time for x2, but as shown in Figure 51 takes longer to dampen
vibrations in the flexible structure.
Figure 52 shows the plant output y and the compensator output u for the four designs for k
= 1.
5
k = 0.95 gives a settling time of 19.4 seconds.
PH
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153
QUALITATIVELY ROBUST FUZZY CONTROLLER
y 2
u
2
1
1.5
0
1
-1
0.5
0
-0.5
0
-2
Design A
Design B
Design #1
Design #3
5
10
Time(sec)
Design A
Design B
Design #1
Design #3
-3
15
-4
0
20
5
10
15
Time(sec)
(a)
20
(b)
Figure 53. Tracking response to unit step, with plant output y, Figure (a), and
compensator output u, Figure (b).
2.2 Scenario 4 results - Step Response
Scenario 4 of the Benchmark tests the step tracking performance of the compensator.
Figure 53 shows graphically the step response for the nominal plant, while the following
tabulates the simulation results for the tracking response to a unit step:
Spring Constant (k)
x2 settling time
0.2
∞
Marrison and Stengel 1
0.5
1.0
2.0
4.0
x 2 settling time
>40
>40
14.5
14.3
15.2
10.6
15.8
34.1
Peak (x2max)
Max. Effort (umax)
Total Effort (Σu15.0)
2.47
1.1
4.8
1.70
1.1
3.3
1.61
1.1
3.2
1.7
1.1
3.6
Spring Constant (k)
Marrison and Stengel 3
0.5
1.0
2.0
4.0
x2 settling time
0.2
∞
∞
x 2 settling time
12.1
13.7
13.1
34.4
Peak (x2max)
Max. Effort (umax)
Total Effort (Σu15.0)
1.84
1.0
3.6
2.8
1.0
3.6
> 40
Fuzzy Controller Design A
0.5
1.0
2.0
4.0
8.0
0.2
∞
21.3
9.5
8.8
2.0
9.4
0.9
9.2
1.2
7.6
2.0
2.06
4.6
3.0
1.05
2.8
1.2
1.05
3.4
1.1
1.05
2.9
1.0
1.07
8.9
4.6
8.0
0.2
∞
∞
Fuzzy Controller Design B
0.5
1.0
2.0
4.0
8.0
∞
8.0
15.4
4.4
9.6
1.5
10.1
0.7
10.1
0.5
15.7
8.3
1.21
1.0
1.7
1.03
1.0
0.9
1.03
1.0
0.8
1.03
1.0
0.8
1.27
4.6
9.3
154
APPENDIX I
QRC CONTROLLER – NO NOISE
Additional, the following table (where emax is the maximum error) shows results for the
tracking response to a unit ramp for the nominal plant (k = 1):
Fuzzy Controller Design A
Fuzzy Controller Design B
Marrison and Stengel, Design 1
Marrison and Stengel, Design 3
ts0.1
ts0.05
emax
41
13.0
13.2
>40
23.9
5.9
14.6
>40
3.9
1.66
4.44
2.75
2.2.1 Plant Noise Response
Plant noise rejection is tested with both an impulse and band-limited white noise disturbance
on m1. The following tabulates the simulation results for an impulse disturbance
on m1:
Spring Constant (k)
0.2
∞
x2 settling time
Marrison and Stengel 1
0.5
1.0
2.0
4.0
> 40
14.5
13.5
14.1
13.7
14.4
26.6
3.3
2.5
2.8
2.2
2.7
2.1
3.0
x 2 settling time
Peak (x2max)
Total Effort (Σu15.0)
Spring Constant (k)
0.2
∞
x2 settling time
Marrison and Stengel 3
0.5
1.0
2.0
4.0
∞
x 2 settling time
> 40
> 40
11.0
12.8
11.9
42.7
Peak (x2max)
Total Effort (Σu15.0)
1.97
8.3
1.5
2.9
1.24
3.0
8.0
0.2
∞
∞
8.0
0.2
∞
∞
Fuzzy Controller Design A
0.5
1.0
2.0
4.0
8.0
0
0.5
0
0.4
0
0.6
0
0.5
22.5
14.3
0.05
7.0
0.05
7.0
0.08
7.0
0.05
7.2
1.15
17.0
Fuzzy Controller Design B
0.5
1.0
2.0
4.0
8.0
14.0
4.6
20.0
5.0
20.0
6.4
26.4
9.4
26.0
18.6
0.55
2.2
0.96
2.4
0.96
2.3
1.32
3.1
1.74
2.16
Additionally, plant noise rejection is tested by injecting band-limited white noise to the plant
input for 30 seconds. The noise sample time is held constant at 0.5 seconds while the power
level was varied. If the plant output was bounded by ±0.1 for 30 seconds the plant is labeled
bounded, if not it is labeled unbounded. The following table shows a B if the output was
bounded and a U if the output was unbounded:
Spring Constant (k)
p = 0.01
p = 0.001
p = 0.0001
p = 0.00001
PH
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P LINDER
Marrison
and Stengel 1
0.5
1.0
2.0
U
U
U
B
U
U
U
B
U
U
U
B
Fuzzy Controller
Design A
0.5
1.0
2.0
B
B
B
B
B
B
B
B
B
B
B
B
155
APPENDIX II
FIS DEFINITIONS
Appendix II
FIS Definitions
This appendix gives a listing of the files used by Matlab's Fuzzy Toolbox, Revision: 1.13.
These files have the extension's fis, short for Fuzzy Inference System (FIS), and are used
by the Fuzzy Toolbox to describe the FIS. Two sets of FIS are described: the first set
contain the original files used by our 1997 ACC paper (Linder and Shafai 1997), described
in the appendices, and the second set contain the newer version of the FIS described in the
Theoretical Results chapter.
Each version of the Fuzzy compensator has a FIS for the fuzzy process model used to
ascertain the qualitative state of the spring and a controller FIS that implements the
stability and tracking behaviors.
PH
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157
ORIGINAL FIS FILES FROM ACC 1997 PAPER
1. Original FIS Files from ACC 1997 Paper
1.1 FIS Definition for Process Model
[System]
Name='Sugeno Spring Observer for MSM'
Type='mamdani'
NumInputs=2
NumOutputs=1
NumRules=17
AndMethod='prod'
OrMethod='max'
ImpMethod='min'
AggMethod='max'
DefuzzMethod='centroid'
[Input2]
Name='delta_delta_error'
Range=[-5 5]
NumMFs=5
MF1='negative':'trapmf',[-5 -5 -0.2 -0.1]
MF2='sm-neg':'trimf',[-0.2 -0.03 0]
MF3='zero':'trimf',[-0.03 0 0.03]
MF4='sm-pos':'trimf',[0 0.03 0.2]
MF5='positive':'trapmf',[0.1 0.2 5.0 5.0]
[Input2]
Name='delta_delta_delta_error'
158
APPENDIX II
FIS DEFINITIONS
Range=[-5 5]
NumMFs=5
MF1='negative':'trapmf',
[-5 -5 -2 -1]
MF2='sm-neg':'trimf',
[-2 -1 0.0]
MF3='zero':'trimf',
[-1 0 1]
MF4='sm-pos':'trimf',
[0.0 1 2]
MF5='positive':'trapmf',
[1 2 5.0 5.0]
[Output1]
Name='spring state'
Range=[-1 1]
NumMFs=7
MF1='stretching_fast_near_zero_accel':'trapmf',
MF2='stretching_near_zero_accel':'trimf',
[-1 -1 -0.3 -0.2]
[-0.3 -0.2 -0.1]
MF3='stretching_at_zero_accel':'trimf',[-0.2 -0.1 0]
MF4='neutral':'trimf', [-0.1 0.0 0.1]
MF5='compressing_at_zero_accel':'trimf',[0 0.1 0.2]
MF6='compressing_near_zero_accel':'trimf', [0.1 0.2 0.3]
MF7='compressing_fast_near_zero_accel':'trapmf',[0.2 0.3 1.0 1.0]
[Rules]
1 0, 4 (1) : 1
2 1, 1 (1) : 1
2 2, 2 (1) : 1
2 3, 4 (1) : 1
2 4, 6 (1) : 1
2 5, 7 (1) : 1
3 1, 2 (1) : 1
3 2, 3 (1) : 1
PH
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159
ORIGINAL FIS FILES FROM ACC 1997 PAPER
3 3, 4 (1) : 1
3 4, 5 (1) : 1
3 5, 6 (1) : 1
4 1, 1 (1) : 1
4 2, 2 (1) : 1
4 3, 4 (1) : 1
4 4, 6 (1) : 1
4 5, 7 (1) : 1
5 0, 4 (1) : 1
1.2 FIS Definition for Controller
[System]
Name='msm_with_observer'
Type='sugeno'
NumInputs=4
NumOutputs=1
NumRules=13
AndMethod='prod'
OrMethod='max'
ImpMethod='prod'
AggMethod='max'
DefuzzMethod='wtaver'
[Input1]
Name='spring state'
Range=[-1 1]
NumMFs=7
MF1='stretching_fast_near_zero_accel':'trapmf',[-1 -1 -0.3 -0.2]
MF2='stretching_near_zero_accel':'trimf',
[-0.3 -0.2 -0.1]
MF3='stretching_at_zero_accel':'trimf',
[-0.2 -0.1 0]
160
APPENDIX II
FIS DEFINITIONS
MF4='neutral':'trimf', [-0.1 0.0 0.1]
MF5='compressing_at_zero_accel':'trimf',[0 0.1 0.2]
MF6='compressing_near_zero_accel':'trimf', [0.1 0.2 0.3]
MF7='compressing_fast_near_zero_accel':'trapmf', [0.2 0.3 1.0 1.0]
[Input2]
Name='position error'
Range=[-5 5]
NumMFs=3
MF1='negative':'trapmf',[-5 -5 -0.03 -0]
MF2='zero':'trimf',[-0.03 0 0.03]
MF3='positive':'trapmf',[0 0.03 5 5]
[Input3]
Name='velocity'
Range=[-5 5]
NumMFs=3
MF1='negative':'trapmf',[-5 -5 -0.02 0.0]
MF2='zero':'trimf',[-0.02 0 0.02]
MF3='positive':'trapmf',[0.0 0.02 5 5]
[Input4]
Name='acceleration'
Range=[-5 5]
NumMFs=2
MF1='small':'trimf',[-0.1 0 0.1]
MF2='tiny':'trimf',[-0.01 0 0.01]
PH
S
P LINDER
161
ORIGINAL FIS FILES FROM ACC 1997 PAPER
[Output1]
Name='control_output'
Range=[-3 3]
NumMFs=7
MF1='zero':'linear',
[0
0
0
0
0]
MF2='big_stop_spring_stretching':'linear',
[0
-2
-4
0
2.9 ]
MF3='big_stop_ spring_compressing':'linear',
[0
-2
-4
0
-2.9]
MF4='small_stop_spring_stretching':'linear',
[0
-0.5
-1
0
0.6]
MF5='small_stop_ spring_compressing':'linear',
[0
-0.5
-1
0
-0.6]
MF6='zero_velocity':'linear',
[0
0
-5.5
0
0]
MF7='zero_position':'linear',
[0
-2.2
-6.5
0
0]
[Rules]
1 0 0 0,
2 (1) : 1
2 0 0 0,
4 (1) : 1
3 0 0 0,
4 (1) : 1
4 0 0 0,
1 (1) : 1
5 0 0 0,
5 (1) : 1
6 0 0 0,
5 (1) : 1
7 0 0 0,
3 (1) : 1
0 0 1 1,
6 (1) : 1
0 0 2 2,
1 (1) : 1
0 0 3 1,
6 (1) : 1
162
APPENDIX II
FIS DEFINITIONS
0 1 0 1,
7 (1) : 1
0 2 0 1,
1 (1) : 1
0 3 0 1,
7 (1) : 1
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2. FIS Files for Final Version of QRC Compensator
2.1 FIS File for Process Model
[System]
Name='Simplified Mamdani Spring Observer for MSM '
Type='mamdani'
NumInputs=2
NumOutputs=1
NumRules=17
AndMethod='prod'
OrMethod='max'
ImpMethod='min'
AggMethod='max'
DefuzzMethod='centroid'
[Input2]
Name='spring_length_estimate'
Range=[-5 5]
NumMFs=5
MF1='negative':'trapmf',[-5 -5 -1 -0.06]
MF2='small_negative':'trapmf',[-0.12 -0.10 -0.02 -0.00]
MF3='zero':'trapmf',[-0.03 -0.00 0.00 0.03]
MF4='small_positve':'trapmf',[0.00 0.02 0.10 0.12]
MF5='positive':'trapmf',[0.06 1 5.0 5.0]
[Input2]
Name='delta_spring_length_estimate'
164
APPENDIX II
FIS DEFINITIONS
Range=[-2 2]
NumMFs=5
MF1='negative':'trapmf',[-5 -5 -0.6 -0.3]
MF2='sm_neg':'trimf',[-0.8 -0.5 -0.02]
MF3='zero':'trimf',[-0.1 0 0.1]
MF4='sm_pos':'trimf',[0.02 0.5 0.8]
MF5='positive':'trapmf',[0.3 0.6 5.0 5.0]
[Output1]
Name='spring_state'
Range=[-1 1]
NumMFs=5
MF1='compressing_fast_with_neutal_spring':'trapmf',[-1 -1 -0.8 -0.2]
MF2='compressing_slowly_with_neutal_spring':'trimf',[-0.8 -0.2 0]
MF3='not_stretching_or_compressing_with_neutal_spring':'trimf',[-0.2 0.0 0.2]
MF4='stretching_slowly_at_zero_accel':'trimf',[0 0.2 0.8]
MF5='stretching_fast_with_neutal_spring':'trapmf',[0.2 0.8 1.0 1.0]
[Rules]
1 0, 3 (1) : 1
2 1, 1 (1) : 1
2 2, 3 (1) : 1
2 3, 3 (1) : 1
2 4, 3 (1) : 1
2 5, 5 (1) : 1
3 1, 1 (1) : 1
3 2, 1 (1) : 1
3 3, 3 (1) : 1
3 4, 5 (1) : 1
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3 5, 5 (1) : 1
4 1, 1 (1) : 1
4 2, 3 (1) : 1
4 3, 3 (1) : 1
4 4, 3 (1) : 1
4 5, 5 (1) : 1
5 0, 3 (1) : 1
2.2 FIS Definition for Controller
[System]
Name='msm_with_observer'
Type='sugeno'
NumInputs=4
NumOutputs=1
NumRules=10
AndMethod='prod'
OrMethod='max'
ImpMethod='prod'
AggMethod='max'
DefuzzMethod='wtaver'
[Input1]
Name='spring state'
Range=[-1 1]
NumMFs=5
MF1='compressing_fast_with_neutal_spring':'trapmf',[-1 -1 -0.8 -0.2]
MF2='compressing_slowly_with_neutal_spring':'trimf',[-0.8 -0.2 0]
MF3='not_stretching_or_compressing_with_neutal_spring':'trimf',[-0.2 0.0 0.2]
MF4='stretching_slowly_at_zero_accel':'trimf',[0 0.2 0.8]
MF5='stretching_fast_with_neutal_spring':'trapmf',[0.2 0.8 1.0 1.0]
166
APPENDIX II
FIS DEFINITIONS
[Input2]
Name='position_error'
Range=[-5 5]
NumMFs=5
MF1='BigNegative':'trapmf',[-5 -5 -2 -0.2]
MF2='negative':'trimf',[-2 -0.04 -0.0]
MF3='zero':'trimf',[-0.04 0 0.04]
MF4='positive':'trimf',[0.0 0.04 2]
MF5='BigPositive':'trapmf',[0.2 2 5 5]
[Input3]
Name='velocity'
Range=[-5 5]
NumMFs=3
MF1='negative':'trapmf',[-5 -5 -0.1 -0.01]
MF2='zero':'trimf',[-0.1 0 0.1]
MF3='positive':'trapmf',[0.01 0.1 5 5]
[Input4]
Name='spring_length'
Range=[-1 1]
NumMFs=4
MF1='small':'trapmf',[-0.25 -0.02 0.02 0.25]
MF2='tiny':'trimf',[-0.01 0 0.01]
MF3='smallBell':'gbellmf',[0.2 0.6 0]
MF4='tinyBell':'gbellmf',[0.06 0.4 0]
%MF4='tinyBell':'gbellmf',[0.04 1.8 0]
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FIS FILES FOR FINAL VERSION OF QRC COMPENSATOR
2.2.1 Rules for Increased Tracking Performance
[Output1]
Name='control_output'
Range=[-3 3]
NumMFs=7
MF1='zero':'linear',
[0
0
0
0
0]
MF2='big_stop_spring_stretching':'linear',
[0
-1.0 -3
0
8.0 ]
MF3='big_stop_ spring_compressing':'linear',
[0
-1.0
-3
0
-8.0]
MF4='small_stop_spring_stretching':'linear',
[0
0
0
0
0.6]
MF5='small_stop_ spring_compressing':'linear',
[0
0
0
0
-0.6]
MF6='zero_small_position':'linear',
[0
-0.25 -2.5
0
0]
MF7='zero_large_pos_position':'linear',
[0
-0.75
-1.2 0 0]
[Rules]
1 0 0 0,
2 (1) : 1
2 0 0 0,
4 (1) : 1
3 0 0 0,
1 (1) : 1
4 0 0 0,
5 (1) : 1
5 0 0 0,
3 (1) : 1
0 1 0 0,
7 (1) : 1
0 2 0 4,
6 (1) : 1
0 3 2 0,
1 (1) : 1
0 4 0 4,
6 (1) : 1
168
APPENDIX II
FIS DEFINITIONS
0 5 0 0,
7 (1) : 1
2.2.2 Rules for Increased Stability
MF1='zero':'linear',
[0
0
0
0
0]
MF2='big_stop_spring_stretching':'linear',
[0
-1.0
-3
0
8.0 ]
MF3='big_stop_spring_compressing':'linear',
[0
-1.0
-3
0
-8.0]
MF4='small_stop_spring_stretching':'linear',
[0
0
0
0
0.6]
MF5='small_stop_spring_compressing':'linear',
[0
0
0
0
-0.6]
MF6='zero_small_position':'linear',
[0
-0.4
-2.1
0
0]
MF7='zero_large_pos_position':'linear',
[0
-0.05
-0.1
0
0]
%MF6='zero_small_position':'linear',
[0
-0.5
-2.2
0
0]
[Rules]
1 0 0 0,
2 (1) : 1
2 0 0 0,
4 (1) : 1
3 0 0 0,
1 (1) : 1
4 0 0 0,
5 (1) : 1
5 0 0 0,
3 (1) : 1
PH
0 1 0 4,
6 (1) : 1
0 2 2 0,
1 (1) : 1
0 3 0 4,
6 (1) : 1
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I took a speed-reading course and read War and Peace in
twenty minutes. It involves Russia.
Woody Allen
170
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