UNIVERSITY OF CINCINNATI
August, 20th
02
_____________
, 20 _____
Amit Shukla
I,______________________________________________,
hereby submit this as part of the requirements for the
degree of:
Doctorate of Philosophy
________________________________________________
in:
Mechanical Engineering
________________________________________________
It is entitled:
Stability Analysis and Design of Servo-Hydraulic Systems
________________________________________________
- A Bifurcation Study
________________________________________________
________________________________________________
________________________________________________
Approved by:
David F. Thompson (Chair)
________________________
Randall J. Allemang
________________________
Edward J. Berger
________________________
Ron L. Huston
________________________
________________________
Stability Analysis and Design of Servo-Hydraulic Systems
- A Bifurcation Study
A dissertation submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
DOCTORATE OF PHILOSOPHY (Ph.D.)
in the Department of Mechanical, Industrial and Nuclear Engineering
of the College of Engineering
2002
by
Amit Shukla
B.E.M.E., MNR Engineering College, India 1996
M.S.M.E., University of Cincinnati, 1998
Committee Chair: Dr. David F. Thompson
Abstract
Design for robust stability is one of the most important issues in nonlinear systems
theory. The validity of linear system design in a small neighborhood is not a sufficient criterion
for systems that undergo parametric variations and have strong nonlinear characteristics. With
rapid growth in the systems theory, the design of nonlinear systems using bifurcation theorybased procedures has been one of the key developments. Servo-hydraulic systems are one of the
most commonly used actuation and control devices, due to their force to weight ratio. They also
are highly nonlinear in nature and hence provide considerable difficulty in the design and analysis
of these systems and their control algorithms.
The goal of this dissertation is to tackle some of the issues of the nonlinear systems
theory with applications to servo-hydraulic systems. The use of bifurcation theory for the design
and analysis of a nonlinear system is illustrated, and a detailed investigation into the dynamics
associated with the servo-hydraulic systems is done. Further, the model decomposition/reduction
strategy for parametric study in the nonlinear system is suggested. The idea of control-induced
bifurcation is introduced and explained in light of servo-hydraulic systems. The servo-hydraulic
system nonlinearities are explained and their effects on the robust stability are highlighted. This
numerical work is also complemented with the experimental results on the servo-hydraulic
circuits. This general procedure for robust stability design and control design, under the influence
of nonlinearities, presented in this work can be used for any nonlinear system. The limitations of
bifurcation theory based tools are also highlighted.
Acknowledgements
This work has been possible due to motivation and support of various individuals who gave me
sound advice and guidance at numerous occasions throughout my years as a graduate student at
the University of Cincinnati. I express my deepest gratitude to my academic advisor Dr. David F.
Thompson, whose excellent guidance and critical comments paved the way for this work as well
as contributed to my development as an academician. He has been a true guide, a mentor and a
friend. I also wish to acknowledge Dr. Randall J. Allemang, Dr. Ronald Huston, and Dr. Edward
J. Berger for being on my dissertation committee and for providing valuable help and guidance.
All of them have shaped my graduate education and life very significantly by their great teaching
and thoughtful concerns.
I am also grateful to the members of UC SDRL, specially, Dr. David Brown, Dr. Jay Kim, Dr.
Allyn Phillips, Dr. Gregory Kremer, Dr. Doug Adams, Dr. Bill Fladung, Dan Lazor, Srinivas
Kowta, Tom Terrell, Jeff Hylok and Bruce Fouts who made my stay at UC most enjoyable. I also
would like to thank Rhonda Christman for her exceptional secretarial help during my 6 years at
UC.
I am also thankful to the National Science Foundation for providing monetary support for this
work and Ford Motor Company (Dr. Gregory M. Pietron) for support and fruitful discussions on
hydraulic system modeling.
This has been a project which would be impossible without unflinching support of my family. I
dedicate this work to my parents. They provided an atmosphere and everything else in my
upbringing for me to reach this point in my life. The support and understanding that my wife gave
me provided extra boost for completion of this work. Her love has been my strength for the past
few years. Her parents were also very cooperative and patient regarding my tenure as a graduate
student.
Table of Contents
Abstract
List of Figures
4
List of Symbols
9
1. Introduction
11
1.1. Background/Motivation
11
1.2. Applications
16
1.3. Organization of the dissertation
17
2. Servo hydraulic drives
20
2.1. Previous work
20
2.2. Fundamentals of hydraulic system modeling
23
2.2.1. Orifice flows
23
2.2.2. Continuity equations
24
2.2.3. Example
25
2.2.4. Valve modeling
26
2.3. Modeling assumptions
33
2.4. Physically realistic models of servo hydraulic system
34
2.5. Modeling of servo-pump actuator system
37
2.6. Modeling of servo-valve actuator system
39
2.7. Servo-valve actuator model with line dynamics
42
3. Linear and nonlinear analysis of servo-hydraulic systems
45
3.1. Linear analysis of servo-hydraulic system –a review example
45
3.2. Experimental results for servo-hydraulic systems – a review
47
3.3. Nonlinearities in the servo-hydraulic systems and their effect on stability
48
1
3.3.1. Effect of flow nonlinearity on the stability
48
3.3.2. Effect of saturation on the stability
50
3.4. Large scale coupled nonlinear systems
50
3.5. Why nonlinear analysis is needed?
52
4. Nonlinear systems analysis and bifurcation theory
53
4.1. Dynamical systems and equilibrium points
54
4.2. Generic bifurcations
57
4.3. Global bifurcations, jumps and non-local behavior
62
4.4. Stability of forced systems-feedback loop nonlinearities and nonlinear feedback
63
4.5. Nonlinear systems analysis tools
64
5. Nonlinear systems analysis –a bifurcation theory based approach
66
5.1. Previous work
66
5.2. Multi-parameter multi-space bifurcation theory
71
5.2.1. Parameter space, eigen-space and state space
74
5.2.2. Constant velocity solutions
76
5.2.3. Robust bifurcation stability analysis
77
5.3. Decomposition of the system
80
5.3.1. Fastest unstable mode with associated parameters
81
5.3.2. Transformations for parametric model decomposition
82
5.4. Control induced bifurcations
83
5.5. Control of bifurcation instabilities
85
5.6. Measure of nonlinearity
86
5.7. Numerical aspects of bifurcation analysis
87
5.7.1. Data noise and nonlinear effects
89
6. Preliminary bifurcation results: numerical
90
2
6.1. Characterization of nonlinear dynamics- computational results
90
6.2. Results for servo-pump actuator system
90
6.2.1. Simulation studies
91
6.2.2. Parametric bifurcation studies
94
6.3. Results for the servo-valve actuator model
101
6.4. Chapter summary
104
7. Development of experimental apparatus
125
7.1. Development of the test stand
125
7.2. Servo-valve model development and refinement
127
7.2.1. Pressure-flow-voltage nonlinear static characteristics
129
7.2.2. Linear transfer function model of the servo-valve
131
7.2.3. Servo-valve actuator model and real-time control schematic
133
7.3. Effect of accumulators
137
7.4. Chapter summary
139
8. Control Studies- Introduction
140
8.1. PD controller
140
8.2. Effect of PD controller on the servo-valve actuator system
141
8.3. Effect of 5th order linear controller on the servo-valve actuator system
161
8.4. Chapter summary
177
9. Conclusions and suggestions for future research
180
10. Bibliography
184
11. Appendix
198
3
List of Figures
2.1
A sample hydraulic system model
25
2.2
A valve schematic
27
2.3
Port flow area schematic of a spool valve
28
2.4
Open valve port area configuration
29
2.5
Closed valve configuration
30
2.6
Port flow area as a function of the valve displacement
30
2.7
A common four way spool valve
31
2.8
Axial and lateral components of flow forces in a valve port
32
2.9
Model 1, servo-pump actuator system
38
2.10
Block diagram of model 1
38
2.11
Servo-valve actuator model schematic with external loads
41
2.12
Pipe line model schematic
43
3.1
Proportional feedback for the linearized servo-pump actuator model
46
3.2
Root locus for varying the proportional feedback gain
46
3.3
Dead zone and saturation nonlinearity-input and response
50
3.4
Parameter space investigation using simulation
51
4.1
Time response, phase plane and eigenplane
55
4.2
Invariant subspaces and manifolds
56
4.3
Saddle node bifurcation plot representing the behavior of equilibrium point
58
4.4
Trans critical bifurcation
59
4.5
Pitch fork bifurcation
59
4.6
Super critical Hopf bifurcation
60
4.7
Soft and hard generation of limit cycles
60
4.8
Transition from stationary to periodic solution, via Hopf Bifurcations
62
4
4.9
Nonlinear system decomposition- nonlinear feedback
63
5.1
Parameter space, state space and eigenspace
75
5.2
Block diagram of the state space system with proportional output feedback law
78
5.3
Decomposition of system for robust stability analysis
80
5.4
Block diagram of plant and controller depicting extended parameter vector
84
5.5
Control induced bifurcation and distance to bifurcation
84
5.6
Parametric step size: large steps may skip some details
87
6.1
Eigenvalues of the nominal open loop servo-pump actuator model
92
6.2
Open loop simulation response of the servo-pump actuator system
93
6.3
Closed loop eigenvalue plot
94
6.4
Unstable response of the servo-pump actuator system for a high feedback gain
95
6.5
Effect of varying orifice diameter on the stability of the system
96
6.6
Effect of varying the constant velocity of the solution
97
6.7
Effect of varying the proportional gain and load mass
99
6.8
Effect of varying proportional feedback gain and orifice diameter
100
6.9
Effect of varying controller parameter
101
6.10
Servo-valve actuator model
102
6.11
Block diagram of closed loop servo-valve actuator model
102
6.12
Eigenvalues of the nominal system with force input and feedback PD
105
6.13
Displacement and velocity of the cylinder for nominal closed loop case –
106
step input
6.14
Displacement of the valve spool for nominal closed loop case – step input
107
6.15
Displacement and velocity of the cylinder for closed loop – ramp input
108
6.16
Displacement of the valve spool for nominal closed loop case – step input
109
6.17
Eigenvalues of the nominal PD control system with ramp input
5
110
6.18
Effect of varying Kp
111
6.19
Effect of varying Kd
112
6.20
Effect of varying line pressure
113
6.21
Effect of varying load mass
114
6.22
Effect of varying proportional gain and constant velocity
115
6.23
Stability boundary in the parameter space of proportional gain and constant
116
velocity
6.24
Time response for parameter values below critical
117
6.25
Time response for parameter values above critical
118
6.26
Effect of proportional gain and load mass
119
6.27
Stability boundary in the parameter space (proportional gain and load mass)
120
6.28
Effect of proportional gain and valve spool diameter
121
6.29
Stability boundary in the parameter space (proportional gain and valve dia.)
122
6.30
Stability boundary in the parameter space (proportional and derivative gain)
123
6.31
Effect of varying position feedback gain and velocity feedback gain
124
7.1
Test stand with real time data acquisition and control hardware
125
7.2
Flapper nozzle servo proportional valve
127
7.3
Servo-valve reduced order model
128
7.4
Pressure differential-voltage input-flow thorough put characteristics
129
7.5
Effective kvalve as a function of the pressure differential across the valve
130
7.6
Servo-valve actuator area characteristic model
131
7.7
Line pressure to valve voltage transfer function for a long hose
132
7.8
Line pressure to valve voltage transfer function for a short hose
133
7.9
Open loop Bode plot of the valve transfer function
134
7.10
Servo-valve actuator model schematic
135
6
7.11
Servo-valve actuator system with real time control hardware/software
135
7.12
Experimental transfer function of the closed loop (PD controller) servo-valve
136
actuator system
7.13
Open loop Bode magnitude – actuator position/valve voltage
138
7.14
Open loop Bode phase – actuator position/valve position
138
7.15
Effect of accumulator size on the frequency of load mass mode
139
8.1
Bode diagram of a PD controller
141
8.2
Eigenvalues of the nominal closed loop system
142
8.3
Effect of variation in proportional gain
145
8.4
Effect of variation in derivative gain
146
8.5
Effect of variation in load mass
147
8.6
Effect of variation of constant velocity of the system
148
8.7
Simulation response of the system for controller parameter just below critical
149
8.8
Simulation response of the system for controller parameter just below critical
150
8.9
Effect of variation on the grid of controller parameters
151
8.10
Effect of parameter variation on a grid of parameters (Kp, M)
152
8.11
Effect of parameter variation on a grid of parameters (Kp, V)
153
8.12
Stability boundary in two-dimensional parameter space (Kp, Kd)
155
8.13
Effect of load mass on the stability boundary in two-dimensional parameter
156
space (Kp, Kd)
8.14
Stability boundary in two-dimensional parameter space (Mload, Kp)
157
8.15
Effect of varying constant velocity on the stability boundary in two-
158
dimensional parameter space (Mload, Kp)
7
8.16
Experimental data of cylinder position with reference for parameter values
159
above critical
8.17
Experimental data of cylinder position with reference for parameter values
160
above critical
8.18
Bode magnitude plot of the compensator
162
8.19
Bode phase plot of the compensator
162
8.20
Nominal closed loop system is stable
163
8.21
Effect of varying Bode gain of the controller
165
8.22
Effect of varying load mass
166
8.23
Effect of varying the constant velocity
168
8.24
Effect of varying compensator gain (100-250) and load mass (50-500lb)
169
8.25
Effect of varying line pressure and load mass
170
8.26
Simulation response of cylinder position and velocity for compensator gain
170
below critical
8.27
Simulation response of cylinder position and velocity for compensator gain
172
above critical
8.28
Experimental response of the cylinder for parameters above critical
172
8.29
Experimental response of the cylinder pressure for parameters above critical
174
8.30
Bifurcation stability boundary in the 2-dimensional parameter space of load
175
mass and Bode gain
8.31
A comparison of the two controllers
176
8.32
Measure of nonlinearity for varying load mass
179
8.33
Measure of nonlinearity for varying Bode gain
179
8
List of Symbols
Symbol
Description
x
State vector
•
State derivative (with respect to time) vector
f (x, t , u, p )
State equations
xo
Equilibrium point
Dx
Jacobian matrix
N
Measure of bifurcation nonlinearity
Φ
Volume flow rate of flow through pipeline
ν
Kinematic viscosity
ℜ
Real space
air
Percentage air in the working fluid
Aport
Port area
c
Damping
Cd
Coefficient of discharge
cr
Clearance radius
cv
Velocity
D
Diameter of spool
d
Orifice diameter
det
determinant
dline
Line diameter
F
Force
G(s)
Compensator
Jx
Jacobian matrix with respect to x
x
9
k
stiffness
Kgain
Feedback gain
Kv
Bode gain
kvalve
Effective discharge coefficient for the servo-valve
M
mass
P
Pressure
p
Parameters
P(s)
Plant
Pline
Line pressure
Q
Flow
qport
Flow thorough port
R
Reference signal
t
time
u
Input
V
Volume
x
Position
Κd
Derivative gain
Κp
Proportional gain
α
Natural frequency modification factor of pipe line modes
β
Damping modification factor of pipe line modes
βεff
Effective Bulk modulus
λ
Eigenvalue
ρ
Fluid density
10
Chapter 1- Introduction
1. Introduction
1.1. Background and motivation
Hydraulic systems and their uses are mentioned in the ancient Roman texts, where they
was used to gain mechanical advantage. Since then there have been immense advances in the
field of hydraulic systems, most importantly during the period of World War II. Servo-hydraulic
systems are used because of their mechanical advantage, response accuracy, self-lubricating and
heat transfer properties of the fluid, relatively large torques, large torque-to-inertia ratios, high
loop gains, relatively high stiffness and small position error. Although the high cost of hydraulic
components and hydraulic power, loss of power due to leakage, inflexibility, nonlinear response
and error-prone low-power operation tends to limit the use of servo-hydraulic drives, they
nevertheless constitute a large subset of all industrial drives and are extensively used in the
transportation and manufacturing industries.
Design of servo-hydraulic systems is a challenge because of the ever-increasing demands
on the performance and economics of the intended systems. Further, these systems have been
optimized in design over so many years that further improvement is harder to achieve using linear
system theory. The current practice places stringent demands on the system by requiring it to be
more reliable and robust in a larger design space. Servo-hydraulic systems may fail due to loss of
power due to wear and/or leakage, clogging of the fluid flow due to contamination, inaccurate or
undesired response and loss of stability. The loss of dynamic stability is in many cases the most
important causes of system failure. In this case, the servo-hydraulic systems undergo large
pressure oscillations under certain operating conditions, which may result in catastrophic failure.
The stability of these systems is their most important asset. Any deviation from the optimal
response is not affordable in the current state of the art industry applications since the
performance dictates the economics of the whole operation and can also result in serious loss of
11
efficiency.
These large amplitude oscillations are noticed in various applications of servo-
hydraulic systems, which range from the aircraft and car industry to heavy machinery to fluid
power and manufacturing industry.
With the advent of cheap computational power, it is possible to perform design studies
which were simply cost prohibitive some time ago. Designers in the hydraulic systems area, and
more generally in mechanical systems, are going for more detailed and accurate designs than ever
before. Design considerations now include static as well as dynamic attributes. System designs
now include the dynamic response and robust stability behavior for better performance. It is easy
to study and design a system under linearity assumptions, but as the size of the design space
increases these assumptions do not hold true. Further, all realistic models of the real world are
nonlinear, involving a large set of forcing and boundary conditions. The system characteristic
with the boundary conditions and initial conditions together make the system exhibit nonlinear
response. These servo-hydraulic systems have inherent nonlinearities and exhibit nonlinear
response and complex dynamic behavior. This nonlinear nature makes such systems difficult to
analyze and design. It is this nonlinear nature which results in unexpected dynamic response
under very similar yet different operating conditions. These dynamic responses can range from a
simple overshoot to large transients to undamped responses to bifurcations to complex chaotic
responses. It is possible to observe some or all of these responses in the servo-hydraulic systems
depending upon the design space and parameter space. These nonlinearities can be asymmetric
which adds to the problem of design.
The servo-hydraulic drives also provide for interesting dynamical behavior due to the
presence of square root nonlinearity and coupling between the hydraulic and mechanical
response. Also, the drives have complex dynamics associated with them due to the presence of
friction, hard limits, complicated port flows, and intricate coupling between various components
in the circuit. A typical servo-hydraulic circuit consists of servo valves, orifices, tubes, volumes,
pumps, solenoid valves, etc. These drives exhibit modes, which range from very slow (drift) to
very fast ones (as associated with the valve spool and flapper dynamics).
12
Presence of nonlinearities in the systems provides for the intricate dynamics exhibited by
most of the mechanical systems. Most of the static nonlinearities can be classified as state
saturation/cut-off, state jump, polynomial nonlinearity, delay discontinuity, motion limiting stops/
impact nonlinearity, or backlash. The dynamic nonlinear behavior due to coupling and interaction
of system states is another area of prime interest as far as dynamic performance is concerned.
These interactions can result in significant modifications in qualitative behavior of the dynamical
system under small perturbations in parameter space. The design of systems with significant
nonlinear behavior for dynamic performance is still not very well developed and documented.
The use of hybrid system models and bifurcation theory as design tool is a reasonable choice for
the design of systems with nonlinearities.
Servo-hydraulic systems are generally designed by linearizing the nonlinearities around a
nominal operating point in conjunction with “rules of thumb” and other established practices. It is
essential to simulate the detailed design, as simulation with a detailed model is one possible mode
of verification of stability and performance. This simulation is very cost prohibitive in the case of
nonlinear systems, even with all the available computing power, since it is impossible to
investigate all the possible cases. For a nonlinear system, the linear superposition principle does
not hold true. This entails that each and every combination of boundary condition and initial
conditions can have a dramatically different response in a nonlinear system.
In nonlinear dynamical systems theory, bifurcation theory is a very powerful
methodology to study structural stability of the system. Multi-parameter bifurcation theory exists
for continuous time systems as well as for discrete time systems. The importance of the
bifurcation theory can be realized by the fact that it provides a tool for the study of dynamical
system without extensive simulations.
Nonlinearities in the system behavior pose problems in understanding, modeling and
identification of the system, but the effect of nonlinearities on the system performance is not
always devastating. Many times, nonlinearities result in increased stability margin and bandwidth,
although they make it hard to predict the response under some new conditions. The most
13
important difference in the linear models and nonlinear models is the presence of multiple
equilibrium solutions and their interactions as the system evolves or modifies itself under the
influence of various attributes – internal or external. Use of bifurcation theory as a tool for design
for nonlinearities is a very significant development because it provides a relatively inexpensive
tool to characterize and classify the dynamical behavior of the system involved. This
characterization is important in the understanding of the nonlinearity and its effect on the total
system, which may be highly coupled in nature.
Most realistic systems have many parameters and it is important to understand the effect
of the each one of those on the qualitative behavior of system dynamics. Multiple parameter
bifurcation theory provides a tool for mapping this effect in a nonlinear sense and hence is very
handy. Further, for efficient parametric study, it is often required to do a decomposition of the
system and reduce the parameter set to a smaller dimension. This is important for all systems and
is of interest in case of servo-hydraulic systems. Perturbation methods provide a good tool for
bifurcation and continuation studies, although the most fundamental restriction of these methods
is that they do not work across a discontinuity. This limits their use in case of mechanical
systems, including servo-hydraulic systems, which can be most easily modeled with hard limits
or discontinuities. It is important to understand that any model is just an imitation of the reality
and hence it is important to determine if the dynamic response obtained is a manifestation of the
model or true nature of the physical system. Servo-hydraulic systems are easily modeled using a
hybrid system model, which couples a number of low order models with a state change criterion
to imitate the complex dynamics accurately. Hybrid system models can generate dynamic
behavior is equivalent to the effect of discontinuity in the higher order model. This gives rise to
the issue of accurately depicting the state transitions as well as distinguishing between the real
and model dynamics.
Further, this design for nonlinearity is to be robust so that parametric uncertainties do not
result in catastrophic failure of the system. This requires the analysis of the system models for the
distance to dynamic failure, which is equivalent to distance to bifurcation (usually measured in
14
parameter space) or rate of bifurcation. Ideally, all the complex dynamics involved in systems
with nonlinearities should be identified and cataloged for use by future designers. For servohydraulic systems, this study has not been reported in any detail in the literature and hence is of
importance as the use of servo-hydraulic systems in industrial settings is tremendous with
demands for more compact and high performance systems every day. Linear models have been
studied in great detail, but the nonlinearities involved are so dominant in the routine operation of
the servo-hydraulic systems that it is inaccurate to ignore them.
This design using a bifurcation theory-based approach is suitable for the analysis,
identification, and control of any undesirable dynamics. This would include any dynamic
response which is a result of lower and higher order bifurcations to any chaotic dynamics. The
ability to control any undesirable modes is important in pushing the envelope of the design space.
The feedback control of any such modes is of interest in the case of servo-hydraulic systems since
these systems are used mostly in closed-loop operation. This gives rise to the idea of controlinduced bifurcations and their stabilizability and controllability. These bifurcations are a result of
feedback and can be avoided by suitably designing the system and the control loop. Moreover,
properly designed feedback control can often stabilize an inherently unstable open loop system.
Feedback coupled with nonlinearities in the system can result in complex dynamics even for
simple systems. The very fact that simple systems with nonlinear coupling can exhibit complex
response is one of the major tenet of chaos theory.
These aspects of mechanical system design for nonlinearities are of importance under
current design practice. This dissertation aims to highlight issues mentioned above and also
provides a comprehensive study of the dynamics of servo-hydraulic mechanisms. The nature of
nonlinearity, control-induced bifurcations, model and dynamical behavior of servo-hydraulic
systems are all important aspects of this work. This research aims to provide significant insights
and concrete design methodologies in these areas. The idea of control of bifurcations as well as
control-induced bifurcation is an important aspect of feedback design and control. It is this idea
that also distinguishes the dynamical behavior of the systems. The system dynamics can then be
15
classified as either an inherent system property or an artifact of modeling or one resulting from
the feedback. Once the dynamics have been characterized then it is possible to modify them by
design or control. The goal of this dissertation is to develop a framework for the nonlinear design
and analysis of servo-hydraulic systems with emphasis on using bifurcation based procedures for
the study of possible qualitative dynamical behavior under various operating conditions including
nonlinear loads and boundary conditions. It is also a goal to develop a framework for
decomposition of large-scale systems for analysis and diagnosis in a nonlinear sense.
Furthermore, the classification of the behavior of nonlinearity -static and dynamic- according to
its effect on the stability of the system dynamics, using the notion of distance to bifurcation, is an
important aspect of this work.
1.2. Applications
Servo-hydraulic systems are concerned with the transmission and control of energy for
the purpose of moving and imposing forces on the machine elements with certain mechanical
efficiency. It is a means of converting, transmitting, controlling, and applying fluid energy to
perform useful work. Throughout the history of mankind, a natural evolution of energy and power
utilization has taken place. The mode of transmission and control of power has also changed from
manual to mechanical to hydraulic and electrical. Combinations of these modes are used in the
industry these days. Torricelli discovered orifice flow laws in 1664, which started the study of the
laws of hydraulics; and more recently hydraulic component standards were developed which
provide a guideline in designing these systems. Servo-hydraulic systems are often preferred over
mechanical systems because they can rotate, push, pull, oscillate and regulate mechanisms in
modern industry. They can transmit power with fast response and can multiply force simply and
efficiently. They can provide a constant force or torque irrespective of the speed changes and are
more reliable.
Servo-hydraulic systems are used in various industries, some of which are agriculture,
automobiles, aircrafts, chemical processing plants, construction, lumber industry, material
16
handling, mining, metal working, military, manufacturing, and transportation industries. This
would include industrial robots and control of machine tools, suspension, brakes and clutches for
automotive vehicles, flight control actuation, and stabilization of wheel set in rail transport, and
active stabilization of large civil engineering structures. Basically, servo-hydraulic systems are
used where the requirement of power is large with compact size. With the advent of good control
algorithms, accurate and precise use of servo-hydraulic mechanisms is very common and control
authority is increased with the use of embedding the control functions in the software with
accompanying sensing and actuation mechanisms. It must be noted that all these systems, when
designed using ad-hoc or trial and error methods, can nevertheless result in unexpected dynamics
due to the highly nonlinear nature of servo-hydraulic systems.
1.3. Organization of the dissertation
The goal of this research is to understand the dynamical behavior of the servo-hydraulic
drives under the effects of various nonlinear operating conditions and system dynamics. Further,
it provides a detailed methodology for design and analysis of systems with nonlinearities using
bifurcation theory-based approaches. It investigates the effect of controller parameters and plant
parameters on the qualitative behavior of the system dynamics. The effect of controller
parameters is very important distinction since the feedback provides an extended envelope for the
design of systems. This research also aims to characterize the nature of nonlinearities for their
effect on the stability behavior of the system.
To achieve this, first the effect of various
parameters has to be understood, and then a decomposition performed on the parameter space to
reduce the problem for efficient handling with the bifurcation-continuation based analysis and
design schemes. Since the bifurcation-continuation based procedures do not work across the
discontinuities, it limits the analysis to only the smooth-continuous regimes of operation. Thus it
is important to understand how a discontinuity, either induced due to modeling or inherent in the
system characteristics, affects the dynamical behavior, especially robust stability.
17
System nonlinearities are either static or dynamic in nature and they tend to be coupled
together to provide intricate dynamics, which is a delight to the theoretician but could present
great difficulty for the design engineer. These nonlinearities result in various system behaviors
including stability bifurcations, which could be as simple as saddle-node or Hopf bifurcations or
as complex as a chaotic response. A detailed study of these behaviors as shown by the servohydraulic mechanisms is one of the goals of this dissertation. To study the contribution of the
nonlinearities, the idea of constant velocity bifurcation analysis is proposed. This helps in
characterizing nonlinearities for their effect on stability behavior. Further, the control of complex
dynamics is also important in the feedback sense, which means it is to be understood when a subcritical bifurcation can be changed to a supercritical bifurcation by a smooth state feedback,
which is equivalent to the local feedback stabilization of the system at the bifurcation point. This
local feedback stabilization becomes nontrivial only when the unstabilizable eigenvalues of the
linearization have zero real parts. This research provides qualitative as well as quantitative
estimates on the regions of stability for the servo-hydraulic systems under various feedback and
boundary conditions. This parameter space investigation provides crucial information for future
analysis and design of these systems. The methodology proposed in this work can be extended to
any system which shows complex dynamics due to the presence of nonlinearities.
This dissertation is organized into nine chapters. Chapter 2 provides a review of the
servo-hydraulic drives and reviews the previous work done in the area. It also provides details of
general modeling issues for servo-hydraulic drives, and then physically realistic models of the
two test systems (servo-pump actuator system and servo-valve actuator system) is given. Chapter
3 compares the linear vs. nonlinear dynamics of the servo-hydraulic systems with emphasis on
the qualitative aspects and hence provides motivation for the need of nonlinear analysis and
design of servo-hydraulic systems. Chapter 4 introduces the basic theory of dynamical systems,
with emphasis on multi-parameter bifurcation theory for use as a design and analysis tool.
Chapter 5 deals with the bifurcation theory and its various details for use as a multi-parameter
design and analysis tool. This chapter focuses on issues including distance to bifurcation and
18
robust stability, model decomposition for analysis, control induced bifurcations, and control of
bifurcation instabilities. It also discusses various numerical issues related to bifurcation analysis.
Chapter 6 provides preliminary results obtained on the servo-pump actuator model and servovalve actuator model. Chapter 7 describes the experimental apparatus development and validation
of the model. Chapter 8 provides some experimental control studies done on the system for
stability analysis. Chapter 9 provides summary and conclusions. Suggestions for future research
are also outlined in this chapter.
19
Chapter 2 – Servo hydraulic drives
2. Servo hydraulic drives
Servo-hydraulic systems are in many instances the most important class of motion and
force control systems used in various industries like manufacturing, transportation and heavy
equipment. They offer advantages from the point of high force and power capacities with
relatively compact actuator sizes. However, hydraulic actuation comes with a penalty in the form
of significant nonlinear dynamics, and basic nonlinearities associated with the servo-valve and
actuators remain even after removing the complicated control loop nonlinearities. This chapter
includes a review of previous work in this area, hydraulic system modeling fundamentals and the
details of two test system models.
2.1. Previous work
Servo-hydraulic systems have been a subject of extensive study. The stability of the
servo-hydraulic systems is a topic of great practical interest, especially with regards to nonlinear
oscillations. The servo-hydraulic system modeling can be found in various references including
Blackburn et. al. (1960), Lewis and Stern (1962), McCloy and Martin (1980), Merritt (1967),
Nightingale (1956). The text by Blackburn (1960) et.al. also provides the nonlinear dynamics of a
hydraulic system with a single stage pressure relief valve. Lewis and Stern (1962) also have an
interesting comparison of the steady state forces and transient forces. Merritt (1980) is a
compilation of practical and empirical information on the hydraulic control system design. Most
of these references give a linearized stability analysis of a single stage pressure relief valve,
although Merritt notes the requirement of a damping orifice in the system for stability reasons.
Some other works on hydraulic system dynamics are Stringer (1976), Viersma (1980), Skaistis
(1988), and Watton (1989). The importance of flow forces in the stability of the servo-hydraulic
system is discussed in most of these works. It is noted that the strength of these flow forces as
20
compared to system stiffness and damping would determine the possibility of limit cycle
oscillations. But most of these discuss the linearized analysis of the systems. An extensive survey
reveals that the most published work in this area deals with the linearized or quasi-linearized
analysis or simulations. Linear analysis tends to overlook the complex dynamics, which is only
possible in the system with nonlinear interaction between the states, and simulation is costprohibitive in a system that does not satisfy the linear superposition principle.
The study of dynamic stability of servo-hydraulic systems has been an area of interest to
various investigators, which includes Noton and Turnbull (1958), who used a graphical method of
isoclines to obtain approximation to a describing function and predict the possibility of
continuous oscillations in a four-way spool valve. Foster and Kulkarni (1968) did a quasi-linear
analysis; Burton (1975) used the describing function approximation to the nonlinearities using the
binomial expansion for a parameter plane technique. Ray (1978) did extensive simulation of the
relief valve to investigate the chattering and limit cycle oscillations due to input pressure
variation. Dokainish and Elemadany (1978) used a harmonic balance method to study the
nonlinear response of the relief valve. Taft and Twill (1978) did a linearized analysis, which they
compared to experiments and simulation results for a three-way under lapped valve. Jen-Kou
Chou (1983), McClamroch (1984), and Cox and French (1986) used describing function-type
methods for limit cycle analysis. McLeod (1985) did a linearized Routh-Hurwitz analysis for
several operating conditions. Karmel (1990) and Margolis (1996) did simulation studies for
hydraulic systems and demonstrated the ability of the systems to exhibit oscillations. Kremer
(1998) did extensive bifurcation based studies of the hydraulic control systems and showed the
emergence of limit cycle oscillation and their parameter dependence. Some recent work in the
area of analysis of servo-hydraulic systems would certainly include Alleyne and Liu (1999) in
which they studied the limitations of force tracking control of the servo-hydraulic systems.
Hwang (1996) investigated the sliding mode control of an electro-hydraulic system. Plummer and
Vaughan (1996) investigated the robust adaptive control of servo-hydraulic systems. Tsao and
Tomizuka (1994) studied the robust, adaptive control of hydraulic-servo for noncircular
21
machining. Yao, Bu, Reedy, and Chiu (1999) verified theoretical results with experimental data
for a robust control of the single-rod hydraulic actuator.
The study of servo-hydraulic systems would consist of the actuator dynamics, valve
dynamics, and any line and load dynamics. Investigators in this area have used various tools and
methodologies available at times to investigate dynamical behavior. This includes Fourier
analysis of the response, investigation of the effect of port flow on stability, estimation of limit
cycle amplitude and frequency, effect of feedback on the valve dynamics, valve chatter and its
causes, effect of high frequency oscillations in pressure at valve input, effect of leakage, study of
super-harmonic and sub-harmonic responses in valve dynamics, effect of actuator seal friction on
stability, effect of saturation in valve travel, impedance matching between load dynamics and
valve dynamics, effect of damping on the stability, and effect of fluid bulk modulus.
Further, the study of servo-hydraulic systems is also done to investigate various controlrelated issues. This would include studies in the area of adaptive control of actuators, neural
networks and Lyapunov-based techniques. This would certainly include Alleyne, Neuhaus, and
Hedrick (1993), Alleyne and Hedrick (1995), Cho and Hedrick (1991), Hwang (1996). Also,
application of feedback linearization techniques has been demonstrated by a number of authors
including Del Re and Isidori (1995) and Vossoughi and Donath (1992), and Richard and
Scavarda (1996) and Thompson, Pryun and Shukla (1999). Kremer and Thompson (1998)
developed a bifurcation-based procedure for designing and analyzing stability of nonlinear
hydraulic control circuits. This work uses the concept of the shortest distance to instability to
investigate robust stability of the hydraulic models. It uses the concept of bifurcation surface to
analyze the robust stability of the nonlinear systems with smooth nonlinearities.
Loads acting on the servo-hydraulic systems can be highly nonlinear in nature. This is of
utmost practical interest. Notable examples of such loads are force tracking control applications,
civil engineering structural testing applications, and system loads with significant friction effects.
These three classes of applications are a very short list of possible cases which may cause
significant nonlinear behavior, but does cover a range of all possible cases that are of interest to
22
industry. Another example would include hardware-in-the-loop simulation of the nonlinear
structural testing specimens, which are of considerable importance for earthquake resistant
designs. McClamroch (1985) outlines stability issues involving hydraulic actuators with coupled
nonlinear structural loads.
2.2.
Fundamentals of hydraulic system modeling
For the purpose of dynamic analysis and control design, hydraulic systems are modeled
using the approximation that the systems can be represented as a collection of lumped volumes
separated by sharp edged orifices. Such low order models are widely used in industrial practice
(such as in the automotive industry) and, with sufficient detail, have been found to yield good
correlation with experimental data. However, model updating/validation is needed in instances
where a highly accurate representation of the physical system is required. The orifice flow is
classified as either being laminar or turbulent, depending on the Reynolds number. Each volume
has to satisfy the continuity equations as laws of conservation of mass and energy holds true.
2.2.1. Orifice flows
To define the dynamics of fluid flow through an orifice, it is important to note that whenever
the pressure differential is large for all operating points of interest, it can be safely assumed that
the flow always has a large enough Reynolds number so that it can be calculated using the
turbulent flow equation (Merritt, 1967). The magnitude of the volume rate of flow (in3/sec)
q port , through the orifice of area Aport , is given by Equation 2.1.
q port = Cd Aport
2
ρ
Pdifferential
(2.1)
This equation is obtained by applying the Bernoulli’s equation to the streamline flow, for
the turbulent flow case where Cd is the discharge coefficient, and Pdifferential is the differential
pressure across the two chambers connected by the orifice. The sign of the Pdifferential determines
23
the direction of the flow and ρ is the density of the fluid. It should be noted that although the
assumption of compressible fluid volumes is made, effective change in density of the fluid is
small. If the orifice is assumed to be sharp edged then the discharge coefficient can be assumed
constant. The value 0.61 is normally used for the discharge coefficient (Merritt, 1967). The flow
through the orifice is normally turbulent, but becomes laminar when the pressure difference
across the orifice is close to zero, corresponding to the low flow rate and hence low Reynolds
number. The laminar and turbulent regimes must be both modeled, as the turbulent flow equation
has an infinite gain (i.e. infinite slope), at zero flow and hence causes numerical problems in
analysis. Generally the transition point is selected around 1 Psi pressure drop and hence the flow
equation can be given as:
q port

2
( Pdifferential ),
0 ≤ Pdifferential ≤ 1 psi
Cd Aport
ρ

=
2
 C A
 d port ρ ( Pdifferential ) sgn( Pdifferential ), Pdifferential > 1 psi

(2.2)
This combined laminar and turbulent flow equation is continuous but not smooth at the transition
point, which can be easily demonstrated. This flow equation can be further used to compute the
pressure derivative relationship for each volume element in the circuit.
2.2.2. Continuity equation
For a fixed volume system, the continuity equation is a manifestation of conservation of
mass; total flow into the system must be equal to the total flow out of the system. Generally for
hydraulic systems this equation can be given as
∑Q
in
= ∑ Qout +
24
dV V •
+ P
dt β
(2.3)
where V is the volume, dV dt is the flow consumed by expansion of the control volume, and
•
V P β is the compressibility flow resulting from pressure changes, and β is the effective bulk
modulus of the fluid. The effective bulk modulus (Merritt, 1967) of the fluid is given by
β eff =
1
1
β airfree
(2.4)
air %
+
1.4( Pvolume + Patm )
where β airfree is a constant equal to the bulk modulus of the fluid with no air entrainment, and
air% is the entrained air percentage by volume in the fluid at the atmospheric pressure Patm .
2.2.3. Example
An example of a basic hydraulic system, which illustrates use of the preceding laws, is
given in the Figure 2.1.
Mass M
Qin
P1
k
P2
Q12
x
c
Figure 2.1: A sample hydraulic system model.
The continuity equations for the two volumes can be written as follows
Qin = Q12 +
V1
β1
P1
dV V
Q12 = 2 + 2 P2
dt β 2
25
(2.5)
dV2
is due to the fact that the volume changes with time as the piston and load mass moves, and
dt
is equal to Aload x . The flow Qin is the input flow for this system, and the flow Q12 is the flow
through the orifice due to the pressure differential between the two chambers, given as

Cd Aorifice

Q12 = 
C A
 d orifice

2
ρ
( P1 − P2 ),
2
ρ
0 ≤ P1 − P2 ≤ 1 psi
(2.6)
( P1 − P2 ) sgn( P1 − P2 ),
P1 − P2 > 1 psi
where Aorifice, is the area of the sharp edged orifice separating the two chambers. The load
dynamics can be defined by considering the force balance using the Newton’s Law of motion and
is given as
mx + cx + kx = ( P1 − P2 ) Aload
(2.7)
Thus, the state space representation of this hydraulic system will have following states:
x
x
 
 P1 
 
 P2 
(2.8)
It is easily seen that the above hydraulic system, while being modeled as a lumped volumes
interconnected with the sharp edged orifices, uses the fundamental concepts of continuity and
orifice flows with Newton’s Law of motion to capture the dynamics of the system. It should be
noted that in this system the nonlinearity due to square root nature of the orifice flow is the
dominant static nonlinearity.
2.2.4. Valve modeling
A valve, as shown in Figure 2.2, has two major components, a valve body housing and
the actuation unit. The valve body houses the valve seat, and the clearance between it and the
valve plug determine the amount of flow through the valve. Thus, the equations of motion can be
26
derived for the valve position and velocity using the force balance. The various forces that act on
the valve body are pressure, inertia, spring, and flow forces. Using Newton’s Law, the equation of
motion can be written as follows:
Figure 2.2: A valve schematic (Courtesy: Bosch Corp. www.bosch.com).
∑F
pressure
+ ∑ Fspring + ∑ Fflow = M valve − spool xvalve− spool
(2.9)
The mass term M valve − spool represents the mass of valve spool and that of the fluid
between the valve lands in the pressure control chamber. The force due to the pressure differential
Fpressure consists of the pressure differential on the valve lands which includes the effect of
variable area on both sides of the valve spool (which are exposed to the line pressure) and since
sump pressure is assumed to be approximately equal to zero. The magnitude of the variable
spring force Fspring depends on the spring rate and the valve spool position. There is also a
constant preload spring force due to the difference between the spring’s free length and the
installed length. The spring rate could be constant or a function of the valve spool displacement,
which makes the total spring force a nonlinear function. The flow forces F flow are the forces
27
which act on the valve spool due to the fluid movement in the various chambers/lands and
through the ports. They are also called flow-induced forces or Bernoulli’s forces (Merritt, 1967).
Pline>Po
X
D
Po
D
X
Figure 2.3: Port flow area schematic of a spool valve.
Port Area: Port area Aport (Figure 2.3), is a nonlinear function of the valve position as given
below:
Aport

 π Dcr ,


2
= π D x 2 + cr ,

2
 A = π  dline  ,


 max
 2 
x≤0
0 ≤ x ≤ xmax
(2.10)
xmax ≤ x
When the valve is closed ( x ≤ 0 ), the flow limiting area is the annular area created by
the radial clearance between the valve spool and the sleeve. As the valve starts to regulate, as in
28
Figure 2.4, the flow area becomes a function of the valve diameter D , valve position x and
radial clearance cr . Since the radial clearance is small relative to the valve diameter, the flow is
approximated as the product of the valve circumference (Figure 2.5) and the length of the port
opening. Finally, a flow saturation limit is imposed by the size of the hydraulic line connecting
the port to the discharge area. The maximum area Amax, (Figure 2.6) is given by the flow
saturation in the line or the hard limit on the valve travel.
Port Flow: To completely define the valve dynamics, a relationship for the net flow analysis
using the continuity equations has to be developed for a control volume. This should take into
account any leakage across the valve spool and the compressible nature of the fluid. The
conservation of mass or continuity equation is given as
∑Q
in
= ∑ Qout +
dV V •
+ P
dt β
(2.11)
Pline>Po
x
Cr
X +C
2
Po
2
r
D
Figure 2.4: Open valve port area configuration.
29
Pline>Po
Cr
Po
D
Figure 2.5: Closed (overlapped) valve port configuration.
Effective Flow Area Aport
Wide Open
Overlapped
π d spool rc _ spool
xmax
Laminar Leakage
xspool
Valve Displacement
Figure 2.6: Port flow area as the function of valve displacement.
30
where V is the volume, dV dt is the flow consumed by expansion of the control volume, and
•
V P β is the compressibility flow resulting from pressure changes, and β is the effective bulk
modulus of the fluid. The valve port is considered as an orifice, whose area is given by Equation
2.10. The flow equation (Equation 2.2) can be further used to compute the pressure derivative
relationship for each volume element in the circuit. Thus the state space description of a servohydraulic system consists of the two position-velocity states for each valve spool and a pressure
state for each independent fluid volume. Further, the relationships that define the mass
conservation and force balance are important to derive the intended state space description of the
servo-hydraulic system.
Pline
L
X
X
Input
Force or
Position
Pa
Pb
Figure 2.7: A common four-way spool-valve.
Flow forces: The equations for static and dynamic flow forces are derived using the general
equation of momentum to a control volume in the valve. The static and steady-state flow forces
are a result of the inherent fluid accelerating property of the valve port and are present during the
flow out of the valve chamber through the port. For a common four-way spool valve, as in Figure
2.7, (with inflow and outflow through the same port, depending on valve position) the static flow
31
force is analogous to a centering spring since it always acts in a direction to close the valve. The
static flow force is due to a pressure gradient in the chamber caused by a higher velocity near the
port.
Faxial θ
Fjet
Figure 2.8: Axial and lateral components of flow forces in a valve port.
The dynamic flow forces are present only if the fluid is accelerating in the chamber. To
determine the static flow force, the jet force due to the flow needs to be calculated and is given by
the mass of the fluid flowing through the flow limiting passage multiplied by the local
acceleration of the fluid near the port. According to the Newton’s third Law, the jet force has an
equal and opposite reaction force. This reaction can be projected into an axial and a lateral
component (Figure 2.8). When the port is circumferential, the net lateral force is negligible. The
jet angle is a function of the valve spool position and radial clearance (Merritt, 1967). This
relationship between the jet angle and the valve spool position and radial clearance is complex
and Kremer (1998) used a polynomial curve-fit to approximate the exact relationship, which will
be used in this work.
Fjet =
ρ q 2port
(2.13a)
Cc Aport
Faxial = − Fjet cos θ
(2.13b)
The dynamic or transient flow is present only when the flow is accelerating through the
chamber and the magnitude of the axial fluid reaction force is given by Newton’s second Law
(Blackburn et.al., 1960).
Fdynamicflow = ρ L
32
dq port
dt
(2.14)
where L is the damping length (Figure 2.7) or the distance between the centers of incoming and
outgoing flows. As the flow is a function of position and pressure differential, the resulting
dynamic flow force has a primary damping term and a secondary term that is proportional to the
pressure derivative and the total dynamic flow force is evaluated as follows:
Fdynamicflow = ρ L
= LCd
dq port
dt
 0

0 ≥ x ≥ xmax 



Pline x 

  π d spool x
0 ≤ x ≤ xmax
2



2 ρ   x 2 + cr




Aport
+

Pdifferential
 2 Pdifferential



(2.15)
The direction of the flow force is determined by the direction of the fluid acceleration.
The net reaction force on the valve will be opposite the direction of pressure difference driving
the flow. If the valve position is defined so that a positive direction corresponds to opening valve
port, then a positive valve velocity means the valve port is opening and the flow rate is
increasing. The positively damped and negatively damped valve corresponds to whether the
dynamic flow force is in the same direction or opposite to the valve velocity. It is easy to show
that a valve is positively damped if the fluid flows out through the metering orifice and is
negatively damped if the inflow is metered.
2.3. Modeling assumptions
Servo-hydraulic systems generally consist of an actuator piston and ram, connected to a
load. The actuator is supplied with the pressurized hydraulic fluid metered by the spool valve.
The spool is driven by an electrical or hydraulic or mechanical input. In other instances, notably
flight control actuation, the fluid supply to the actuator may be driven directly by a reversible
servo pump. Several simplifying assumptions are made to help transform the physical system into
33
a mathematical model. Some of those assumptions, which apply to most of the servo-hydraulic
systems, are as follows:
•
Energy losses due to bends, fittings and sudden changes in flow cross sections are negligible.
•
Sump pressure is negligible and cavitation is not modeled.
•
Ports are symmetric circumferentially.
•
Servo-hydraulic systems can be accurately modeled using lumped parameters.
•
The transition from laminar to turbulent flow in orifice occurs at a pressure difference of 1
Psi.
The assumption on the transition pressure eliminates the need for calculating flow
transition based on the Reynolds number calculations (Kremer, 1998). Leakage flows are
assumed negligible except for the annular flow for a closed bypass port. Further, this research
also includes models of short tubes and pipelines for including flow dynamics and their effects on
the model stabilities. Saturation in valve travel and flows is also considered in this study. For a
physically realistic model of servo-hydraulic systems, the effect of variable bulk modulus,
discharge coefficients, transition grooves, short tubes and other hydraulic subsystem models
should also be added as required. It is to be noted that there are other ways to model the systems
behavior, for example the transition from laminar to turbulent zone could be substituted by a
variable discharge coefficient (Viall and Zhang, 2000).
These models are restricted such that :
•
Flow behavior is calculated in a macro sense
•
Cavitation is not modeled
•
Supply reservoirs, for example providing Pline in servo-valve actuator model, are
constant pressure and infinite capacity.
2.4. Physically realistic models of general servo-hydraulic system
The servo-hydraulic system can be modeled as a set of coupled nonlinear ordinary differential
equations, as suggested by the example given in Equations (2.5)- (2.8). The servo-hydraulic
34
system when modeled as a set of lumped volumes connected through a set of sharp-edged orifices
result in two position states (i.e. position and velocity) for each valve spool, two position states
for each load (i.e. position and velocity) and one pressure state for each volume. The servohydraulic systems have two primary units, a control valve and an actuation unit. The dynamics of
the servo-hydraulic system are captured using a set of ordinary differential equations and involve
the valve dynamics and the actuator dynamics coupled with the flow-pressure dynamics of the
system subject to the external input and other operating conditions.
The state space description of the nonlinear system is helpful as it facilitates the analysis
using various nonlinear dynamics techniques. It requires the system to be modeled as a finite
number of coupled, first order ordinary differential equations which can be written as follows in
the vector notation:
dx
≡ x = f (x, t , u , p)
dt
(2.16)
where x ∈ ℜn is an n-dimensional state vector and f (x, t , u , p ) is a smooth function defined as
n
f : ℜn × ℜ× ℜnu × ℜ p → ℜ n and depends on time t, inputs u, and parameters p, where nu is the
number of inputs and np is the number of parameters. If f (x, t , u , p ) does not depend explicitly
on the input it is called unforced, and if it does not depend upon the time then it is time invariant.
A set of equations is autonomous if they are both unforced and time invariant. It is obvious that
higher order differential equations can be decomposed into more than one, first-order differential
equations. Thus the state space vector for a servo-hydraulic system with n-valves and m-pressure
volumes is as follows:
35
x1
xn
=
=
f1 ( x1 ,...., xn )
f n ( x1 ,...., xn )
 x1 




 xn _ valve  Position


 _____ 
 v1 


x=
 Velocity
v

 n _ valve 
 _____ 
 P  Pressure
 1 




 Pm _ vol 
(2.17)
The typical equation of motion for valve consist of a force balance as given below:
•
v n −valve =
1
( ∑ Fpressure + ∑ Fspring + ∑ Fshear + ∑ Fflow + ∑ Fbias )
Mv
(2.18)
and the typical continuity equation for a pressure state equation is given as:
Pvolume =
β volume 
Vvolume
 ∑ Qvolume,in − ∑ Qvolume ,out −

d (Vvolume ) 

dt

(2.19)
where volume refers to the control volume being analyzed. Calculation of the flow estimates for a
given control volume is the most difficult task at hand in most cases but can be accomplished by
defining the flows with respect to the metering entities in the circuit. The models used in this
research were selected specifically to highlight the effect of servo-hydraulic system nonlinearity
and classify the dynamics related to actuator, valve, pipeline and load. This helps in breaking
down the interconnection between the different components of any typical servo-hydraulic
system, and their individual contributions to the total system dynamics. It is important to realize
the interconnection between these components and the effect of coupling between the hydraulic
and mechanical modes. The primary idea of selecting these systems was to isolate the
nonlinearities in orifice, flow forces, feedback, loadings, and line dynamics including vibroacoustic interactions in the system.
36
2.5.
Modeling of servo-pump actuator system
Figure 2.9 illustrates is a model of the valveless servo-hydraulic system being driven by a
reversible servo-pump. This is an “academic” model, developed for study of fundamental
nonlinear dynamic characteristics. This model, denoted as Model 1, is thought to be of lowest
practical order and complexity so as to yield meaningful dynamic behavior. The parameter
choices are basically arbitrary but include variables that are easily controllable and can be varied
numerically and physically for study. Similar models for flight control actuation are developed by
Raymond and Chenoweth (1993). It is assumed that the flows are such that the inlet pressure
does not go below zero, which is assumed to be the sump pressure. The line dynamics are
represented in an approximate sense as the volumes separated by an orifice, which is the primary
nonlinearity in the system, apart from the loading. The other assumptions as highlighted in the
previous section hold true. The nominal system parameters (in English units) are given in Table
1.
Table 1: Nominal system parameters for servo-pump model (Model 1).
Symbol
kl
Cl
M
Aload
β
air
Vanm
Vbnm
V1nm
V2nm
d1
d2
Aleak
ρ
Description
Load spring stiffness
Load damping
Load mass
Piston area
Bulk modulus
Percentage air
Volume A, nominal
Volume B, nominal
Volume 1, nominal
Volume 2, nominal
Orifice 1 (diameter)
Orifice 2 (diameter)
Leakage area
Fluid density
37
Nominal Value
1000
4
200
7
22000
0.05
35
35
10
10
0.1
0.1
0.001
0.0000795
Units
lbf/in
lbfsec/in
lbf
in2
psi
in3
in3
in3
in3
in
in
in2
lbf in2/sec2
qp
Figure 2.9: Model 1, servo-pump actuator model with two orifices.
x(position)
u=pump flow
Reference input
(position)
-
x = f (x, u )
kgain
Figure 2.10: Block diagram of Model 1.
The open loop system representation in the state space form is given as follows:
Mx + cx + kx = A( Pa − Pb )
Pa =
Pb =
P1 =
P2 =
β ( Pa )
Va ( x)
(2.20)
[ qa1 − qleak − Aload x ]
β ( Pb )
 − q2b + qleak + Ap x 
Vb ( x) 
β ( P1 )
V1
[ −qa1 + q ]
β ( P2 )
V2
[ −q2b − q ]
38
(2.21)
qleak = ( (2 / ρ ))Cd Aleak ( ( Pa - Pb )) sgn( Pa - Pb )
qa _1 = ( (2 / ρ ))Cd Aa _1 ( Pa - P1 )) sgn( Pa - P1 )
q2 _ b = ( (2 / ρ ))Cd A2 _ b ( ( P2 - Pb )) sgn( P2 - Pb )
(2.22)
(2.23)
Flows are very important functions of the states and inputs in servo-hydraulic systems
and must be accurately modeled for the precise analysis and design. This representation does not
show laminar to turbulent transition region for flows, which is used in the model. Generally this
system (servo-pump actuator model) has an equilibrium solution at zero flow, which is of not
much interest as far as the dynamic stability analysis is concerned. As such the use of a constant
velocity solution modeled in the perturbation coordinates with respect to a reference trajectory is
exploited to highlight fully the effect of orifice nonlinearity in the system. This model is expected
to show simple bifurcation behavior in open and closed loop dynamics.
2.6.
Modeling of servo-valve actuator system
In the following section, the more common arrangement for a servo-hydraulic drive,
consisting of a servo valve and a cylinder (i.e., actuator) is discussed. The open loop system input
is valve spool position, or force, depending on whether the spool valve dynamics are modeled or
not. This model, referred to as Model 2 (Figure 2.11) has combined dynamics of a servo-valve
with the actuator that are connected to each other by a set of orifice restrictions and volumes.
These volumes represent the associated volumes in the system apart from the actuator cylinder
and valve land volumes. The assumptions defined previously hold true and the model is analyzed
for stability behavior. In a closed loop sense, the system has proportional feedback with position
or force input. For nonlinear system characterization purposes it is essential to investigate other
points on the trajectory apart from the zero flow conditions and hence the analysis is done at the
constant velocity solution perturbations. The servo-valve actuator system can be modeled as
follows using a set of nonlinear differential equations with nominal parameters as given in Table
2.
39
Mx + cx + kx = A( Pa − Pb ) + f d
mxv + cxv + kxv = f v
P1 =
P2 =
Pa =
Pb =
β ( P1 )
V1
[ q1 − qa1 ]
β ( P2 )
V2
[ q2 b − q2 ]
(2.24)
β ( Pa )
 qa1 − qleak − Ap x  ,
Va ( x) 
Pa ≥ 0
β ( Pb )
 −q2b + qleak + Ap x  , Pb ≥ 0
Vb ( x) 
qleak = (2 ρ )Cd Aleak
Pb − Pa sgn( Pa − Pb )
and for
xv > 0
q1 = 2 ρ Cd Aport
Pline − P1 sgn( Pline − P1 )
(2.25)
q2 = 2 ρ Cd Aport P2
qa1 = 2 ρ Cd Aa1 P1 − Pa sgn( P1 − Pa )
q2b = 2 ρ Cd Ab 2 Pb − P2 sgn( Pb − P2 )
(2.26)
The equations can be suitably modified for the negative valve spool travel. At the
maximum travel of the valve spool the system is reduced to one with fixed position and zero
valve spool velocity and acceleration. State variables for the problem include position and
velocity states for the valve spool and the piston/actuator as well as the pressure states for the four
volume chambers. Hence the state vector can be written as:
 xv
x
 v
x

x
 P1

 P2
P
 a
 Pb
40












(2.27)
Figure 2.11: Servo-valve actuator model (Model 2) schematic with external restoring and
dissipative loads.
Table 2: Nominal system parameters for the servo-valve actuator model.
Symbol
k
c
M
Aload
β
air
Vanm
Vbnm
V1nm
V2nm
d1
d2
Aleak
ρ
Pline
cv
kv
mv
Description
Load spring stiffness
Load damping
Load mass
Piston area
Bulk modulus
Percentage air
Volume A, nominal
Volume B, nominal
Volume 1, nominal
Volume 2, nominal
Orifice 1 (diameter)
Orifice 2 (diameter)
Leakage area
Fluid density
Line pressure
Valve damping
Valve spring stiffness
Valve mass
41
Nominal Value
1000
4
200
5
22000
0.05
50
50
20
20
0.1
0.1
0.001
0.0000795
450
5
3500
0.2
Units
lbf/in
lbfsec/in
lbf
in2
psi
in3
in3
in3
in3
in
in
in2
lbf in2/sec2
psi
lbfsec/in
lbfsec/in
lbf
2.7.
Servo-valve actuator model with line dynamics
The line dynamics in the servo-hydraulic systems are important, as it is the leading cause
of limiting the system bandwidth. The effect of line dynamics could be serious on the stability of
the system as it gives a water hammer type response every time the flow direction in the circuit
changes. To analyze this, the system needs to include appropriate dynamic models of the
transmission lines and their effect on the system dynamics should be computed. The line
dynamics in lower accuracy models presented in Sec. 2.4 and 2.5 are modeled using the volumes
(V1 and V2) (Figure 2.9, 2.11) with no internal dynamic properties.
Theoretical models of a single transmission line carrying compressible fluid assume that
the wall of the transmission line is rigid, flow through the line is laminar, temperature is constant,
and flow is one-dimensional. The following model can be given by using mass balance and
momentum balance (Yang and Tobler, 1991), respectively:
∂P 2  ∂u ∂v v 
+c ρ  + +  = 0
∂t
 ∂x ∂r r 
 ∂ 2u 1 ∂u 
∂u 1 ∂P
+
=µ 2 +
∂t ρ ∂x
r ∂r 
 ∂r
(2.28)
where c is the velocity of sound. This model takes into account the velocity profile with time t,
and position x. v is the radial fluid velocity, and u is the axial fluid velocity. The volume rate of
flow is given as follows:
r
Φ ( x, t ) = ∫ 2π rudr
(2.29)
0
These models can be approximated by modal approximations with first two modes accounted for
in each subsystem. The modal approximation (Yang and Tobler, 1991) for the transmission line
(Figure 2.12) between valve and actuator is given as follows:
42
Pa
Qa
Pb
Qb
D
Figure 2.12: Pipe line model schematic.
 2(−1)i +1 λci

Dnα 2

β
λci2
 2
+
+
s
s
8
 Pbi  
α
α2
=

Q 
2s
 ai  

Z o Dnα 2

2
 s 2 + 8 β s + λci
α
α2

0
 •  
 Pbi  =  (−1)i +1 λ
ci
 •  −
2
Qai  
Z oα


2Z
α

−( o )
β
λci2 
Dn 2
s +8 s + 2 
α
α  Pa 
 
i +1
2(−1) λci
 Qb 
2

Dnα

β
λci2
2

s +8 s+ 2
α
α

s+
8β

(−1)i +1 Z o λci 
 0
  Pbi  
8β
 Q  +  2
−
  ai  
α
2
 Z o Dnα
−
2Z o 
Dn   Pa 
 
 Q
0  b

(2.30)
(2.31)
where Equation 2.30 can be reduced to its state space realization as given in Equation 2.31 and Pa
and Qa denote pressure and flow at the inlet of the pipeline. Pb and Qb denotes the pressure and
flow at the outlet of the pipeline. The subscript i in the pressure and flow states denotes the
contribution of the ith mode related to that state. Also, dissipation parameter, Dn and line
impedance constant Zo are given in Equation 2.32.
43
Dn =
lν
co r 2
Zo =
ρ o co
Ao
λci = π (i − 0.5) / Dn i = 1, 2,3...
βe
Ao = π r 2
ρo
where co =
β e = effective bulk modulus
(2.32)
l =line length
ν =kinematic viscosity
ρ o = density
Also, the values of natural frequency modification factor (α), and damping modification factor
(β) are given in Table 3. All other parameters are associated with the hydraulic oil (Mobil DTE
24) and a 3/8 in. diameter pipeline is used.
Table 3. Natural frequency (α) and damping modification factor (β) for the two-mode
approximation of pipe line dynamics.
Mode #
α
β
1
1.06
2.31
2
1.05
3.38
44
Chapter 3- Linear and nonlinear analysis of servo-hydraulic systems
3. Linear and nonlinear analysis of servo-hydraulic systems
The analysis and design of servo-hydraulic system requires full understanding of the
impact of the nonlinearities present in the system and their effect on the system response. It is
essential to know the deviations that result in response from its true nonlinear behavior, due to
linear approximation of the full nonlinear system. The nonlinearities such as saturation, and flow
nonlinearities cause significant changes in the stability behavior of the system. This chapter
includes a comparison of linear and nonlinear analysis of servo-hydraulic systems with a review
of the effects of nonlinearities on the stability properties of the system.
3.1.
Linear analysis of servo hydraulic systems –a review example
An a priori (i.e., fixed parameter) linearized model of the servo-pump actuator system, as
given by Model 1, (Figure 2.10) with pump flow as input and the actuator position as the output,
at zero flow, has the following open loop transfer function:
P( s) =
X (s)
N ( s)
= 6
5
4
q pump ( s ) s + c5 s + c4 s + c3 s 3 + c2 s 2 + c1s + c0
(3.1)
where
N ( s ) = 1.932s 4 + 3.914*103 s 3 + 2.053*106 s 2 + 7.165*107 s − 6.2*10−5
and
c5 = 2.0335*103
c4 = 1.095*106
c3 = 7.7475*107
c2 = 1.30189*106
c1 = 2.0*10−4
c0 = 0
45
(3.2)
After closing the loop around the plant (as defined by Equation 3.1) using a proportional feedback
gain, the resultant system is as shown in Figure 3.1 and the closed loop transfer function is given
is Equation 3.3.
kP( s )
kN ( s )
= 6
5
4
1 + kP( s ) s + c5 s + c4 s + c3 s 3 + c2 s 2 + c1s + c0 + kN ( s )
qpump
kP(s)
-
x
Figure 3.1: Proportional feedback for the linearized servo-pump actuator model.
Figure 3.2: Root locus for varying proportional feedback gain.
46
(3.3)
The root locus obtained by varying the proportional negative feedback gain (Figure 3.1) is shown
in Figure 3.2.
This linearized analysis shows that the value of feedback gain at which this system
becomes unstable is k=687. However, the true value as obtained by the full nonlinear analysis of
the system via the bifurcation theory-based methods reveal that feedback gain at which the system
loses stability at a zero flow case is k=480. Details of this analysis are given in following
chapters. This clearly demonstrates that linear models fail to capture some of the critical
dynamics and stability behavior as compared to the true nonlinear system behavior. This is even
more evident in the global behavior of the nonlinear systems as linear approximations are only
valid in a local sense.
3.2 Experimental results for servo-hydraulic system – a review
Experimental results (Merritt, 1967) show that the effect of flow forces is to reduce the
open loop gain of the servos and also affect the stability of the system. The gain of the servo is a
function of the forcing frequency and it is largest near the natural frequency; this is the worst
possible condition with regards to loop stability, and the flow forces tend to increase the resonant
amplitude in the region of the natural frequency. With smaller inertia loads, viscous damping has
a pronounced effect. If the natural frequency lies within the operational bandwidth, it may be
necessary, from considerations of stability, to reduce the flow forces. If the load is predominantly
of the viscous or spring nature, the stability of the system is not so drastically affected as then the
inertia load predominates. Destabilizing effect of the flow force can be greatly reduced by
introduction of a viscous load, perhaps in the form of a dashpot or by the use of an orifice. The
presence of a leak tends to reduce the positional accuracy of the system, but the increase in gain
thus made possible may more than compensate for this. If the flow forces are eliminated, then the
electro-hydraulic gain is very high; this is undesirable as most of the nonlinearities are in this part
of the feedback loop.
47
The analysis of a linear servo system, excluding leakage effects, indicates that there are
no threshold conditions (McCloy and Martin, 1980). The system responds to the smallest input
signals and disturbances, while the time taken for similar movement is constant irrespective of the
overall amplitude scale of movement. In practice, however, a hydraulic servo has nonlinearities
which detract from this ideal, and experimental results show that as the input amplitude is
reduced, the output waveform is not in exact proportion to the input signal.
It is thus noted that the true experimental result does not match with the linear system
model behavior and hence the servo-hydraulic system should be modeled as a full nonlinear
system. It is thus assumed that the stability results obtained using the linear second order model
can differ in some cases from the actual behavior of the nonlinear system depending upon the
contribution of the nonlinearity to the stability behavior of the system at any given operating
point. It is the goal of this research that the trends observed in the experimental results are
replicated using the bifurcation theory-based approach. Thus model refinement to match the
experimental behavior is a critical and important step of this research.
3.3 Nonlinearities in servo-hydraulic systems and their effect on stability
A servo-hydraulic system has strong nonlinear characteristics which include friction,
saturation, flow nonlinearity, and various dynamic nonlinearities like hydro-mechanical coupling.
The state space representation of a servo-hydraulic model clearly shows the coupling between
mechanical (displacement and velocity) states to hydraulic (pressure) states. Beside the square
law relationship between the pressure and flow, in reality a servo-hydraulic system shows other
significant non-ideal behaviors such as flow deadband, saturation, nonlinear opening of valve
orifice, and friction (Ziaei and Sepehri, 2000). These nonlinear effects tend to cause the actual
response of the servo-hydraulic systems to deviate from the idealized linear response. This does
not mean that all of these nonlinearities are “bad”; some of them tend to provide stability to the
system response, especially if they are dissipative in nature. Thus, the difference between the
linear approximation and the actual nonlinear system response and stability is most visible for
very low or very high end of the performance bandwidth. Various other details which add to the
48
nonlinearity of the problem are leakage flows, flow forces – static and dynamic, viscous shear
forces, variable bulk modulus, flow transitions – laminar to turbulent, effect of hydraulic volumes
and the flexible line dynamics.
These nonlinearities provide difficulties in designing globally robust control algorithms
since they are only valid in a local sense, especially if any linear approximations are used.
Further, these nonlinearities can result in complex dynamics like limit cycle oscillations which
are not possible in linear systems. These systems are known to undergo Hopf bifurcations and
resultant pressure oscillations can be damaging to the successful operation of the pump and/or
other components in the servo-hydraulic circuit.
3.3.1 Effect of flow nonlinearity on the stability of servo-hydraulic system
Not all of these nonlinearities are harmful; actually some of them provide stability to the
system by dissipating energy. Friction is one of them and helps in increasing the size of the
stability region by dissipating energy (Feeney and Moon, 1993), but also results in the wear of the
system, which changes the characteristics of the system and hence can result in the loss of control
efficiency. Similarly, the variable bulk modulus results in increased stability of the system when
very small volume, such as those related to the valve lands, are dominating in the dynamics.
On the other hand, the flow characteristic (Equation 2.2) results in numerical instabilities
(Chou, 1983), as transition between laminar and turbulent regimes of flow is only a numerical
approximation.
49
Response
Response
Input
Input
Deadzone Nonlinearity
Saturation Nonlinearity
Figure 3.3: Dead zone and saturation nonlinearity – input and response.
3.3.2 Effect of saturation on the stability of servo-hydraulic system
Practical hydraulic control systems have a limit on the flow (e.g., due to pump capacity)
and the motion of the valve in the bore. These saturations present significant nonlinear effects in
the response of the system. Flow saturations can decide the bandwidth of the whole system, while
the saturation in position will result in an impact type response in the system (Merritt, 1967). This
causes a very high frequency oscillation (Wang, 2000) and provides a little uncertainty in the
behavior after the saturation (Figure 3.3) has occurred. Both of these nonlinearities result in state
transitions and change the model completely. In fact, even the current bifurcation analysis tools
also fail when the continuous system goes through state transitions due to saturation or due to
other effects (e.g., state-event systems with a variable number of degrees of freedom). Further,
the dead zone or saturation type nonlinearities, which have neutral stable mode in which the
escape condition is not unique, do not have unique equilibrium and hence cannot be analyzed
using the standard techniques from passivity and Lyapunov theory.
3.4 Large-scale coupled nonlinear systems
The servo-hydraulic systems are a typical example of a large-scale coupled nonlinear
system with nominal operation such that the nonlinearities are excited and hence dominantly
contribute to the dynamical behavior exhibited. These systems exhibit a range of dynamical
50
behavior from simple harmonic/sub-harmonic response to frequency distortion (an early
indication of the violation of the linearity assumptions), to parametric oscillations as a sign of
Hopf-bifurcations, to grazing bifurcations (as seen when there is a hard limit), to a chaotic
response or catastrophe which is simply an extreme case of operational failure. The details of
these various dynamical effects will be discussed in the later chapters of this work and can be
found in many standard texts on dynamical systems like Seydel (1992), Wiggins (1991),
Guckenheimer and Holmes (1991). The nonlinearity is a natural occurrence and provides for the
existence of the multiple equilibriums, which interact with each other in a local sense due to
parametric variation and hence cause these interesting dynamics. Dynamic behavior similar to
servo-hydraulic mechanisms can be found in the other natural or man-made systems which have
strongly nonlinear characteristics or coupling between various attributes of the system.
Response 3
P2
Response 2
New Parameter
?
Response 1
Response 4
P1
Figure 3.4: Parameter space investigation using simulation.
51
3.5 Why nonlinear analysis is needed?
In the case of a nonlinear system, an exhaustive simulation or experiment may uncover
some modes of failure but still may not uncover all the possible unstable modes, as the parametric
space investigation required for any such study is simply not practical due to computation cost
and time involved (Figure 3.4). So, a methodology for the nonlinear system analysis for robust
stability analysis is of utmost importance. In the linear case, various parametric analysis tools are
available and are used regularly by the scientific community. For the systems which have strong
nonlinearities guiding their dynamic behavior, linearity assumptions do not hold true even for a
small neighborhood of parameter space. The use of bifurcation-based procedures for the analysis
of systems is very important in designing for nonlinearities.
52
Chapter 4- Nonlinear systems analysis and bifurcation theory
4
Nonlinear systems analysis and bifurcation theory
A nonlinear-coupled system is mathematically represented by the set of differential
equations x = f (x, t , u , p ) where f can depend on time t, input u, and parameters p and is
continuous in the time domain. This chapter reviews the fundamental theory of dynamical
systems and generic local bifurcations. This chapter further explores the concept of stability of
forced systems and nonlinear feedback. A review of the nonlinear system analysis tools is also
given in this chapter.
Equivalent representation of the discrete system can be done using maps. The flow of the
state is evolved as defined by the above set of differential equations by simply integrating over
time. The fundamental theorem of existence-uniqueness provides the basis of the uniqueness of
these flows, or solutions or trajectories. According to the theorem, let E be an open subset of
ℜn containing xo and assume that f ∈ C1 (E) , then there exists an a>0 such that the initial value
problem
x = f ( x)
x(0) = xo
(4.1)
has a unique solution x(t ) on the closed interval [-a,a]. The proof of this theorem can be seen in
any standard text on dynamical systems or differential equations including Perko (1996). Further,
if there exists a δ > 0 such that for all x o ∈ Nδ (x o ) , where Nδ is compact set in the
neighborhood of the initial condition, the above initial value problem has a unique solution
x(t , x o ) with x ∈ C1 (G) where G = [− a, a] × Nδ (x o ) ⊂ ℜn +1 and x(t , x o ) is a twicecontinuous differentiable function of t for each value of xo . Also, if E is an open subset of ℜn + m
containing the point (x o , po ) where x o ∈ℜ n and po ∈ ℜm and assume that f ∈ C 1 ( E ) , then
there exists an a>0 and δ > 0 such that for all x ∈ Nδ (x o ) and p ∈ Nδ ( po ) , the above initial
value
problem
has
a
unique
solution
x(t , xo , po )
with
x ∈ C1 (G )
where
G = [− a, a] × Nδ (xo ) × Nδ ( p o ) . These relations define the uniqueness and existence of the
solution for each and every combination of initial condition, boundary condition, input and
53
parameter values. The flow of the vector field f (x) is a set of mappings φt such that
φt (xo ) = φ (t , xo ) . The first step in the analysis of nonlinear systems is to compute the
equilibrium points and to describe the dynamics of the system near those points.
4.1 Dynamical systems and equilibrium points
A point x o ∈ℜ n is called an equilibrium point or critical point of the initial value
problem (4.1) if f (x o ) = 0 . Physically, equilibrium represents a situation without life. It may
mean no motion of a pendulum, no reaction in a reactor, no nerve activity, no flutter in airfoil, no
laser operation, or no circadian rhythms of biological clocks. This equilibrium point or critical
point or singular point of the vector field is called a sink, source or a saddle (Figure 4.1)
depending on the sign and values of eigenvalues of the matrix Dx f (x o ) , which is the Jacobian
 ∂f 
. The Jacobian is also the coefficient of the
 ∂x  x= xo
of the system, and is defined as Dx f (xo ) = 
first order approximation of the full nonlinear system. Using Taylor’s theorem, the first order
approximation of the full nonlinear system can be written as:
f (x) = Dx f (x o )(x - xo ) +
(4.2)
The state space of this system can be divided into stable, unstable and center subspaces
(Figure 4.2) that define the existence of the stable, unstable and center manifolds of appropriate
dimensions as given by the Stable Manifold Theorem which is a very important result for
dynamical systems theory. It states that if the matrix Dx f (x o ) has k eigenvalues with negative
real parts and n-k eigenvalues with positive real parts, then there exists a k-dimensional
differentiable manifold S tangent to the stable subspace E s of the first order linear approximation
of the nonlinear system at x o such that t ≥ 0, φt ( S ) ⊂ S and for all x o ∈ S , lim φt (x 0 ) = 0 and
t →∞
54
Asymptotically Stable
x
x
x
Unstable
Limit Cycle
x
x
x
t
t
x
x
t
x
x
x
x
Im
Im
Im
Re
X
x
x
x
X
t
X
Re
X
Figure 4.1: Time response, phase plane and eigenplane.
55
X
X
Re
there exists an n-k dimensional differentiable manifold U tangent to the unstable E u at x o such
that t ≤ 0, φt (U ) ⊂ U and for all x o ∈ U , lim φt (x 0 ) = 0 .
t →−∞
Wc
Ec
Eu
Wu
Ws
Es
Figure 4.2: Invariant Subspaces and Manifolds: Stable (s), Unstable (u) and Center (c).
The Center Manifold Theorem (Perko, 1996) states that if there are m eigenvalues with the
zero real parts then there exists an m-dimensional center manifold, which is tangent to the center
subspace E c which is invariant under the flow φt . The Hartman-Grobman theorem
(Guckenheimer and Holmes, 1983) is another very important result in the qualitative theory of
ordinary differential equations. A fixed point is hyperbolic if none of the eigenvalues of the
system Jacobian matrix have zero real part. Similarly, a non-hyperbolic fixed point has at least
one eigenvalue with the zero real part. The theorem shows that near a hyperbolic fixed point, x0,
the nonlinear system x = f (x) has the same qualitative structure as the linear system x = Ax
with A = Dx f (x 0 ) . This allows us to use advanced techniques of stability analysis like
continuation-type bifurcation methods. The Hartman-Grobman theorem shows that in the
neighborhood of the hyperbolic critical point, the qualitative behavior of the nonlinear system is
determined by its linear part. The Local Center Manifold theorem (Guckenheimer and Holmes,
1983) extends this result for nonhyperbolic critical points. It states that the qualitative behavior of
the nonlinear system around non-hyperbolic critical points can be approximated by the qualitative
56
behavior of the nonlinear system on the center manifold. Since the dimension of the center
manifold is less than n, this simplifies the problem of determining the qualitative behavior of the
system near a nonhyperbolic critical point. This complexity in analysis is further reduced by the
use of normal forms (Nayfeh, 1998) to approximate the nonlinear component of the system.
Peixoto’s theorem (Perko, 1996) implies that typically a two-dimensional vector field will contain
only sinks, saddles, sources, and repelling and attracting closed orbits in its invariant sets.
4.2 Generic bifurcations
Systems of physical interest have parameters that appear in the defining equations. As
these parameters are varied, changes may occur in the qualitative nature of the solutions for those
parameter values (Guckenheimer and Holmes, 1996). These changes are called bifurcations and
the parameter values are called bifurcation values. In parameter regions consisting of structurally
unstable systems, the detailed changes in the topological equivalence class of a flow can be
exceedingly complicated. Since the analysis of such bifurcations in second order systems is
generally performed by studying the vector field near the degenerate (bifurcating) equilibrium
point or closed orbit, and bifurcating solutions are also found in a neighborhood of that limit set,
these bifurcations are referred to as local. Local bifurcations result due to changes in the
qualitative dynamics close to the critical points.
The term bifurcation was originally used by Poincare to describe the “splitting” of
equilibrium solutions in a family of differential equations. If x = f (x, t , u , p ) is the system of
differential equations describing the system depending on the m-dimensional parameter vector p,
then as p varies, the implicit function theorem (Wiggins, 1990) implies that these equilibria are
described by smooth functions of p away from those points at which the Jacobian derivative of
f (x, t , u, p ) with respect to x has a zero eigenvalue.
57
x
Stable branch
Unstable branch
p
Figure 4.3: Saddle node bifurcation plot representing the behavior of equilibrium point
x0 with respect to a parametric variation p.
It is of particular interest that there are identifiable classes of bifurcations which occur
repeatedly in nature. Ideally it would be nice to have a classifications of bifurcations which
produced a specific list of possibilities for each model depending on the size of state space and
parameter space and any symmetries or similar attributes it might have. There has been much
work done in the area of these classification schemes, which are based on the transversality in
differential topology. The transversality theorem (Seydel, 1997) implies that when two manifolds
of dimensions k and l meet in an n-dimensional space, then in general, their intersection would be
a manifold of dimension (k+l-n). It should be noted that non-transversal intersections can be
perturbed to transversal ones, but transversal intersections retain their topology under
perturbations. The codimension of an l-dimensional sub-manifold of n-space is (n-l).
Codimension of the bifurcation is the smallest dimension of a parameter space which contains the
bifurcations in the persistent way. At the bifurcation value, some of these transversality
conditions for structural stability are violated, and these determine the type of bifurcations that
occur. The simplest bifurcations of equilibrium are those with codimension one (Guckenheimer
and Holmes, 1983). These are generically called saddle node (Figure 4.3), trans-critical (Figure
4.4), pitchfork (Figure 4.5) and Hopf bifurcations (Figure 4.6).
58
x
p
Stable branch
Unstable branch
Figure 4.4: Trans critical bifurcation.
A generic saddle node bifurcation (Kuznetsov, 1995) is qualitatively like a family of
equations x = p − x 2 in the direction of the zero eigenvector, with hyperbolic behavior in the
complementary directions. The transversality conditions control the non-degeneracy of the
behavior with respect to the parameter and the dominant effect of the quadratic nonlinear term.
There may be global changes in a phase portrait associated with a saddle node bifurcation. The
bifurcation of one-parameter families at equilibrium with a zero eigenvalue can be perturbed to
saddle-node bifurcations (Wiggins, 1990). Thus, one expects that the zero eigenvalue bifurcations
encountered, in applications, will be saddle-nodes. If the exchange of stability occurs, then the
transcritical bifurcation or pitchfork bifurcations happen depending on the transversality
condition. The direction of bifurcation and the stability of the branches are determined by the sign
of the transversality conditions, and the bifurcations are called subcritical and supercritical
depending on the sign.
x
Stable branch
Unstable branch
p
Figure 4.5: Pitchfork bifurcation (super critical).
59
x
2
Periodic Orbits
Equilibria
µp
x1
µ=µ
p=p0 o
Figure 4.6: Super critical Hopf bifurcation, with bifurcation parameter p.
x
x
Periodic
Oscillations
Periodic
Oscillations
p
p
Figure 4.7: Soft and hard generation of limit cycles.
60
The generic transcritical bifurcation (Kuznetsov, 1995) is given as x = px − x 2 and a
generic pitchfork bifurcation is given as x = px − x 3 . A stationary bifurcation does not give rise
to such interesting phenomenon (Figure 4.6,4.7); the branching solutions show no more life than
the previously known solutions. The full richness of the nonlinear world is not found at the
equilibrium. The type of bifurcation, which connects the equilibrium point with the periodic
motion, is called Hopf bifurcation (Figure 4.8). Near Hopf bifurcations, there is only one periodic
solution for each value of the parameter vector.
The dimensions of the stable and unstable manifolds of parameter space do change if the
eigenvalues of the Jacobian of f (x, t , u , p ) cross the imaginary axis at the equilibrium solution.
This qualitative change in the local flow near the equilibrium point must be marked by some
other local changes in the phase portraits not involving fixed points. The Hopf bifurcation
theorem states that the contents, of the qualitative nature of the system, remain unchanged if the
higher order term are added to the system near origin. The theorem (Hassard et. al., 1981) is
given below:
Suppose that the system x = f (x, p ), x ∈ℜn , p ∈ℜ has an equilibrium point (x o , po ) at
which the following properties are satisfied: Dx f (x o , po ) , the Jacobian at the equilibrium point
in state space and parameter space, has a simple pair of pure imaginary eigenvalues and no other
eigenvalues with zero real parts. This implies that there is a smooth curve of equilibria
(x( p), p) with x( po ) = xo . The eigenvalues λ ( p), λ ( p) of Dx f (xo , po ) which are imaginary
d
(Re λ ( p)) | p = po = d ≠ 0 then there is a unique
at p = po vary smoothly with p . If moreover,
dt
three-dimensional manifold passing through (x o , po ) in ℜn × ℜ and a smooth system of
coordinates for which the Taylor expansion of degree three on the center manifold is given by:
x = (dp + a ( x 2 + y 2 )) x − (ω + cp + b( x 2 + y 2 )) y
y = (ω + cp + b( x 2 + y 2 )) x + (dp + a( x 2 + y 2 )) y
61
(4.3)
If a ≠ 0 then there is a surface of periodic solutions in the center manifold which has
quadratic tangency with the eigenspace of λ ( p ), λ ( p ) agreeing to second order with the
paraboloid p = −(a / d )( x 2 + y 2 ) . If a < 0 , then these periodic solutions are stable limit cycles,
while if a > 0 the periodic solutions are repelling.
x≠0
Periodic
x≠0
x=0
Hopf
Figure 4.8: Transition from stationary to periodic solutions, via Hopf bifurcation.
In practice, computations of the bifurcations of periodic orbits from a defining system of
equations are substantially more difficult than those of equilibria because the equations must be
integrated near the periodic orbit to find the Poincare return map before further analysis can be
done.
4.3 Global bifurcations, jumps and non-local behavior
Global bifurcations are such that they result in a qualitative change in the orbit structure
of an extended region of the phase space. The complete theory of global bifurcations is far from
known, because techniques for the global analysis of the orbit structure of dynamical systems are
under development (Wiggins, 1991).
There are instances when systems suddenly change to a different qualitative behavior and
this does not include the local bifurcations. An example would be Lorenz attractor, which
undergoes phase switching. It is not possible to study these jumps in the phase space by using
62
continuation type methods. These non-local dynamics results in global bifurcations thus
sometimes leading to failure of the system.
4.4 Stability of forced systems-feedback loop nonlinearities and nonlinear feedback
Nonlinear systems can, sometimes, be decomposed as a linear part and a nonlinear part
(Figure 4.9). This decomposition can be restrictive in a sense that many nonlinear interactions
cannot be decoupled as such. This would appear to be true in servo hydraulic systems since it is
not thought possible to decouple the hydro-mechanical interaction in a meaningful way.
In general, nonlinearities in systems take many different forms, one possible feature is
that they produce more than one frequency in the system response while being excited by a single
frequency. This is also known as frequency distortion. External harmonic forces generate higher
and lower harmonics in the total response of nonlinear systems. The internal feedback due to the
nonlinearity is equivalent to an internal force. The forced response is at the excitation frequency
and the free response is at the higher and/or lower harmonic of that frequency. This could be
explained by the idea of feedback of the nonlinear force to the linear system. Natural frequencies
of vibration change with the amplitude of the response. The ideas of feedback of nonlinear force
have been used for system analysis and design for a very long time, and were used in the
identification and experimental analysis by Adams and Allemang (1999).
SYSTEM
Input
Linear
Plant
Output
Nonlinear
Plant
Controller
Figure 4.9: Nonlinear system decomposition: Nonlinear feedback.
63
The nonlinear systems like servo-hydraulic systems are most often used in a feedback
control loop for better operational bandwidth, control and performance. This results in an
additional feedback loop being closed around the nonlinear plant (Figure 4.9). This transforms the
state space and parameter space, and does result in the qualitative change in the stability behavior
of the system in closed loop; generally, it is used to increase the bandwidth and performance and
hence increase the stability margin. As is often the case in feedback control systems, feedback
could also be a cause of unstable dynamics in the closed loop; thus feedback gain should be
treated as a parameter for bifurcation analysis. This treatment would help identify the rate of
bifurcation in state space or equivalently, the closest distance to bifurcation in parameter space,
which includes the feedback gain as a parameter hence providing an estimate of the rate with
respect to the feedback control. It is just a matter of looking at the projections on the respective
direction in the extended parameter space for studying the effect of various parameters on the
bifurcation stability of the system. This idea does not require the decoupling of the linear and
nonlinear part of the system model and hence is not restrictive as the idea of nonlinear feedback
force analysis.
4.5 Nonlinear system analysis tools
The nonlinear system analysis has been a focus of theoreticians, mathematicians,
physicists and engineers alike for a long time. Early development was due to the interest in
celestial dynamics. Linearized analysis in time domain or frequency domain was the most
convenient tool to study the dynamics of the system. Frequency domain analysis techniques are
commonly used for the stability analysis. The servo-hydraulic systems analyzed in this
dissertation are complex multiple degree-of-freedom systems that contain multiple nonlinearities.
Most of the literature deals with specialized cases rather than the design and analysis of general
nonlinear systems because there is no general theory available for the analysis of nonlinear
systems. The harmonic balance method (Kremer, 1998) is a frequency domain approach for
64
calculating the steady state response of a system. Assuming that there is a certain periodic
solution to the system, the time domain equations are converted into the frequency domain
algebraic equations. This can be used to handle multiple degrees of freedom with multiple
nonlinearities. The describing function technique is a special form of harmonic balance method
and is a graphical first order technique to handle a single static nonlinearity. Various
generalization of describing function theory has been developed by Swern (1981), Tamura (1981)
and Mees (1972), Jones and Billings (1991) including many others. General harmonic balance
methods are required to analyze general nonlinear systems with multiple nonlinearities, multivariable systems and multiple harmonic responses. All harmonic balance methods are special
cases of Galerkin’s method. Nonlinearities are converted into polynomial approximations or
describing function approximations and then studied in the time domain.
Absolute stability theory guarantees the global asymptotic stability of feedback control
systems with dynamic linear time-invariant system in the forward path and a memoryless
nonlinearity in the feedback path. Common methods are Lyapunov’s direct method (Showronski,
1990), Popov’s criterion and the circle criterion. There are differences between the analysis
methods depending upon the allowable class of nonlinearities.
Continuation methods are used to generate the branch tracing and path following.
Haselgrove (1960) and Klopfenstein (1961) contributed basic work in the area of continuation. In
1960s, continuation method were introduced in the analysis of engineering and scientific
problems of civil engineering, flow research, chemical reactions, solidification, and combustion.
An important class of continuation methods is homotopy for the computation of one solution.
Other predictor-corrector methods are available and are in use widely. Continuation methods
differ in predictor, parameterization strategy, corrector and step-length control.
65
Chapter 5-Nonlinear systems analysis - a bifurcation theory based
approach
5
Nonlinear systems analysis - a bifurcation theory based approach
Nonlinear systems exhibit various dynamic behaviors, which includes limit cycles and
other types of bifurcation behavior. It is important to catalogue the types of behavior a system
like servo-hydraulic systems can undergo for better analysis and design. Further, to fully
categorize the dynamics of a nonlinear system, the effect of parameters on the states and stability
properties of the system should be studied. The effect of parametric variations along the
directions on parameter space on the qualitative change in stability behavior is an important
aspect of bifurcation analysis. This provides insight into the stability robustness of the system.
This chapter reviews the previous work in the area of nonlinear systems analysis and then
provides insights into the multi-parameter multi-space bifurcation theory. This chapter also
introduces the notion of constant velocity solutions and control induced bifurcations and explains
the use of eigenspace, parameter space and state space in the analysis. This chapter imparts
contributions to the areas of system decomposition, robust stability analysis and numerical
bifurcation analysis.
5.1 Previous work
The area of nonlinear system analysis and design has been investigated for many years
and the techniques have evolved from time to time depending upon the availability of resources
and the need of the technical community. The list of previous work which is of greatest relevance
would start with Venkatasubramanian et al. (1995), which highlights the idea of stability
boundaries or bifurcation surfaces in the parameter space and methods of constructing them.
Further, Dobson (1993) developed a method for computation of the locally closest distance and
direction to bifurcation surface using the idea of intersection of the hyper-surface with the
direction vector in the parameter space. Kremer and Thompson (1998) demonstrated the
66
effectiveness of bifurcation theory in the design of hydraulic control systems using this technique.
The use of the graphical Hopf theorem for limit cycle analysis was demonstrated by various
investigators including Mees and Chua (1979), Mees (1981), and Moiola and Chen (1993, 1996).
The application of the QFT (Quantitative Feedback Theory) technique to a fluid-elastic system
for suppression of nonlinear dynamics was considered by Yu, Bajaj, and Nwokah (1993). Berns,
Moiola and Chen (1998) developed a feedback control of limit cycle amplitudes using higher
order harmonic balance methods using degenerate Hopf bifurcation theory. Feng and Liew
(2000) studied modal interactions in parametrically excited systems with zero-one internal
resonance using the Melnikov functions.
Scheidl and Manhartsgruber (1998) analyze the dynamic behavior of partially singularly
perturbed servo-hydraulic system of 10th order is investigated using the center manifold theorem.
This work uses the center manifold theory in Fenichel’s format is used to study the behavior of
the drive system in the case of periodic motions. The phase space is decomposed into sub-spaces
with sufficient differentiability properties. Approximate analytical solutions are given and
transition layers are estimated by asymptotic expansion. The stability of these transition layers is
also investigated and the theoretical results compared with the numerical solutions. The local
analysis of continuous bifurcations in the n-dimensional piecewise smooth dynamical systems are
explored in Bernardo et al. (1999). The classical development in the area of border collision
bifurcations for piecewise smooth systems was done by Feigin (1994).
The models of hybrid systems, which undergo state transitions, and their bifurcation
stability behavior, have been studied by Chen and Aihara (1998).
Michel and Hu (1999)
proposed the stability theory of a general hybrid dynamical system, which used the idea of
Lyapunov stability and Lagrange stability theorems. Chellaboina et al. (2000) developed an
invariance principle for nonlinear hybrid and impulsive dynamical systems, which have left continuous flows. This idea is of importance since the bifurcation theory analysis across the
discontinuities or singularities is still not well developed. Zhai et al. (2000) studied the stability
analysis of hybrid linear systems, which are composed of stable and unstable subsystems using
67
average dwell time approach. Schinkel, Wang and Hunt (2000) proposed a design of robust and
stable controller for hybrid systems. Demir (2000) developed theory and efficient numerical
methods for nonlinear perturbation and noise analysis of oscillators described by a system of
differential-algebraic equations. Olivar et al. (2000) studied discontinuous converter dynamics
models and generated geometrical and topological features related to smooth and non-smooth
bifurcations.
In the analysis of Hopf bifurcations, the work of Hassard et al. (1981) gives a detailed
analysis procedure for Hopf bifurcations. Banerjee et al. (2000) studied power electronic having
piecewise smooth structure for bifurcation stability. Mohrenschildt (2000) used automata theory
for the analysis of stability of hybrid systems and extended the notion of controllability and
observability to hybrid systems. Guckenhiemer and Williams (2000) studied asymptotic analysis
of subcritical Hopf-homoclinic bifurcation in the modified Hodgkin-Huxley neuron model, which
exhibited existence of Hopf bifurcation close to the homoclinic bifurcation and an asymptotic
theory for the scaling properties of the interspike intervals in singularly perturbed systems. He
(2000) developed a method coupling homotopy techniques and perturbation techniques to
develop a systematic approach for defining the slowly varying perturbation parameter. Basso et
al. (1997) derived frequency domain conditions for the existence of limit cycle for an unforced
feedback interconnection of linear and nonlinear systems. This idea was again derived on the
principles of harmonic balance and control theory.
Various investigators have studied specific cases of the nonlinearity using one or more
techniques at hand. Natsiavas and Verros (1999) investigated the dynamics of a strongly
nonlinear oscillator with asymmetric damping, which is equivalent to Coulomb friction. Salenger
et al. (1999) studied the transition of a damped nonlinear oscillator using Pade approximations.
Goncalves et al. (2000) studied the global stability analysis of relay feedback systems using
quadratic Lyapunov functions for the associated Poincare map. Kayihan and Doyle (2000)
studied the effect of friction on the process control valve in a distributed control system and fault
diagnosis and tracking techniques are presented. This work provides insights into the effect of
68
friction on the control valve. Andreaus and Casini (2000) provided a nonlinear dynamical study
of a single degree-of-freedom oscillator with a hysteretic motion limiting stop. This work
included a comparison of response for soft and hard stops and demonstrated the existence of
stable orbit and periodic doubling routes to chaos. Khinast and Luss (2000) developed qualitative
characterization of the parameter space using continuation procedures for adiabatic reverse-flow
reactor in which the loci of the singular points for forced periodic features is computed. The
concept of nonlinearity measure (Helbig et al. 2000) and its computation (Harris et al. 2000)
provide a quantification of open loop and closed loop nonlinearity of the system. The
methodology is demonstrated on an operating point analysis of the continuously operated stirred
tank reactor. A slightly different approach to measuring the control-law nonlinearity is to define
an optimal degree of nonlinear compensation in the controller, a system property distinct from
open loop nonlinearity (Stack and Doyle, 1997). The idea of interpolated cell mapping, a
technique for parameter space investigation, to obtain initial condition map corresponding to
multiple existing periodic motions, is also used to study the bifurcation in escape equation models
(Raghothama and Narayanan, 2000). The possibility of studying the nonlinear behavior of the
physical systems by small feedback action require bounds on the possible energy changes of the
system, for control of the forced response in case of nonlinear response (Fradkov, 1999). Orsi et
al. (1998) developed sufficient conditions for a nonlinear dynamical system to posses an
unbounded solution. The stability of systems with hysteresis nonlinearities, parametric
uncertainty and finite dimension unmodeled dynamics is such that equilibrium is generally not
unique (Jonsson, 1998).
Nayfeh (1999) and others have studied the dynamics of cubic nonlinear vibration
absorbers and demonstrated the existence of nonlinear resonance and bifurcations. The patterns of
bifurcations governing the escape of periodically forced oscillations from a potential well over a
smooth potential barrier with varying system parameter can help decide the relationship between
the optimal escape and nonlinear resonance (Stewart et al., 1995). Contraction analysis
69
(Lohmiller and Slotine, 1998) is another tool which is used for the analysis of nonlinear systems.
This generalizes the idea of eigenvalue analysis to the nonlinear systems.
Analysis of the control of chaos has been another area of significant research. Aston and
Bird (1997) demonstrated a methodology to extend the basin of attraction related to the maximum
parameter perturbation using the idea developed by Ott et al. (1990). Perturbation methods have
been used extensively for the analysis of nonlinear systems; more recently, the work has been
focused towards the analysis of the singularly perturbed nonlinear systems, including discrete
systems (Bouyekhf and Moudni, 1997). Hayase et al. (2000) studied the micro-stick-slip
vibrations in the servo-hydraulic systems, and suggested that by compensating for the servo-valve
nonlinearity using feedback, some micro-stick-slip vibrations could be eliminated. The addition
of dither frequency to eliminate stick and reduce the dead-band is well known. Lee and Kim
(1999) developed a hydraulic servo cylinder with mechanical feedback and also studied its
stability.
Actuator saturation is a cause of nonlinear behavior in the servo-hydraulic systems.
Gokcek et al. (2000) addressed the issue of disturbance rejection in control systems with
saturating actuators using the method of stochastic linearization. Herman and Franchek (2000)
studied the design and experimental validation of the engine idle speed controller for a V-8 fuelinjected engine using actuator saturation. Existence of saddle-node bifurcations in a relief valve is
well known (Maccari, 2000) and is often studied using the asymptotic perturbations. The
existence of sub-harmonic and combination resonance is also demonstrated. Magnusson (2000)
demonstrated a methodology to analyze bifurcation points by asymptotic expansion in the
bifurcation equations, which are obtained using the Lyapunov-Schmidt decomposition. Feckan
and Gruendler (2000) provided a functional analytic approach together with Lyapunov-Schmidt
method to study bifurcations from homoclinic and periodic orbits for singular differential
equations. The effect of magnitude saturation in the control of bifurcations is motivated by the
problems in the area of rotating stall in the compression stages of gas turbines (Wang and
Murray, 1999). The design of feedback control laws to achieve desirable size of the region of
70
attraction when the noise is modeled as a closed set of initial conditions in the phase space is
proposed by Murray (1999). The dynamics of the actuator is sufficiently fast and the output
feedback controller tends to stabilize the origin of the actual system, and the actuator dynamics
for output feedback stability purpose is not required to be faster than the system dynamics
(Aldhaheri and Khalil, 1996; Khalil, 2000). The effect of actuator saturation on the performance
of PD-controlled servo systems is to result in the drop in performance at the maximum gain
(Goldfrab and Sirithanapipat, 1999).
Numerical continuation is a very important methodology for analysis of nonlinear
systems for bifurcation stability and significant work certainly includes that of Allgower and
Georg (1990) and Seydel (1994). The application of numerical continuation is found in various
areas of engineering including the aeroelastic stability problems (Gee, 2000). Continuous wavelet
transforms are also used to perform Hopf bifurcation analysis on the aeroelastic problems
(Mastrodddi and Bettoli, 1999). Analyses of systems for aeroelastic stability includes multiple
nonlinearities including a free play nonlinearity (Kim and Lee, 1996) and cubic restoring force
(Lee and Jiang, 1999). A detailed study of nonlinear aeroelastic analysis of airfoils for
bifurcations and chaos was done by Lee et al. (1999) and provides an extensive detail on the
nonlinearities, structural and aerodynamic, found in aeroelastic systems, their solution techniques
and bifurcation associated with them including the effect of gust in longitudinal atmospheric
turbulence.
5.2 Multi-parameter multi-space bifurcation theory
The results on co-dimension one bifurcations for dynamical systems can be extended to
n-dimensional systems as a result of the center manifold theorem (Wiggins, 1990). The
dependence of the system on the parameters is of interest and the direction of the bifurcation
study in the parameter space is of importance as it guides the existence of various types of
bifurcations in the system. It is important to note that higher-order dynamics do not affect the
qualitative behavior of the phase space in a local sense, but this cannot be guaranteed globally.
71
For a nonlinear system dependent on a parameter x = f (x, p ), x ∈ℜn , p ∈ℜm , this
could be extended into a system with following state equations x = f (x, p ), p = 0 and then the
center manifold theorem can be applied. The center manifold is tangent to the parameter plane as
well as the eigenspace (Kuznetsov, 1995). All essential events near the bifurcation parameter
value occur on the invariant manifold and are captured by the projection of the system on the
center manifold. Essential dynamics include any and all qualitative changes associated around the
equilibrium point.
The saddle-node and Hopf bifurcations occur for real eigenvalues and
complex conjugate eigenvalues crossing the real axis, respectively. The center manifold is not
unique in either the fold or Hopf cases, but the bifurcating equilibrium or the cycle belong to any
of the center manifolds. In the fold bifurcation case, the manifold is unique near the saddle and
coincides with its unstable manifold if it exists. The uniqueness is lost at the stable node.
Similarly, in the Hopf bifurcation case (Hassard et al. 1981), the manifold is unique and coincides
with the unstable manifold of the saddle-focus until the stable limit cycle, where the uniqueness
breaks down. Periodic solutions or limit cycles also go through the fold, flip and Naimark-Sacker
bifurcation depending upon the multiplier value (Berns et al. 1998). If two limit cycles (stable
and unstable), denoted by two points on the Poincare map, collide and disappear on an attracting
invariant manifold, this results in a fold bifurcation of cycles. Further, in a period doubling
bifurcation, a cycle of period two appears for the map, while the fixed point changes stability.
One-parameter systems undergo saddle-node and saddle-saddle bifurcations on the plane with
one or more homoclinic orbits. Several other exotic bifurcations like, nontransversal homoclinic
orbit to a hyperbolic cycle or homoclinic orbit to a nonhyperbolic limit cycle or blue-sky
catastrophe occur in multi-parameter systems, and is beyond the scope of the present work
(Wiggins, 1990). In engineering, our interest primarily lies in the study of loss of stability of a
system via the co-dimension one bifurcations in the applications presented here.
In a multi-parameter system, there is a bifurcation curve/plane in the parameter space
along which the system has an equilibrium point exhibiting the same bifurcation (Seydel, 1994).
72
A smooth scalar function ψ = ψ (x, p ) can be constructed in terms of Jacobian matrix ∂f ∂x .
This results in a system:
f (x, p) = 0
ψ (x, p) = 0
(5.1)
which, generically defines a curve Γ passing through the equilibrium point (x o , po ) in ℜn + m .
Γ defines the equilibria satisfying the defining bifurcation condition and the standard projection
of Γ onto the parameter space p-plane results in the corresponding bifurcation boundary. This
function ψ = ψ (x, p ) for fold bifurcations is, defining the curve of equilibria having at least one
zero eigenvalue, given below:
ψ = ψ (x, p ) = det(
∂f (x, p)
)
∂x
(5.2)
and for the Hopf bifurcation is:
ψ = ψ (x, p) = det(2
where
∂f (x, p)
∂x
I)
(5.3)
denotes the bi-alternate product (Wiggins, 1990) of two matrices. If more than one
parameter is varied simultaneously to track a bifurcation curve Γ , then the following events
might occur to the monitored non-hyperbolic equilibrium at some parameter values:
•
Extra eigenvalues can approach the imaginary axis and thus change the dimension of the
center manifold.
•
Some of the nondegeneracy conditions for the co-dimension one bifurcation can be
violated.
73
This essentially means that, in higher dimension parameter space any combination of
lower-order bifurcation can be seen like, Hopf-Hopf, Bogdanov-Takens (double zero) or foldHopf bifurcations (Seydel, 1994).
Singularity theory is concerned with the local properties of smooth functions near a zero
of the function. It provides a classification of various cases of bifurcations, based on
codimension, since codimension k-manifolds in a smooth function space can be algebraically
described by imposing conditions on derivatives of the function. However, problems arise in the
study of degenerate local bifurcations. Fundamental work of Takens (1974), Langford (1979) and
Guckenheimer (1981) has shown that that arbitrarily near these degenerate bifurcation points,
complicated global dynamics such as invariant tori and Smale horseshoe may arise. These
phenomena cannot be described and/or detected using the singularity theory technique.
5.2.1
Parameter space, eigenspace and state space
The dynamics of the nonlinear problem that can be represented in the form
x = f (x, p, u, t ) has three spaces: state space, eigenspace, and parameter space (Figure 5.1),
which allow us to study the system. The state space consists of all the states required to represent
the system and is of n-dimensions. The parameter vector includes all system constants that are
used to define the system dynamics and may vary quasi-statically. Parameters define the system
properties to help model the system and do not include system states. The dimensionality of the
parameter space is equal to the number of parameters in the system. It is important to note that the
dimensionality of both of these systems is a result of the modeling and hence are up to designer to
decide. In the case of closed loop systems with feedback control parameters to be determined, the
feedback gain and other controller parameters can be included in the parameter vector to generate
74
x2
x
Domain of Attraction
xo(po)=0
x1
t
Eigen Space
X
p2
Im
x
x
x
Stability boundary
local to xo(po)
Re
Xx x x
Xx x
Parameter Space
x
po
λ(po)
Figure 5.1: Parameter space, state space and eigen space.
75
p1
an extended parameter space. This extended parameter space helps in the study of control of
bifurcations and control-induced bifurcations.
5.2.2
Constant velocity solutions
Considering a general forced system with known input u*(t), x = f (x, p, u , t ) the
stability of solution x(t) in the neighborhood of the known solution x∗ (t) can be reduced to
consideration of a related unforced system. Let x(t ) = x ∗ (t ) + ξ (t ) , then ξ (t ) represents a
perturbation of the original solution. It is sufficient to investigate the stability behavior of the
perturbed solution to discuss the stability of the original solution in a local sense.
•∗
i
x + ξ = f (x∗ , p, u , t ) + { f (x∗ + ξ , p, u, t ) − f (x∗ , p, u, t )}
(5.4)
This could be reduced to
i
ξ = { f (x∗ + ξ , p, u, t ) − f (x* , p, u , t )} = h(ξ , p, u, t )
(5.5)
which could be approximated to
i
ξ = h(ξ , p, u, t ) ≈ A(t )ξ
(5.6)
when higher order terms are negligible. This perturbation is effective in capturing the dynamic
changes, due to parameter variation, as the Jacobian approximation is calculated at each point in
the parameter space. If the solution x∗ is on a trajectory or flow where the piston or the valve
spool is moving at a constant velocity for a given servo-hydraulic system, that is defined as the
constant velocity solutions. The advantage of analysis at a point on the trajectory other than zero
velocity solution is in the excitation of the significant nonlinear response of the system (i.e, zero
input corresponds to the zero flow/zero velocity condition). Further, the zero flow case for
hydraulic systems is not of interest. For successful identification, detection and design for any
nonlinear effects, it is essential for the model to show full nonlinear behavior. The above
methodology provides for analyzing stability in terms of bifurcation of an equilibrium point, since
constant velocity does not qualify as an equilibrium condition in the original state space
76
coordinates, as described in Chapter 2. Then the nonlinear effects could be properly accounted for
and design could include the robustness towards the nonlinear effects of parametric variations.
The analysis along the direction of changing velocity gives an insight into the effect of
nonlinearity on the bifurcation stability of the system. This coupled with the feedback gain, can
provide significant insight into the area of closed loop stability boundary in the parameter space.
The mapping of stability boundary in the parameter space assists in the robustness analysis, as a
given distance to instability in parameter space is an equivalent measure of the fastest loss of
stability mode in eigenvalue space.
It should be noted that although the loss of stability of a periodic orbit is of theoretical
interest, the design engineer is predominantly interested in the mechanisms by which a stable
system loses stability via a co-dimension one bifurcation, like saddle-node or Hopf bifurcation.
First emergence of bifurcation phenomenon along a direction in parameter space is a sign of
failure of the nonlinear system under study, and hence needs redesign.
5.2.3
Robust bifurcation stability analysis
The robust stability analysis involves the study of parametric variation due to uncertainty
or due to wear and use. The effect of parametric variation could be severe in case of large coupled
nonlinear systems as it can initiate a bifurcation; subsequent bifurcations even lead to a chaotic
response and dramatic failure of the system. Robust stability analysis tools in linear systems are
well understood and quite well developed. However, the understanding of nonlinearity and their
effects on the robustness is still a matter of prime investigation, as nonlinear response is hard to
predict unless it is already characterized and included into the design (Bhattacharya, et.al. 1995).
It is important to realize that nonlinear response could be dramatically different at various points
in the parameter space and state space, and to design for all points in parameter space is nearly
impossible. However, in a closed loop operation, the design could be modified to provide desired
response locally, and could even be kept safely far-off from the stability boundary, if the
bifurcation surface in the extended parameter space is known. Thus the robustness could be
77
maintained and analyzed along the various directions in parameter space. The robustness analysis
problem can be formulated as a study of effect of perturbation around a nominal point with
unknown direction of perturbation. Thus, the importance of bifurcation theory to study stability
robustness in a highly nonlinear system like servo-hydraulic systems is apparent.
The nonlinear system can be written as follows using the Taylor series approximation
x = f (x, p, u, t ) = Dx f (x o )(x - xo ) +
(5.8)
Then in the closed loop sense, as shown in Figure 5.2, with a linear output feedback and
proportional gain k, the nonlinear state space decomposition becomes, where xo is assumed to be
an equilibrium point:
x = Dx f (x o )(x - xo ) +
(5.9)
x = Ax + Bu; x = x - xo
+
0
(u − u0 ) +
0
A = Dx f (x, u ) x
B=
∂f
∂u
(5.10)
o , uo
(5.11a)
(5.11b)
x0 ,u0
y = Cx
(5.11c)
u = ky
u
B
+
x
∫
x
C
y
A
k
Figure 5.2: Block diagram of the state space system with proportional output feedback law.
78
The equation (5.11c) is the output equation, in conjugation with the proportional
feedback law. This has appropriate assumptions about the feedback law and the input and output
observability. In a sufficiently small neighborhood of the operating point, the stability of the
nonlinear system can be understood by looking at the corresponding linear approximations,
provided it is calculated for each parametric perturbation. Then the closed-loop stability can be
investigated locally by the analysis of stability of the matrix M (Bhattacharya, et.al., 1995 and
Lungo and Paolone, 1997):
M
A + BkC
(5.12)
The parametric perturbation due to uncertainty or any other reason is p = p o + ∆p ,
where p o is the nominal parametric vector and ∆p is the perturbation. Then:
M ( p ) = M ( p o + ∆p) = M ( p o ) + ∆M ( p o , ∆p)
(5.13)
The parameter space stability robustness radius in a linear sense is a measure of the size
of resulting perturbation on the effective system under the given parametric perturbation. Thus
the estimate of ∆M ( p o , ∆p ) provides an important measure on the robustness of the system via
the eigenvalue analysis. This is equivalent to the gain margin in a one-parameter space of
feedback gain. This is directly related to the distance from the bifurcation surface and not just the
stability boundary on the eigenspace. It is important to realize that in a local sense, any additive
or multiplicative parametric perturbation results in a nonlinear closed loop transfer function,
which has the same number of unstable eigenvalues as the unperturbed system, meaning that
stability is dependent on the size of the perturbation given that the unperturbed system is stable.
The perturbed system is stable given the fact that the perturbation is small enough so as not to
cross the imaginary axis on the nonlinear eigenlocus plots, developed using the bifurcation theory
methods.
79
5.3 Decomposition of the system
Any real-life mechanical system is “large” in scale, which means it can only be modeled
using both a huge state vector and a long list of parameters, or as a group of small, coupled
subsystems. The idea of using low-order interconnected models to represent complex dynamics is
very important since it gives the designer more control in understanding the “troublesome”
dynamics of the large system and thus helping in parametric reduction for efficient parameter
space investigation of stability robustness properties.
The dimensionality of the parameter vector used for bifurcation analysis can complicate
the possibility of finding all the global modes of instability (Figure 5.3). Thus, the smaller the
parameter vector is, the easier it is to detect and characterize the dynamic behavior associated
with the system. This decomposition of the large-scale system is based on two assumptions:
firstly, the low-order models can represent all the relevant dynamics and secondly, certain
parameters would contribute to certain nonlinear modes of the system more than the other
parameters. This parametric contribution can be assessed using modal decomposition in the
nonlinear system. Modal decomposition can numerically be done via singular value
decomposition of the Jacobian matrix to study the participation of each parameter to every modes
exhibited by the system. The idea of stability robustness gives rise to the issue of fastest unstable
mode; there could be more than one fast unstable mode, which then results in higher- order
bifurcations.
Bifurcation
Parameter
Space
Extended Parameter
Space
Parameters
Associated
with Fast
Dynamics
80
Robust
Stability in
Parameter
Space
Figure 5.3: Decomposition of system for robust stability analysis.
It is also important to decouple the fast modes from the slow modes for ease of analysis,
as then perturbation or asymptotic analysis could be easily applied to the system. The fast
dynamics of the system are generally responsible for the associated loss of qualitative stability; on
the other hand, slow modes tend to provide sluggish nature to the system dynamics and hence
create numerical problems (i.e., in the use of continuation-based procedure for the analysis of
nonlinear dynamics).
5.3.1
Fastest unstable mode with associated parameters
The rate of change of stability behavior is more important in the parametric variation
studies than the stability at a particular operating point. The easiest measure of the rate of stability
bifurcation is the distance to the bifurcation surface (Dobson, 1996) in the associated parameter
space. This distance to bifurcation surface is computed on the relevant parameter space as a norm
of the distance between the operating point and the closest point on the bifurcation surface. The
possibility of computing the distance to instability is complicated by the fact that the bifurcation
surface could be composed of many bifurcating modes/hyper surfaces.
The nonlinear system could be approximated as follows:
x = Dx f (x o )(x - xo ) +
x = Ax + Bu; x = x - xo
∂f
∂u
(5.14b)
+
0
(u − u0 ) +
0
A = Dx f (x, u ) x
B=
(5.14a)
o , uo
(5.14c)
(5.14d)
x0 ,u0
Or, for a very small parametric variation, hence for computing the parametric sensitivity of the
eigenvalues, this nonlinear system can be approximated as x = Ax , with system matrix A (n × n
matrix) being approximated by the system Jacobian. If λi and U i are the eigenvalues and the
corresponding eigenvectors, where i = 1, 2,3, 4..., n , it is well known that:
81
AU i = λiU i ; i = 1, 2,3,.., n
AT V j = λ jV j ; j = 1, 2,.., n
T
(5.15)
T
Also, Vi U i = 1;V j U i = 1; i ≠ j as the left and right eigenvectors are orthogonal. Differentiating
with respect to parameter p, it is easy to show that the first order eigenvalue parametric sensitivity
is:
∂λ
T ∂A
= Vi
Ui
∂p
∂p
(5.16)
This is used to estimate the local direction of fastest eigenvalue change and hence is useful in
parametric decomposition of the model for bifurcation stability analysis. Similarly, second order
sensitivity equations can be derived. But since the eigenvalue calculations are valid in a local
sense, the second order eigenvalue sensitivity calculations are prone to errors and are not
recommended.
5.3.2
Transformations for parametric model decomposition
The full nonlinear system can exhibit dynamics dominated by certain modes and related
parameters. Thus the identification of the fastest unstable mode for a direction in extended
parameter space, and mapping of the corresponding parameters affecting the bifurcation stability
of those modes, is an important step in efficient analysis of the nonlinear system. The system
stability in open or closed loop is given by the eigenvalues of the Jacobian matrix.
x = f (x, p, t , u )
J x = ∂f (x, p, t , u ) / ∂x
(5.17)
J p = ∂f (x, p, t , u ) / ∂p
Using singular value decomposition
J x = ∂f (x, p, t , u ) / ∂x = U x *S *V ' x
J p = ∂f (x, p, t , u ) / ∂p = U p *S *V ' p
(5.18)
where U i and Vi are corresponding left and right eigenvectors, which provide information about
the contribution of each mode to the dynamics of the system at that operating point. This gives
insight into the modal composition of the system for a given set of parameters, initial conditions,
82
boundary conditions and inputs. For analyzing the effect of parameter variation on the system
dynamics, specifically modal contributions of the parameter, so that model decomposition can be
performed, let the extended parameter list be perturbed on a suitable scale.
p = p o + ∆p
(5.19a)
Then the Jacobians
J x = ∂f (x, p, t , u ) / ∂x = U x *S *V ' x
J p = ∂f (x, p, t , u ) / ∂p = U p *S *V ' p
(5.19b)
are estimated again for the new (perturbed) parameter values. By comparing the value of U i and
Vi for the original and perturbed solutions, the effect of each parameter on a particular mode is
understood. This is a local estimate for decomposition of the model so as to reduce the
dimensionality of the parameter space for the bifurcation analysis. This decomposition should be
performed at each perturbation of the parameter vector to effectively capture all the significant
dynamics of the system. Similar reduction of system dynamics has been investigated by Lall and
Marsden (1999).
5.4 Control induced bifurcations
In the parameter space, the computation of closest distance and direction to the
bifurcation provides insight into the stability robustness of the operating point (Dobson, 1996). In
the closed loop, this robustness measure provides a handle on the effect of the feedback gain and
other control loop parameters on the stability of the bifurcation (Figure 5.4). The projection of
this direction along which the shortest distance to bifurcation lies along the individual parameter
axes provides comparative measure of the components of rate of bifurcation. The system stability
is more robust with respect to perturbations of certain parameters of the system if the projection
of the distance to instability is larger for that parameter in comparison to the others. This idea is
one of the important contributions of this work.
83
x = f (x, p, u , t )
y = Cx
u
y
Controller
Figure 5.4: Block diagram of the plant and controller depicting extended parameter vector
composed of plant parameters and controller parameters.
Feedback Gain
po
d = d1 + d 2 + d 3
d
d2
Parameter 2
Parameter 1
d3
Bifurcation Surface
d1
Figure 5.5: Control induced bifurcation and distance to bifurcation.
The distance to the stability boundary in the parameter space is a measure of the robust
bifurcation stability. If a parameter is varied along a direction in parameter space (Figure 5.5) and
results in the loss of stability via a bifurcation, then stability behavior is sensitive to that
parameter. If the emergence of bifurcation is due to variation in controller parameter, then the
resulting bifurcation is a control induced bifurcation. Robustness is one of the most important
issues in control design. There are perturbations, which do not change the qualitative nature of
local bifurcation behavior of the closed loop system, but only change the amplitude of the
bifurcation equilibrium. In practice, perturbation might not be mathematically generic so
uncertainties must be quantified to analyze the bifurcation of closed loop system under those
84
perturbations. If the perturbations are small enough, then the perturbed bifurcation equilibrium
and the nominal bifurcation equilibrium are close, although the qualitative bifurcation behavior
may be altered by the perturbation.
5.5 Control of bifurcation instabilities
The geometric structure of stabilizability of finite dimensional linear time invariant
systems has been well known (Alvarez, 1996). Many techniques, such as pole placement and
solving linear algebraic Lyapunov functions, can be used to design a stabilizing feedback. For a
finite dimensional nonlinear systems, the question of feedback stabilization becomes much more
difficult and subtle. The two basic ideas in the literature are derived from Lyapunov stability
theory.
The first idea is based on Lyapunov’s second method that uses control Lyapunov
functions. The advantage of this technique is that global stabilizability can be tackled. The
drawback is that the method is too general and solving for control Lyapunov functions can be
difficult for many systems. Artstein’s theorem states that the necessary and sufficient condition
for smooth feedback stabilization is the existence of a smooth control Lyapunov function. For
nonlinear systems, stabilizability via continuous feedback is not guaranteed even if for each point
in a neighborhood of the equilibrium there exists a smooth control law steering the system to
equilibrium (Brockett, 1983).
The second idea is based on the Lyapunov direct method and Poincare normal form
theory. The basic observation is that if the linearized system is stabilizable, then the full nonlinear
system is also stabilizable by the same linear feedback. If the linearization is unstabilizable and
there is an uncontrollable eigenvalue with positive real part, then the full nonlinear system is not
stabilizable via a continuous feedback (Tesi et.al., 1995). So the only challenging problem in
local feedback stabilization is the critical case: all the linearly unstabilizable eigenvalues are on
the imaginary axis. In this case, the system can be reduced to the local center manifold that is
tangent to the eigenspace associated with the unstabilizable critical eigenvalues. Normal form
85
transformation is then used to analyze the stability of the reduced system. The advantage of this
method is that local stabilizability for a fairly large class of systems can be analyzed. The
drawback is that it is a local technique so it is not applicable to issues of global stabilizability
(Seron, et.al., 1999) .
The local feedback stabilization is closely related to control of bifurcations in that if the
system is locally stabilizable at the bifurcation point, then the bifurcation for the closed loop
system is supercritical. Although the qualitative local bifurcation behavior for the closed loop
system is robust to parametric uncertainties, the location for the bifurcation, the center manifold,
the nominal equilibrium and the periodic orbits are all perturbed away. If the system is
unstabilizable, then the Hopf bifurcation for the closed loop system is also robust, i.e., the
bifurcation remains sub-critical under perturbations in the control gains and parametric variations.
5.6 Measure of bifurcation nonlinearity
The deviation of the system from its linear approximation close to an operating point has
been a focus of study. This would include works of several researchers including Nayfeh and
Balachandran (2000), Khalil (1994), Guckenhiemer and Holmes (1986) and Helbig, Marquardt
and Allgower (2000). The primary disadvantage of linearized analysis is its lack of capturing the
full nonlinear behavior even locally. The nonlinear effects tend to dominate the system dynamics
as the system starts to operate close to the edge of the stability boundary. These nonlinearites can
be a manifestation of the inherent system dynamics as well as the control law. Considering the
open loop system only can be misleading to estimate the effect of nonlinear effects of feedback
control law. It is important to include the effect of all parameters in the extended parameter
space to fully capture the effects of nonlinearity. It is important to quantify the difference
between a linear system model and a nonlinear system model so as to capture the effect of
nonlinearity on the stability boundary. To incorporate this idea of a quantitative estimate of the
nonlinear behavior of the system, a notion of the measure of bifurcation nonlinearity is
86
proposed. The measure of bifurcation nonlinearity, N of a nonlinear system (Equation 5.8),
which depends on parameters p and states x is defined as follows:
N = Dx (x o ( p ), p ) − Dx (x o ( po ), p )
(5.20)
2
This is a two-norm of the difference in Jacobians (DX) evaluated along a direction in the
parameter space. This gives an estimate of the nonlinear nature of the full system as any
parametric studies are conducted to estimate the stability boundary. This measure (Equation
5.20) is proposed to capture the essence of difference between the dynamics of a nonlinear system
as compared to the dynamics of its linearized approximation about an equilibrium point. This
measure effectively captures the effect of local parametric variations on the equilibrium point and
the overall system dynamics.
5.7 Numerical aspects of bifurcation analysis
The numerical bifurcation analysis primarily involves location of equilibrium, their
continuation with respect to parameters, and for the detection, analysis, and continuation of limit
cycles and their associated bifurcations (Fujii and Ramm, 1997). Numerical issues related to path
following are many and of concern to investigate the stability behavior of the system. One
frequent problem is undesired jumping between the bifurcation branches. This jump is an
indication of the closed paths on the state space and is in a sense a “failure”.
x
x
p
p
Figure 5.6: Parametric step-size: large steps (left) may skip some details (right).
87
Another question is, if the two or more branches in the bifurcation curve intersect. In such
situation, the issue can be clarified by the increasing the resolution near the bifurcation point on
each bifurcation branch, in the eigen-locus. This resolution or step-size is important (Figure 5.6)
since an overly big step-size could result in overlooking the bifurcation, which might originate
and terminate in a very small region of the parameter space.
The calculation of equilibrium point requires a solution of a nonlinear algebraic set of
equations of the form f (x, p, t , u ) = 0 . For stable equilibrium this solution can be obtained by
integrating or by solving the above set of equations. In general for an equilibrium that is neither
stable nor repelling, the location problem can only be solved provided the position of equilibrium
is known approximately, because one typically starts with an equilibrium found analytically or by
integration at fixed parameter values and then “continues” it by small stepwise variations of
parameters. The standard procedure that generates the sequence of points that converge to
equilibrium in very general conditions is Newton’s method (Seydel, 1997). This in its most
elementary form has a recurrence relation given by:
x j +1 = x j + η j
(5.21)
and the displacement is given by the solution of the following algebraic equation:
J (x j ) η j = − f (x j )
(5.22)
The continuation of equilibrium is the process of estimating the curve in the space of (x,
p). Computation of this equilibrium curve defines the dependence of the equilibrium on the
parameter. Most of the continuation algorithms for bifurcation analysis implement predictorcorrector methods and include three basic steps performed (Allgower, 1990) repeatedly:
•
Prediction of the next point
•
Correction
•
Step-size control
88
5.7.1
Data noise and nonlinear effects
In the experimental or numerical analysis of the nonlinear systems the data noise can be a
significant hindrance, it could distort the qualitative behavior significantly by introducing
frequency distortions (Skaistis, 1988). Some nonlinear effects have a similar feel so it provides
additional difficulty in the characterization and analysis, since all real systems are limited in some
sense and at certain accuracy level it is impossible to distinguish between the data noise and the
true system response. The broadband excitation response looks very similar to chaotic response.
89
Chapter 6- Preliminary Results: numerical investigation
6
Results: numerical investigation
This chapter provides a summary of the results obtained via bifurcation stability analysis
of the two test systems for various parametric variations. This provides initial proof of concept as
well as gives better insight into the stability behavior of the two test systems.
6.1 Characterization of nonlinear dynamics – computational results
The simulation and bifurcation study is done on the two test servo-hydraulic systems
(servo-pump actuator system (Figure 2.9) and servo-valve actuator system (Figure 2.11)). These
hydraulic systems are defined in the previous chapters. This preliminary study reveals the
inherent dynamics of the system in open and closed loop configurations. This section discusses a
detailed insight into the models as obtained by the preliminary stability analysis and hence
identifies the various influential parameter sets for extended investigations. The study of these
preliminary models is essential to facilitate the identification of key parameters as well as
appropriate sizing of the various components for experimental design. This includes studies in the
control-induced bifurcations and the analysis at constant velocity solutions.
6.2 Results for servo-pump actuator system
The servo-pump actuator is characterized by significant nonlinear behavior due to two
sharp-edged orifices, which restrict the flow to the actuator. The other parameters of significant
interest are load mass, load stiffness, stroke length of piston, operating or line pressure, piston
dimensions, pump flow, and feedback proportional gains.
This system is modeled in perturbation coordinates for bifurcation analysis to study the
effect of change of parameters. The effect of parametric variation on the stability of the system is
analyzed by studying the eigenvalues of the Jacobian matrix. Also, the effect of zeros on the
90
dynamics of the system should not be overlooked in this approach. Zeros of the system affect the
controller design and closed loop performance.
6.2.1
Simulation studies
The open loop eigenvalue plot of the nominal system is shown in Figure 6.1. This shows
that the nominal system is stable, as expected, for low enough velocity and pump flow. Also, the
nominal system has two eigenvalues at origin in open loop configuration. Simulation response of
open loop model is type 1, which is physically realistic. A type 1 system is defined to be one
which has a one free integrator, and correspondingly one zero eigenvalue. A type 1 system (open
loop) has a ramp response to a step input. This system has two eigenvalues at the origin and one
of them is cancelled by a system zero. The system dynamics is fourth order while the system is
modeled as sixth order. Eigenvalues of the system are :
-320.32, -330.60, 0.0000, -0.0000, -21.39 + 87.32i, -21.39 – 87.32i
Zero of the nonlinear system are:
-8.4959e+012, -3.4134e+002, -3.2276e+002, -4.5749e+001, 0.00
The rank of the observability matrix is 5 and that of the controllability matrix is 4, while
the rank of the system matrix, A, or Jacobian is 4. A system is said to be minimal (Brogan, 1990)
linearly if it is both controllable and observable. A system is controllable if the controllability
matrix is full rank and is observable if the observability matrix is full rank too. The zeros of the
system can be computed via the full nonlinear model by assigning pump flow to be the input and
position to be the output with proportional feedback. A reduced-order representation of the
nonlinear system model is not thought to be possible globally. Locally the linearized model is
uncontrollable and unobservable. The nonminimal nature of the system model results in a
singular Jacobian and thus leads to numerical problems in the computation of the equilibrium
point. A methodology is developed for state space reduction of the models near the steady state
conditions, based on the physics of the system. This model also shows the two complex conjugate
91
modes associated with the load mass. These modes are stable for this operating condition and
nominal point. The frequency of the complex conjugate mode is of particular interest during
experimental design of the test apparatus such that the system dynamics is within the control
bandwidth of the servo-valve and can be effectively analyzed experimentally. Slow real modes
are associated with the hydraulic fluid drain. This hydraulic fluid drain is estimated by analyzing
a decoupled piston draining the pressurized fluid to exhaust through an orifice. Simulation
response of the open loop nominal system is given in Figure 6.2. This response is stable as
demonstrated by the eigenvalue analysis. The operating condition used for these studies is a
constant velocity of 1 in./sec.
Figure 6.1: Eigenvalues of the nominal open loop servo-pump actuator model.
92
0.8
4
0.6
2
0.4
0.2
0
250
200
P1
6
velocity
Position
1
0
150
-2
0
0.5
1
-4
0
0.5
time
1
100
time
200
200
150
0
0.5
1
1
0.5
1
Pb
100
Pa
250
P2
250
50
0.5
time
150
0
0
100
150
0
0.5
time
time
1
100
0
time
Figure 6.2: The open loop simulation response of the servo-pump actuator system at the
nominal shows a type 1 response for a step input.
93
6.2.2
Parametric bifurcation studies
This preliminary model is studied for parametric variations and their effect on the system
stability. The closed loop response is stable for small gains as shown in Figure 6.3. At a high
proportional feedback gain the system response starts to become unstable. This result is shown in
the Figure 6.3. The simulation response demonstrates instability of the system (Figure 6.4).
80
60
40
Im aginary
20
0
-20
-40
-60
-80
-80
-60
-40
-20
0
Real
20
40
60
80
Figure 6.3: Closed loop (proportional position feedback with kgain=20.0) shows a stable
response.
94
0.4
0.03
0.3
0.02
6000
P1
0.04
veloc ity
P osition
0.5
0.2
0.1
0.01
4000
2000
0
0
0.05 0.1 0.15
0.05 0.1 0.15
time
160
60
165
155
20
Pb
170
40
0.1
0.15
time
80
Pa
P2
0.05
time
160
150
145
155
140
0
0.05
time
0.05
time
0.1
0.15
0
0.1
time
Figure 6.4: Unstable response of the servo-pump actuator system for a high feedback gain
(kgain=500).
95
150
130
110
90
70
50
Im aginary
30
10
-10
-30
-50
-70
-90
-110
-130
-150
-200-180-160-140-120-100-80 -60 -40 -20 0 20 40 60 80 100
Real
Figure 6.5: Effect of varying orifice diameter (0.2 in to 0.5 in).
96
120
100
80
60
Im aginary
40
20
0
-20
-40
-60
-80
-100
-120
-120 -100 -80 -60 -40 -20
0 20
Real
40
60
80
100 120
Figure 6.6: Effect of varying the constant velocity (0.5 to 3.0 in/sec) of the solution.
97
It is possible to drive the system unstable by increasing the orifice diameter. The
emergence of Hopf bifurcation and the saddle node bifurcation are evident (Figure 6.5). Effect of
varying the constant velocity of the system at a constant feedback gain value also demonstrates
the sensitivity of the load mass mode to become unstable via a Hopf bifurcation. As expected the
system tends to become unstable for a higher velocity of piston and hence high servo pump flow
case (Figure 6.6). The effect of varying the proportional gain and the load mass on the stability
behavior of the system is seen in Figure 6.7. It is again visible that this parametric variation can
result in Hopf bifurcation of the system and the load mass mode is the one that becomes unstable
via a Hopf bifurcation. This two dimensional grid is useful in demonstrating the sensitivity
variation in the parameter space.
The effect of varying proportional feedback gain and the orifice diameter on the stability
of the system is shown in Figure 6.8. The emergence of complex dynamics is associated with the
interaction of the hydraulic and mechanical modes. This necessitates the use of advanced
continuation type techniques for the study of the stability behavior. The sensitivity of parametric
variation to the stability of the system may be different for various directions in the parameter
space. It is a very important to study this sensitivity to help reduce the parameter space for further
analysis in any nonlinear system. The effect of parametric variations along a direction in the
parameter space are shown in Figure 6.9.
98
Figure 6.7: Effect of varying the proportional gain (20 to 400) and the load mass (20 to 220 lbf)
on the stability behavior of the system.
99
Figure 6.8: Effect of varying proportional feedback gain (20 to 400) and the orifice diameter (0.2
to 0.5 in) on the stability of the system.
100
Figure 6.9: Effect of parameter variation along a direction (Kp+Kd*0.01+M).
6.3 Results for servo-valve actuator model
The study of the servo-valve actuator model (Figure 6.10) involves the study of following
issues:
1. Model without valve dynamics (i.e., valve position input)
2. Model with valve dynamics (i.e., valve force input)
3. Effect of parametric variation on the stability of the system
These results will be used to further characterize the model and investigate the dynamics
associated with this system. The servo-valve actuator model with valve dynamics is
predominantly used in this study.
101
Figure 6.10: Servo-valve actuator model.
Reference
+
Error
Controller
Kp+Kd s
Valve
Input
Plant
x = f (x, p, u , t )
-
Figure 6.11: Block diagram of closed loop servo-valve actuator model. The plant input is
valve spool force.
102
In this work the bifurcation analysis is done for a fixed proportional controller and for a fixed PD
controller. The operating condition used is a constant velocity of 1 in./sec. This nominal closed
loop servo-valve actuator model with a PD controller (Kp= 100, Kd= 10) shows (Figure 6.12) four
complex conjugate modes associated with the load mass and the servo-proportional valve. These
modes are stable for this operating condition. Real axis eigenvalues are associated with hydraulic
fluid modes. The simulation response of the cylinder with step input shows a stable response
(Figure 6.13). The valve response (Figure 6.14) is much faster than the cylinder response. Further
it is possible with the nominal closed loop system to drive the cylinder at constant velocity
(Figure 6.15) by means of a reference input consisting of ramp function.
As indicated previously, the analysis of nonzero constant velocity/flow operating points
is conducted to evaluate the effect of parameters on the bifurcation behavior, which would not be
evident at zero velocity flow conditions. The response of the valve is shown in Figure 6.16 and
the corresponding eigenvalues are shown in Figure 6.17. The effect of varying proportional gain
(Kp) is to drive the system unstable via a Hopf bifurcation (Figure 6.18). The effect of changing
the derivative gain (Kd) is shown in Figure 6.19. The effect of changing line pressure is to lower
the frequency of the load mass mode and close to the unstable condition (Figure 6.20). Figure
6.21 shows the effect of the variation of load mass. The frequency of the load mass mode is
lowered as the load mass is increased.
The effect of varying two parameter sets is studied to gain further insight into the nature
of the stability boundary in the parameter space. The distance from the stability boundary is a
measure of robust bifurcation stability of the system. The effect of varying Kp (200-400) and
constant velocity of the cylinder motion (0.5 to 1.5 in/sec) is shown in Figure 6.22. Figure 6.23
shows the stability boundary in the space of Kp and constant velocity. Figure 6.24 shows the
simulation response of the stable system with parameters below critical value. The response
undergoes a limit cycle oscillation for parameter values above a critical value (Figure 6.25).
103
The effect of varying proportional gain (200-400) and load mass (100 – 700 lbf) is shown
in Figure 6.26. The stability boundary in the parameter space (proportional gain and load mass) is
shown in Figure 6.27. The effect of varying proportional gain (200-400) and valve spool diameter
(0.4-1.5 in) is shown in Figure 6.28. The stability boundary in the parameter space (Kp and valve
spool diameter) is shown in Figure 6.29. The effect of varying Kp and Kd as parameters in the
extended parameter space is shown in Figure 6.30. This demonstrates the fact that variation in the
controller parameters can also result in loss of stability via bifurcation. The effect of these
parametric studies is to explore the stability boundary and hence understand the robust bifurcation
stability of the system. Figure 6.31 shows the behavior of the system for variation along Kp+Kd*0.1 in the parameter space.
6.4 Chapter summary
The effect of various parameters on the stability of the system is studied to demonstrate
the existence of Hopf and saddle-node bifurcations for the two test systems (servo-pump actuator
and servo-valve actuator system). The aim is to understand the complicated quantitative and
qualitative behavior of the two systems. These studies demonstrate the nature and sensitivity of
the stability boundary in the parameter space. The idea is to conduct initial exploration of the
parameter space for the servo-hydraulic systems and to identify the parameters which affect the
stability boundary. Moreover, this is an attempt to catalog effect of these parameters on the
stability boundary. The servo-valve actuator system is subsequently implemented as an
experimental test stand, and these preliminary studies are useful in the design of the experimental
apparatus. This development is the topic of the next chapter.
104
Figure 6.12: Eigenvalues of the nominal system with force input and feedback PD controller for
servo-valve actuator model. One high frequency real mode at 2530 is not shown.
105
Figure 6.13: Displacement and velocity of the cylinder for nominal closed loop PD controlled
stable case –step input.
106
Figure 6.14: Displacement of the valve for closed loop PD controlled stable case.
107
Figure 6.15: Displacement and velocity of the cylinder for closed loop PD controller with ramp
input.
108
Figure 6.16: Displacement of the valve for closed loop PD controller constant velocity case with
ramp input.
109
Figure 6.17: Eigenvalues of nominal system (PD control) with ramp input.
110
Figure 6.18: Effect of varying Kp (50 to 500). Hopf bifurcation occurs at approx. Kp=500.
111
Figure 6.19: Effect of varying Kd (10 to 200). Saddle node bifurcation occurs at approx. Kd=200
as shown in the magnified portion of the plot (below).
112
Effect of varying line pressure
300
Fixed PD-controller
200
Im aginary
100
0
-100
-200
-300
-600
-500
-400
-300
Real
-200
-100
0
Figure 6.20: Effect of varying line pressure (300 to 1200 Psi).
113
Figure 6.21: Effect of varying load mass (200 to 600 lbf).
114
Figure 6.22: Effect of varying proportional gain and constant velocity.
115
Figure 6.23: Stability boundary in the parameter space of proportional gain and constant
velocity.
116
Figure 6.24: Time response for parameters values (proportional gain and constant velocity)
below critical.
117
Figure 6.25: Time response for parameters values (proportional gain and constant velocity)
above critical.
118
Figure 6.26: Effect of proportional gain and load mass.
119
Figure 6.27: Stability boundary in the parameter space (proportional gain and load mass).
120
300
200
100
0
-100
-200
-300
-400
-600
-500
-400
-300
-200
-100
0
100
Figure 6.28: Effect of varying proportional gain and valve spool diameter.
121
Figure 6.29: Stability boundary in the parameter space (proportional gain and valve spool
diameter).
122
Figure 6.30: Stability boundary in the parameter space (proportional and derivative gain).
123
Figure 6.31: Effect of varying position feedback gain and velocity feedback gain.
124
Chapter 7- Experimental apparatus development
7
Introduction
The development of the experimental apparatus and the methodology for model validation
and analysis with appropriate real-time control hardware/software is described in this chapter.
Further, a full nonlinear model is developed for the experimental hardware implemented system,
and is refined and validated using various methodologies. The various experimental issues related
to control bandwidth and signal to noise ratio are investigated. Individual components as well as
the full servo-valve actuator system are both validated. This chapter describes the details of this
development and validation.
7.1 Development of test stand
An experimental test stand (Figure 7.1) is developed for verification of model and numerical
results obtained for the stability analysis of the servo-valve actuator system. The test stand will
also be used to conduct experimental investigations of the geometric effects of the various control
structures. The test stand consists of a reaction stand for supporting the electro-hydraulic cylinder.
The electro-hydraulic cylinder used in this test stand is made by Parker Hannifin Corporation
(model number 3LXLTS34A7). These cylinders are equipped with MTS Temposonics linear
differential transducer (LDT) (model number LHTRB00U00701V0) to measure cylinder position.
The servo-valve used for this analysis is a flapper-nozzle type servo-proportional valve (Merritt,
1967) made by Parker Hannifin Corporation (model number DY05AFCNA5). A sample
schematic of a similar valve is shown in Figure 7.2. The hydraulic power supply consists of the
gear pump installed on the Parker electro-hydraulic trainers. To reduce the effect of pump noise
in the line pressure, a diaphragm type accumulator (charged to 400 Psi) is used. A high-pressure
inline filter is used to guarantee the cleanliness of the hydraulic fluid. A Tektronix PS280 power
supply is used power the servo-valve current driver card (Parker BD101-24) as well as the LDT
on the cylinders. A Matlab based real time data acquisition and control hardware and software
125
(WINCON 3.2), developed by Quanser Computing (www.quanser.com) is used to provide closed
loop control with various control structures which can be implemented in Simulink via the Matlab
real-time workshop. The pressure signal is measured by means of PCB Piezotronics static
pressure sensor (model number 1502A02FJ1 KPSIS). The data acquisition and real time control
system is setup in a client-server (two PC) configuration to provide maximum computational
power. The overall test stand setup is shown in Figure 7.1. The reaction support for the cylinder
was designed to withstand any static as well as dynamic loads (Krutz, 2001). This test stand was
also designed to have no significant structural modes within the anticipated control bandwidth (up
to 40 Hz).
Figure 7.1: Test stand with real time data acquisition and control hardware.
126
Figure 7.2: Flapper nozzle servo-proportional valve (www.moog.com).
7.2 Servo-valve model validation and refinement
A flapper-nozzle servo valve (Figure 7.2) consists of a flapper-nozzle stage and a
cylindrical spool stage. The electromagnetic torque motor drives the flapper, which is connected
to the housing by a spring. The flapper deflection controls the oil flow through the nozzles, thus
controlling the pressures on both sides of the spool. This creates a pressure differential across the
spool and results in movement of the spool. This movement results in deflection of the
mechanical feedback spring. The nonlinearities in this valve can be categorized as follows:
•
Nonlinear torque motor
•
Flapper Dynamics
•
Nonlinear flow through nozzles
•
Nonlinear flow forces on flapper
•
Pressure-flow dynamics
•
Spool dynamics
•
Clearance in feedback spring
127
•
Nonlinear port flow in spool
A flapper-nozzle servo-valve can be modeled by first principles (Merritt, 1967), but preliminary
simulation studies showed that flapper dynamics had no significant effect on the system dynamics
below 800 Hz. A reduced order model of the servo-valve is developed. This reduction is based on
the physics of the system and is developed iteratively based on the experimental data. The aim is
to approximate the full nonlinear dynamic characteristics of the servo-valve by decomposing it in
two parts (Figure 7.3). Based on the simulation studies, it was decided that the two parts consist
of a linear transfer function from command voltage (V(s)) given to the valve amplifier card to the
spool position (X(s)) of the valve and a nonlinear pressure-flow-voltage static (i.e., algebraic)
characteristic of the form q = q ( xvalve , ∆P ) , where q is the flow with a valve position input of
xvalve and pressure differential ∆P . The servo-valve has a flow throughput based on the command
voltage and the pressure differential across the servo-valve. This is referred to as pressure-flowvoltage characteristics. As described in the previous chapters, this is a highly nonlinear function.
A command voltage generates a position of the valve spool, which results in a flow through the
servo-valve.
Main Spool
Position
Valve
Command
Voltage
V(s)
X(s)
Linear Transfer Function
Model
Pressure-Flow-Voltage
Model
Valve Position/Voltage Input
Static Characteristics
Figure 7.3: Servo-valve reduced order model.
128
Flow
q
7.2.1
Pressure-flow-voltage nonlinear static characteristics
In order to capture the nonlinear nature of the pressure-flow-voltage characteristics of the
servo-valve several, static tests were conducted. These static tests involved studying response of
the servo-valve for static input under a dummy load (needle valve). The data obtained is shown in
Figure 7.4.
Figure 7.4: Pressure differential-voltage input-flow throughput characteristics of the servovalve.
It is observed from the experimental data that the valve flow, q (gal./min.) characteristics can be
modeled (Kowta, 2002) as follows:
q = kvalveV
∆P
(7.1)
where V is the valve input voltage, kvalve is a function of valve discharge coefficient, hydraulic
fluid density, nominal valve spool diameter and valve spool position. To model these flow
129
characteristics, kvalve is assumed to be a function of valve input voltage (thus a function of valve
spool position). This approach is similar to previous studies in which the variable discharge
coefficient of the servo-valve based on the valve position was used (Viall and Zhang, 2000).
Figure 7.5: Effective kvalve as a function of the pressure differential across the valve.
The variation of kvalve with the valve opening is shown in Figure 7.6. The valve is symmetrical
and has no measurable dead-band. The data is limited by the lower limit of the flow meter
available, which has a range of 0.2 to 2 gal./min.. Based on the experimental observations (Figure
7.4), following fifth-order polynomial model for the effective kvalve is proposed:
kvalveV = a5V 5 + a4V 4 + a3V 3 + a2V 2 + a1V 1 + a0
a0 = 0;
a1 = 0.00298482488548
a2 = 0.00048272373205; a3 = 0.00009829412559
a4 = -0.00001264772476; a5 = 0.00000038527429
The resultant model matches well with the experimental data as shown in Figure 7.6.
130
(7.2)
Figure 7.6: The servo-valve area characteristics model.
7.2.2
Linear transfer function model of the servo valve
The transfer function for the valve input voltage to the valve spool position could be obtained
from the flapper-nozzle servo valve model from first principles, the details of which are given in
many references including Schothorst (1997). The servo-valve (Parker DYO5AFCNA5) does not
have a measurable spool position output signal so an experiment was designed to evaluate the
frequency response of the servo-valve in terms of valve voltage input to load pressure output. An
orifice load (using a needle valve) is connected to the servo-valve with a short hose. Then sine
input tests were performed to capture the transfer function between the servo-valve voltage input
and load pressure output. The effect of the length of hose is clearly visible (Figure 7.7, 7.8) in
terms of the decrease in the corner frequency of the system as the hose length is increased. No
other dynamics are observed in the range of 0-60 Hz. At higher frequencies, the signal to noise
ratio is very low.
131
___ Model
-*- Experiment
Figure 7.7 Load pressure/valve voltage transfer function for a long (12 ft) hose.
132
___ Model
-*- Experiment
Figure 7.8: Load pressure/valve voltage transfer function for a short (2 ft) hose.
133
The linear transfer function (voltage input to spool position output) representation of the valve
(based on the Parker catalog information) is thus chosen as follows:
X (s) =
k
s 2
s
(1 +
) (1 +
)
2π 80
2π 900
V (s)
(7.3)
where k is 0.005 in./V. and is shown in Figure 7.9.
Figure 7.9: The open loop Bode plot of the valve transfer function model.
7.2.3
Servo-valve actuator model and real-time control schematic
The servo-valve actuator model schematic is given in Figure 7.10. The system construction is
also given in Figure 7.11. The closed loop transfer function of the servo-valve actuator system as
observed experimentally is given in Figure 7.12.
134
PLANT
Controller
Valve Dynamics
Xv(s)
U(s)
Reference
E(s)
+
R(s)
G(s)
Pvalve(s)
-
Fluid-structure
interaction and
actuator dynamics
Xsool(s)
Figure 7.10: Servo-valve actuator model schematic.
System
D/A Converter
Servo-valve
Hydraulic
Power Unit
Hydraulic
Cylinder
Load Mass
Pressure Sensor
Position Sensor
PC with Real Time Control
Hardware/Software
Figure 7.11:Servo-valve actuator system with the real-time control hardware/software.
135
Figure 7.12: Experimental transfer function of the closed loop (PD controlled) servo-valve
actuator system.
136
Figure 7.12 shows two modes (8Hz, and 25.5 Hz) related to the model. The 8 Hz mode is the
load mass mode and the 25.5 Hz mode is the first pipeline mode. The servo-valve actuator model
with first two modes of the pipeline dynamics also shows the similar closed loop Bode plot. Thus
the model seems to match the experimental observations. The details of this model are given in
the Appendix. A linearization of this model for a nominal system is also presented in the
Appendix. The open loop response of the numerical model of the servo-valve actuator system is
shown in Figure 7.13 and 7.14. This will be used in the next chapter to design higher order
controllers.
7.3 Effect of accumulators
The analysis of the servo-valve actuator system shows that the frequency of the complex load
mass mode is around 47 Hz. Depending upon the control bandwidth of the servo-valve actuator
system, this frequency is very high for the experimental stability analysis and control. Thus
accumulators are added to reduce the effective stiffness of the system (Figure 7.1). These are
diaphragm type accumulators (0.5 liters volume, charged to 150 Psi). The effect of adding
accumulators is to create additional volume in the cylinder and to modify the effective bulk
modulus of the electro-hydraulic cylinder. The effect of size of the accumulator used to reduce
the frequency of the load mass mode is shown in Figure 7.15. The temperature effects and other
nonlinear effects of the dynamics induced due to the diaphragm are neglected.
It is clearly visible that the by adding the accumulators if is possible to reduce the
frequency of the load mass mode so that it is sufficiently with in the control bandwidth of the
system. The control bandwidth of the servo-valve is significantly high (up to 80 Hz, per valve
specifications as given by Parker (Parker Catalog: HY14-1483-B1/USA)), but due to the
pipelines in between the servo-valve and the actuator (cylinder), the effective system control
bandwidth is around 25 Hz.
137
Figure 7.13: Open loop Bode magnitude: Actuator position/valve voltage.
Figure 7.14: Open loop Bode phase plot: Actuator position/valve voltage.
138
7.4 Chapter summary
In this chapter, the design of a servo-hydraulic system test-stand is presented with details
about selected components. A detailed experimental study is also presented for development of a
physically-based model for this test stand. A reduced-order model of the flapper nozzle valve is
developed. This model is then validated via the experiments described. The effect of
accumulators on the load mass mode is studied; these are useful in decreasing the frequency of
the load mass mode so as to place this mode within the system control bandwidth requirements.
After this model development and refinement, a detailed study of stability properties, effect of
feedback, and experimental evaluation of the stability boundaries of the system in parameter
space is to be conducted. This is given in Chapter 8.
Figure 7.15: Effect of the size of accumulator on the frequency of the load mass mode.
139
Chapter 8- Control studies
8
Introduction
A model for the experimental test stand is developed and validated in the previous chapter.
The aim of this chapter is to study the stability behavior of the servo-valve actuator system (with
pipe line model) under the influence of different linear control structures. In this chapter a
controller is designed based upon a linearized plant model. Two types of controllers are
investigated; a PD (proportional plus derivative) controller and a fifth-order linear controller are
used in this study. Then the nonlinear bifurcation stability of the closed loop system under the
effect of each of these controllers is investigated. All of the studies done in this chapter are with
feedback of cylinder position signal measured by the LVDT. The effect of PD control on the
stability of the system is studied under constant velocity operation. Further, a higher order
controller is designed via loop shaping methods. This controller is then implemented in the
system and stability analysis is conducted. It is shown that the stable region of the parameter
space may be enlarged by the use of an higher order controller as compared to the simple PD
controller. It is the principal aim of this work to characterize the nonlinear bifurcation stability of
the servo-valve actuator system under the linear feedback structures. Both experimental and
numerical results are presented.
8.1
PD controller
A PD controller is very frequently used in industrial control systems. This is a logical first
step in analyzing the effect of feedback on the system stability. It is a non-model-based controller
which requires very little prior knowledge of the system plant; as such these controllers are often
tuned online. Most systems benefit from a phase lead in some suitable frequency range. The
transfer function of the PD controller is Gc ( s ) = K p + K d s where Kp is the proportional gain and
Kd is the derivative gain. The PD controller may be tuned using Ziegler-Nichols rules (D’azzo
and Houpis, 1966). If the model of the plant is available then further mathematical analysis can be
140
done to tune the controller parameters. A Bode diagram of a sample PD controller is given in
Figure 8.1. The effect of a PD controller on the plant is to add a phase lead as seen from its Bode
diagram. These controllers are universally used because of the ease of implementation and
variability.
Figure 8.1: Bode diagram of a PD controller given by Gc=(s+10)
8.2
Effect of PD controller on the servo-valve actuator system
The effect of the PD controller on the stability behavior of the servo valve actuator system is
studied, both experimentally and numerically. The controller parameters (Kp and Kd) are part of
the extended parameter space and hence can be used in design and analysis of the closed loop
operation of the system. The control structure is implemented both numerically and
experimentally according to the schematic shown in Figure 7.11. The reference signal used for all
of these studies is a constant velocity ramp (for constant velocity operation of the system) for
reasons discussed in Sec. 5.2.2. The nominal closed loop eigenvalues of the servo-valve
141
Figure 8.2: Eigenvalues of the nominal closed loop system (Kp = 100 and Kd =1). Some
stable high frequency modes are not shown.
142
Table 4: Eigenvalues and damping of the nominal closed loop system.
Eigenvalue
Damping (%)
0
-
-1.48
-
-1.23e+001 + 1.17e+002i
1.05e-001
-1.23e+001 - 1.17e+002i
1.05e-001
-1.24e+001 + 1.18e+002i
1.05e-001
-1.24e+001 - 1.18e+002i
1.05e-001
-1.83e+001 + 3.53e+002i
5.19e-002
-1.83e+001 - 3.53e+002i
5.19e-002
-1.83e+001 + 3.53e+002i
5.18e-002
-1.83e+001 - 3.53e+002i
5.18e-002
-2.56e+001 + 4.58e+001i
4.88e-001
-2.56e+001 - 4.58e+001i
4.88e-001
-5.02e+002
-
-5.03e+002
-
-5.65e+003
-
143
actuator system for a constant velocity of 0.2 in/sec with Kp=100 and Kd=0.1 and a load mass (M)
of 100 lbs shown in Figure 8.2. The nominal closed loop system is stable. The complex modes (as
shown in Figure 8.2) are contributed by the load mass and line dynamics. A list of nominal
eigenvalues and damping is given in Table 4.
The system has a non-minimal state space representation as discussed previously in Sec.
6.2 and has a pole-zero cancellation at the origin. The effect of increase in the controller
parameters on the stability of the system is to result in Hopf bifurcation. The effect of variation in
the proportional gain (Kp) is shown in Figure 8.3. This shows the emergence of Hopf bifurcation
in the system at Kp=150 for a derivative gain Kd of 1. This is also experimentally verified and the
critical frequency of this system is 78 radians/sec. This critical frequency is also observed
experimentally. The effect of variation in Kd is shown in Figure 8.4. Hopf bifurcation occurs at a
critical value of Kd=63. The effect of variation in the load mass is to reduce the frequency of the
load mass mode, as seen in Figure 8.5. The Hopf bifurcation occurs at 400 lb for the nominal
controller parameters. The effect of increasing the constant steady state velocity of the system is
to increase the flow gain and hence is similar to increasing the proportional gain as seen from the
Figure 8.6. The Hopf bifurcation occurs at a constant velocity of 2.0 in/sec. The simulation
responses of the closed loop system for two sets of controller parameters, below and above
critical value, are shown in Figure 8.7, 8.8. This is in agreement with the numerical bifurcation
result. Also, the critical frequency (12.4 Hz) of the limit cycle oscillations is as predicted by the
bifurcation analysis. The effect of variation of parameter on a two dimensional grid is studied.
This gives insight into the bifurcation stability boundary. The effect of variation of controller
parameters (Kp, Kd) is shown in Figure 8.9. The effect of variation of mass and proportional gain
is shown in Figure 8.10. The effect of increasing the load mass is to reduce the critical frequency
of the bifurcating mode. The effect of variation of constant velocity and the proportional gain of
the system is shown in Figure 8.11.
144
Figure 8.3: The effect of variation of proportional gain. Hopf bifurcation occurs at Kp=150,
Kd=1.
145
Figure 8.4: The effect of variation of derivative gain. Hopf bifurcation occurs at Kp=100, Kd=63.
146
Figure 8.5: The effect of variation of load mass. Hopf bifurcation occurs at M=400 lb.
147
Figure 8.6: The effect of variation of the constant velocity of the system. Hopf bifurcation occurs
at V=2.0 in/sec.
148
Figure 8.7: Simulation response of the system for controller parameters just below critical
(Kp=140, Kd=1).
149
Figure 8.8: Simulation response of the system for controller parameters just above critical
(Kp=155, Kd=1).
150
Figure 8.9: Effect of parameter variation on a grid of controller parameters (Kp, Kd).
151
Figure 8.10: Effect of parameter variation on a grid of parameters (Kp, M)
152
.
Figure 8.11: Effect of parameter variation on a grid of parameters (Kp, V).
153
The stability boundary is sensitive to the change in parameters. This sensitivity is also a nonlinear
function of the parameters. Some insight into this behavior can be obtained by looking at the
bifurcation boundary in a multi parameter space. Figure 8.12 shows the Hopf bifurcation surface
as the control parameters (Kp, Kd) are varied. Experimental results are also shown on the same
figure. The experimental studies were started at a stable nominal in the parameter space and then
directional searches were conducted for estimating the stability boundary. The experimental data
is at an offset because the limit cycle oscillation amplitude has to be above noise level (5 Psi
oscillations –peak to peak) to be detected during the experiment. Thus it is seen that the
experimental data matches the bifurcation result very satisfactorily. Figure 8.13 shows the relative
movement of the stability boundary in the parameter space of the controller as load mass is
changed. The nonlinear sensitivity of the stability boundary is clearly evident from the figure.
Figure 8.14 shows the stability boundary in the two-dimensional parameter space of load
mass and proportional gain (Kp). The experimental observations are also depicted in the figure,
which match the numerical results. Figure 8.15 demonstrates the level cuts of stability boundary
in the two-dimensional parameter space of load mass and proportional gain (Kp) as the constant
velocity of the solution is varied. Figure 8.16, 8.17 show the experimental data (cylinder position
and cylinder chamber pressure) for a set of parameters above the critical value (Kp=200, M=100,
Kd=1, V=0.2 in/sec). It is clearly visible that small oscillations in the position of the cylinder
cause large oscillations in the cylinder pressure chambers. These large pressure oscillations are
one of the leading causes of system failure.
154
Figure 8.12: Stability boundary in two-dimensional parameter space (Kp, Kd) for load
mass=370lb - experimental and numerical result.
155
Figure 8.13: Effect of load mass on the stability boundary in two-dimensional parameter space
(Kp, Kd) - numerical result .
156
Figure 8.14: Stability boundary in two-dimensional parameter space (Mload, Kp) for constant
velocity =0.2 in/sec and Kd=0.1- experimental and numerical result.
157
Figure 8.15: Effect of varying constant velocity on the stability boundary in two-dimensional
parameter space (Mload, Kp) for Kd=0.1- numerical result.
158
.
Figure 8.16: Experimental data of cylinder position with reference for parameter values above
critical (Kp=200 Kd=1 M=100lb).
159
Figure 8.17: Experimental data of cylinder pressure for parameter values above critical (Kp=200
Kd=1 M=100lb)-pressure in both chambers of the cylinder is presented.
160
8.3
Effect of fifth-order linear controller on the servo-valve actuator system
The study of the PD controller in the preceding section showed that some control of the
stability boundary is possible by a simple feedback structure, and it also demonstrated the effect
of feedback on the stability of the system. A higher order (fifth-order) controller for a linear plant
model based upon an operating condition of a 2 volt (10% of full scale) command input to the
servo-valve with a load mass of 100 lbs and a line pressure of 450 Psi is designed to improve the
stability of the servo-valve actuator system. Based upon preliminary studies, it was decided that
the compensated system should have at least 30o phase margin (D’azzo and Houpis, 1966) and a
gain margin (D’azzo and Houpis, 1966) of 3 db or more with an open loop bandwidth of 8 Hz.
Classical compensator design is a trial-and-error process which is discussed in detail by several
authors including D’azzo and Houpis (1966). Using Bode analysis and design (D’azzo and
Houpis, 1966) the following fifth-order compensator is developed:
C (s) =
1.433e015 s 4 + 1.054e017 s3 + 5.396e019 s 2 + 2.391e021 s + 1.445e023
(8.1)
(s 9 + 5615 s8 + 1.042e007 s 7 + 9.839e009 s 6 + 5.478e012 s5 + 1.909e015 s 4
+ 4.221e017 s 3 + 5.751e019 s 2 + 4.402e021 s + 1.445e023)
The poles of the compensator are as follows:
-3.1280e+003
-5.6497e+002
-3.9273e+002 +7.8390e+001i
-3.9273e+002 -7.8390e+001i
-3.9502e+002
-2.1400e+002 +8.3589e+001i
-2.1400e+002 -8.3589e+001i
-1.6932e+002
-1.4444e+002
The zeros of the compensator are as follows:
-1.2946e+001 +1.8229e+002i
-1.2946e+001 -1.8229e+002i
-2.3811e+001 +4.9526e+001i
-2.3811e+001 -4.9526e+001i
161
Figure 8.18: Bode magnitude plot of the compensator.
Figure 8.19: Bode phase plot of the compensator.
162
Figure 8.20: Nominal closed loop system is stable (Kv=100), some high frequency stable
eigenvalues are not shown.
163
Table 5: Eigenvalue and damping of the closed loop nominal system.
Eigenvalues
-5.82e-005
-5.15e+000
-1.22e+001 + 3.43e+002i
-1.22e+001 - 3.43e+002i
-1.23e+001 + 3.43e+002i
-1.23e+001 - 3.43e+002i
-1.83e+001 + 1.05e+003i
-1.83e+001 - 1.05e+003i
-1.83e+001 + 1.05e+003i
-1.83e+001 - 1.05e+003i
-2.63e+001 + 4.73e+001i
-2.63e+001 - 4.73e+001i
-6.56e+001
-1.40e+002 + 1.17e+002i
-1.40e+002 - 1.17e+002i
-2.82e+002 + 1.75e+002i
-2.82e+002 - 1.75e+002i
-4.56e+002 + 1.52e+002i
-4.56e+002 - 1.52e+002i
-5.10e+002
-5.77e+002 + 5.10e+001i
-5.77e+002 - 5.10e+001i
-3.13e+003
-5.65e+003
Damping (%)
3.57e-002
3.57e-002
3.58e-002
3.58e-002
1.75e-002
1.75e-002
1.75e-002
1.75e-002
4.86e-001
4.86e-001
7.67e-001
7.67e-001
8.50e-001
8.50e-001
9.48e-001
9.48e-001
9.96e-001
9.96e-001
-
164
Figure 8.21: Effect of varying Bode gain of the controller (Kv). Hopf Bifurcation occurs at
Kv=220, M=100 lb, V=0.2 in/sec.
165
Figure 8.22: Effect of varying load mass. Hopf Bifurcation occurs at Kv=100, M=600 lb, V=0.2
in/sec.
166
Bode gain of a system transfer function of type -m (with m poles at s=0) is defined as the
DC gain of sm P(s), when the transfer function is represented in Bode form. Critical Bode gain is
the value of the Bode gain when the system loses stability via bifurcation. This fifth-order
controller is designed to increase the critical Bode gain of the servo-valve actuator system as
compared to the PD controlled case. The aim is to increase the stable region of the extended
parameter space by shifting the stability boundary further out and thus extending the envelope of
the design in closed loop. The Bode magnitude and phase of the compensator is given in Figure
8.18 and 8.19. Nominal eigenvalues of the closed loop system are stable (Figure 8.20) as shown
in Table 5. The effect of varying Bode gain is to drive the system unstable (Figure 8.21) with a
Hopf bifurcation occurring at Kv=220. The effects of increasing load mass is to reduce the
frequency of the associated mode, and eventually result in loss of stability via Hopf bifurcation
(Figure 8.22). Effect of varying the constant velocity of the system is shown in Figure 8.23. Hopf
bifurcation occurs at a velocity of 2.8 in/sec. Two-dimensional grids in parameter space are
investigated for stability behavior. Figure 8.24 shows the effect of variation in load mass and
compensator gain. Figure 8.25 shows the effect of line pressure and load mass. Simulation of the
nonlinear model also verifies the bifurcation stability result. Figure 8.26 and 8.27 are the
simulation results obtained for compensator gain below critical value and compensator gain
above critical value, respectively. Figure 8.28 shows the experimental cylinder position response
for compensator gain above critical. The oscillation in cylinder chamber pressure (Figure 8.29)
has an amplitude of nearly 80 Psi, which can be very detrimental to the system performance and
could eventually lead to failure. Under this unstable operating condition, the flexible pipelines
also vibrate violently causing large acoustic emissions. The critical frequency of the system is
around 11 Hz. It is found that the load mass mode is the bifurcating mode. Figure 8.30 shows
level cuts of the stability boundary of the system as velocity is varied.
167
Figure 8.23: Effect of varying the constant velocity.
168
Figure 8.24: Effect of varying compensator gain (100-250) and load mass (50-500 lb).
169
Figure 8.25: Effect of varying line pressure and load mass.
170
Figure 8.26: Simulation response of cylinder position and velocity for compensator gain below
critical (Kv=100 M=370 lb).
171
8.27: Simulation response of cylinder position and velocity for compensator gain above critical
(Kv=220 M=370 lb).
172
Figure 8.28: Experimental response of the cylinder for parameters above critical (Kv=220).
173
Figure 8.29: Experimental pressure response in the cylinder chambers for parameter above
critical (Kv=220).
174
Figure 8.30: Bifurcation stability boundary in the 2-dimensional parameter space of load mass
and Bode gain. Level cuts of constant velocity solution are shown. Experimental data matches
closely with the numerical result.
175
Figure 8.31: A comparison of the two controllers – PD and fifth-order for a constant velocity
case (0.2 in/sec).
176
Table 6: Comparison of critical parameter values
Parameter
PD controller case
Bode gain
190
Fifth-order
controller case
220
Constant velocity (in/sec)
2.0
2.2
Load mass (lbs)
400
560
Table 7: Comparison of critical parameter values for linear analysis and bifurcation analysis
for a fifth –order controller case.
Critical parameter
Linear analysis
Bifurcation analysis
Bode gain
68
220
8.4 Comparison of control studies
It is noted that as the constant velocity of the solution is increased, the system becomes
less stable. The experimental observation of the critical gains is in agreement with the numerical
result obtained via the bifurcation analysis.
The fifth-order compensator was designed to improve the stability margin of the servovalve actuator system over the PD controlled system in closed loop operation. Figure 8.31 shows
that the higher-order controller indeed results in a larger stability margin as compared to the PD
controlled case. A similar comparison of a few parameters is presented in the Table 6. Critical
parameter values of the system for the two controllers (PD and fifth-order) are compared and it is
clearly visible from this comparison (Table 6) that the fifth-order controller manages to enlarge
the stable region of the servo-valve actuator system for better closed-loop stability margin.
The measure of bifurcation nonlinearity as defined is Sec. 5.6 is estimated for the servovalve actuator system with the fifth-order compensator. It is seen from Figure 8.32 and 8.33 that
the increase in the nonlinear nature of the system is accurately captured by this metric. It should
177
be noted even further that the linear Bode analysis of the servo-valve actuator system suggests
that the fifth-order controller for a gain margin of 3 db in conjunction with the nominal Kv of 1
will have a critical gain of at least 68 for load mass of 100 lb. But in actual experimental set up it
is observed that the critical gain of the fifth-order controller is 220 for the same operating
condition (Table 7). This opens a further range of stability margin which can be utilized for
operating the system at the edge of the stability boundary. This range will not be known if the
designer was to trust only the linearized analysis. Thus not only the nonlinear analysis provides
better estimates of the stability boundary but also provides increased robustness and control
bandwidth. Thus it is clear that with a design of higher order controllers stability margin can be
increased even with simple linear controllers.
178
Figure 8.32: Measure of bifurcation nonlinearity as the load mass is varied for the servo-valve
actuator system with the fifth-order compensator.
Figure 8.33: Measure of bifurcation nonlinearity as the load mass is varied for the servo-valve
actuator system with the fifth-order compensator.
179
Chapter 9- Conclusions and suggestions for future research
The stability analysis and design of nonlinear systems is studied in this research with an
emphasis on the nonlinear dynamic behavior of servo-hydraulic systems and their components.
The concept of constant velocity solutions is proposed and demonstrated. The idea of extended
parameter space is introduced. A mathematical model is developed. This model includes modal
approximation of the pipe line dynamics between the servo-valve and actuator. It was found that
the inherent nature of servo-hydraulic system models is such that near a stationary point it
becomes non-minimal. This non-minimal nature of the model is highlighted. Bifurcation theory is
used to analyze nonlinear stability of the servo-hydraulic system. The three test cases (servopump actuator system, servo-valve actuator system, physically reduced flapper nozzle servovalve actuator system) are investigated in this dissertation. Bifurcation stability results for all the
three cases are given in this dissertation. An experimental test stand is developed to implement a
servo-valve actuator system. The refinement of the model is an iterative process limited by the
measurement sensors available and the signal to noise ratio characteristics of the system. Once
the mathematical model validation is conducted, further control and stability studies can be
accomplished. The existence of saddle-node and Hopf bifurcation in the servo-hydraulic systems
is demonstrated both numerically and experimentally in this work.
The effect of feedback and different control structures is evaluated. The effect of
controllers on the stability of the servo-hydraulic system is also an important aspect of this work.
The idea of control-induced bifurcations is proposed and demonstrated in this work. A stabilizing
controller and a stable nominal point is calculated for each specified control structure. Then the
bifurcation stability boundary for the system with the control structure is investigated. The
stability behavior of a simple PD controller is first studied both numerically and experimentally.
The primary output feedback structure used in this study is position feedback. A fifth-order
180
controller is also designed for a nominal point using Bode design methods to increase the critical
Bode gain for the overall system. It is demonstrated that the fifth-order controller does extend the
stability boundary out thus giving more stability margin for exploitation. This leads to extension
of the stability region as compared to the PD controller. The results obtained are validated both
by nonlinear simulation and via the experimental observations on the test stand. This work is a
qualitative and quantitative study of the effect of feedback on the stability of the servo-hydraulic
systems. Following are the primary conceptual contributions of this work:
1. Development of the idea of constant velocity solutions for servo-hydraulic system
analysis.
2. Development of the concept of control induced bifurcations.
3. Introduction of the notion of extended parameter space.
4. Proposed a measure of bifurcation nonlinearity.
The development of the test stand and its analytical model paves the way for future research in
this area. This test stand and model can be used in several future studies. These studies can be
related to stability analysis, closed loop performance, robust control design, optimal control
studies and/or evaluation of various other feedback control structures. The effect of other
boundary conditions and configurations can also be evaluated in future. Following are some of
the proposed directions of future research:
1. The effect of various hardware modifications such as hard plumbing, larger hydraulic
power supply, and different valve and actuator configurations can be investigated for
its effect on the system stability.
2. The effect of pressure feedback can be studied for bifurcation stability and control.
3. The effect of other control structures can be investigated which could include
feedback linearization.
4. A QFT based robust controller design can be used to investigate its effect on the
stability of the system.
181
5. Other nonlinear controllers (neural-network based controllers) can be investigated.
6. Codimension-1 bifurcations are the subject of this study, however further bifurcations
of higher co-dimensions can be investigated.
7. Development of hybrid system models of the servo-hydraulic systems to include hard
nonlinearities can be investigated. The effect of these hard nonlinearities on the
system performance can be investigated.
8. A development of the comprehensive methodology for bifurcation analysis of the
hybrid systems is very highly desirable and a very significant direction of future
research.
9. A global methodology for parametric space and state space decomposition for the
nonlinear systems can be investigated. Any development along these lines would be
useful to the entire nonlinear systems community.
10. Optimization based strategies can be investigated to develop control structures with
multiple objectives such as performance, bifurcation stability margin, and robustness.
This would require computation of optimally minimum distance to instability as well
as the robustness bounds.
This work provides foundation for further research in the area of nonlinear systems and
more specifically in the area of servo-hydraulic systems by providing a basic ground work using
experimental and numerical investigations. These studies, when employed for design for
nonlinearity, are useful in optimal use of resources available to the design and controls engineer.
This leads to new explorations and also, opens further vistas for the efficient use of control effort
as well as the stability region. Nonlinearity, when exploited properly, can be used to increase the
system performance and stability margin. This design for nonlinearity is the general direction of
future research. Nonlinearity in the systems should be evaluated as a design parameter. The
nonlinear nature of the system should be quantified (e.g., by using the measure for bifurcation
182
nonlinearity). These estimates can then be exploited for their potential during the design as well
as operation of the intended system.
183
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Appendix-1:
Servo-valve actuator mode with pipeline dynamics and fifth order
complex controller – Matlab implementation
function [xdot] =SVA_TLD(t,x)
global x_valve
global param;
global omega;
%Kd=param(1);
%Kd=param(2);
vref=param(3);
mload=param(4)/32.2/12;
Pline=param(2); %450;
Kv=param(1);
%
xload=x(4);
dxload=x(5);
%
% % P1=x(6);
Pa=x(6);
Pb=x(7);
xref=vref*t; %0.2*t+0.05*sin(2*pi*omega*t);%0.2*t+
vref=vref; %0.2+(0.05*(cos(2*pi*omega*t))*2*pi*omega);%0.2; %
% CONTROLLER STATES
con_num
=[0,0,0,0,0,1.433055185448249e+015,1.053510289988146e+017,5.395545716638774e+
019,2.391325547474389e+021,1.445311090374986e+023];
con_den
=[1,5.615220397075729e+003,1.042386415118273e+007,9.839272314838625e+009,5.4
78011475094872e+012,1.909445984590581e+015,4.220774524445812e+017,5.750710
724984177e+019,4.401930654600736e+021,1.445311090374986e+023];
[A_con,B_con,C_con,D_con]=tf2ss(con_num,con_den);
x_con=x(end-8:end);
con_input=Kv*(xref-xload);
dx_con=A_con*x_con+B_con*con_input;
u_volt=C_con*x_con+D_con*con_input;
198
% **************************************
% SERVO VALVE
% **************************************
% states....
% x1v=x(1);
% x2v=x(2);
% x3v=x(3);
% transfer function..
% num= 7.143846147141077e+006;
% den=conv([1 2*pi*80],[1 2*pi*80]);
% den=conv(den,[1 2*pi*900]);
num= 1;%7.143846147141077e+006;
den=conv([1/(2*pi*80) 1],[1/(2*pi*80) 1]);
den=conv(den,[1/(2*pi*900) 1]);
% state space model....
[A,B,C,D]=tf2ss(num,den);
xdot(1:3,1)=A*[x(1);x(2);x(3)]+B*[u_volt];
x_valve=C*[x(1);x(2);x(3)]+D*u_volt ;
% line dynamics output......
P1qa=[1 0 1 0; 0 1 0 1]*[x(8);x(9);x(10);x(11)];
P1=P1qa(1);
qa=P1qa(2);
qa=-qa;
P2qb=[1 0 1 0; 0 1 0 1]*[x(12);x(13);x(14);x(15)];
P2=P2qb(1);
qb=P2qb(2);
% area characteristic.. and flow calculations........
%valve volt for area ch
V_volt= x_valve;
%area,cd characteristics
area_curve=[0.00000038527429 -0.00001264772476 0.00009829412559
0.00048272373205 0.00298482488548 0] *3.85; %0.00003070588999];
if abs(V_volt)<=10.0,
Cd_area_rho=polyval(area_curve,abs(V_volt));
else
Cd_area_rho=polyval(area_curve,10);
end
if V_volt>=0.0,
if abs(Pline-P1)<=1,
qv1=Cd_area_rho*(abs(Pline-P1))*sign(Pline-P1);
else
qv1=Cd_area_rho*sqrt((abs(Pline-P1)))*sign(Pline-P1);
199
end
if abs(P2)<=1,
q2v=Cd_area_rho*(abs(P2))*sign(P2);
else
q2v=Cd_area_rho*sqrt((abs(P2)))*sign(P2);
end
else
if abs(P1)<=1,
qv1=-Cd_area_rho*(abs(P1))*sign(P1);
else
qv1=-Cd_area_rho*sqrt((abs(P1)))*sign(P1);
end
if (Pline-P2)<=1,
q2v=Cd_area_rho*(abs(Pline-P2))*sign(P2-Pline);
else
q2v=Cd_area_rho*sqrt((abs(Pline-P2)))*sign(P2-Pline);
end
end
% **************************************
% PIPE LINE DYNAMICS
% **************************************
% state space
rho=0.0000795;; % density
ro=3/8/2;
%E =4.716260998743000e+003;
E=22000;
% E=250;
co=sqrt(E/rho);%sound velocity of fluid
mu=0.0496000992001984 ;%2*1.55*0.001/0.78e-4;
% mu =0.02388;
L=6*12;
om=mu/ro^2;
Dn=L*mu/(co*ro*ro);
Zo=rho*co/(pi*ro*ro);
lambda1=pi/(2*Dn);
lambda2=3*pi/(2*Dn);
alpha1=1.06;
alpha2=1.04;
beta1=2.31;
beta2=3.38;
% alpha1=1.07;
% alpha2=1.05;
% beta1=1.76;
% beta2=2.94;
200
%inlet side
A10=[0
Zo*lambda1 ;...
-lambda1/(Zo*alpha1*alpha1) -8*beta1/alpha1];
B10=[0 -2*Zo/Dn; 2/(Zo*Dn*alpha1*alpha1) 0];
A20=[0
-Zo*lambda2 ;...
lambda2/(Zo*alpha2*alpha2) -8*beta2/alpha2];
B20=[0 -2*Zo/Dn; 2/(Zo*Dn*alpha2*alpha2) 0];
A_tl0=[A10 zeros(2,2);zeros(2,2) A20];
B_tl0=[B10' B20']';
diff_mat=[1 -8*Zo*Dn; 0 1];
Hn=-1*(inv(A10)*B10+inv(A20)*B20);
G=inv(Hn)*diff_mat;
xdot(8:11,1)=om*(A_tl0*[x(8);x(9);x(10);x(11)]+B_tl0*G*[Pa;-qv1]);
xdot(12:15,1)=om*(A_tl0*[x(12);x(13);x(14);x(15)]+B_tl0*G*[Pb;q2v]);
% **************************************
% **************************************
% M C K of piston
Cd=0.61;
k=0;
c=10;
Aleak=pi*0.001*0.001/4;
if abs(Pa-Pb)<=1,
qleak=Cd*Aleak*sqrt(2/rho)*(abs(Pa-Pb))*sign(Pa-Pb);
else
qleak=Cd*Aleak*sqrt(2/rho*abs(Pa-Pb))*sign(Pa-Pb);
end
%cross sectional area of piston
Ap1=pi/4*2.5^2;% lower volume inch^2
Ap2=pi/4*(2.5^2-1); % upper volume inch^2
% Volumes of each chamber, in^3
Va=3.5*Ap1;% lower volume
Vb=3.5*Ap2;% upper volume
% Variable volume
Va=Va+Ap1*x(4);
Vb=Vb-Ap2*x(4);
beta_airfree=22000;
percentage_air=0.05;
P_atm= 14.4;
201
beta_airfree=22000;
percentage_air=0.05;
P_atm= 14.4;
% % % Calculate the variable Beta.
% beta_eff1=1/(1/beta_airfree + percentage_air/(1.4*(Pa+P_atm)));
% beta_eff2=1/(1/beta_airfree + percentage_air/(1.4*(Pb+P_atm)));
% % beta_eff1=1/(1/beta_eff1a + 1/(1.4*(150+P_atm)));
% % beta_eff2=1/(1/beta_eff2a + 1/(1.4*(150+P_atm)));
beta_eff1a=1/(1/beta_airfree + percentage_air/(1.4*(Pa+P_atm)));
beta_eff2a=1/(1/beta_airfree + percentage_air/(1.4*(Pb+P_atm)));
beta_eff1=1/(1/beta_eff1a + 1/(1.4*(150+P_atm)));
beta_eff2=1/(1/beta_eff2a + 1/(1.4*(150+P_atm)));
%
%load states
xdot(4)=x(5);
xdot(5)=(-k*x(4)-c*x(5)+Ap1*Pa-Ap2*Pb-mload*32.2*12)/mload;
xdot(6)=beta_eff1/Va*(qa-qleak-Ap1*x(5));
xdot(7)=beta_eff2/Vb*(qleak+Ap2*x(5)-qb);
xdot=[xdot;dx_con];
% *************************************
% change the state vector.........
% *************************************
%xdot=[dxv;dxload;ddxload;dPa;dPb;dlined1;dlined2];
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