Stability Margin of a Feedback System with Structured and Unstructured Uncertainties

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

STABILITY MARGIN OF A FEEDBACK SYSTEM WITH STRUCTURED AND UNSTRUCTURED UNCERTAINTIES Y.C. Soh* and Y.K. Foo** *School oJ Electrical and Electronic Enf!,ineering, N(UlYWI!!, Technological University, Nanyang Avenlle, Singapore 2263 **Leun Wah Electric Co., 487 Tanglin Halt Road. Sinl!(Jl'ore 11314

where

INTRODUCTION

An important control problem is the computation of the largest allowable magni tude of all admissible perturbations which will not de-stabilize the feedback system . This is commonly known as the problem of stability margin calculation, and is exemplified by the well-known gain margin and phase margin of Bode. Some recent works that can be classified under this category include the works of Safonov (1982), Gaston and Safonov (1988) , Barmish (1984), and Foo and Soh {1991, 1993). This approach is useful because it provides a basis for comparing the "goodness" of different designs. In order to capture all admissible uncertainties, we need to include both the low-order parametric uncertainties and high-order unstructured per turbations/unmode I led dynamics in the uncertainty models. In this paper, we consider the problem of computing the perturbation bound for the parametric uncertainty while imposing an unstructured perturbation on the whole plant set. We assume that the structured parametric uncertainty is affine linear in ~, the vector representing the uncertain system parameters. For the unstructured part, we assume that a bounding function is given . Our problem is to compute the largest allowable bound on ~ which will preserve the robust stability of the perturbed system.

N{s)

n n{s) + foes) + }: a . f.{s) i=l 1 1

(2)

D{s)

n des) + goes) + }: a.g. Cs) i=l 1 1

(3)

aCne jw)

known proper rational functions analytic in the CRHP, a are uncertain real system parameters and i

a{') denotes the maximum singular value . Note that in the above formulation,

f i Cs) or gi Cs) can be

zero. We grouped the uncertain real parameters as an nx1 uncertainty vector ... a ]T

(5)

n

Thus pes) as defined in (1) can be represented as P{~, s) to indicate its dependence on ~. We shall assume that each component of ~ is of the form = 1, 2, ... n

(6)

where '\ are known posi tive scalars and a denotes the stability bound for the structured uncertainty which we aim to maximise while maintaining the robust stability of the feedback system. We shall assume that the closed-loop system is robustly stable with a nominal plant model characterized by ~ = Q (or equivalently, a = 0). Suppose that the controller which robustly stabilizes the nominal plant P(~=Q.s) admits a coprime factorization (Desoer et aI, 1980). i.e. C{s) = [D (s)]-l N Cs) c

PROBLEX FORMUL4.TIONS

Consider a Single-input single-output feedback system as shown in figure where pes) is a perturbed system and C{s) is a fixed controller.

It

(7)

c

is well-known that C{s) if and only if

will

also

stabilize

P{~ , s)

Suppose that the perturbed system pes) is under joint parametric and unstructured perturbations of the form N{s) [D{s)]-l

(4)

and where foes), goes), fits). gi{s) and w(s) are

There are some recen t works on robus t s tabi I i ty analysis of feedback systems with joint parametric uncertainty and unstructured perturbation. For example, Fan et al (1991) have considered the use of structured singular values to solve the problem of joint parametric uncertainty and unmodel1ed dynamics, but their computation is potentially inexact . Wei and Yedavalli (1989) have derived sufficient conditions for robust stabilizability of perturbed SISO plants. However. the approach adopted suffers from the assumption that the whole set of uncertain plants must be of minimum phase . Foo and Soh (1993) have examined the case where the structured parametric uncertainty is characterised by a 2-norm. In this case, one needs to solve a fourth order polynomial at each frequency.

pes)

d{jw)] ~ Iw{jw) I

is stably invertible for all we define

(I)

387

~

satisfying (6) . If

Equation (15) has a simple geometrical interpretation . It implies that the perturbed feedback system is robustly stable if and only if the normal i sed Nyqu i s t image of the sys tern wi th just the parametric uncertainty (a parpolygon) does not enter a circular region which denotes the normalised Nyquist image of the unstructured perturbation. This is illustrated in figure 2 where r(w) denotes the radius of R(jw) . Note that a necessary condition for robust stability of the feedback system is that r(w) < 1 for all w.

(8)

then we have the following lemma .

Le.ma 1:

Suppose that C(s) of (1) and (8) robustly stabilizes the nominal plant P(ll,=Q.s). Then C(s) also robustly stabilizes P(ll,.s) for all ~ satisfying (6) if and only if at each w € [~. m]. n

I[N (jw){f (s)+ ~ aif . (jw)}] c 0 i=1 1

Before we proceed . we shall show that the Nyquist

n

+ [D (jw){g (s)+ ~ aigi(jw)}]1 c 0 i=1 > Iwc(jw)w(jw) I

image of ll,TH(s) is indeed a parpolygon. First. we define the image of each component of H(s) at each s = jw. w € ffi as

(9)

Hi(jw) = Fi(jw)/Fo(jw) for all!!, satisfying (6).

= ~i(w)exp(jBi(w» ; i

Prool: The feedback system is stable under the perturbations if and only if

; ~i(w) ~ 0

= 1.

2 .••• n

(16)

Lesaa 2 :

Consider the normalised function ll,TH(s) where !!, and H(s) are defined in (6) and (13) respectively. Let Bi(w) denote the phase angle of

has no zero in the CRHP . But

Hi(jw) as defined in (16) . For each s = jw. w € ffi.

n

the image of ll,TH(s) is a 2n-sided parallel parpolygon . symmetrical with respect to the origin. and whose sides are at angles Bi(w); i = 1. 2 .•••

G(s) = N (s){n(s)+f (s)+ ~ aifi(s)} c 0 i=1 n

n. with respect to the horizontal axis .

+ D (s){d(s)+g (s)+ ~ aigi(s)} . c 0 i=1 Since the nominal closed-loop plant is robustly stable. this implies that the Nyquist locus of G(s) has the correct number of encirclements when a = O. Now. continuous perturbation in ~ (or equival;ntlY. a) can only resul t in a continuous change in the Nyquist locus . Thus . the number of encirclements will not change if and only if

Thus.

n

= exp(-jBk(w»

n

[N (jw){f (jw)+ ~ aifi(jw)}] c 0 i=l

1

1

+ a Bk(w) le

Therefore.

n

+ [D (jw)(g (jw)+ ~ aigi(jw)] c 0 i=l

is

independent

ne [

jW)] Dc(jw)] d(jw) # 0

+ [Nc(jw)

~ a ' ~i(w)exp(jB.(w»

i=1 i#k

~.

(10)

So .

the

must

be

top

boundary

of

Or

horizontal.

equivalently. !!,TH(jw) has an edge with angle Bk(w)

The lemma thus follows immediately (Vidyasagar and Kimura. 1986). AAA

with respect to the horizontal . Thus. ll,TH(jw) is a 2n-sided parpolygon . It is obvious that ll,TH(jw) is symmetrical with respect to the origin. AAA

1lIE MAIN RESULTS In this section. we shall make use of lemma 1 to develop a simple procedure to compute the largest allowable bound for a. To this end. we define

Clearly.

the size of

the parpolygon is directly

proportional to a. For a given a € ffi+ and w € ffi. the distance of the bottom (and hence the top) horizontal boundary of ll,TH(jw)exp(-jBi(w» E(s) = [F 1 (s)

F (s) ... Fn(s)]T 2

(12)

!!(s) ~ E(s) [F (s)]-l o

origin.

that is. max{Im{ll,TH(jw)exp(-jBi(w»}} can !!, be readily computed as

(13)

and

1. 2 .••• n

(14)

n

hi(w)

k:l IIm{exp(j[Bk(w)-Bi(w)])6~k(w)}1 (18)

n

N (s){f (s)+ ~ aifi(s)} + D (s){g (s)+ ~ aigi(s)} c 0 i=1 c 0 i=1

So. the problem of finding the largest allowable a which

T

= Fo(s) + ! E(s). and IFo(jw) + !TE(jw)I

>

>

IR(jw)I

>

solves

(15)

for ~

all

w

involves

the

computation of a real a(w) > 0 at each frequency w such that the two Nyquist images just touch each other . There are three possible cases . For simplicity of expression. we shall suppress the dependence on w in subsequent discussions .

IWc(jw)w(jw)I

• 11 + !TE(jw)(F (jw»-ll o _11 + !T!!(jw)I

(11)

where

where Fi = Nc(s)fi(s)+Dc(s)gi(s). Then. n

from the

IW (jw)w(jw)(F (jw»-11 c o (15)

388

Case 1: The edge touches the circle This is illustrated in figure 3. In this case. the

Procedure 1) Set

Nw (stopping frequency)

A

value of a can be calculated via

Aw (incremental frequency} amax : = Na (an overbound for a max )

A

ab

i + r =

sin~i

:

sin~i

>r

w := 0

Le.

(19)

2)

1 r := Iw (jw)w(jw)F- (jw) 1 c

9

where hi is given by (18) with 9

and

Case 2 : The vertex touches the circle This case is illustrated in figure 4. In this case. the point of intersection in ~ between two adjacent edges is given by

a(x+jy)

3)

:= 9

mod

i

Rearrange

o

where

~i

~ ~i

11'

i = I, .. . n

•

as

in ascending order, i . e.

~i

< ~i+1' i = I, 2 •.. . M-I

where M denotes the number of distinct If M = 1.

if ~i = .../2

hi

i

~i

specified by (16) .

i

0

:= arg{Hi(jw)} := arg{Fi(jw)lFo(jw)}

~i

Goto Step 6 .

A

x "{

.'+1 c + -c

1 i - -i m mi+1- i

y = {

4)

if ~i+1 = .../2

a:= O. z : = O. 2 : = 0 i := 1

Do while 2 = 0 and i otherwise

mi +1x + ci+1

if ~i = .../2

mix + c i

otherwise

~

M

n

hi : = ! IIm{exp(-j~i)6k~(jw)}1 k=l If ~i ~ .../2. mi : = tan~i ' c i : =

-hi/cos~i

a : = (sin~i-r)/hi

If a

~

a.

a :=

A

a,

Z

A

So, the required a can be determined by solving

:=

A

If a > 0 and a > a, i : = i+1 End Do If z = O. Go to Step 7

2 : =-1

n

5)

If

Z

= 1.

ho:= k:1 IIm{exp(-j~M)6k~(jw)} I,

lbat is.

= x ± J(x 2-(1-r2)(x 2+y2» (X2+y2) if ~2-1 = Tr/2

From which we obtain

~ = min {at .a2 : at or a 2

€

if ~2

R+}

x - J(x 2-(1-r2)(x 2+y2» (X 2+y2)

otherwise if ~2-1 = Tr/2

Case 3 : Degenerate case In this case the parpolygon is just a straight-line in~. If sin~i > r. then the straight-line will not

End Do

: = (x 1-x2 )2 + (Y1-Y2)2 o b 1 := ~(xl-~) + Y2(Yl-Y2) -(x 1-x2 ) X := -b 1/bo If X € [0. 1], Go To Step 8 Dok=I.2 2 ~ = xk - (1-r )(xk+Yk) b

touch the circle . See figure 5 . Hence we can set a = CD and this can be ignored . If

sin~i

~

= 11'/2

r . as illustrated in figure 6 . we solve

for

If ~ ~ 0,

Then ~ : = (~-Jdk)l(xk+Yk) Else ~ : = Na

End Do

lbat is.

a := min {a 1 ,a2 } Go To Step 8

= x ± J(x 2-(1-r 2 )(x2+y2» (X 2+y2)

6)

and we obtain

If

sin~1

>r.

a:=

Na' Go To Step 8

n

v : = ! IRe{exp( -j~I)6k~(jw)}1 k=1 x : = v cos~1

.a

a = min {at 2 : at or a 2 € IR+} x - J(x 2-(1-r2)(x 2+y2» (X 2+y2) With the above derivations. following procedure to compute a

we . max

y := v propose

sin~1

; = x - J(x 2-(I-r2)(x 2+y2» (X2+y2) Go To Step 8

the

389

aJNa..USIOOS

~

In conclusion. we have presented a simple procedure for computing the maximum robust stability bound of a feedback system which is subjected to both a low order parametric uncertainty and a h i gh frequency perturbation . In the proposed procedure. the computation at each frequency is very simple and can be implemented readily .

M if "I'i = -.r/2 if "I'i+l = -.r/2

otherwise

otherwise Figure 1.

d := x~ - (l-r2}(x~+y~) If d ~ O. Then a

:=

(xi-Jd}/(x~+y~)

Else a ._ N

a

If 0


~

If a > a i ._ i+l End Do 8}

If a

a.

a:= a

> O.

< a max •

2 : =-1

Figure 1 .

Figu r e 3.

J+R

HR

a

:= a

max

w := w + A

w

Go To Step 2

If w ~ Nw .

o

Stop In

Remark

the

above

procedure.

step

Figu r e 5.

4

calculates the maximum a when the infini te line representing an edge touches the circle. Step 5 confirms if it is the edge and not the extension of the edge that touches the circle. Otherwise. one

References Barmish. B. R. (1984) . Invariance of the strict Hurwitz property for polynomials with perturbed coefficients. IEEE Trans . Automat. Contr . . 29. pp . 935-936.

computes the maximum a from the two adjacent vertices . Step 6 considers the degenerate case . We

Desoer . C. A.. R. W. Liu. J . Murrayand R. Sacks (1980) . Feedback system design: the fractional representation approach to analysis and synthesis. IEEE Trans . Automat. Contr .. 25 . pp . 399-412 .

only need to compute the maximum a from the vertices with step 7 if all the infinite lines that represent the edges do not give a positive a . 0

Fan. K. H. . A. L. Tits and J . C. Doyle (1991) . Robustness in the presence of joint parametric uncertainty and unmodeled dynamics . IEEE Trans . Automat. Contr .. 36. pp. 25-38 .

Remark 2 It is clear from the procedure that. we need not search through all the edges (or vertices) in order to compute a . 0 Remark 3 : The above procedure can also be used to handle the case when the parametric uncertainties is described by I-norm. i.e. 1I!!lIl < a. In this

Foo. Y. K. and Y. C. Soh (1991). Maximal perturbation bound for Hurwitz property of analytic functions . IEEE Trans . Automat . Contr. . 37 . pp. 615-618 .

case. the additional computation is to identify all the exposed vertices at each frequency. 0 the

We sunmarise resul t.

the derivations wi th

Theor.,. 1:

Let amax be computed from the above

Foo. Y. K. and Y. C. Soh (1993). Robust stability bound for systems with structured and unstructured perturbations. To appear in IEEE Trans . Auto . Contr.

following

de Caston. R. R. E. and M. G. Safonov (1988) . Exact calculation of the multioop stability margin. IEEE Trans. Automat . Contr . . 33. pp. 156-171.

procedure . then the feedback system characterized by (1}-(8) is robustly stable for all !! such that lail

< amaxo i .

is such that

i = 1. 2.

I;.1 I

= a

n . Furthermore. if !!

Safonov. M. G. (1982) . Stability margins of diagonally perturbed multivariable feedback systems . Proc. lEE Pt-D. 129. pp . 251-256 .

0i' i = 1. . .. n. then the

max closed-loop system has some poles on the jw-axis. Proof :

Figur e 6.

Vidyasagar. M. and H. Kimura (1986) . Robust controllers for uncertain linear multivariable system. Automatica. 22. pp . 85-94.

A consequence of lemma 1 .

Remark 4 : The proposed procedure can be readily modified to compute the robustness margin with respect to the unstructured uncertainties when the size of parametric uncertainty is fixed. 0

Wei. K. and R. K. Yedavalli (1989). Robust stabilizability for linear systems with both parameter variation and unstructured uncertainty. IEEE Trans. Automat. Contr . . 34. pp . 149-156.

390