Robust Control Analysis Considering Real Parametric Perturbations Based on Sign Definite Conditions

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

ROBUST CONTROL ANALYSIS CONSIDERING REAL PARAMETRIC PERTURBATIONS BASED ON SIGN DEFINITE CONDITIONS T. Kimura and S. Hara Department of Systems Science, Tokyo Institute of TechnoLogy, Nagatsuta, MUiori-ku, Yokohama, Kanagawa 227, Japan

Abstract. Convenient tests for D-stability, H co-norm constraints and gain/phase margin constrains of one-parameter systems are proposed in this paper. These tests are on the basis of Sign Definite Condition and the Euclidean algorithm in matrix form, which can be carried out in finite number of steps. Key Words. Euclidean algorithm; parametric perturbations; performance bounds; pole placement; polynomials; robust control; stability criteria

1.

form.

INTRODUCTION

Considering the literature of LTI systems subject to real parametric perturbations, Kharitonov has shown in [Kharitonov 1979] that an interval polynomial is Hurwitz stable if and only if the four extremal polynomials are all stable. This implies that the stability of the interval polynomial can be easily checked out. Motivated this result, several extensions have been proposed considering not only stability but also design performance indices such as D-stability [Bartlett et . al. 1988], Hoo norm constraints and gain/phase margins [Keel et. al. 1991] . In these articles, it has been shown that LTI systems subject to parametric perturbations maintain their performances if and only if a finite number of segment systems keep them, where segment systems mean the systems with only one varying parameter . Therefore, similarly to the Kharitonov 's stability test result , convenient tests for these performance indices are required for robust control [Bialas 1985] .

2.

PRELIMINARIES

One of the key ideas of this paper IS a sign definite condition defined as follows [Kimura and Hara 1991] : Definition 1 : A continuous function f(x) : R-+ R is sign definite in the interval [a, b], denoted by f(x) E No[a, b], if f(x) preserves its sign in [a, b] 01'

does not cross zero in [a, b] .

Remark: The sign definite condition f(x) E N ora , b] can be transformed to the condition fe z ) E No[O,00] by a bilinear transformation z = -(x - a)/(x - b). Then, if f(x) is polynomial in x , the sign definite condition can be readily checked by Routh-Hurwitz type criterion [Siljak 1971] . The following fact related to the Euclidean algorithm is another key idea: Let us consider simultaneous equations

The purpose of this paper is to provide some useful tests for these performance indices so that they can be carried out in a finite number of steps. These tests are based on sign definite conditions (SDC). The basic idea is the same as in our previous work [Kimura and Hara 1991] . Our objective here is to derive a simpler version of the SDCs corresponding to 1) D-stability, 2) H 00norm and 3) gain and phase margin constraints for one-parameter systems. This derivation will be carried out on the basis of the corrected version of the check condition for the SDC and the application of the Euclidean algorithm in matrix

f(x, A) { g(X,A)

o o

(1)

where f(x, A) and g(x , A) are multivariate polynomials in x and t respectively. For convenience, we represent f(x, A) and g(x, A) as

f(x,A) g(x, A)

= fo(A)xn + fl(A)X n- 1 + ... + fn(A)

= go(A)Xn + gl(A)X n- 1 + .. . + gn(A) (2)

We omit A from f i(A) and gi(A) ; i simplify the notation from now on. 19

= 0, . .. , n to

Under the above notation, we have the following lemma.

In addition, we assume that the domains of the variables are specified as x E [£, x) and A E [X,~ . Furthermore, we assume that al) the order of f(x, A) w.r.t. x is n (i.e. fa E No[4,~]) and a2) the order of g(X,A) w.r.t. x is less than or equal to the order of f( x, A).

Lemma 1 Consider the simultaneous equations (1) with the assumptions a1) and a2). Then the simultaneous equations (1) do not hold for all x E [~, x) and A E [~,~) if and only if

We introduce the following notation to state a key lemma: Let R., be a 2n x 2n-matrix defined as

fa go R., ..-

h g1 fa go

(5) 01'

fn gn

v AO E Z(det R.,),

c2)

f1 g1

(3)

where

Rn fa go

h

fn gn

g1

1'k(X) :=

x k- 1f(x) x k- 1g(x) Xk - 2 f(x) x k- 2 g(x)

It g1 fo go

f1 g1

Eucrdf(x, A),g(X , A), x) ; k = 1,2, ... ,n

0

Remark: The well-known Euclidean algorithm in fractional form is not applicable to multivariate case.

3.

(4)

f(x) g(x)

f1 g1

(7) (8)

EucR[f(x, A),g(X, A), x)

This lemma can be proved immediately from Theorem 10.1 in [Takagi 1930).

3.1.

fa go

.-

1'i;(x)

where f; and gi (i = 0, ... , n) are the coefficients of f(x , A) and g(x, A) defined in (2). Let Rk (k = 1, . .. , n) be the principle minors of R., with the size 2k x 2k; k = 1, .. . ,n and 1'k(X) be 2k x 2k (k = 1, . .. , n )-matices defined by

fo go

(6)

det 1'm(>'o,R" )(x) \>'=>'0 E N o[£, x)

fn gn

PERFORMANCE CRITERIA

D-STABILITY

In this subsection, we consider the D-stability constraints for a one-parameter polynomial n

We denote Rn and 1'k(X) associated with f(X,A) and g(x, A) as

f(s, A)

= I: f;(,\)si

(9)

i=O

EUCR[f(X, A), g(x, A), x) Eucrk[f(X, A),g(X, A), x)

Rn 1'k(X) ; k = 1, ... ,n

where J;(A) (i = 0, . .. ,n) are polynomials in A E [~, ~). The D-stable domain V is given by its boundary oV which is expressed as

oV := T}(w)

Knowing that f; and gi (i = 0,1, .. . , n) are polynomials in A and det R., is a polynomial in A, we denote the set of the zeros of det Rn in [4,~) as Z(detRn), that is,

det Rkl>.=>.o

(10)

From the continuity of the variation of zeros of a polynomial w.r .t. the variation of its coefficients, f( s, A) is D-stable for all A E ~, ~l if and only if f(s,~) is D-stable and

Let us consider the series of determinants det RI 1>'=>'0' det R 2 1>.=>.0, ... , det R.,1>.=>.0 and let m be the greatest number such that the corresponding determinant det Rm is not equal to zero . This index m is uniquely determined if AO and Rn are given. Therefore, we denote this as m(Ao, R.,), i.e.,

1

wE R

where T}(w) and ~(w) are rational functions in w . Note that several important domains such as the interior part of wedge-shape regions, conics (parabolas, ellipses, hyperbolas) and n-th order curves can be expressed in this form.

Z( det R.,) := {Aa E [~,~) 1 det R., 1>'=>'0 = O}

m(>.o, R.,) := max!;{k = 1, ... , n

+ j~(w);

i= 0 ;

f(s)sE8D

Vs E oV, v t E [~, ~l

(11)

or equivalently

{

i= O} 20

fr(w, >.) fi(W, A)

o o

(12)

hold for all w E R and A E [~, 1], where fr (w, A) and f;(w, A) are real and imaginary part of f(s, >.) at the boundary oD =:: 7J(w) + j~(w), i.e.,

f(S)'E&D

=:: :

fr(w, >.)

+ ifi(W, >.)

where

h(W,A)

Here we note that the SDC for the double-variate polynomial (20) is equilibrate to the SDC for the double-variate polynomial defined by

f(w, t) Theorem 1 Considering D-stable domain defined above, a one-parameter polynomials f( s, >.) ; A E ~, 1] is D-stable if and only if f( s , ~) is D-stable and

>.

+ 1t

numerator of h(w, =--1 ) t+

Theorem 2 Consider a double-variate real polynomial f(w, t) and assume that its order w.r.t. t is two. Then, f(w, t) is written as

or

det rm(Ao,Rn)(w)IA=Ao E No[-oo, 00]

: =::

with w E Rand t E [0, ocl Then, since the order of f( w, t) w.r .t t is two, we can check whether the segment system O(s, >.) is satisfies the Hco-norm constraint by using the following Theorem 2.

(14)

c2) v AO E Z(det Rn),

(3(jw, A)(3(-jW, A) _,20:(jw, A)cx( - jw , >.)

(13)

Applying Lemma 1 to the simultaneous equations (12), we immediately obtain the following theorem:

cl) det Rn E N o[4, 1)

.-

(15)

f(w, t)

=:::

a(w)t 2 + b(w)t

+ c(w)

(21)

and we assume that a(w) and b(w) have no common real zero. Then,

hold, where EucR[fr(w, A), f;(w, A) ,W] (16)

Pl : f(w ,t):/; 0 for all wE R, t E [0,00]

Eucrdfr(w , >.) , f;(w, A) , w] (17) ; k =:: 1,2, .. . , n

hold if and only if Cl) f(O, t) E No[O, 00]

and fr(w , A) and fi(W, >.) are defined in (13). C2-1) C2-2)

Considering parametric perturbed systems with poly topic characteristic polynomials in the coefficient space, it has been shown that these perturbed systems are D-stable in the face of parameter uncertainties if and only if all corresponding segment systems are also D-stable [Bartlett et. al. 1988). Therefore, we can check the D-stability of the parametric perturbed systems by using Theorem 1.

c(w) E No[-oo, 00] a(w) has no common real zero with odd multiplicity D(w)

C3-1)

E

or C3-2)

3 wo

ER

s. t.

b2(w) - 4a(w)c(w) No[- oo, oo] D(wo) =:: 0

===> b(wo) > 0 0

3.2.

Hco-NORM CONSTRAINT Outline of the proof: Assuming a(w) :/; 0 vw E R, the constraint PI is equivalent to

Consider a stable SISO LT! system which have one varying parameter>' E [~, Xl

O(s, A)

=::

P2 : h(w , u) :=::

(18)

(3(s, >.)/o:(s, A)

where O:(S,A) and (3(s , >') are polynomials in s and>' . Here we assume that the order of 0:( s, >.) and (3(s, >.) w.r.t A are less than or equal to one.

where

p(w)

Then, it is obvious that H co-norm constraint

1I0(s, A)II

<, ; v>. E [~, X]

is satisfied if and only if 10(00,0)1

h(w, >.)

:/; 0 for all w E R, >.

:=::

b(w)/a(w),

q(w) :=:: c(w)/a(w)

Then , we see that the conditions Cl), C2-l) and C3-l,2) are necessary and sufficient condition for P2, equivalently Pl.

(19)

< , and

E [4,1]

u 2 +p(w)u+q(w):/;0 Itw E R , Itu E [0,00)

Next , we investigate the singular case . Suppose 3 wo s.t. a(wo) =:: O. Then , at this Wo, the parabola

(20) 21

/2(w, u) changes its shape into a line b(wo)u + c(wo) . Combining condition C2-2), we confirm the validity of the theorem by investigating the following two cases: case 1) The multiplicity of zeros associated with a(w) is odd. case 2) The multiplicity of zeros associated with a(w) is even. o Remark: According to Keel et. al. (1991), Hoonorm constraints for the interval systems are satisfied if the corresponding segment systems satisfy the constraints. Thus, applying Theorem 2, we can check whether the interval systems satisfy the H oo-norm constraints.

3.3.

Since the proof of the theorem associated with phase margin constraint is almost the same, so it is omitted. 0 Remark: For one-parameter systems, the SDCs (22) become double-variate SDCs. Though the computation may be complicated, the doublevariate SDCs can be checked in finite number of steps by using the SDC criterion [7].

4.

Convenient tests for 1) D-stability, 2) Hco-norm constraints and 3) gain/phase margin constrains of one-parameter systems have been proposed. These tests are on the basis of Sign Definite Condition and the Euclidean algorithm in matrix form, which can be carried out in finite number of steps. Therefore, the proposed tests for the performance indices 1), 2) and 3) are feasible .

GAIN AND PHASE MARGIN CONSTRAINTS

The stability margins can be reduced to SDCs by using Lemma 1 as follows:

Finally, we wish to express our appreciation to Prof. S.P.Bhattacharyya for his useful comments.

Theorem 3 Consider an SISO LTI open-loop system C(s) = n(s)/d(s) and partition COw) into COw) (gr(w) + jgj(w»/d(w) nr(w)/dr(w) + inj(w)/dj(w). Then , C(s) satisfies the gain margin (Jm,...,.M) , ...,.m ::; 1,...,.M ;::: 1 (or the phase margin 0 ::; cjJ ::; 7r) constraint in the negative unity feedback case if and only if

Tg(-X) (Tp(-X)

E

N o[-lhm, _l/...,..~f]

E

No[-l,cos(-7r+cjJ)])

5.

(22)

detEucR[fl(w,t),/2(w),w] (23) det EUCR[h(w), f4(w, t) , w])(24) and

:: .-

:= :=

nr(w) - dr(w)t nj(w)

(25)

2

g;(W)+g](W)-d (W») (26) nr(w) - dr(w)t

Outline of the proof: By following the corresponding gain margin constraint based on the Nyquist locus, we see that the gain margin constraint is identical to the simultaneous equations

h(w ,t) /2(w)

.-

nr(w) - dr(w)t nj(w)

REFERENCES

Bartlett, A.C ., Hollot, C.V. and Lin, H. (1988). Root Locations of an Entire Polytope of Polynomials, It Suffices to Check the Edges, Math. Contr., Signals, Syst., 1,61-71. Bialas, S. (1985). A Necessary and Sufficient Condition for the Stability of Convex Combinations of Stable Polynomials or Matrices, Bulletin of the Polish Academy of Sciences Tech . Sciences, 33, 473-480. Hara, S., Kimura, T . and Kondo, R. (1991). Hoo Control System Design by a Parameter Space Approach, Froc. of MTNS-91, Kobe 287-292, Keel, L.H., Shaw, J and Bhattacharyya, S.P. (1991). Robust Control of Interval Systems, Proc. of MTNS-91 Kobe Kharitonov, V.L. (1979). Asymptotic Stability of an Equilibrium Position of a Family of Systems of Linear Differential Equations, Differential Equations, 14, 1483-1485 Kimura, T. and Hara, S. (1991). A Robust Control System Design by a Parameter Space Approach Based on Sign Definite Condition, Proc. of KACC-91, Seoul, Korea Siljak, D.D. (1971). New Algebraic Criterion for Positive Realness, J. of the Franklin Inst ., 291, 109-120 Takagi, T . (1930). In: Lecture of Algebra (in Japanese), Kyoritsu Pub., Japan.

holds, where

fl(w,t) /2(w)

CONCLUSION

(27)

are not satisfied for all w E Rand t E [-lhm, -lhM]. Applying Lemma 1 to (27), we conclude that gain margin constraint is satisfied if the corresponding to two SDCs cl) and c2) are satisfied. Simple investigation leads that the second condition can be removed and hence we have the theorem. 22