Computation of QFT bounds for robust tracking speci'cations P.S.V. Nataraj ∗
Systems and Control Engineering Group, Department of Electrical Engineering, Indian Institute of Technology,
Bombay 400 076, India
Abstract
An algorithm is proposed for generation of QFT controller bounds to achieve robust tracking speci'cations. The proposed
algorithm uses quadratic constraints and interval plant templates to compute the bounds, and presents several improvements over
existing QFT tracking bound generation algorithms. The proposed algorithm (1) guarantees robustness against template inaccuracies,
(2) guarantees robustness against phase discretization, (3) provides a posteriori error estimates, (4) is computationally e8cient,
achieving a reduction in !ops and execution time, typically by 1–2 orders of magnitude. The algorithm is demonstrated on an
aircraft example having 've uncertain parameters.
Keywords: Interval analysis; Quantitative feedback theory (QFT); Robust control; Robust performance; Tracking characteristics
1. Introduction
In the bound generation step of Horowitz’s quantitative feed back theory (QFT) design procedure (Horowitz,
1991), the plant template is used to translate the given
robustness speci'cations into domains in the Nichols
chart where the controller gain-phase values are allowed
to lie. These domains de'ne what are commonly known
as QFT controller bounds.
Various approaches to the bound generation problem
are available. For speci'c plant structures such as interval plants, the bounds can be generated using the techniques in Fialho, Pande, and Nataraj (1992) and Zhao
and Jayasuriya (1994). For more general plant structures,
early bound generation algorithms (East, 1981; Houpis &
Lamount, 1988; Longdon & East, 1979; Nataraj, 1994;
Wang, Chen, & Wang, 1991; Yaniv, 1990) used geometrical and=or search-based techniques to compute the
bounds. However, these algorithms are typically rather
slow. Recently, more e8cient numerical algorithms based
on quadratic constraints have been proposed to automate
this step (Chait, Borghesani, & Zheng, 1995; Chait &
Yaniv, 1993a, b; Rodrigues, Chait & Hollot, 1997). In
these algorithms, at a given design frequency and controller phase, the QFT bounds are computed by solving
certain quadratic equations for each plant in the template.
We shall refer to these algorithms as quadratic constraints
(QC) algorithms.
In this paper, we present further advancements and
improvements in QC algorithms for robust tracking
speci'cations. These are discussed below.
1. Guaranteed robustness against template inaccuracies: A key property of the proposed algorithm is that
the computed tracking bounds are guaranteed to be robust against template inaccuracies, in the sense that regardless of the accuracy of the interval plant template
used, the generated bounds can never lead to violation
of the given robust tracking speci'cations (assuming,
of course, that in the subsequent loop-shaping step of
QFT, a controller that respects the generated bounds
is synthesized).
2. Guaranteed robustness against phase discretization:
In QC algorithms and in the QFT toolbox (Borghesani, Chait, & Yaniv, 1995), the tracking bounds are
generated only at user-selected controller phases in
the range [ − 2; 0]. Therefore, at non-selected phases,
any blips or dips in the bounds would be missed,
328
and might lead to violation of the given robust tracking speci'cations at these phases. To overcome this
problem, the tracking bounds are generated in the
proposed algorithm in such a way that they are
guaranteed to be valid over (entire) controller phase
intervals.
3. Error estimates: The proposed algorithm can construct inner as well as outer enclosures of the exact
tracking bounds. Such enclosures directly provide a
posteriori the maximum possible error in the generated tracking bounds.
4. Computational e6ciency: The proposed algorithm
is computationally e8cient, achieving a reduction in
!ops typically by a few orders of magnitude compared
to QC algorithms.
We assume in this paper that the reader is familiar
with general concepts of bound generation for QFT
tracking speci'cations, such as those found in Horowitz
(1991).
2. Background
2.1. Robust tracking speci:cations
Consider a single-input–single-output linear time invariant plant G(s) and controller K(s) embedded in a
two-degree-of-freedom structure. Suppose there is uncertainty in some or all of the plant parameters so that we
have a plant family {G(s)}.
The QFT robust tracking speci'cation follows from
the requirement that the plant output should follow a
given desired output (as in a servo system) where, due
to the plant uncertainty, the desired output is bounded
between upper and lower time functions (Yaniv, 1999).
In the frequency domain, this requirement can be speci'ed as follows. At each frequency !, for any plant
in the plant family the magnitude of the closed loop
transfer function T (s) from the setpoint to the output is
bounded by
20 log10 ((!)) 6 20 log10 |T ( j!)| 6 20 log10 (
(!));
(1)
where (!) and (!) are positive real valued functions
of ! (for details of modeling the functions (!) and (!)
from given lower and upper time functions, see D’Azzo
& Houpis, 1995; Horowitz, 1993).
For any plant family elements Gi (s) and Gk (s), we can
rewrite (1) as
20 log10 |Ti ( j!)| − 20 log10 |Tk ( j!)|
6 20 log10 (
(!)) − 20 log10 ((!))
or (henceforth we shall drop the arguments)
|KGi =(1 + KGi )|
6 , :
|KGk =(1 + KGk )|
(2)
Therefore, to achieve the QFT robust tracking speci'cations, we need to 'nd a controller K(s) such that
at each frequency !, the above inequality is satis'ed for
all pairs of plant elements Gi (s) and Gk (s) in the plant
family {G(s)}.
2.2. Interval plant templates
In the sequel, we let ! be an arbitrary but 'xed frequency. The plant template at ! is de'ned as G ,
{G( j!)}. Usually, instead of the exact template G we
have an approximating template generated by an appropriate existing template generation algorithm. The algorithms in Nataraj and Sardar (2000) and Sardar and
Nataraj (1997)
generate the so-called interval plant template Galg , G, where each G , (; G) is a 2-dim. interval vector, with denoting the angle interval and G the
magnitude interval. Each G, therefore, forms an angle–
magnitude rectangle (or a box) in the Nichols chart. A
key property of the interval template Galg is its inclusion
property.
Theorem 2.1 (Nataraj and Sardar (2000) and Sardar and
Nataraj (1997)) G ⊆ Galg .
Denition 2.1. Gbnd will denote the collection of all the
boundary rectangles of Galg .
We make the following standing assumption.
Assumption 1. G and Galg are closed and simply connected sets. Furthermore, the closed loop system is
robustly stable with respect to these sets.
2.3. Interval quadratic equation
With respect to the interval plant template Galg , we can
write the QFT robust tracking spec in (2) as
|KGi =(1 + KGi )|
6¿1
|KGk =(1 + KGk )|
for all Gi ; Gk ∈ Galg :
It follows from Assumption 1 and application of the maximum modulus theorem that
KGi
| 1+KG
|
i
KGk
| 1+KG
|
k
⇔
6
∀Gi ; Gk ∈ Galg
KGi
|
| 1+KG
i
KGk
| 1+KG
|
k
6
∀Gi ; Gk ∈ Gbnd :
(3)
329
Hence, at , the allowable controller gain range, or simply, the controller gain range, Kik for the pair of boundary rectangles Gi ; Gk is
Kik , [0; kmin ] [kmax ; ∞):
(8)
The above result is given in Chait, Borghesani, and Zheng
(1995) for discrete controller phases, but we can extend it
in a straightforward way for the case of interval controller
phases.
Let K = kej , where k is the controller gain and the
controller phase interval. In the sequel, unless otherwise
stated, we shall assume to be an arbitrary but 'xed
controller phase interval.
Further, let each rectangle G ∈ Gbnd be also expressed
in polar form as G = Gej , where G denotes the magnitude interval and the phase interval. Consider an
arbitrary but 'xed pair of boundary rectangles Gi ; Gk ∈
Gbnd . Then, by substituting these quantities in (3), squaring both sides and simplifying, we get the quadratic
inequality
1
Gi2 Gk2 1 − 2
k2
Gk Gi2
2
+2 Gk Gi cos(i + ) −
cos(k + ) k
2
G2
(4)
+ Gk2 − 2i ¿ 0:
The quantities kmax and kmin are known as the upper and
lower bounds on the controller gain at for the considered pair.
Therefore, the controller gain range K at over the
entire set of boundary boxes Gbnd is
K,
Kik :
(9)
all pairs i; k of boundary boxes
From (3), this is the same as the controller gain range
at for the entire interval template Galg . Similar to the
boundary values kmin and kmax in (8), the boundary values of K are called as the bounds on the controller gain
at for the entire interval template. Note that when
both the (upper and lower) bounds on the controller gain
exist then the tracking bounds become multiple-valued
at that .
Making the following substitutions (note that each Xi
below is an interval)
1
1
X1 , ; X2 ,
; X3 , i + ; X4 , k + Gi
Gk
(5)
3. Main developments
We see that the terms inside {·} on the RHS of
(6) and (7) correspond to the ranges of the following
and solving for k in (4) with the equality sign we get

2 
2 

X
cos
X
X
cos
X
1
X


2
4
2
4
2

− X1 cos X3 +
− X1 cos X3 − 1 − 2
X12 − 2 



2
2

;
kmax , max


1




1
−




2

2 
2 

X
cos
X
X
cos
X
1
X


2
4
2
4
2

− X1 cos X3 −
− X1 cos X3 − 1 − 2
X12 − 2 



2
2

:
kmin , min


1




1
−




2
(6)
(7)
functions:
x
f(x1 ; x2 ; x3 ; x4 ) ,
2
cos x4
2
− x1 cos x3 +
x cos x
2
4
h(x1 ; x2 ; x3 ; x4 ) ,
2
− x1 cos x3 −
x cos x
2
4
2
− x1 cos x3
1
1− 2
2
1
− 1− 2
x12
x2
− 22
;
x
2
cos x4
2
− x1 cos x3
1
1− 2
2
1
− 1− 2
x12
x2
− 22
;
(10)
330
where x1 ; x2 ; x3 ; x4 are real numbers in the region
X = {(x1 ; x2 ; x3 ; x4 ) | x1 ∈ X1 ; : : : ; x4 ∈ X4 };
where X1 ; : : : ; X4 are closed and bounded intervals on the
real line Xi = [ai ; bi ]; ai 6 bi ; i = 1; 2; 3; 4.
Once we compute the ranges of f and h for each pair
of boundary rectangles of the interval template, we can
readily 'nd the controller gain range using (6) – (9).
The main problem is then of 'nding the ranges of f and
h in the said region. We can pose this problem, of course,
as a standard optimization problem over the bounded region X , and solve it using any one of a variety of classical
optimization methods available in the optimization literature. However, we seek relatively more e8cient methods that can exploit the speci'c nature of our functions
f and h.
We can obtain the ranges of f and h with relatively
much less eOort using the powerful concept of monotonicity applied to the whole of X or to some suitably
subdivided regions (such as subboxes) of X . By combining the concept of monotonicity with the tool of subdivision, we can 'nd the ranges of f and h over X , directly
and exactly, with no need for any initial guess values
or algorithmic iterations (as in classical optimization
methods).
We explain monotonicity and subdivision, using the
function f.
3.1. Monotonicity
Let the partial derivative of f w.r.t. xi be denoted
as fi . Then, it is well known from calculus (see, for
instance, Taylor, 1955) that if fi does not contain
both positive and negative values, then f is monotone
(non-increasing or non-decreasing) in the xi direction and
attains its extremum on those two parts of the boundary
of X where xi attains its extremal values. That is to say,
if fi (x1 ; x2 ; x3 ; x4 ) ¿ 0 (resp. 6 0) for all x ∈ X , then
min f (resp. max f) occurs in @ai X and max f (resp.
min f) occurs in @bi X , where for Xi = [ai ; bi ]; ai 6 bi ;
i = 1; 2; 3; 4;
@ai X , {(x1 ; : : : ; xi−1 ; ai ; xi+1 ; : : : ; x4 ) | xj ∈ Xj ; j = i};
@bi X , {(x1 ; : : : ; xi−1 ; bi ; xi+1 ; : : : ; x4 ) | xj ∈ Xj ; j = i}
Furthermore, if f is monotone in all directions x1 ; : : : ; x4
on a box X , then we can intersect those parts of the
boundary of X where the extremum of f are known to
lie, and obtain the range of f using just two function
evaluations of f.
To illustrate the idea, suppose f is monotone in
all directions x1 ; : : : ; x4 on a given box X , with, say,
f1 6 0; f2 ¿ 0; f3 6 0; f4 ¿ 0 for all x ∈ X . Then,
max f
Table 1
Vanishing points of partial derivatives
Partial derivative
f1 ; h1
Solution for vanishing points
√
cos x4 ± xˆ1 = x2 cos x3 2
cos2 x3 − 2 + 1
√
cos x3 ± −
cos2 x4 + 2 − 1
f2 ; h2
xˆ2 = x1 cos x4
f3 ; h3 ; f4 ; h4
xˆ3 ; xˆ4 ∈ {n; n = 0; ±1; ±2; : : :}
occurs in @a1 X; @b2 X; @a3 X; @b4 X , and min f occurs in
@b1 X; @a2 X; @b3 X; @a4 X . Moreover, we can intersect these
parts of the boundary of X where the extrema of f are
known to lie and 'nd that max f occurs at
@b 2 X
@a3 X
@b4 X = (a1 ; b2 ; a3 ; b4 )
xQ , @a1 X
(11)
and that min f occurs at
@a 2 X
@b3 X
@a4 X = (b1 ; a2 ; b3 ; a4 ):
x , @b 1 X
(12)
We note that xQ and x are just points.
Thus, if the special hypothesis (that f is monotone in
all variables) holds for X = (X1 ; X2 ; X3 ; X4 ) itself, then we
can 'nd the range of f using just two evaluations of f,
one each at the points xQ and x.
3.2. Combining monotonicity with subdivision
In general, the special hypothesis of the preceding
subsection will not hold for X itself. In this case, we
can subdivide X into several parts in the form of
subboxes, such that f is monotone in all variables on every subbox. Then, since (by construction) f is monotone
in all variables on each subbox, we can apply the above
monotonicity-based range 'nding technique to each subbox and 'nd the range of f on each subbox, easily and
exactly, using just two evaluations of f. Lastly, we can
take the union of the ranges of f over all the subboxes
and obtain the range of f over the original box X .
We next, therefore, examine how to subdivide X into
subboxes such that f is monotone in all variables on
every subbox.
First, using symbolic software, we 'nd the expressions
for all the points at which the partial derivatives f1 ; : : : ; f4
vanish in the respective intervals X1 ; : : : ; X4 . This is done
by setting the partial derivative w.r.t. each variable to
zero, and solving for the independent variable. Table 1
gives the resulting expressions for the vanishing points
(in this table, , cos2 x4 − 2 cos2 x3 + 2 − 1).
Now, let X̂1 , {x̂1; j1 ; j1 = 1; : : : ; N1 } denote the set
of all points (say, N1 in number) in the interval X1 at
which the partial derivative f1 (x1 ; x2 ; x3 ; x4 ) vanishes. We
331
sort these points (without loss of generality) such that
x̂1; 1 6 x̂1; 2 6 · · · 6 x̂1; N1 . If we subdivide the interval X1
using these points into N1 + 1 (not necessarily equal)
subintervals, we obtain
X1 = [a1 ; x̂1; 1 ] [x̂1; 1 ; x̂1; 2 ] · · · [x̂1; N1 ; b1 ]
=: X1; 1
X1; 2
···
X1; N1 +1 :
(13)
We do likewise for the other intervals X2 ; X3 ; X4 ; to get
the sets X̂2 ; X̂3 ; X̂4 ; and then subdivide accordingly these
intervals. This gives
X2 = [a2 ; x̂2; 1 ] [x̂2; 1 ; x̂2; 2 ] · · · [x̂2; N2 ; b2 ]
= X2; 1
X2; 2
···
X2; N2 +1 ;
(14)
X3 = [a3 ; x̂3; 1 ] [x̂3; 1 ; x̂3; 2 ] · · · [x̂3; N3 ; b3 ]
= X3; 1 X3; 2 · · · X3; N3 +1 ;
(15)
X4 = [a4 ; x̂4; 1 ] [x̂4; 1 ; x̂4; 2 ] · · · [x̂4; N4 ; b4 ]
= X4; 1 X4; 2 · · · X4; N4 +1 :
(16)
We see that this subdivision process creates several subboxes, given by (X1; j1 ; X2; j2 ; X3; j3 ; X4; j4 ), where
j1 ∈ {1; : : : ; N1 + 1}; : : : ; j4 ∈ {1; : : : ; N4 + 1}. We can
prove (cf. Appendix A) that these subboxes are indeed
such that f is monotone in all variables on each of
them. Therefore, we 'nd the exact range of values of
f on each subbox with just two function evaluations
(cf. Appendix A), and obtain the range of f over X
as the union of all these ranges. We proceed similarly and
'nd the range of h over X .
Lastly, we obtain the controller gain range K at by
following (6) – (9).
3.3. Proposed algorithm
We put the above ideas based on monotonicity and subdivision into an algorithm for generating tracking bounds
at a given frequency !. Suppose we are given the tracking
speci'cation and the set of boundary rectangles Gbnd at
a given !. Then, the proposed algorithm runs as follows.
Algorithm TSI. {Bound Generation for Tracking Speci'cations Using Interval Controller Phases}
1. Select a controller phase interval in the range
◦
[ − 2; 0] having a width, say, of 5 : Choose a pair
of boundary rectangles from Gbnd . Denote the pair as
(i ; Gi ) and (k ; Gk ).
2. Construct the intervals X1 ; : : : ; X4 as X1 = 1=Gi ;
X2 = 1=Gk ; X3 = i + ; X4 = k + .
3. Subdivide the intervals X1 ; : : : ; X4 as in (13) – (16) using the points where the partial derivatives vanish (cf.
Table 1).
4. On each subbox, 'nd points xQ and x as illustrated in
(11) and (12). Evaluate f at xQ and x to get the exact
range of f on each subbox. Do likewise for h.
5. Find kmax in (6) as the maximum of f values over all
the subboxes. Likewise, 'nd kmin in (7) as the minimum of h values over all the subboxes.
6. From kmax and kmin , obtain Kik in (8). (Note that this
controller gain range is only for the considered pair
i; k of boundary rectangles.)
7. Repeat the above over all pairs of boundary rectangles
in the interval plant template.
8. Obtain the controller gain range K at (for the entire
interval plant template) as in (9).
9. Repeat the entire procedure at all other controller
phase intervals , so as to cover the entire phase
range [ − 2; 0].
End Algorithm
4. Properties of the algorithm
4.1. Guaranteed robustness against phase
discretization
Theorem 4.1. For a given ; Gbnd ; and ; Algorithm
TSI indeed generates the controller gain range K in (9).
Proof. See Appendix A.
Remark 4.1. Theorem 4.1 states that K given by
Algorithm TSI is guaranteed to be valid over the entire
interval , rather than at a discrete phase value. Thus,
this property of Algorithm TSI overcomes the loss of
robustness associated with the phase discretization
process in QC and other existing QFT bound generation
algorithms.
4.2. Guaranteed robustness against template
inaccuracies
The following result brings out another key property
of Algorithm TSI—it guarantees that the gain range K
computed in (9) is always an inner enclosure of the exact
range, irrespective of the accuracy of Galg .
Theorem 4.2. (At ) Let Kexact denote the exact allowable range of controller gain for satisfying the tracking
speci:cations w.r.t. the exact template G. Recall that K
is the controller gain range K computed by Algorithm
TSI . Then; K ⊆ Kexact .
332
Proof. The proof follows from the inclusion property
given in Theorem 2.1 and an application of the maximum
modulus theorem as in (3).
4.3. Error estimates
outer
be deWe introduce some more quantities. Let kmax
'ned by the RHS in (6) but with max replaced by min.
outer
be de'ned by the RHS in (7) but with min
Let kmin
be de'ned by the RHS in
replaced by max. Let Kouter
ik
outer outer
; kmin in place of
(8) but with these new quantities kmax
outer
be de'ned by the RHS
kmax ; kmin , respectively. Let K
in place of Kik .
in (9) but with Kouter
ik
Then, using standard set inclusion and set intersection
properties, we can show that Kouter is an outer enclosure
of the exact gain range, i.e., K ⊆ Kexact ⊆ Kouter .
Therefore, using the outer and inner enclosures Kouter
and K, we can immediately obtain the maximum possible
error at any particular phase as the diOerence between the
corresponding inner and outer values. Thereby, we can
obtain a posteriori the maximum possible overall error in
the generated tracking bounds.
5. Aircraft example
We next demonstrate Algorithm TSI on an illustrative
example. The longitudinal motion of the aircraft in open
loop has the transfer function (Thomspon & Nwokah,
1994)
G(s) =
k(1 + s=z)
;
s(1 + s=p)(1 + (2&=!n )s + s2 =!n2 )
k ∈ [0:2; 20]; z ∈ [0:5; 0:75]; p ∈ [1; 10];
!n ∈ [5; 6]; & ∈ [0:8; 0:9]:
From aerodynamic data supplied by Blakelock, the ranges
for the uncertain parameters are identi'ed. The nominal
values are as follows: k0 = 2; z0 = 0:5; p0 = 10; !n0 = 6;
&0 = 0:8. Suppose we want to generate the tracking
bounds for the above plant at ! = 0:1, with the tracking
speci'cation as = 1:03.
We perform the computations on a PC=Pentium-II
333 MHz machine. The various steps are detailed below.
25
magnitude (dB)
Remark 4.2. Thus, even for very coarse interval template approximations, there is never any danger of
generating bounds that could lead to violation of the
robust tracking properties. Clearly then, the inner enclosure property is crucial for ensuring the robustness of the
resulting closed loop system against template inaccuracies. This is in contrast with QC and other existing
algorithms, which have the potential to cause such loss
of robustness (Rodrigues, Chait, & Hollot, 1997).
30
20
15
10
5
-92
-90
-88
-86
angle (degrees)
-84
-82
-80
Fig. 1. Gbnd comprising 47 boundary rectangles, for the plant family
in aircraft example (the rectangles are so small that they are seen just
as lines in the plot). The stars are template points generated using
the QFT toolbox. ! = 0:1.
Interval plant template generation: We generate
the interval template Galg of the plant at ! = 0:1 using
the template generation algorithm in Nataraj and Sheela
(2001a). This gives a Galg comprising 70 rectangles
in 0:82 s using 85; 480 Tops.
Boundary extraction: From the interval template Galg ,
all the boundary rectangles Gbnd are extracted using the
algorithm in Nataraj and Sheela (2001b). This gives
Gbnd comprising 47 boundary rectangles in just 0:02 s
using 88 Tops. These boundary rectangles are plotted
in Fig. 1.
Template generation using QFT toolbox: For comparison, we also generated the plant template by simply rastering each uncertain parameter at the minimum,
mean and maximum values (as is the common practice
in QFT). With such a rastering, the QFT toolbox gives
a template consisting of 35 = 243 plants (also shown in
Fig. 1, as stars).
Tracking bounds, discrete phases: We apply Algorithm TSI to Gbnd to generate the tracking bounds
on the nominal loop transmission function Lo (0:1j) =
K(0:1j)Go (0:1j)—as the plots on Lo (·) are usually preferred to those on K(·) in QFT designs. The resulting
plot of tracking bounds on Lo (0:1j) is shown in Fig. 2
as the upper solid line. In this example, Algorithm TSI
requires about 46 times less !ops than the QFT Toolbox
algorithm, whereas it executes about 31 times faster.
Outer tracking bounds, discrete phases: We also
generate the outer enclosures of the exact tracking
bounds and plot them in Fig. 2 as the lower solid
line. Note that the upper line in this 'gure corresponds to the inner enclosure K(0:1j) Go (0:1j), while
the lower solid line corresponds to the outer enclosure K outer (0:1j) Go (0:1j). It is guaranteed (cf. Section
333
ranges computed in the algorithm. The computational ef'ciency and increased speed of the proposed algorithm
have been brought out through a practical example.
70
60
50
Appendix A. Proof of Theorem 4.1
dB
40
(Cf. Step 3 of Algorithm TSI): Consider an arbitrary
subbox Xsubbox created by the subdivision process in this
step. In the notation of Eqs. (13) – (16), we can express
Xsubbox as
30
20
Xsubbox , (X1; j1 ; X2; j2 ; ; X3; j3 ; X4; j4 );
10
0
-350
-300
-250
-200
-150
-100
-50
0
Degrees
Fig. 2. Inner (upper solid line) and outer (lower solid line) tracking
bounds generated by Algorithm TSI for the illustrative example at
! = 0:1. The stars denote the tracking bound generated using the QFT
toolbox.
4.3) that the exact tracking bounds (which are usually unknown) on Lo (0:1j) always lie between these
inner and outer enclosures. To verify this point, we
obtain the bounds using the QFT toolbox and plot
them in this 'gure—as stars. We see that the tracking bounds generated by the QFT toolbox are indeed
enclosed by the inner and outer enclosures. The maximum possible error in the generated tracking bounds
is the maximum of the diOerences between the corresponding inner and outer enclosures over the entire
phase range. From the 'gure, we see that this is about
1:5 dB.
Tracking bounds, interval phases: Using Algorithm
TSI, we generate the tracking bounds for interval con◦
troller phases, each of 5 phase width (plot omitted). The
required !ops and execution time are given in Table 2,
which includes the time and !ops for extraction of template boundary rectangles. Although these are now very
slightly larger than those of the discrete phase case with
Algorithm TSI, they are still much less than those of the
QFT toolbox algorithm.
6. Conclusions
An algorithm for generating QFT bounds to achieve
robust tracking speci'cations has been presented. The
proposed algorithm can generate tracking bounds over intervals of controller phase values, as opposed to discrete
controller phase values in existing algorithms. The robustness of the tracking bounds is always guaranteed, even
for interval templates of poor accuracy. The maximum
possible error in the generated results can be obtained a
posteriori using outer and inner enclosures of the gain
(17)
where j1 ∈ {1; : : : ; N1 + 1}; : : : ; j4 ∈ {1; : : : ; N4 + 1}. Because we have subdivided at all the vanishing points of
fi , for every i = 1; 2; 3; 4; we must have either
(a) fi (x) 6 0;
or
(b) fi (x) ¿ 0
for all x belonging to this subbox. Thus, f is monotone in
all variables on the arbitrary Xsubbox and hence, on every
subbox created in this step.
(Cf. Step 4): Consider again an arbitrary subbox
Xsubbox created in the above step. Suppose that on this
subbox equation (a) above holds for i ∈
S; and equation
(b) but not (a) holds for i ∈ T , where S T = {1; 2; 3; 4}.
Then, f attains its maximum value on this subbox at the
point xQ whose ith component is given by
right endpoint of the interval Xi; ji for i in T;
xQi =
for i in S
left endpoint of the interval Xi; ji
(18)
and its minimum value on the same subbox at the point
x whose ith component is given by
left endpoint of the interval Xi; ji
for i in T;
xi =
right endpoint of the interval Xi; ji for i in S:
(19)
Therefore, we can 'nd the exact range of values of f on
this subbox with just two function evaluations (at points
xQ and x). Since Xsubbox is an arbitrary subbox, the assertion
holds for every subbox. Identical argument holds for the
function h.
(Cf. Step 5): In this step, the maximum of f over all the
subboxes is found. From (6) and (10), this maximum is
nothing but the upper bound kmax on controller gain at for the considered pair of boundary boxes. Likewise, by
referring to the same equations, we see that the minimum
of h over all the subboxes as found in this step is just the
lower bound kmin on controller gain at for the same
pair. Thus, this step gives the bounds kmax and kmin for
the considered pair of boundary rectangles.
334
(Cf. Step 6): This step follows (8) to construct the
controller gain range Kik from the bounds kmax and kmin
for the same pair at .
(Cf. Step 7): The above steps are repeated over all
possible pairs of boundary rectangles to get the set of
controller gain ranges {Kik } at .
(Cf. Steps 8 and 9): Step 8 implements (9) to get the
controller gain range at for the entire interval plant
template. Step 9 produces the controller gain range over
the entire controller phase range of [ − 2; 0].
References
Borghesani, C., Chait, Y., & Yaniv, O. (1995). The quantitative
feedback theory toolbox for MATLAB. The MathWorks Inc.
USA.
Chait, Y., Borghesani, C., & Zheng, Y. (1995). Single loop QFT
design for robust performance in the presence of non-parametric
uncertainties. Transactions of the ASME Journal of Dynamic
Systems, Measurement and Control, 117, 420–424.
Chait, Y., & Yaniv, O. (1993a). Direct control design in sampled-data
uncertain systems. Automatica, 29(2), 365–372.
Chait, Y., & Yaniv, O. (1993b). Multi-input=single-output
computer-aided control design using quantitative feedback theory.
International Journal of Robust and Nonlinear Control, 3(1),
47–54.
D’Azzo, J. J., & Houpis, C. H. (1995). Linear control system analysis
and design (4th ed.). New York: McGraw-Hill.
East, D. J. (1981). A new approach to optimum loop synthesis.
International Journal of Control, 34(4), 731–748.
Fialho, I. J., Pande, V., & Nataraj, P. S. V. (1992). Design of feedback
systems using Kharitonov’s segments in Quantitative Feedback
Theory. Proceedings of the :rst QFT symposium, Dayton, OH,
USA (pp. 457– 470).
Horowitz, I. M. (1991). Survey of quantitative feedback theory
(QFT). International Journal of Control, 53(2), 255–291.
Horowitz, I. M. (1993). Quantitative feedback design theory (QFT).
Boulder, CO: QFT Publications.
Houpis, C. H., & Lamount, G. B. (1988). ICECAP-QFT Users
Manual. Wright-Patterson: AFB.
Longdon, L., & East, D. J. (1979). A simple geometrical technique
for determining loop frequency bounds which achieve prescribed
sensitivity speci'cations. International Journal of Control, 30(1),
153–158.
Nataraj, P. S. V. (1994). A MATLAB based toolbox for synthesis of
lumped linear and nonlinear and distributed systems. IEEE=IFAC
symposium on computer aided control system design, Tucson, AZ
(pp. 513–518).
Nataraj, P. S. V., & Sardar, G. (2000). Template generation for
continuous transfer functions using interval analysis. Automatica,
36, 111–119.
Nataraj, P. S. V., & Sheela, S. (2001a). A template generation
algorithm using parallel function evaluations and adaptive
subdivisions, under review.
Nataraj, P. S. V., & Sheela, S. (2001b). An algorithm for extraction
of boundary rectangles from interval QFT templates, under review.
Rodrigues, J. M., Chait, Y., & Hollot, C. V. (1997). An e8cient
algorithm for computing QFT bounds. Transactions of the ASME
Journal of Dynamic Systems, Measurement and Control, 119(3),
548–552.
Sardar, G., & Nataraj, P. S. V. (1997). A template generation
algorithm for non-rational transfer functions in QFT designs.
Proceedings of the 36th IEEE conference on decision and control,
San Diego, USA (pp. 2684 –2689).
Taylor, A. E. (1955). Advanced calculus. Boston: Ginn and Company.
Thomspon, D. F., & Nwokah, O. D. I. (1994). Analytical loop
shaping methods in quantitative feedback theory. Transactions
of the ASME Journal of Dynamic Systems, Measurement and
Control, 116, 169–177.
Wang, G. C., Chen, C. W., & Wang, S. H. (1991). Equation for loop
bound in quantitative feedback theory. Proceedings of the IEEE
conference on decision and control, England (pp. 2968–2969).
Yaniv, O. (1990). QFT software, Israel.
Yaniv, O. (1999). Quantitative feedback design of linear and
nonlinear control systems. Boston: Kluwer Academic Publishers.
Zhao, Z., & Jayasuriya, S. (1994). On generation of QFT bounds
for general interval plants. Transactions of the ASME Journal of
Dynamic Systems, Measurement and Control, 116(4), 618–627.
Paluri S.V. Nataraj obtained his Ph. D.
from IIT, Madras, India in process dynamics and control in 1987. He then worked
in the CAD center at IIT Bombay for
about one and a half years before joining
the faculty of the systems and control engineering group at IIT Bombay in 1988.
His current research interests are in the
areas of robust stability and control, nonlinear system analysis and control, and
reliable computing.