Small- Theorems with Frequency-Dependent Uncertainty Bounds
Small- Theorems with Frequency-Dependent Uncertainty Bounds A. L. Titsy Department of Electrical Engineering and Institute for Systems Research University of Maryland College Park, MD 20742 Phone: +1 301 405-3669 Fax: +1 301 405-6707 Internet: [email protected]
V. Balakrishnan School of Electrical and Computer Engineering Purdue University West Lafayette, IN 47907-1285 Phone: +1 765 494-0728 Fax: +1 765 494-3371 Internet: [email protected]
August 3, 1999 Abstract We present conditions, some necessary and some sucient, valid under weak assumptions, for robust stability and uniform robust stability of uncertain linear timeinvariant systems with linear time-invariant uncertainties that are block-diagonal, with known frequency-dependent norm bounds on the diagonal blocks. Small- theorems with frequency-independent uncertainty bounds are recovered as special cases.
Keywords: Robust stability, Uncertain systems, Robust control, Small- theorem,
Frequency-dependent uncertainty bounds.
This research was in part supported by NSF's Engineering Research Center No. NSFD-CDR-88-03012, and in part by the Oce of Naval Research under contract No. N00014-97-1-0640. y Corresponding author.
1 Introduction A popular paradigm for modeling control systems with uncertainties is illustrated in Figure 1. Here P is the transfer function of a stable linear system, and is a stable operator that represents the \uncertainties" that arise from various sources such as modeling errors, neglected or unmodeled dynamics or parameters, etc. Often, the uncertainty is assumed to possess various additional properties. Common examples are that is structured (i.e., diagonal or block-diagonal), that it is linear time-invariant or real-constant etc. Such control system models have found wide acceptance in robust control; see for example [Doy82, Saf82, PD93, Lev96, wJDG96]. We will henceforth refer to the system in Figure 1 as the P { interconnection, and denote it by I (P; ). The Small- Theorem is the foundation of the structured singular value approach towards assessing the stability of systems a ected by structured uncertainties. Speci cally, the theorem gives a necessary and sucient condition on the \-norm" of the plant transfer function P for I (P; ) to be stable for all structured 2 H1 of a given maximum size (H1-norm). Various versions of the small- theorem exist in the literature, and it was shown in [TF95] that some commonly quoted versions of this theorem are in fact incorrect. The situation is even less clear when the norm-bound on is frequency-dependent. To the authors' knowledge, no formal extension of the small- theorem to this case is available in the open literature. Proofs informally mentioned within the research community are usually based on the construction, for each uncertainty block, of an H1 function, invertible in H1, whose magnitude on the unit circle (in the discrete-time case) is equal to the given uncertainty bound wi(z). Such proofs assume that the uncertainty bounds possess some regularity properties (implying their boundedness), and are bounded away from zero. In this article, we rst formally state and prove such a small- theorem (Theorem 1). Stability of I (P; ) for all in the uncertainty set of interest is usually referred to as \robust stability." Another concept that has received scant attention in the literature is that of \uniform robust stability." I (P; ) is uniformly robustly stable over a given uncertainty set if not only the transfer matrix of interest remains in H1 as ranges over that set, but furthermore its norm remains bounded by a quantity independent of . The major part of the article is devoted to deriving conditions for robust stability and uniform robust stability of the P { interconnection, when the boundedness and regularity assumptions on the uncertainty bounds are relaxed. Note that, in particular, the boundedness assumption on the wis may be overly restrictive, as often the uncertainty is modeled 1
-
?1
P
6
Figure 1: The P { interconnection modeling an uncertain system.
2
as being arbitrarily large at high frequency. We rst show that if mere stability rather than uniform stability is sought, then a sucient \small-" condition holds without any boundedness or regularity assumptions on the wis. We also derive a necessary condition for uniform robust stability where the assumption that the wis are bounded away from zero is relaxed; however, certain regularity conditions on how fast the wis can approach zero become necessary in order to have a nontrivial set of uncertainties. Finally, we show that for certain uncertainty models, a necessary and sucient condition for robust stability can be obtained, without any boundedness assumption on the wis. Known small- theorems with frequencyindependent uncertainty bounds are recovered as special cases and stated as Corollaries 1 through 4.
2 Preliminaries We will focus our attention for the most part on discrete-time systems, as the technical details become somewhat simpler here; we discuss continuous-time systems brie y in x5. Let R+ be the set of nonnegative real numbers. Let D denote the closed unit disk fz : z 2 C ; jzj 1g, D c = C n D its complement, D c the closure of D c , and @ D the unit circle fz : z 2 C ; jz j = 1g. Given a matrix M 2 C nn , let M denote its complex conjugate transpose, (M ) its largest singular value, and (M ) its smallest singular value. H1(D c ) denotes the set of functions that are bounded and analytic in D c . For compactness of notation, we will also use H1(D c ) to denote matrix-valued functions whose entries are in H1(D c ). Finally, the H1 norm of H 2 H1 (D c ) is denoted kH k1, and de ned as
kH k1 = zsup (H (z )): 2Dc
Let R, S and F be nonnegative integers, not all zero, and let n, r1 ; : : : ; rR , s1 ; : : : ; sS and f1; : : : ; fF be positive integers with n = P ri + P si + P fi. De ne nr = P ri and P P nc = si + fi . Let ?r be the subspace of real nr nr matrices de ned by
?r = fdiag( 1Ir1 ; : : : ; RIr ) : R
i 2 Rg;
(1)
let ?c be the subspace of complex nc nc matrices de ned by
n
?c = diag( 1Is1 ; : : : ; S Is ; ?1; : : : ; ?F ) : i 2 C ; ?i 2 C f f i
S
and let
? = fdiag(?r; ?c) : ?r 2 ?r; ?c 2 ?c g; 3
i
o
;
(2) (3)
and
B? = f? 2 ? : (?) 1g: The structured singular value of a matrix M 2 C nn with respect to a block structure ? is de ned to be (M ) = 0 if there is no ? 2 ? such that det(I + ?M ) = 0, and
(M ) = min f(?) : det(I + ?M ) = 0g ?2?
?1
otherwise. The description above corresponds to the so-called \standard mixed-" framework. The quantity plays a central role in the stability analysis of systems with real and complex structured perturbations; see, e.g., [wJCD98, x8.12]. We will make use of various sets of functions in H1(D c ) that will be assumed to have continuous extension on D c .1 In that context, we will often abuse notation and use the same symbol for a function in H1(D c ) and the corresponding extension. Let c be de ned by
c = fc 2 H1(D c ) : c (z) 2 ?c 8z 2 D c ; c has a continuous extension on D c g; and let be de ned by
= f : = diag(r; c ); r 2 ?r; c 2 c g :
(4)
In this paper, our concern is to extend the standard mixed- analysis to the case when the complex uncertainties have frequency-dependent upper bounds. To this end, de ne 8 9 W = diag( I ; : : : ; I ; w I ; : : : ; w I ; w I ; : : : ; w I ) for > > 1 r R r 1 s S s S +1 f S + F f 1 1 1 < = W = >W : some nonnegative real numbers i , i = 1; : : : ; R and some functions> : (5) : ; wi : @ D ! R+ , i = 1; : : : ; (S + F ) Of special interest are those W that are conjugate-symmetric, i.e., that satisfy R
S
F
W (z ) = W (z ) for all z 2 @ D :
Given W 2 W , de ne
n
BW = 2 : (z) (z) W (z)2 for all z 2 @ D
o
:
(6)
For future use, also de ne, for W 2 W ,
BW R = f 2 BW : is real on the real axisg; In [Tit95], a small- theorem is obtained, in the case of constant scaling, without such assumption on P and . 1
4
and
BW RR = f 2 BW : is real-rationalg; as well as (corresponding to W (z) = I for all z 2 @ D ), B = f 2 : ((z)) 1 8z 2 @ D g; BR = f 2 B : is real on the real axisg;
and
BRR = f 2 B : is real-rationalg: Consider now I (P; ) where P 2 P , with P = fP 2 H1(D c ) : P has a continuous extension on D c g;
and with assumed to lie in . For every such P and such that I (P; ) is well-posed,2 we will denote by G the following transfer function, associated with I (P; ):
1 0 ?1 (I + P )?1 ( I + P ) A: G = @ ?1 ?1 P (I + P )
(I + P )
(7)
Given S , if G is well de ned and belongs to H1(D c ) for all 2 S or, equivalently, if (I + P ) is invertible in H1(D c ) for all such , we will say that I (P; S) is robustly stable. If I (P; S) is robustly stable, and if moreover sup kGk1 < 1;
2S
(8)
we will say that I (P; S) is uniformly robustly stable. The second de nition is inspired from [KGP87].
Remark: Whenever W is bounded, (8) is clearly equivalent to sup k(I + P )?1 k1 < 1:
2S
(9)
When W is unbounded, however, (9) is weaker than (8). In such a case, it is not clear that (8) is the \right" de nition for uniform robust stability. Indeed, the question of which nodes in the block-diagram of Figure 1 are \physical" depends on the uncertainty model from which this blockdiagram is derived (additive, multiplicative, etc.). Still, most results derived below remain true when (8) is replaced with (9). When this is not the case, it is pointed out. 2
This simply means that the equations describing the interconnection have a unique solution.
5
When W (z) = In for all z 2 D c , the situation considered here is the standard complex- problem, and a number of precise mathematical statements have been made relating robust stability, uniform robust stability and in this case; see [TF95] for example. For general W , however, no such statements can be found in the literature. The objective of this paper is to explore this issue. We rst note that under some assumptions on W , we can provide necessary and sucient conditions for uniform robust stability of the P { interconnection by appealing to standard results. We have the following theorem.
Theorem 1 Let P 2 P and let W = diag(1 Ir1 ; : : : ; R IrR ; w1 Is1 ; : : : ; wS IsS ; wS +1If1 ; : : : ; wS +F IfF ) 2 W :
(10)
Suppose for each i = 1; : : : ; (S + F ), wi : @ D ! R+ is continuously di erentiable, and nowhere vanishing. Then, I (P; BW ) is uniformly robustly stable if and only if
sup (W (z)P (z)) < 1:
(11)
z 2@ D
In addition, if W is conjugate-symmetric, then I (P; BW R) is uniformly robustly stable if and only if (11) holds. Finally, if condition (11) does not hold, then I (P; BW ) is not even robustly stable.3
Proof: The main step in the proof is to apply Lemma 6 (in Appendix A) to construct analytic functions i : D c ! C , i = 1; : : : ; (S + F ), with i; ?i 1 2 H1(D c ), satisfying j i(rz)j ?! wi(z) as r # 1 for every z 2 @ D . Let = diag(1 Ir1 ; : : : ; RIr ; 1Is1 ; : : : ; S Is ; S+1If1 ; : : : ; S+F If ): R
S
F
Then, with P~ = P , condition (11) is equivalent to the condition that sup (P~ (z)) < 1:
z 2@ D
(12)
A standard result in analysis4 states that this condition is equivalent to I (P~ ; B) being robustly stable, and sup~ 2B k(I + ~ P~ )?1k1 < 1: Moreover, if condition (12) does not hold, then I (P; ) is unstable for some ~ 2 B.
However, I (P; BW R) can still be robustly stable, even when P is real-rational, and even when W (z ) = I for all z 2 @ D : see [TF95]. 4 In particular, this result is alluded to at the end of [TF95]. It is obtained in this paper at the end of x4 (Corollary 1) as a special case of our general result, Theorem 4, proved from rst principles. 3
6
.. ...................................... .. ~ P (z ) ... .. .. .. .. .... .. P (z) ( z ) . .. .......................................... ?1 6...................................... .. .. .. .. . .. (z) (z)?1 .... .. .. .. .. ~ (z) .... ....................................... ~ is. Figure 2: The P~ {~ interconnection. I (P; ) is stable if and only if I (P~ ; )
7
~ we see that ~ 2 B if and only if 2 BW . Moreover, Now, de ning = , ~ is stable if and only if I (P; ) is stable (see Figure 2). Also note that P = ~ P~ . I (P~ ; ) Therefore, we conclude that condition (11) is equivalent to I (P; BW ) being robustly stable, and sup2B k(I +P )?1k1 < 1: Moreover, if condition (11) does not hold, then I (P; ) unstable for some 2 BW . The claim concerning I (P; BW R) follows similarly, in view of the fact that, if W is conjugate-symmetric, is real on the real axis. This completes the proof. W
Clearly, condition (11) is sucient for uniform robust stability of I (P; BW RR) as well. However, the proof of Theorem 1 cannot be easily modi ed to address the question of whether condition (11) is necessary for uniform robust stability of I (P; BW RR). Moreover, the uniform robust stability condition is stated in Theorem 1 under fairly strong assumptions on wi . In particular, it is required that wi be bounded, which is unnatural as often uncertainty bounds are known reliably only over certain frequency \bands". (This is specially true in the continuous-time case, where the uncertainty is typically modeled as being arbitrarily large at high frequency.) In the rest of the paper, we explore the possibility of addressing some of these issues. A direct approach becomes necessary however.
3 Irreducible representation of uncertainty balls It is in general not necessary that (W (z)P (z)) < 1 for all z 2 @ D in order for I (P; BW ) to be robustly stable: consider the trivial case where some wi is discontinuous at some point and takes a large value there but is small everywhere else. A less trivial situation in which I (P; BW ) can be robustly stable even though (12) is violated arises when for some i, wi is nonzero, but there is no nonzero function in H1(D c ) whose magnitude on @ D lies below wi. Finally, it is clear that, unless W is conjugate-symmetric, I (P; BW R) and I (P; BW RR) can be robustly stable even when (12) does not hold. These observations motivate the following de nitions. We say that W = diag(1 Ir1 ; : : : ; R IrR ; w1 Is1 ; : : : ; wS IsS ; wS +1If1 ; : : : ; wS +F IfF ) 2 W
(13)
is H1(D c ){irreducible if (i) wi, i = 1; : : : ; (S + F ) is lower semicontinuous, and (ii) for every wi, i 2 f1; : : : ; (S + F )g that is not identically zero, the corresponding uncertainty sub-ball contains a nonzero element, i.e., for i = 1; : : : ; (S + F ), there exists some nonzero 8
c;i 2 H1(D c ), continuous on D c , such that c;i(z) c;i (z) wi(z)2 I 8z 2 @ D :
(14)
We say that W is real H1(D c ){irreducible if it is H1(D c ){irreducible, and W is conjugatesymmetric. We say that it is real-rational H1(D c ){irreducible if it is real H1(D c ){irreducible, and for every wi that is not identically zero, there exists a nonzero real-rational c;i such that (14) holds.
Remark: When the wis are lower semicontinuous, H1(D c ){irreducibilit y of W reduces to the following question: Given wi : @ D ! R+ , lower semicontinuous, does there exist a nonzero 2 H1(D c ), with a continuous extension on D c , whose magnitude on the unit circle lies under wi?
The answer turns out to be \yes" if wi vanishes nowhere on @ D . And, in cases when wi (z0 ) = 0 for some z0 2 @ D , the answer depends on how \fast" wi approaches zero as z approaches z0 . More speci cally, wi is required to satisfy a log-integrability condition; in such a case, can always be picked to be real on the real axis, see Lemma 7 (in Appendix A). In the case when is required to be rational as well, wi (z ) has to bounded below by a function of the form kjz ? z0 j?2N , around z0 , for some integer N and some k > 0.
Conditions for robust stability and uniform robust stability of I (P; BW ) will be stated in terms of bounds W that are H1(D c ){irreducible, real H1(D c ){irreducible, or real-rational H1(D c ){irreducible. This incurs no loss of generality, as we show next. Let W of the form (13). Corresponding to every wi, let wi be the lower envelope of wi, i.e., let it be such that its epigraph is the closure of the epigraph of wi. Then, de ne ~ = diag(1 Ir1 ; : : : ; R Ir ; w~1Is1 ; : : : ; w~S Is ; w~S+1If1 ; : : : ; w~S+F If ); W (15) R
S
F
where
8 < 0 if there exits no nonzero c;i 2 H1(D c ) such that (14) holds, w~i = : wi otherwise.
Next, de ne ^ = diag(1 Ir1 ; : : : ; R IrR ; w^1Is1 ; : : : ; w^S IsS ; w^S+1If1 ; : : : ; w^S+F IfF ); W where
(16)
w^i (z ) = minfw~i (z ); w~i (z )g; z 2 @ D :
Finally, de ne W y = diag(1 Ir1 ; : : : ; R IrR ; w1y Is1 ; : : : ; wSy IsS ; wSy +1 If1 ; : : : ; wSy +F IfF );
9
(17)
where
8
< wy = i
0 if there exits no nonzero c;i 2 H1(D c ), real-rational, that (14) holds, : w^i otherwise.
It is readily veri ed that W~ , W^ , and W y are H1(D c )-irreducible, real H1(D c )-irreducible, and real-rational H1(D c )-irreducible, respectively; note in particular that for each i, wi is lower semicontinuous. Then, we have the following result.
Proposition 1 Let W 2 W . Then, BW~ = BW , BW^ R = BW R, BW RR = BW RR. y
Proof: Suppose 2 BW . With W as in (10), we have r r diag(21Ir1 ; : : : 2R Ir ); c;i (z) c;i(z) wi(z)2 I 8z 2 @ D ; i = 1; : : : ; S + F; (18) R
where c;i is the ith diagonal block of c. We show that c;i(z) c;i wi(z)2 I 8z 2 @ D : Let z0 2 @ D , let i 2 f1; : : : ; S + F g, and pick a sequence fzi;k g1 k=1 @ D , converging to z0 , such that wi(zi;k ) ! wi(z0 ) as k ! 1. Letting z = zi;k in (18), letting k ! 1, and invoking continuity of on @ D , we get c;i(z0 )c;i (z0) wi(z0 )2I; as claimed. The remaining claims are readily proved.
4 Main Results In the proof of Theorem 2 below we will make use of the following form of Nyquist's criterion, which applies without real-rational assumption.5
Lemma 1 Let F 2 H1(D c ), with a continuous extension on D c , be such that (I + F ) is not invertible in H1(D c ). Then there exist a scalar 2 (0; 1] and a point z~ 2 @ D such that det(I + F (~z)) = 0.
It is a simpli ed version of a result given in [Tit95] in the more general case where existence of a continuous extension of F on D c is not assumed. 5
10
Proof:
Let G be de ned by G(z) = F (1=z). Thus G is analytic and bounded in the interior int(D ) of D and admits a continuous extension on D . Since (I + F ) is not invertible in H1(D c ), g = det(I + G) cannot be bounded away from 0 on int(D ). Thus there must exist z^ 2 D such that g(^z) = 0. If z^ 2 @ D the proof is complete, with = 1 and z~ = 1=z^. Thus assume now that jz^j < 1. Consider the closed path (circle) : [?; ) ! D given by
() = ej . Since g is analytic and bounded in int(D ) and has no zero in the range of , it follows from Cauchy's Principle of the Argument (see, e.g. [Rud74, Theorem 10.43]) that the origin has a nonzero index with respect to the compound map g (i.e., g \encircles" the origin at least once). Consequently if, for every 2 [0; 1], we de ne h : [0; 2) ! C by h () = det(I + F ( ()))
then h0 (=1) and h1 (= g ) are not homotopic with respect to the punctured plane. Thus there exists 2 (0; 1) such that the range of h contains the origin, i.e., for some z^0 2 @ D , det(I + G(^z0 )) = 0. The claim follows, with z~ = 1=z^0.
Theorem 2 Let P 2 P and W 2 W . If (W (z )P (z )) < 1
8z 2 @ D ;
(19)
then I (P; BW ) is robustly stable, and thus so are I (P; BW R) and I (P; BW RR). Moreover, suppose that W is bounded. Then, if
sup (W (z)P (z)) < 1;
z 2@ D
(20)
then I (P; BW ) is uniformly robustly stable, and thus so are I (P; BW R) and I (P; BW RR).
Remark: Note that Theorem 2 applies with no irreducibility assumptions on W .
Proof: Use contradiction to prove the rst claim. Thus, let 2 BW be such that I (P; ) is unstable, i.e., such that (I + P ) is not invertible in H1(D c ). Since P and are continuous on D c , it follows from Lemma 1 that there exist 2 (0; 1] and z^ 2 @ D such that det(I + (^z )P (^z)) = 0: Since 2 BW , (^z ) = ?^ W (^z) for some ?^ 2 B?. It follows that (W (^z )P (^z )) 1;
11
contradicting (19). Thus the rst claim holds. Concerning the second claim, again proceed by contradiction: Suppose that given any > 0, there exist 2 BW and z 2 @ D such that (I + (z )P (z)) < :
(21)
Thus there exists a matrix E, with (E) < , such that (I + (z )P (z )(I + E )?1 ) = 0;
Since it follows that
(z)(z) W (z)2;
(22)
(W (z )P (z)(I + E )?1 ) 1:
(23)
Since W and P are bounded, there exists a sequence fk g, with k # 0 as k ! 1, a point ~ such that z ! z~ and W (z ) ! W~ as k ! 1. Since () z~ 2 @ D , and a diagonal matrix W is upper semicontinuous, it then follows from (23) that k
k
~ P (~z)) 1: (W If () is continuous at W~ P (~z) then it follows that, given any < 1 there exists k such that (W (zk )P (zk )) ;
contradicting (20). Thus () must be discontinuous at W~ P (~z). It follows from [PP93, Lemma 5.1] that ~ P (~z))rr) = (W~ P (~z)) 1; ?r ((W where ?r denotes the structured singular value with respect to block-structure ?r and subscript \rr" refers to the top left nr nr submatrix. Now, (W~ P (~z ))rr = W~ rrPrr(~z ) = Wrr(~z )Prr(~z ) = (W (~z)P (~z))rr where we have used the fact that Wrr is constant (see (5)), thus continuous. Thus ?r ((W (~z )P (~z ))rr) 1;
implying that
(W (~z )P (~z )) 1;
12
a contradiction.
Remark: Clearly, if W is unbounded, boundedness of kGk1 over BW does not follow from (20). Note however that, in the 1-block case, (20) implies boundedness of k(I + P )?1 k1 over BW without boundedness assumption on W . Indeed, then becomes the largest singular value
and, if (20) holds, then, for some < 1,
8 2 BW ; z 2 @ D ;
((z )P (z )) < 1
implying that
(I + (z )P (z )) 1 ? > 0
8 2 BW ; z 2 @ D :
Since, as per the rst statement in the theorem, robust stability does hold, it follows that k(I + P )?1k1 1 ?1 ; proving the claim. On the other hand, it may be worth stressing that, in the block-structured case, not even k(I + P )?1 k1 need be bounded under (20) when W is unbounded. The reason is that, in such case, uncertainties of arbitrarily large size (as allowed by an unbounded W ) can leave the system stable while rendering k(I + P )?1 k1 arbitrarily large. This happens, for example, if the unbounded uncertainty is purely multiplicative, so the nominal plant P sees no feedback. Then, stability is no longer an issue; however, k(I + P )?1 k1 can be made arbitrary large by a suitable choice of . A speci c example is illustrated in Figure 3. Here,
3 2 0 0 5; P =4 1 0
and is a \repeated complex scalar uncertainty", i.e., in our notation, n = 2, R = 0, F = 0, S = 1 and s1 = 2. (Note that similar examples can be constructed with non-repeated uncertainties as well.) Suppose that
8 < max(j tan(=2)j; 1)I; W (z ) = : 0;
= ej ; 2 (?; ); z = ?1:
z
Then, it is readily checked that W is H1 (D c )-irreducible (indeed, real-rational H1(D c )-irreducible), and that det (I + (z )P (z )) = 1 for all z 2 @ D and 2 BW , so that sup (W (z )P (z )) = 0:
z 2@ D
Thus, I (P; BW ) is robustly stable. However, consider the system with the uncertainty = I , where 1 2(z + 1) (z ) = p ; 2 ((1 + )z + (1 ? ))2
13
with > 0. Then it is readily veri ed that 2 BW for all > 0. And,
2 3
1 0 k(I + P )?1 k1 =
4 5
> k k1 = 2p2(11 ? 2) ;
? 1 1
which can be made arbitrarily large by selecting small enough.
We now turn to necessary conditions. In view of the development thus far, the following question assumes central importance: Under what conditions on W is it the case that violation of (20) implies the existence of a destabilizing 2 BW ? Also of interest are conditions on W under which, when (20) is violated, there exists a destabilizing in BW R or BW RR. We now show that all that is required for the existence of appropriate destabilizing is that W be irreducible in the appropriate sense. We begin with the following four lemmas, proved in appendices (except for Lemma 4).
Lemma 2 Let W 2 W , W lower semicontinuous, let P 2 P , and suppose that sup (W (z)P (z)) 1:
z 2@ D
(24)
Then, given any > 0, there exists z^ 2 @ D , ?^ 2 B? satisfying
det(I + (1 + )?^ W (^z)P (^z )) = 0
(25)
with either ?^ real or z^ 62 f?1; 1g.
Lemma 3 6 Let M 2 C nn and suppose det(I + ?M ) = 0 for some ? 2 ?. Then there exists
?0 = diag( 1Ir1 ; : : : ; RIr ; 1Is1 ; : : : ; S Is ; ?01; : : : ; ?0F ) 2 ?; R
S
with (?0 ) = (?) and rank(?0i ) 1, such that
det(I + ?0M ) = 0: A proof of this result can be found in [TF95], embedded in the proof of Theorem 1. It is reproduced here for the reader's convenience. 6
14
e1 e2
r
P
66
u1 u2
-
0 u? ru ? -
6-
A A A
6
Figure 3: When W is unbounded, uncertainties of arbitrarily large size can leave the system stable while rendering the H1 norm of the transfer function from u to e arbitrarily large.
15
Lemma 4 Let fM 2 C nn :
> 0g be such that
det(I + (1 + )M) = 0: Then
lim (I + M ) = 0: #0
Lemma 5 7 Let z^ 2 @ D n f?1; 1g, and let ? = diag( 1Ir1 ; : : : ; RIr ; 1 Is1 : : : : ; S Is ; ?1; : : : ; ?F ) 2 B?; R
S
be such that rank(?i ) 1. Then there exists 2 BRR such that (^z ) = ?.
Theorem 3 Let P 2 P and W 2 W . Suppose that sup (W (z)P (z)) 1:
z 2@ D
(26)
Then, if W is H1(D c ){irreducible (respectively, real H1(D c ){irreducible and real-rational H1(D c ){irreducible), then I (P; BW ) (respectively, I (P; BW R) and I (P; BW RR)) is not uniformly robustly stable.
Proof: Suppose that (26) holds. In view of Lemmas 2, 3, and 4, there exist families fz 2 @ D : > 0g and f? 2 B? : > 0g such that lim (I + ? W (z )P (z)) = 0 #0
(27)
and, for each > 0, either (i) ? is real or (ii) z 62 f?1; 1g and rank(?;i) 1. Now let ^ 2 BW such that either (I + ^ P ) is not invertible C > 0. We show that there exists in H1(D c ) or k(I + ^ P )?1k1 > C , thus proving the claim. Invoking (27), let z^ 2 @ D and ?^ 2 B? be such that (I + ?^ W (^z )P (^z )) < 1=(2C ): Either ?^ is real or z^ 62 f?1; 1g and rank(?^ i) 1. If ?^ is real, let ^ = ?^ and the proof of the rst statement is complete. Thus, suppose z^ = ej^ 62 f?1; 1g and rank(?^ i ) 1. Without loss of generality suppose that ^ 2 (0; ). To complete the proof we construct a family f : 2 (0; maxf^; ? ^g)g in the appropriate ball (BW , BW R, BW RR) such that This result is standard. It can be found in [TF95] for the continuous-time case. Some inaccuracies in [TF95] have been corrected in the presentation here. 7
16
(^z ) converges to W (^z)?^ as ! 0. Thus let 2 (0; maxf^; ? ^g). Then, will be of the form
(z) = (z)diag Ir1 ; : : : ; Ir ; 1(z)Is1 ; : : : ; S (z)Is ; S+1(z)If1 ; : : : ; S+F (z)If ~ (z); R
S
F
where the factors in the right-hand side, all in , are speci ed now. Express W as W = diag(1 Ir1 ; : : : ; R IrR ; w1 Is1 ; : : : ; wS IsS ; wS +1If1 ; : : : ; wS +F IfF ):
First, for every i 2 f1; : : : ; (S + F )g such that wi(^z ) = 0, let i be identically zero. For such i, the corresponding entries in the rst and third factors are now arbitrary. Now for the rst factor. Invoking Lemma 7 (in Appendix A), let = diag(1 Ir1 ; : : : ; R Ir ; '1Is1 ; : : : ; 'S+F If ); R
F
respectively in BW , BW R, and BW RR, be such that 'i(^z) 6= 0 for every i such that wi (^z ) 6= 0. Concerning the second factor, rst let i =
min j'i(ej )j?1wi(ej ) ^ ^
2[?;+]
for every i 2 f1; : : : ; (S + F )g such that wi(^z ) 6= 0. (Reduce the value of as necessary for 'i to be invertible in the required range.) Since wi is lower semicontinuous, the \min" is achieved, and moreover i 1 whenever wi(^z) 6= 0. To handle the cases when is required to belong to BW R or BW RR, for every i 2 f1; : : : ; (S + F )g such that wi(^z ) 6= 0, let i ?1 ), with f i de ned by be the real-rational function in H1(D c ) given by i (z) = fi( zz+1 fi (s) =
ai s2 + bi s + ci
(28)
where ai, bi, and ci, with bi; ci > 0, are real numbers selected in such a way that
ji(^z )j = i ji(z)j 8z 6= z^ and
ji (z)j 1 8z 62 fej : 2 [?^ ? ; ?^ + ] [ [^ ? ; ^ + ]g: (29) In the case when is merely required to belong to BW , use the Poisson integral formula in Lemma 6 (in Appendix A) to construct, for all i 2 f1; : : : ; (S + F )g such that wi(^z ) = 6 c 0, a function i in H1(D c ), with continuous extension i z?on1 D and iwhose magnitude isj a continuously di erentiable function on @ D , equal to f z+1 with f as above for z 2 fe : 17
2 [^ ? ; ^ + ]g, and less than or equal to one for z 62 fej : 2 [^ ? ; ^ + ]g. Taking into
account the fact that, in the cases when is required to belong to BW R or BW RR, the wi's are assumed to be conjugate-symmetric, it is readily checked that, in all three cases,
j'i(z)i (z)j wi(z) 8z 2 @ D i = 1; : : : ; (S + F ) and
j'i(^z )i (^z )j ! wi(^z ) as ! 0:
Finally, for i = 1; : : : ; (S + F ), let i be the phase of 'i(^z )i (^z ). To complete the proof in all three cases, we invoke Lemma 5 to construct the third factor ~ 2 BRR such that ~ (^z ) = ?~ = diag(Inr ; e?j1 Is1 ; : : : ; e?j
S F
+
IsS+F )?^ :
Remark: As noted in Footnote 3, condition (26) does not imply the existence of a destabilizing in BW R (let alone in BW RR), even when when W (z ) = I for all z 2 @ D . However, under additional regularity assumptions on W , a destabilizing in BW can be constructed, as can be shown by appropriately modifying the proof of Theorem 3. Continuous di erentiability of W is one such regularity assumption. Note that mere continuity of W is not sucient, as seen from the scalar example p(z ) = 1, w(z ) = 1=(1 + jj) for z = ej , 2 [?; ), where I (p; Bd ) is robustly stable since H1(C + ) functions that take the value one at z = 1 and stay under w(z ) on @ D must have a discontinuous phase at z = 1 and thus cannot be in . On the other hand, whenever the right-hand side in (26) is strictly larger than one, then of course, under the respective irreducibility assumptions, destabilizing s do exist in all three balls. (As a matter fact, this follows from Theorem 5 below.)
Theorems 2 and 3 can be combined to yield a necessary and sucient condition for uniform robust stability which improves on Theorem 1.
Theorem 4 Let P 2 P , and let W 2 W be bounded. Suppose that W is H1(D c ){irreducible. Then I (P; BW ) is uniformly robustly stable if and only if sup (W (z)P (z)) < 1:
z 2@ D
(30)
Suppose moreover that W is real H1(D c ){irreducible (respectively, real-rational H1 (D c ){ irreducible). Then I (P; BW R) (respectively, I (P; BW RR)) is uniformly robustly stable if and only if (30) holds.
18
A special case of major importance is that where the uncertainty bounds are frequencyindependent, i.e., W = I . This result is well known but, to our knowledge, neither a precise statement nor a proof are available in the open literature.
Corollary 1 Given any P 2 P , the following are equivalent: 1. I (P; B) is uniformly robustly stable; 2. I (P; BR) is uniformly robustly stable; 3. I (P; BRR) is uniformly robustly stable; 4. supz2@D (P (z)) < 1.
For completeness let us also note (see [TF95] and [Tit95, Remark]) that, for P 2 P ,8 sup (P (z)) = sup (P (z))
z 2@ D
z 2D c
so that a fth equivalent statement is sup (P (z)) < 1:
z 2Dc
It may be somewhat unsatisfying that the necessary and sucient condition obtained in Theorem 4 requires boundedness of W . This situation however is unavoidable regardless of whether uniform robust stability or mere robust stability is sought. Indeed (i) as shown in the remark following the proof of Theorem 2, boundedness of W is required to guarantee suciency of (30) for uniform robust stability and (ii) as shown in the remark following Theorem 3, some regularity (implying boundedness) of W is required to guarantee necessity of (30) for (mere) robust stability (and even then, only for BW ). We conclude this section by showing that, if \open" uncertainty balls are considered instead, then an appropriate 8 When n = 0, () is continuous and thus D c can be replaced by D c . This is not so in the general case r
however. A counterexample (see [TF95]) is given by
"
#
1 (z + 1)=z P (z ) = ; ?(z + 1)=z 1 with R = 1 and r1 = 2 (\repeated-real"). For such a structure (P (z )) is the spectral radius of P (z ) if P (z ) has a real eigenvalue, and 0 otherwise. It is readily checked that P (z ) has a real eigenvalue if and only if (z + 1)=z is pure imaginary, and the only z 2 D c where this occurs is z = ?1, which is not it D c .
19
condition (nonstrict upper bound) is indeed necessary and sucient for robust stability, without boundedness assumption on W .9 Speci cally, let B W be de ned by
B W = f : 2 BW for some < 1g; and similarly de ne B W R and B W RR from BW R and BW RR. Note that when W is H1(D c ){irreducible (respectively, real H1(D c ){irreducible and real-rational H1(D c ){ irreducible), then for every wi, i = 1; : : : ; (S + F ) that is not identically zero, the corresponding \open" uncertainty sub-ball also contains a nonzero element. We will assume (without loss of generality, in view of Proposition 1) appropriate irreducibility properties for W .
Theorem 5 Let P 2 P , let W 2 W , and suppose that W is H1(D c ){irreducible. Then, I (P; B W ) is robustly stable if and only if (W (z )P (z )) 1
8z 2 @ D :
(31)
Suppose moreover that W is real H1(D c ){irreducible (respectively, real-rational H1 (D c ){ irreducible). Then, I (P; B W R) (respectively, I (P; B W RR)) is robustly stable if and only if (31) holds.
Proof: We use contradiction to prove suciency. An argument similar to that used in the proof of the rst claim in Theorem 2 shows that, if I (P; ) is unstable for some 2 B W , then there exists 2 (0; 1] and z^ 2 @ D such that det(I + (^z )P (^z)) = 0: Since, for some < 1, (^z ) (^z ) W (^z)2, we have that (^z ) = ?^ W (^z) for some ?^ 2 B? and thus that (W (^z)P (^z )) > 1; a contradiction. We now use contradiction to prove necessity. Thus suppose that, for some z^ 2 @ D , (W (^z)P (^z )) > 1: Then there exists ?^ 2 ?, with (?^ ) < 1, such that det(I + ?^ W (^z)P (^z)) = 0: For the special case of frequency-independent uncertainty bound, this result was obtained in [TF95] (in the continuous-time case). 9
20
It remains to construct 2 B W (respectively, B W R, B W RR) such that (^z ) = ?^ W (^z). Let > 1, < 1 be such that ( ?^ ) . Clearly, ?^ 2 ? and det(I + ( ?^)( ?1 W (^z))P (^z)) = 0: The remainder of the construction is analogous to, but signi cantly simpler than, the construction used in the proof of Theorem 3. Thus, will be of the form
~ z); (z) = (z)diag Ir1 ; : : : ; Ir ; 1(z)Is1 ; : : : ; S (z)Is ; S+1(z)If1 ; : : : ; S+F (z)If ( R
S
F
where the factors in the right-hand side, all in , are speci ed now. Express W as W = diag(1 Ir1 ; : : : ; R IrR ; w1 Is1 ; : : : ; wS IsS ; wS +1If1 ; : : : ; wS +F IfF ):
First, as in the proof of Theorem 3, for every i 2 f1; : : : ; (S + F )g such that wi(^z ) = 0, let i be identically zero; the corresponding entries in the rst and third factors are now arbitrary. Now, for the rst factor, again following the proof of Theorem 3, in all three cases, let = diag(1 Ir1 ; : : : ; R Ir ; '1Is1 ; : : : ; 'S+F If ); R
F
respectively in B W , B W R, and B W RR, be such that 'i(^z ) 6= 0 for every i for which wi (^z ) 6= 0. Concerning the second factor, rst let i = j'i (^z )?1 j ?1 wi(^z );
for every i 2 f1; : : : ; (S + F )g such that wi(^z ) 6= 0. Then i 1 whenever wi(^z ) 6= 0 and, since the wi's are lower semicontinuous and since > 1, there is a neighborhood N (^z ) of z^ such that, for every i 2 f1; : : : ; (S + F )g such that wi(^z ) 6= 0, i j'i (z )j?1 wi (z )
8z 2 N (^z ) \ @ D :
Now pick i exactly like i was picked in the proof of Theorem 3 (with i replacing i ), except for replacing (29) with
ji(z)j 1 8z such that i > j'i(z)j?1 wi(z): Finally, ~ 2 BW RR is constructed as in the proof of Theorem 3, with ?^ replacing ?^ . In the case of frequency-independent bounds, the discrete-time counterpart of the result proved in [TF95] is recovered. 21
Corollary 2 Given any P 2 P , the following are equivalent: ) is robustly stable; 1. I (P; B ) is robustly stable; 2. I (P; BR 3. I (P; BRR ) is robustly stable; 4. (P (z)) 1 8z 2 @ D . With reference to the comment following Corollary 1, a fth equivalent statement is (P (z )) 1
8z 2 D c :
5 Synopsis of the continuous-time case We brie y present the continuous-time version of the results obtained in x4 here. Some notation rst. Let R+ be the set of nonnegative real numbers. Let C + be the open right half of the complex plane. Let Re = R [ f1g be the one-point compacti cation of R. H1(C + ) denotes the set of functions that are bounded and analytic over C + . For compactness of notation, we will also use H1(C + ) to denote matrix-valued functions, whose entries are in H1(C + ). With ?r and ?c as de ned in (1){(3), let c be de ned by
c = fc 2 H1(C + ) : c (s) 2 ?c 8s 2 C + ; c admits a continuous extension on C + [ f1gg; and let be de ned as in the discrete-time case by = f : = diag(r; c ); r 2 ?r; c 2 c g : We next de ne
9 8 W = diag( I ; : : : ; I ; w I ; : : : ; w I ; w I ; : : : ; w I ) for > > 1 r R r 1 s S s S +1 f S + F f 1 1 1 = < WCT = >W : some nonnegative real numbers i, i = 1; : : : ; R and some functions> ; : ; wi : Re ! R+ , i = 1; : : : ; (S + F ) R
S
F
(32)
and PCT as
PCT = fP 2 H1(C + ) : P has a continuous extension on C + [ f1gg: Given W 2 WCT, de ne
n
BW = 2 : (j!)(j!) W (!)2 for all ! 2 Re 22
o
;
(33)
BW R = f 2 BW : is real on the real axisg; and
BW RR = f 2 BW : is real-rationalg: For the case when W (!) = I for all ! 2 Re , these balls are again denoted by B, BR, and BRR, respectively. We then have the following theorems.
Theorem 6 Let P 2 PCT and W 2 WCT . If (W (! )P (j! )) < 1
8! 2 Re ;
(34)
then I (P; BW ) is robustly stable and thus so are I (P; BW R) and I (P; BW RR). Moreover, suppose that W is bounded. Then, if
sup (W (!)P (j!)) < 1;
(35)
!2Re
then I (P; BW ) is uniformly robustly stable and thus so are I (P; BW R) and I (P; BW RR).
Note that whenever nr = 0, Re in (35) can equivalently be replaced by R (even if W is discontinuous at 1: use Lemma 9). In parallel with the discrete-time case, we will say that W is H1(C + ){irreducible if every wi , i = 1; : : : ; (S + F ) is lower semicontinuous on Re , and for every wi , i = 1; : : : ; (S + F ) that is not identically zero, the corresponding uncertainty sub-ball contains a nonzero element, i.e., there exists some nonzero c;i 2 H1(C + ), continuous on C + [ f1g, such that c;i (j!)c;i (j!) wi(!)2I 8! 2 Re :
(36)
We will say W is real H1(C + ){irreducible if it is H1(C + ){irreducible, and W is even. We will say W is real-rational H1(C + ){irreducible if it is real H1(C + ){irreducible, and for every wi that is not identically zero, there exists a nonzero real-rational c;i such that (36) holds.
Remark: Similarly to the discrete-time case, when the wi s are lower semicontinuous on Re , H1(C + )-irreducibility amounts to a certain log-integrability condition on those wis that are not identically zero. Speci cally, in the continuous-time case, min 0; (log w (!))=(1 + !2 ) must be i
integrable on the real line. (To see this, make use of a bilinear transformation between D c and C + .)
23
Theorem 7 Let P 2 PCT and W 2 WCT . Suppose that sup (W (!)P (j!)) 1:
!2Re
(37)
Then, if W is H1(C + ){irreducible (respectively, real H1(C + ){irreducible and real-rational H1(C + ){irreducible), then I (P; BW ) (respectively, I (P; BW R) and I (P; BW RR)) is not uniformly robustly stable.
Note that the theorem remains true (but is generally weaker) if, in (37), Re is replaced by R. Again, when nr = 0, such a change leaves (37) una ected. Theorems 6 and 7 yield the following necessary and sucient condition, which also holds with Re replaced with R whenever nr = 0.
Theorem 8 Let P 2 PCT, let W 2 WCT , bounded, and suppose W is H1(C + ){irreducible (respectively, real H1(C + ){irreducible, real-rational H1(C + ){irreducible). Then, I (P; BW ) (respectively, I (P; BW R), I (P; BW RR)) is uniformly robustly stable if and only if sup (W (!)P (j!)) < 1:
!2Re
(38)
Corollary 3 Given any P 2 PCT , the following are equivalent: 1. I (P; B) is uniformly robustly stable; 2. I (P; BR) is uniformly robustly stable; 3. I (P; BRR) is uniformly robustly stable; 4. sup!2Re (P (j!)) < 1.
A fth equivalent statement is
sup (P (s)) < 1:
s2C + [f1g
Finally, as in the discrete-time case, we give a necessary and sucient condition for robust stability that holds without boundedness assumption on W . We de ne
B W = f : 2 BD for some < 1g; and B W R and B W RR similarly. 24
Theorem 9 Let P 2 PCT, let W 2 WCT, and suppose W is H1(C + ){irreducible (respectively, real H1(C + ){irreducible, real-rational H1(C + ){irreducible). Then, I (P; B W ) (respectively, I (P; B W R), I (P; B W RR)) is robustly stable if and only if (W (j! )P (j! )) 1
8! 2 Re :
(39)
Corollary 4 [TF95] Given any P 2 PCT, the following are equivalent: ) is robustly stable; 1. I (P; B ) is robustly stable; 2. I (P; BR 3. I (P; BRR ) is robustly stable; 4. (P (j!)) 1 8! 2 Re .
A fth equivalent statement is (P (s)) 1
8s 2 C + [ f1g:
6 Concluding remarks Both necessary conditions and sucient conditions have been obtained for (uniform) robust stability in the presence of structured uncertainty with frequency-dependent bounds. Two necessary and sucient conditions were also obtained. For the rst one (uniform robust stability) the frequency bound is required to be bounded, and it was shown that the result does not hold without such condition. The second one (robust stability over an \open" ball) holds with essentially no assumption on the uncertainty ball: indeed, irreducibility is a property not of the uncertainty set, but of its representation. Thus our goal of providing small- theorems that hold under \weak" assumptions is achieved.
Acknowledgment. The authors wish to thank Carlos Berenstein, as well as Dan Luecking
and other contributors to sci.math.research, for their valuable help in connection with Lemmas 6 and 7.
25
References [Doy82]
J. Doyle. Analysis of feedback systems with structured uncertainties. IEE Proc., 129-D(6):242{250, November 1982.
[Gar81]
J. B. Garnett. Bounded Analytic Functions. Academic Press, 1981.
[KGP87] P. P. Khargonekar, T. T. Georgiou, and A. M. Pascoal. On the robust stabilization of time-invariant plants with unstructured uncertainty. IEEE Trans. Aut. Control, AC-32:201{207, 1987. [Koo80]
P. Koosis. Introduction to Hp spaces. Cambridge University Press, 1980.
[Lev96]
W. S. Levine, editor. The Control Handbook. CRC Press, Boca Raton, Florida, 1996.
[PD93]
A. Packard and J. Doyle. The complex structured singular value. Automatica, 29(1):71{109, 1993.
[PP93]
A. Packard and P. Pandey. Continuity properties of the real/complex structured singular value. IEEE Trans. Aut. Control, AC-38:415{428, 1993.
[Rud74]
W. Rudin. Real and Complex Analysis. McGraw-Hill, New York, 1974.
[Saf82]
M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D:251{256, November 1982.
[TF95]
A.L. Tits and M.K.H. Fan. On the Small- Theorem. Automatica, 31(8):1199{ 1201, 1995.
[Tit95]
A.L. Tits. The small theorem without real-rational assumption. In Proceedings of the American Control Conference, pages 2870{2872, 1995.
[wJCD98] K. Zhou with J. C. Doyle. Essentials of Robust Control. Prentice Hall, New Jersey, 1998. [wJDG96] K. Zhou with J. Doyle and K. Glover. Robust and Optimal Control. Prentice Hall, 1996.
26
Appendix A: Some results from complex analysis Lemma 6 Let
w : @ D ! R+ be a continuously di erentiable function, with log w being integrable over @ D . De ne the function : D c ! C by
Z 2 + e?j d: 1 log w(e?j ) zz ? (z) = 2 e?j 0
Let (z) = e
(z ) .
(40)
Then:
2 H1(D c ), and admits a continuous extension on D c . j (rej )j converges to w(ej ) as r # 1. Moreover:
If w is conjugate-symmetric, i.e., w(z ) = w(z) on @ D , then is real on the real axis. If w is bounded away from zero over @ D , ?1 2 H1(D c ).
Proof:
This lemma represents an application of the standard Poisson integral formula (see for example, [Gar81, Ch. I, x3]) to construct a function , analytic in D c , such that Re (rej ) ! log w as r # 1. Continuity of (and consequently ) over D c results from the fact that Im (rej ) ! w~ as r # 1, where w~ : @ D ! R is a continuous function (this is guaranteed by the continuity of the derivative of w; see for example, [Koo80, pp. 25{26]). Next, if w is conjugate-symmetric, it is easily veri ed that (z) = (z), whenever z = z , so that , and hence are real-valued on the real axis. Finally, when w is bounded away from zero over @ D , ?1 is analytic in D c , and bounded over D c , so that ?1 2 H1(D c ). We next state a \nontriviality" condition for the set of uncertainties.
Lemma 7 Suppose that w : @ D ! R+ is lower semicontinuous. Let
8 9 c < extension on D = Bw = : 2 H1(D c ) : has a continuous (z) (z) w2I for all z 2 @ D ; ;
and
Bw R = f 2 Bw : is real on the real axisg :
Then the following conditions are equivalent: 1. Bw contains an element besides 0.
27
2. Bw R contains an element besides 0. 3. min(0; log w) is integrable over @ D . Moreover, when any of these conditions hold, there exists an element of the form I in Bw R, with : C ! C , satisfying for z 2 @ D , j (z)j > 0 whenever w(z) > 0.
Proof: Clearly, (2) implies (1). We now show that (1) implies (3). Suppose some nonzero 2 Bw . Let be some nonzero entry of . Then, log jj is integrable over @ D (see for
example, [Gar81, Ch. II, Thm 4.1]). Since w is measurable (being lower semicontinuous) and since w(z) j(z)j for z 2 @ D , min(0; log w) is integrable over @ D as well. Next, we show that (3) implies (2). Suppose that min(0; log w) is integrable over @ D . De ne w^ by w^(z) = min(w(z); w(z )). Then, min(1; w^) is lower semicontinuous and logintegrable over @ D , so that it is is a limit from below of continuous functions which are logintegrable over @ D . Let g : @ D ! R+ be a continuous function, that is conjugate-symmetric, bounded above by min(1; w^), with g being log-integrable. Then g can be uniformly approximated from below by a continuously di erentiable log-integrable function h : @ D ! R+ , also conjugate-symmetric. Then, a function 2 H1(D c ) can be constructed using Lemma 6 with a continuous extension over D c , real on the real axis, and with j (z)j = h(z) for all z 2 @ D . Clearly, I 2 Bw R. Finally, for every z such that w(z) > 0, the functions g and h in the above construction can be chosen to satisfy g(z) > 0 and h(z) > 0 respectively, and the last statement of the lemma follows.
Appendix B. Proof of Lemma 2 The following two lemmas will be used. The rst lemma is a direct consequence of a result of Packard and Pandey [PP93, Lemma 5.1]. It generalizes Lemma 1 of [TF95].
Lemma 8 Let M 2 C nn be such that (M ) > 0 and is discontinuous at M . Then there exists a real matrix ? 2 ? such that (?) = 1=(M ) and det(I + ?M ) = 0. Proof: Let Mrr 2 C nr nr be the top left submatrix of M . From Lemma 5.1 in [PP93], it
follows that discontinuity of at M implies that ?r (Mrr) = (M ), where ?r (Mrr) denotes the structured singular value of Mrr with respect to block structure ?r. Since (M ) > 0, 28
there exists ?r 2 ?r with (?r) = 1=(M ) such that det(I +?rMrr) = 0. It is readily checked that the matrix ? 2 ? de ned by ? = diagf?r; 0nc g satis es the required properties.
Lemma 9 Let M 2 C nn and let i 1, i = 1; : : : ; n. Then, given any block structure ?, (diag( i )M ) (M ):
Proof: If (M ) = 0, the result holds trivially. Thus, suppose (M ) > 0, and let ? 2 ?, with (?) = 1=(M ) satisfy det(I + ?M ) = 0. Let ?0 = ?diag( i)?1 . Since i 1 for all i it follows that (?0) (?). Moreover, det(I + ?0 diag( i)M ) = det(I + ?M ) = 0: The claim follows. Proof of Lemma 2 Let z 0 2 @ D such that (W (z 0 )P (z 0 ))
p3 1 : 1+
There are two cases: either () is continuous at W (z0 )P (z0) or it is discontinuous at that point. First suppose it is discontinuous. In that case, it follows from Lemma 8 that there is p a real ?0 2 ?, with (?0) 3 1 + such that det(I + ?0W (z0 )P (z0)) = 0 i.e., (25) holds with ?^ = 1+1 ?0 2 B?, real, and z^ = z0 . Suppose now that () is continuous at W (z0 )P (z0). If z0 62 f?1; 1g, then the claim follows directly from the de nition of , with z^ = z 0 . Thus suppose that z 0 = 1 (the proof is similar when z 0 = ?1), i.e., 1 : (W (1)P (1)) p3 1+ By continuity, there exists 0 2 (0; ) such that 1 (W (1)P (ej )) 8 2 [0; 0 ); 2 = 3 (1 + ) implying, since is positively homogeneous, that ! 1 1 j 8 2 [0; 0): W (1)P (e ) p3 (1 + ) 1+ 29
Now, since the (diagonal) entries wi of W are lower semicontinuous, there exists ^ 2 (0; 0) such that, for all i, 1 w (1): wi (ej^) p3 1+ i It follows from Lemma 9 that 1 : (W (ej^)P (ej^)) (1 + ) The claim follows readily, with z^ = ej^.
Appendix C. Proof of Lemma 3 Express ? as
? = diag( 1Ir1 ; : : : ; RIr ; 1Is1 : : : : ; S Is ; ?1; : : : ; ?F ); R
S
and let k k denote the Euclidean norm in C n . Let u 2 C n , with kuk = 1, be such that (I + M ?)u = 0: Express u as [^uT ; uT1 ; : : : ; uTF ]T , with u^ 2 C n
r
Ps
+
i
and ui 2 C f , and let i
?0 = diag( 1Ir1 ; : : : ; RIr ; 1Is1 : : : : ; S Is ; ?01; : : : ; ?0F ) R
with, for i = 1; : : : ; F ,
S
8 < ?iuiui =kuik2 if kuik 6= 0; 0 ?i = : 0 otherwise:
Then clearly (?0 ) = (?) and, for i = 1; : : : ; F , rank(?0i) 1. Moreover (I + M ?0 )u = u + M ?u = (I + M ?)u = 0; and thus det(I + M ?0 ) = 0, proving our claim.
Appendix D. Proof of Lemma 5 For i = 1; : : : ; S , let i be an all-pass, stable transfer function de ned by 1 + i z i (z ) = ai z + i 30
where ai 2 [?1; 1] and i 2 (?1; 1] are such that i (^z) = i. It is readily checked that, since z^ 62 f?1; 1g, this is always possible. Next, for i 2 f1; : : : ; F g, let i viwi (i 2 [0; 1]) be the singular value decomposition of ?i and let vij and wij be the j th entries of vi and wi, respectively. Let xi = [x1i ; : : : ; xfi ]T and yi = [yi1; : : : ; yif ]T be real-rational vector functions, with xi analytic and bounded in D c and yi analytic and bounded in the interior of D , de ned by z + ij 1 + ij z ; yij (z ) = bij xji (z ) = aij z+ 1+ z i
i
ij
ij
where aij ; bij 2 [?1; 1], ij ; ij 2 (?1; 1] are such that xji (^z ) = vij ;
yij (^z ) = wij ;
j = 1 : : : ; fi
Let i(z) = i xi(z)yi (1=z)T . Finally, let (z) = diag( 1Ir1 ; : : : ; RIr ; 1(z)Is1 ; : : : ; S (z)Is ; 1(z); : : : ; F (z)): S
R
Clearly, 2 BRR and (^z ) = ?.
31
List of Figures 1 2 3
The P { interconnection modeling an uncertain system. . . . . . . . . . . . 2 ~ is. . . . . 7 The P~ {~ interconnection. I (P; ) is stable if and only if I (P~ ; ) When W is unbounded, uncertainties of arbitrarily large size can leave the system stable while rendering the H1 norm of the transfer function from u to e arbitrarily large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
32
V. Balakrishnan School of Electrical and Computer Engineering Purdue University West Lafayette, IN 47907-1285 Phone: +1 765 494-0728 Fax: +1 765 494-3371 Internet: [email protected]
August 3, 1999 Abstract We present conditions, some necessary and some sucient, valid under weak assumptions, for robust stability and uniform robust stability of uncertain linear timeinvariant systems with linear time-invariant uncertainties that are block-diagonal, with known frequency-dependent norm bounds on the diagonal blocks. Small- theorems with frequency-independent uncertainty bounds are recovered as special cases.
Keywords: Robust stability, Uncertain systems, Robust control, Small- theorem,
Frequency-dependent uncertainty bounds.
This research was in part supported by NSF's Engineering Research Center No. NSFD-CDR-88-03012, and in part by the Oce of Naval Research under contract No. N00014-97-1-0640. y Corresponding author.
1 Introduction A popular paradigm for modeling control systems with uncertainties is illustrated in Figure 1. Here P is the transfer function of a stable linear system, and is a stable operator that represents the \uncertainties" that arise from various sources such as modeling errors, neglected or unmodeled dynamics or parameters, etc. Often, the uncertainty is assumed to possess various additional properties. Common examples are that is structured (i.e., diagonal or block-diagonal), that it is linear time-invariant or real-constant etc. Such control system models have found wide acceptance in robust control; see for example [Doy82, Saf82, PD93, Lev96, wJDG96]. We will henceforth refer to the system in Figure 1 as the P { interconnection, and denote it by I (P; ). The Small- Theorem is the foundation of the structured singular value approach towards assessing the stability of systems a ected by structured uncertainties. Speci cally, the theorem gives a necessary and sucient condition on the \-norm" of the plant transfer function P for I (P; ) to be stable for all structured 2 H1 of a given maximum size (H1-norm). Various versions of the small- theorem exist in the literature, and it was shown in [TF95] that some commonly quoted versions of this theorem are in fact incorrect. The situation is even less clear when the norm-bound on is frequency-dependent. To the authors' knowledge, no formal extension of the small- theorem to this case is available in the open literature. Proofs informally mentioned within the research community are usually based on the construction, for each uncertainty block, of an H1 function, invertible in H1, whose magnitude on the unit circle (in the discrete-time case) is equal to the given uncertainty bound wi(z). Such proofs assume that the uncertainty bounds possess some regularity properties (implying their boundedness), and are bounded away from zero. In this article, we rst formally state and prove such a small- theorem (Theorem 1). Stability of I (P; ) for all in the uncertainty set of interest is usually referred to as \robust stability." Another concept that has received scant attention in the literature is that of \uniform robust stability." I (P; ) is uniformly robustly stable over a given uncertainty set if not only the transfer matrix of interest remains in H1 as ranges over that set, but furthermore its norm remains bounded by a quantity independent of . The major part of the article is devoted to deriving conditions for robust stability and uniform robust stability of the P { interconnection, when the boundedness and regularity assumptions on the uncertainty bounds are relaxed. Note that, in particular, the boundedness assumption on the wis may be overly restrictive, as often the uncertainty is modeled 1
-
?1
P
6
Figure 1: The P { interconnection modeling an uncertain system.
2
as being arbitrarily large at high frequency. We rst show that if mere stability rather than uniform stability is sought, then a sucient \small-" condition holds without any boundedness or regularity assumptions on the wis. We also derive a necessary condition for uniform robust stability where the assumption that the wis are bounded away from zero is relaxed; however, certain regularity conditions on how fast the wis can approach zero become necessary in order to have a nontrivial set of uncertainties. Finally, we show that for certain uncertainty models, a necessary and sucient condition for robust stability can be obtained, without any boundedness assumption on the wis. Known small- theorems with frequencyindependent uncertainty bounds are recovered as special cases and stated as Corollaries 1 through 4.
2 Preliminaries We will focus our attention for the most part on discrete-time systems, as the technical details become somewhat simpler here; we discuss continuous-time systems brie y in x5. Let R+ be the set of nonnegative real numbers. Let D denote the closed unit disk fz : z 2 C ; jzj 1g, D c = C n D its complement, D c the closure of D c , and @ D the unit circle fz : z 2 C ; jz j = 1g. Given a matrix M 2 C nn , let M denote its complex conjugate transpose, (M ) its largest singular value, and (M ) its smallest singular value. H1(D c ) denotes the set of functions that are bounded and analytic in D c . For compactness of notation, we will also use H1(D c ) to denote matrix-valued functions whose entries are in H1(D c ). Finally, the H1 norm of H 2 H1 (D c ) is denoted kH k1, and de ned as
kH k1 = zsup (H (z )): 2Dc
Let R, S and F be nonnegative integers, not all zero, and let n, r1 ; : : : ; rR , s1 ; : : : ; sS and f1; : : : ; fF be positive integers with n = P ri + P si + P fi. De ne nr = P ri and P P nc = si + fi . Let ?r be the subspace of real nr nr matrices de ned by
?r = fdiag( 1Ir1 ; : : : ; RIr ) : R
i 2 Rg;
(1)
let ?c be the subspace of complex nc nc matrices de ned by
n
?c = diag( 1Is1 ; : : : ; S Is ; ?1; : : : ; ?F ) : i 2 C ; ?i 2 C f f i
S
and let
? = fdiag(?r; ?c) : ?r 2 ?r; ?c 2 ?c g; 3
i
o
;
(2) (3)
and
B? = f? 2 ? : (?) 1g: The structured singular value of a matrix M 2 C nn with respect to a block structure ? is de ned to be (M ) = 0 if there is no ? 2 ? such that det(I + ?M ) = 0, and
(M ) = min f(?) : det(I + ?M ) = 0g ?2?
?1
otherwise. The description above corresponds to the so-called \standard mixed-" framework. The quantity plays a central role in the stability analysis of systems with real and complex structured perturbations; see, e.g., [wJCD98, x8.12]. We will make use of various sets of functions in H1(D c ) that will be assumed to have continuous extension on D c .1 In that context, we will often abuse notation and use the same symbol for a function in H1(D c ) and the corresponding extension. Let c be de ned by
c = fc 2 H1(D c ) : c (z) 2 ?c 8z 2 D c ; c has a continuous extension on D c g; and let be de ned by
= f : = diag(r; c ); r 2 ?r; c 2 c g :
(4)
In this paper, our concern is to extend the standard mixed- analysis to the case when the complex uncertainties have frequency-dependent upper bounds. To this end, de ne 8 9 W = diag( I ; : : : ; I ; w I ; : : : ; w I ; w I ; : : : ; w I ) for > > 1 r R r 1 s S s S +1 f S + F f 1 1 1 < = W = >W : some nonnegative real numbers i , i = 1; : : : ; R and some functions> : (5) : ; wi : @ D ! R+ , i = 1; : : : ; (S + F ) Of special interest are those W that are conjugate-symmetric, i.e., that satisfy R
S
F
W (z ) = W (z ) for all z 2 @ D :
Given W 2 W , de ne
n
BW = 2 : (z) (z) W (z)2 for all z 2 @ D
o
:
(6)
For future use, also de ne, for W 2 W ,
BW R = f 2 BW : is real on the real axisg; In [Tit95], a small- theorem is obtained, in the case of constant scaling, without such assumption on P and . 1
4
and
BW RR = f 2 BW : is real-rationalg; as well as (corresponding to W (z) = I for all z 2 @ D ), B = f 2 : ((z)) 1 8z 2 @ D g; BR = f 2 B : is real on the real axisg;
and
BRR = f 2 B : is real-rationalg: Consider now I (P; ) where P 2 P , with P = fP 2 H1(D c ) : P has a continuous extension on D c g;
and with assumed to lie in . For every such P and such that I (P; ) is well-posed,2 we will denote by G the following transfer function, associated with I (P; ):
1 0 ?1 (I + P )?1 ( I + P ) A: G = @ ?1 ?1 P (I + P )
(I + P )
(7)
Given S , if G is well de ned and belongs to H1(D c ) for all 2 S or, equivalently, if (I + P ) is invertible in H1(D c ) for all such , we will say that I (P; S) is robustly stable. If I (P; S) is robustly stable, and if moreover sup kGk1 < 1;
2S
(8)
we will say that I (P; S) is uniformly robustly stable. The second de nition is inspired from [KGP87].
Remark: Whenever W is bounded, (8) is clearly equivalent to sup k(I + P )?1 k1 < 1:
2S
(9)
When W is unbounded, however, (9) is weaker than (8). In such a case, it is not clear that (8) is the \right" de nition for uniform robust stability. Indeed, the question of which nodes in the block-diagram of Figure 1 are \physical" depends on the uncertainty model from which this blockdiagram is derived (additive, multiplicative, etc.). Still, most results derived below remain true when (8) is replaced with (9). When this is not the case, it is pointed out. 2
This simply means that the equations describing the interconnection have a unique solution.
5
When W (z) = In for all z 2 D c , the situation considered here is the standard complex- problem, and a number of precise mathematical statements have been made relating robust stability, uniform robust stability and in this case; see [TF95] for example. For general W , however, no such statements can be found in the literature. The objective of this paper is to explore this issue. We rst note that under some assumptions on W , we can provide necessary and sucient conditions for uniform robust stability of the P { interconnection by appealing to standard results. We have the following theorem.
Theorem 1 Let P 2 P and let W = diag(1 Ir1 ; : : : ; R IrR ; w1 Is1 ; : : : ; wS IsS ; wS +1If1 ; : : : ; wS +F IfF ) 2 W :
(10)
Suppose for each i = 1; : : : ; (S + F ), wi : @ D ! R+ is continuously di erentiable, and nowhere vanishing. Then, I (P; BW ) is uniformly robustly stable if and only if
sup (W (z)P (z)) < 1:
(11)
z 2@ D
In addition, if W is conjugate-symmetric, then I (P; BW R) is uniformly robustly stable if and only if (11) holds. Finally, if condition (11) does not hold, then I (P; BW ) is not even robustly stable.3
Proof: The main step in the proof is to apply Lemma 6 (in Appendix A) to construct analytic functions i : D c ! C , i = 1; : : : ; (S + F ), with i; ?i 1 2 H1(D c ), satisfying j i(rz)j ?! wi(z) as r # 1 for every z 2 @ D . Let = diag(1 Ir1 ; : : : ; RIr ; 1Is1 ; : : : ; S Is ; S+1If1 ; : : : ; S+F If ): R
S
F
Then, with P~ = P , condition (11) is equivalent to the condition that sup (P~ (z)) < 1:
z 2@ D
(12)
A standard result in analysis4 states that this condition is equivalent to I (P~ ; B) being robustly stable, and sup~ 2B k(I + ~ P~ )?1k1 < 1: Moreover, if condition (12) does not hold, then I (P; ) is unstable for some ~ 2 B.
However, I (P; BW R) can still be robustly stable, even when P is real-rational, and even when W (z ) = I for all z 2 @ D : see [TF95]. 4 In particular, this result is alluded to at the end of [TF95]. It is obtained in this paper at the end of x4 (Corollary 1) as a special case of our general result, Theorem 4, proved from rst principles. 3
6
.. ...................................... .. ~ P (z ) ... .. .. .. .. .... .. P (z) ( z ) . .. .......................................... ?1 6...................................... .. .. .. .. . .. (z) (z)?1 .... .. .. .. .. ~ (z) .... ....................................... ~ is. Figure 2: The P~ {~ interconnection. I (P; ) is stable if and only if I (P~ ; )
7
~ we see that ~ 2 B if and only if 2 BW . Moreover, Now, de ning = , ~ is stable if and only if I (P; ) is stable (see Figure 2). Also note that P = ~ P~ . I (P~ ; ) Therefore, we conclude that condition (11) is equivalent to I (P; BW ) being robustly stable, and sup2B k(I +P )?1k1 < 1: Moreover, if condition (11) does not hold, then I (P; ) unstable for some 2 BW . The claim concerning I (P; BW R) follows similarly, in view of the fact that, if W is conjugate-symmetric, is real on the real axis. This completes the proof. W
Clearly, condition (11) is sucient for uniform robust stability of I (P; BW RR) as well. However, the proof of Theorem 1 cannot be easily modi ed to address the question of whether condition (11) is necessary for uniform robust stability of I (P; BW RR). Moreover, the uniform robust stability condition is stated in Theorem 1 under fairly strong assumptions on wi . In particular, it is required that wi be bounded, which is unnatural as often uncertainty bounds are known reliably only over certain frequency \bands". (This is specially true in the continuous-time case, where the uncertainty is typically modeled as being arbitrarily large at high frequency.) In the rest of the paper, we explore the possibility of addressing some of these issues. A direct approach becomes necessary however.
3 Irreducible representation of uncertainty balls It is in general not necessary that (W (z)P (z)) < 1 for all z 2 @ D in order for I (P; BW ) to be robustly stable: consider the trivial case where some wi is discontinuous at some point and takes a large value there but is small everywhere else. A less trivial situation in which I (P; BW ) can be robustly stable even though (12) is violated arises when for some i, wi is nonzero, but there is no nonzero function in H1(D c ) whose magnitude on @ D lies below wi. Finally, it is clear that, unless W is conjugate-symmetric, I (P; BW R) and I (P; BW RR) can be robustly stable even when (12) does not hold. These observations motivate the following de nitions. We say that W = diag(1 Ir1 ; : : : ; R IrR ; w1 Is1 ; : : : ; wS IsS ; wS +1If1 ; : : : ; wS +F IfF ) 2 W
(13)
is H1(D c ){irreducible if (i) wi, i = 1; : : : ; (S + F ) is lower semicontinuous, and (ii) for every wi, i 2 f1; : : : ; (S + F )g that is not identically zero, the corresponding uncertainty sub-ball contains a nonzero element, i.e., for i = 1; : : : ; (S + F ), there exists some nonzero 8
c;i 2 H1(D c ), continuous on D c , such that c;i(z) c;i (z) wi(z)2 I 8z 2 @ D :
(14)
We say that W is real H1(D c ){irreducible if it is H1(D c ){irreducible, and W is conjugatesymmetric. We say that it is real-rational H1(D c ){irreducible if it is real H1(D c ){irreducible, and for every wi that is not identically zero, there exists a nonzero real-rational c;i such that (14) holds.
Remark: When the wis are lower semicontinuous, H1(D c ){irreducibilit y of W reduces to the following question: Given wi : @ D ! R+ , lower semicontinuous, does there exist a nonzero 2 H1(D c ), with a continuous extension on D c , whose magnitude on the unit circle lies under wi?
The answer turns out to be \yes" if wi vanishes nowhere on @ D . And, in cases when wi (z0 ) = 0 for some z0 2 @ D , the answer depends on how \fast" wi approaches zero as z approaches z0 . More speci cally, wi is required to satisfy a log-integrability condition; in such a case, can always be picked to be real on the real axis, see Lemma 7 (in Appendix A). In the case when is required to be rational as well, wi (z ) has to bounded below by a function of the form kjz ? z0 j?2N , around z0 , for some integer N and some k > 0.
Conditions for robust stability and uniform robust stability of I (P; BW ) will be stated in terms of bounds W that are H1(D c ){irreducible, real H1(D c ){irreducible, or real-rational H1(D c ){irreducible. This incurs no loss of generality, as we show next. Let W of the form (13). Corresponding to every wi, let wi be the lower envelope of wi, i.e., let it be such that its epigraph is the closure of the epigraph of wi. Then, de ne ~ = diag(1 Ir1 ; : : : ; R Ir ; w~1Is1 ; : : : ; w~S Is ; w~S+1If1 ; : : : ; w~S+F If ); W (15) R
S
F
where
8 < 0 if there exits no nonzero c;i 2 H1(D c ) such that (14) holds, w~i = : wi otherwise.
Next, de ne ^ = diag(1 Ir1 ; : : : ; R IrR ; w^1Is1 ; : : : ; w^S IsS ; w^S+1If1 ; : : : ; w^S+F IfF ); W where
(16)
w^i (z ) = minfw~i (z ); w~i (z )g; z 2 @ D :
Finally, de ne W y = diag(1 Ir1 ; : : : ; R IrR ; w1y Is1 ; : : : ; wSy IsS ; wSy +1 If1 ; : : : ; wSy +F IfF );
9
(17)
where
8
< wy = i
0 if there exits no nonzero c;i 2 H1(D c ), real-rational, that (14) holds, : w^i otherwise.
It is readily veri ed that W~ , W^ , and W y are H1(D c )-irreducible, real H1(D c )-irreducible, and real-rational H1(D c )-irreducible, respectively; note in particular that for each i, wi is lower semicontinuous. Then, we have the following result.
Proposition 1 Let W 2 W . Then, BW~ = BW , BW^ R = BW R, BW RR = BW RR. y
Proof: Suppose 2 BW . With W as in (10), we have r r diag(21Ir1 ; : : : 2R Ir ); c;i (z) c;i(z) wi(z)2 I 8z 2 @ D ; i = 1; : : : ; S + F; (18) R
where c;i is the ith diagonal block of c. We show that c;i(z) c;i wi(z)2 I 8z 2 @ D : Let z0 2 @ D , let i 2 f1; : : : ; S + F g, and pick a sequence fzi;k g1 k=1 @ D , converging to z0 , such that wi(zi;k ) ! wi(z0 ) as k ! 1. Letting z = zi;k in (18), letting k ! 1, and invoking continuity of on @ D , we get c;i(z0 )c;i (z0) wi(z0 )2I; as claimed. The remaining claims are readily proved.
4 Main Results In the proof of Theorem 2 below we will make use of the following form of Nyquist's criterion, which applies without real-rational assumption.5
Lemma 1 Let F 2 H1(D c ), with a continuous extension on D c , be such that (I + F ) is not invertible in H1(D c ). Then there exist a scalar 2 (0; 1] and a point z~ 2 @ D such that det(I + F (~z)) = 0.
It is a simpli ed version of a result given in [Tit95] in the more general case where existence of a continuous extension of F on D c is not assumed. 5
10
Proof:
Let G be de ned by G(z) = F (1=z). Thus G is analytic and bounded in the interior int(D ) of D and admits a continuous extension on D . Since (I + F ) is not invertible in H1(D c ), g = det(I + G) cannot be bounded away from 0 on int(D ). Thus there must exist z^ 2 D such that g(^z) = 0. If z^ 2 @ D the proof is complete, with = 1 and z~ = 1=z^. Thus assume now that jz^j < 1. Consider the closed path (circle) : [?; ) ! D given by
() = ej . Since g is analytic and bounded in int(D ) and has no zero in the range of , it follows from Cauchy's Principle of the Argument (see, e.g. [Rud74, Theorem 10.43]) that the origin has a nonzero index with respect to the compound map g (i.e., g \encircles" the origin at least once). Consequently if, for every 2 [0; 1], we de ne h : [0; 2) ! C by h () = det(I + F ( ()))
then h0 (=1) and h1 (= g ) are not homotopic with respect to the punctured plane. Thus there exists 2 (0; 1) such that the range of h contains the origin, i.e., for some z^0 2 @ D , det(I + G(^z0 )) = 0. The claim follows, with z~ = 1=z^0.
Theorem 2 Let P 2 P and W 2 W . If (W (z )P (z )) < 1
8z 2 @ D ;
(19)
then I (P; BW ) is robustly stable, and thus so are I (P; BW R) and I (P; BW RR). Moreover, suppose that W is bounded. Then, if
sup (W (z)P (z)) < 1;
z 2@ D
(20)
then I (P; BW ) is uniformly robustly stable, and thus so are I (P; BW R) and I (P; BW RR).
Remark: Note that Theorem 2 applies with no irreducibility assumptions on W .
Proof: Use contradiction to prove the rst claim. Thus, let 2 BW be such that I (P; ) is unstable, i.e., such that (I + P ) is not invertible in H1(D c ). Since P and are continuous on D c , it follows from Lemma 1 that there exist 2 (0; 1] and z^ 2 @ D such that det(I + (^z )P (^z)) = 0: Since 2 BW , (^z ) = ?^ W (^z) for some ?^ 2 B?. It follows that (W (^z )P (^z )) 1;
11
contradicting (19). Thus the rst claim holds. Concerning the second claim, again proceed by contradiction: Suppose that given any > 0, there exist 2 BW and z 2 @ D such that (I + (z )P (z)) < :
(21)
Thus there exists a matrix E, with (E) < , such that (I + (z )P (z )(I + E )?1 ) = 0;
Since it follows that
(z)(z) W (z)2;
(22)
(W (z )P (z)(I + E )?1 ) 1:
(23)
Since W and P are bounded, there exists a sequence fk g, with k # 0 as k ! 1, a point ~ such that z ! z~ and W (z ) ! W~ as k ! 1. Since () z~ 2 @ D , and a diagonal matrix W is upper semicontinuous, it then follows from (23) that k
k
~ P (~z)) 1: (W If () is continuous at W~ P (~z) then it follows that, given any < 1 there exists k such that (W (zk )P (zk )) ;
contradicting (20). Thus () must be discontinuous at W~ P (~z). It follows from [PP93, Lemma 5.1] that ~ P (~z))rr) = (W~ P (~z)) 1; ?r ((W where ?r denotes the structured singular value with respect to block-structure ?r and subscript \rr" refers to the top left nr nr submatrix. Now, (W~ P (~z ))rr = W~ rrPrr(~z ) = Wrr(~z )Prr(~z ) = (W (~z)P (~z))rr where we have used the fact that Wrr is constant (see (5)), thus continuous. Thus ?r ((W (~z )P (~z ))rr) 1;
implying that
(W (~z )P (~z )) 1;
12
a contradiction.
Remark: Clearly, if W is unbounded, boundedness of kGk1 over BW does not follow from (20). Note however that, in the 1-block case, (20) implies boundedness of k(I + P )?1 k1 over BW without boundedness assumption on W . Indeed, then becomes the largest singular value
and, if (20) holds, then, for some < 1,
8 2 BW ; z 2 @ D ;
((z )P (z )) < 1
implying that
(I + (z )P (z )) 1 ? > 0
8 2 BW ; z 2 @ D :
Since, as per the rst statement in the theorem, robust stability does hold, it follows that k(I + P )?1k1 1 ?1 ; proving the claim. On the other hand, it may be worth stressing that, in the block-structured case, not even k(I + P )?1 k1 need be bounded under (20) when W is unbounded. The reason is that, in such case, uncertainties of arbitrarily large size (as allowed by an unbounded W ) can leave the system stable while rendering k(I + P )?1 k1 arbitrarily large. This happens, for example, if the unbounded uncertainty is purely multiplicative, so the nominal plant P sees no feedback. Then, stability is no longer an issue; however, k(I + P )?1 k1 can be made arbitrary large by a suitable choice of . A speci c example is illustrated in Figure 3. Here,
3 2 0 0 5; P =4 1 0
and is a \repeated complex scalar uncertainty", i.e., in our notation, n = 2, R = 0, F = 0, S = 1 and s1 = 2. (Note that similar examples can be constructed with non-repeated uncertainties as well.) Suppose that
8 < max(j tan(=2)j; 1)I; W (z ) = : 0;
= ej ; 2 (?; ); z = ?1:
z
Then, it is readily checked that W is H1 (D c )-irreducible (indeed, real-rational H1(D c )-irreducible), and that det (I + (z )P (z )) = 1 for all z 2 @ D and 2 BW , so that sup (W (z )P (z )) = 0:
z 2@ D
Thus, I (P; BW ) is robustly stable. However, consider the system with the uncertainty = I , where 1 2(z + 1) (z ) = p ; 2 ((1 + )z + (1 ? ))2
13
with > 0. Then it is readily veri ed that 2 BW for all > 0. And,
2 3
1 0 k(I + P )?1 k1 =
4 5
> k k1 = 2p2(11 ? 2) ;
? 1 1
which can be made arbitrarily large by selecting small enough.
We now turn to necessary conditions. In view of the development thus far, the following question assumes central importance: Under what conditions on W is it the case that violation of (20) implies the existence of a destabilizing 2 BW ? Also of interest are conditions on W under which, when (20) is violated, there exists a destabilizing in BW R or BW RR. We now show that all that is required for the existence of appropriate destabilizing is that W be irreducible in the appropriate sense. We begin with the following four lemmas, proved in appendices (except for Lemma 4).
Lemma 2 Let W 2 W , W lower semicontinuous, let P 2 P , and suppose that sup (W (z)P (z)) 1:
z 2@ D
(24)
Then, given any > 0, there exists z^ 2 @ D , ?^ 2 B? satisfying
det(I + (1 + )?^ W (^z)P (^z )) = 0
(25)
with either ?^ real or z^ 62 f?1; 1g.
Lemma 3 6 Let M 2 C nn and suppose det(I + ?M ) = 0 for some ? 2 ?. Then there exists
?0 = diag( 1Ir1 ; : : : ; RIr ; 1Is1 ; : : : ; S Is ; ?01; : : : ; ?0F ) 2 ?; R
S
with (?0 ) = (?) and rank(?0i ) 1, such that
det(I + ?0M ) = 0: A proof of this result can be found in [TF95], embedded in the proof of Theorem 1. It is reproduced here for the reader's convenience. 6
14
e1 e2
r
P
66
u1 u2
-
0 u? ru ? -
6-
A A A
6
Figure 3: When W is unbounded, uncertainties of arbitrarily large size can leave the system stable while rendering the H1 norm of the transfer function from u to e arbitrarily large.
15
Lemma 4 Let fM 2 C nn :
> 0g be such that
det(I + (1 + )M) = 0: Then
lim (I + M ) = 0: #0
Lemma 5 7 Let z^ 2 @ D n f?1; 1g, and let ? = diag( 1Ir1 ; : : : ; RIr ; 1 Is1 : : : : ; S Is ; ?1; : : : ; ?F ) 2 B?; R
S
be such that rank(?i ) 1. Then there exists 2 BRR such that (^z ) = ?.
Theorem 3 Let P 2 P and W 2 W . Suppose that sup (W (z)P (z)) 1:
z 2@ D
(26)
Then, if W is H1(D c ){irreducible (respectively, real H1(D c ){irreducible and real-rational H1(D c ){irreducible), then I (P; BW ) (respectively, I (P; BW R) and I (P; BW RR)) is not uniformly robustly stable.
Proof: Suppose that (26) holds. In view of Lemmas 2, 3, and 4, there exist families fz 2 @ D : > 0g and f? 2 B? : > 0g such that lim (I + ? W (z )P (z)) = 0 #0
(27)
and, for each > 0, either (i) ? is real or (ii) z 62 f?1; 1g and rank(?;i) 1. Now let ^ 2 BW such that either (I + ^ P ) is not invertible C > 0. We show that there exists in H1(D c ) or k(I + ^ P )?1k1 > C , thus proving the claim. Invoking (27), let z^ 2 @ D and ?^ 2 B? be such that (I + ?^ W (^z )P (^z )) < 1=(2C ): Either ?^ is real or z^ 62 f?1; 1g and rank(?^ i) 1. If ?^ is real, let ^ = ?^ and the proof of the rst statement is complete. Thus, suppose z^ = ej^ 62 f?1; 1g and rank(?^ i ) 1. Without loss of generality suppose that ^ 2 (0; ). To complete the proof we construct a family f : 2 (0; maxf^; ? ^g)g in the appropriate ball (BW , BW R, BW RR) such that This result is standard. It can be found in [TF95] for the continuous-time case. Some inaccuracies in [TF95] have been corrected in the presentation here. 7
16
(^z ) converges to W (^z)?^ as ! 0. Thus let 2 (0; maxf^; ? ^g). Then, will be of the form
(z) = (z)diag Ir1 ; : : : ; Ir ; 1(z)Is1 ; : : : ; S (z)Is ; S+1(z)If1 ; : : : ; S+F (z)If ~ (z); R
S
F
where the factors in the right-hand side, all in , are speci ed now. Express W as W = diag(1 Ir1 ; : : : ; R IrR ; w1 Is1 ; : : : ; wS IsS ; wS +1If1 ; : : : ; wS +F IfF ):
First, for every i 2 f1; : : : ; (S + F )g such that wi(^z ) = 0, let i be identically zero. For such i, the corresponding entries in the rst and third factors are now arbitrary. Now for the rst factor. Invoking Lemma 7 (in Appendix A), let = diag(1 Ir1 ; : : : ; R Ir ; '1Is1 ; : : : ; 'S+F If ); R
F
respectively in BW , BW R, and BW RR, be such that 'i(^z) 6= 0 for every i such that wi (^z ) 6= 0. Concerning the second factor, rst let i =
min j'i(ej )j?1wi(ej ) ^ ^
2[?;+]
for every i 2 f1; : : : ; (S + F )g such that wi(^z ) 6= 0. (Reduce the value of as necessary for 'i to be invertible in the required range.) Since wi is lower semicontinuous, the \min" is achieved, and moreover i 1 whenever wi(^z) 6= 0. To handle the cases when is required to belong to BW R or BW RR, for every i 2 f1; : : : ; (S + F )g such that wi(^z ) 6= 0, let i ?1 ), with f i de ned by be the real-rational function in H1(D c ) given by i (z) = fi( zz+1 fi (s) =
ai s2 + bi s + ci
(28)
where ai, bi, and ci, with bi; ci > 0, are real numbers selected in such a way that
ji(^z )j = i ji(z)j 8z 6= z^ and
ji (z)j 1 8z 62 fej : 2 [?^ ? ; ?^ + ] [ [^ ? ; ^ + ]g: (29) In the case when is merely required to belong to BW , use the Poisson integral formula in Lemma 6 (in Appendix A) to construct, for all i 2 f1; : : : ; (S + F )g such that wi(^z ) = 6 c 0, a function i in H1(D c ), with continuous extension i z?on1 D and iwhose magnitude isj a continuously di erentiable function on @ D , equal to f z+1 with f as above for z 2 fe : 17
2 [^ ? ; ^ + ]g, and less than or equal to one for z 62 fej : 2 [^ ? ; ^ + ]g. Taking into
account the fact that, in the cases when is required to belong to BW R or BW RR, the wi's are assumed to be conjugate-symmetric, it is readily checked that, in all three cases,
j'i(z)i (z)j wi(z) 8z 2 @ D i = 1; : : : ; (S + F ) and
j'i(^z )i (^z )j ! wi(^z ) as ! 0:
Finally, for i = 1; : : : ; (S + F ), let i be the phase of 'i(^z )i (^z ). To complete the proof in all three cases, we invoke Lemma 5 to construct the third factor ~ 2 BRR such that ~ (^z ) = ?~ = diag(Inr ; e?j1 Is1 ; : : : ; e?j
S F
+
IsS+F )?^ :
Remark: As noted in Footnote 3, condition (26) does not imply the existence of a destabilizing in BW R (let alone in BW RR), even when when W (z ) = I for all z 2 @ D . However, under additional regularity assumptions on W , a destabilizing in BW can be constructed, as can be shown by appropriately modifying the proof of Theorem 3. Continuous di erentiability of W is one such regularity assumption. Note that mere continuity of W is not sucient, as seen from the scalar example p(z ) = 1, w(z ) = 1=(1 + jj) for z = ej , 2 [?; ), where I (p; Bd ) is robustly stable since H1(C + ) functions that take the value one at z = 1 and stay under w(z ) on @ D must have a discontinuous phase at z = 1 and thus cannot be in . On the other hand, whenever the right-hand side in (26) is strictly larger than one, then of course, under the respective irreducibility assumptions, destabilizing s do exist in all three balls. (As a matter fact, this follows from Theorem 5 below.)
Theorems 2 and 3 can be combined to yield a necessary and sucient condition for uniform robust stability which improves on Theorem 1.
Theorem 4 Let P 2 P , and let W 2 W be bounded. Suppose that W is H1(D c ){irreducible. Then I (P; BW ) is uniformly robustly stable if and only if sup (W (z)P (z)) < 1:
z 2@ D
(30)
Suppose moreover that W is real H1(D c ){irreducible (respectively, real-rational H1 (D c ){ irreducible). Then I (P; BW R) (respectively, I (P; BW RR)) is uniformly robustly stable if and only if (30) holds.
18
A special case of major importance is that where the uncertainty bounds are frequencyindependent, i.e., W = I . This result is well known but, to our knowledge, neither a precise statement nor a proof are available in the open literature.
Corollary 1 Given any P 2 P , the following are equivalent: 1. I (P; B) is uniformly robustly stable; 2. I (P; BR) is uniformly robustly stable; 3. I (P; BRR) is uniformly robustly stable; 4. supz2@D (P (z)) < 1.
For completeness let us also note (see [TF95] and [Tit95, Remark]) that, for P 2 P ,8 sup (P (z)) = sup (P (z))
z 2@ D
z 2D c
so that a fth equivalent statement is sup (P (z)) < 1:
z 2Dc
It may be somewhat unsatisfying that the necessary and sucient condition obtained in Theorem 4 requires boundedness of W . This situation however is unavoidable regardless of whether uniform robust stability or mere robust stability is sought. Indeed (i) as shown in the remark following the proof of Theorem 2, boundedness of W is required to guarantee suciency of (30) for uniform robust stability and (ii) as shown in the remark following Theorem 3, some regularity (implying boundedness) of W is required to guarantee necessity of (30) for (mere) robust stability (and even then, only for BW ). We conclude this section by showing that, if \open" uncertainty balls are considered instead, then an appropriate 8 When n = 0, () is continuous and thus D c can be replaced by D c . This is not so in the general case r
however. A counterexample (see [TF95]) is given by
"
#
1 (z + 1)=z P (z ) = ; ?(z + 1)=z 1 with R = 1 and r1 = 2 (\repeated-real"). For such a structure (P (z )) is the spectral radius of P (z ) if P (z ) has a real eigenvalue, and 0 otherwise. It is readily checked that P (z ) has a real eigenvalue if and only if (z + 1)=z is pure imaginary, and the only z 2 D c where this occurs is z = ?1, which is not it D c .
19
condition (nonstrict upper bound) is indeed necessary and sucient for robust stability, without boundedness assumption on W .9 Speci cally, let B W be de ned by
B W = f : 2 BW for some < 1g; and similarly de ne B W R and B W RR from BW R and BW RR. Note that when W is H1(D c ){irreducible (respectively, real H1(D c ){irreducible and real-rational H1(D c ){ irreducible), then for every wi, i = 1; : : : ; (S + F ) that is not identically zero, the corresponding \open" uncertainty sub-ball also contains a nonzero element. We will assume (without loss of generality, in view of Proposition 1) appropriate irreducibility properties for W .
Theorem 5 Let P 2 P , let W 2 W , and suppose that W is H1(D c ){irreducible. Then, I (P; B W ) is robustly stable if and only if (W (z )P (z )) 1
8z 2 @ D :
(31)
Suppose moreover that W is real H1(D c ){irreducible (respectively, real-rational H1 (D c ){ irreducible). Then, I (P; B W R) (respectively, I (P; B W RR)) is robustly stable if and only if (31) holds.
Proof: We use contradiction to prove suciency. An argument similar to that used in the proof of the rst claim in Theorem 2 shows that, if I (P; ) is unstable for some 2 B W , then there exists 2 (0; 1] and z^ 2 @ D such that det(I + (^z )P (^z)) = 0: Since, for some < 1, (^z ) (^z ) W (^z)2, we have that (^z ) = ?^ W (^z) for some ?^ 2 B? and thus that (W (^z)P (^z )) > 1; a contradiction. We now use contradiction to prove necessity. Thus suppose that, for some z^ 2 @ D , (W (^z)P (^z )) > 1: Then there exists ?^ 2 ?, with (?^ ) < 1, such that det(I + ?^ W (^z)P (^z)) = 0: For the special case of frequency-independent uncertainty bound, this result was obtained in [TF95] (in the continuous-time case). 9
20
It remains to construct 2 B W (respectively, B W R, B W RR) such that (^z ) = ?^ W (^z). Let > 1, < 1 be such that ( ?^ ) . Clearly, ?^ 2 ? and det(I + ( ?^)( ?1 W (^z))P (^z)) = 0: The remainder of the construction is analogous to, but signi cantly simpler than, the construction used in the proof of Theorem 3. Thus, will be of the form
~ z); (z) = (z)diag Ir1 ; : : : ; Ir ; 1(z)Is1 ; : : : ; S (z)Is ; S+1(z)If1 ; : : : ; S+F (z)If ( R
S
F
where the factors in the right-hand side, all in , are speci ed now. Express W as W = diag(1 Ir1 ; : : : ; R IrR ; w1 Is1 ; : : : ; wS IsS ; wS +1If1 ; : : : ; wS +F IfF ):
First, as in the proof of Theorem 3, for every i 2 f1; : : : ; (S + F )g such that wi(^z ) = 0, let i be identically zero; the corresponding entries in the rst and third factors are now arbitrary. Now, for the rst factor, again following the proof of Theorem 3, in all three cases, let = diag(1 Ir1 ; : : : ; R Ir ; '1Is1 ; : : : ; 'S+F If ); R
F
respectively in B W , B W R, and B W RR, be such that 'i(^z ) 6= 0 for every i for which wi (^z ) 6= 0. Concerning the second factor, rst let i = j'i (^z )?1 j ?1 wi(^z );
for every i 2 f1; : : : ; (S + F )g such that wi(^z ) 6= 0. Then i 1 whenever wi(^z ) 6= 0 and, since the wi's are lower semicontinuous and since > 1, there is a neighborhood N (^z ) of z^ such that, for every i 2 f1; : : : ; (S + F )g such that wi(^z ) 6= 0, i j'i (z )j?1 wi (z )
8z 2 N (^z ) \ @ D :
Now pick i exactly like i was picked in the proof of Theorem 3 (with i replacing i ), except for replacing (29) with
ji(z)j 1 8z such that i > j'i(z)j?1 wi(z): Finally, ~ 2 BW RR is constructed as in the proof of Theorem 3, with ?^ replacing ?^ . In the case of frequency-independent bounds, the discrete-time counterpart of the result proved in [TF95] is recovered. 21
Corollary 2 Given any P 2 P , the following are equivalent: ) is robustly stable; 1. I (P; B ) is robustly stable; 2. I (P; BR 3. I (P; BRR ) is robustly stable; 4. (P (z)) 1 8z 2 @ D . With reference to the comment following Corollary 1, a fth equivalent statement is (P (z )) 1
8z 2 D c :
5 Synopsis of the continuous-time case We brie y present the continuous-time version of the results obtained in x4 here. Some notation rst. Let R+ be the set of nonnegative real numbers. Let C + be the open right half of the complex plane. Let Re = R [ f1g be the one-point compacti cation of R. H1(C + ) denotes the set of functions that are bounded and analytic over C + . For compactness of notation, we will also use H1(C + ) to denote matrix-valued functions, whose entries are in H1(C + ). With ?r and ?c as de ned in (1){(3), let c be de ned by
c = fc 2 H1(C + ) : c (s) 2 ?c 8s 2 C + ; c admits a continuous extension on C + [ f1gg; and let be de ned as in the discrete-time case by = f : = diag(r; c ); r 2 ?r; c 2 c g : We next de ne
9 8 W = diag( I ; : : : ; I ; w I ; : : : ; w I ; w I ; : : : ; w I ) for > > 1 r R r 1 s S s S +1 f S + F f 1 1 1 = < WCT = >W : some nonnegative real numbers i, i = 1; : : : ; R and some functions> ; : ; wi : Re ! R+ , i = 1; : : : ; (S + F ) R
S
F
(32)
and PCT as
PCT = fP 2 H1(C + ) : P has a continuous extension on C + [ f1gg: Given W 2 WCT, de ne
n
BW = 2 : (j!)(j!) W (!)2 for all ! 2 Re 22
o
;
(33)
BW R = f 2 BW : is real on the real axisg; and
BW RR = f 2 BW : is real-rationalg: For the case when W (!) = I for all ! 2 Re , these balls are again denoted by B, BR, and BRR, respectively. We then have the following theorems.
Theorem 6 Let P 2 PCT and W 2 WCT . If (W (! )P (j! )) < 1
8! 2 Re ;
(34)
then I (P; BW ) is robustly stable and thus so are I (P; BW R) and I (P; BW RR). Moreover, suppose that W is bounded. Then, if
sup (W (!)P (j!)) < 1;
(35)
!2Re
then I (P; BW ) is uniformly robustly stable and thus so are I (P; BW R) and I (P; BW RR).
Note that whenever nr = 0, Re in (35) can equivalently be replaced by R (even if W is discontinuous at 1: use Lemma 9). In parallel with the discrete-time case, we will say that W is H1(C + ){irreducible if every wi , i = 1; : : : ; (S + F ) is lower semicontinuous on Re , and for every wi , i = 1; : : : ; (S + F ) that is not identically zero, the corresponding uncertainty sub-ball contains a nonzero element, i.e., there exists some nonzero c;i 2 H1(C + ), continuous on C + [ f1g, such that c;i (j!)c;i (j!) wi(!)2I 8! 2 Re :
(36)
We will say W is real H1(C + ){irreducible if it is H1(C + ){irreducible, and W is even. We will say W is real-rational H1(C + ){irreducible if it is real H1(C + ){irreducible, and for every wi that is not identically zero, there exists a nonzero real-rational c;i such that (36) holds.
Remark: Similarly to the discrete-time case, when the wi s are lower semicontinuous on Re , H1(C + )-irreducibility amounts to a certain log-integrability condition on those wis that are not identically zero. Speci cally, in the continuous-time case, min 0; (log w (!))=(1 + !2 ) must be i
integrable on the real line. (To see this, make use of a bilinear transformation between D c and C + .)
23
Theorem 7 Let P 2 PCT and W 2 WCT . Suppose that sup (W (!)P (j!)) 1:
!2Re
(37)
Then, if W is H1(C + ){irreducible (respectively, real H1(C + ){irreducible and real-rational H1(C + ){irreducible), then I (P; BW ) (respectively, I (P; BW R) and I (P; BW RR)) is not uniformly robustly stable.
Note that the theorem remains true (but is generally weaker) if, in (37), Re is replaced by R. Again, when nr = 0, such a change leaves (37) una ected. Theorems 6 and 7 yield the following necessary and sucient condition, which also holds with Re replaced with R whenever nr = 0.
Theorem 8 Let P 2 PCT, let W 2 WCT , bounded, and suppose W is H1(C + ){irreducible (respectively, real H1(C + ){irreducible, real-rational H1(C + ){irreducible). Then, I (P; BW ) (respectively, I (P; BW R), I (P; BW RR)) is uniformly robustly stable if and only if sup (W (!)P (j!)) < 1:
!2Re
(38)
Corollary 3 Given any P 2 PCT , the following are equivalent: 1. I (P; B) is uniformly robustly stable; 2. I (P; BR) is uniformly robustly stable; 3. I (P; BRR) is uniformly robustly stable; 4. sup!2Re (P (j!)) < 1.
A fth equivalent statement is
sup (P (s)) < 1:
s2C + [f1g
Finally, as in the discrete-time case, we give a necessary and sucient condition for robust stability that holds without boundedness assumption on W . We de ne
B W = f : 2 BD for some < 1g; and B W R and B W RR similarly. 24
Theorem 9 Let P 2 PCT, let W 2 WCT, and suppose W is H1(C + ){irreducible (respectively, real H1(C + ){irreducible, real-rational H1(C + ){irreducible). Then, I (P; B W ) (respectively, I (P; B W R), I (P; B W RR)) is robustly stable if and only if (W (j! )P (j! )) 1
8! 2 Re :
(39)
Corollary 4 [TF95] Given any P 2 PCT, the following are equivalent: ) is robustly stable; 1. I (P; B ) is robustly stable; 2. I (P; BR 3. I (P; BRR ) is robustly stable; 4. (P (j!)) 1 8! 2 Re .
A fth equivalent statement is (P (s)) 1
8s 2 C + [ f1g:
6 Concluding remarks Both necessary conditions and sucient conditions have been obtained for (uniform) robust stability in the presence of structured uncertainty with frequency-dependent bounds. Two necessary and sucient conditions were also obtained. For the rst one (uniform robust stability) the frequency bound is required to be bounded, and it was shown that the result does not hold without such condition. The second one (robust stability over an \open" ball) holds with essentially no assumption on the uncertainty ball: indeed, irreducibility is a property not of the uncertainty set, but of its representation. Thus our goal of providing small- theorems that hold under \weak" assumptions is achieved.
Acknowledgment. The authors wish to thank Carlos Berenstein, as well as Dan Luecking
and other contributors to sci.math.research, for their valuable help in connection with Lemmas 6 and 7.
25
References [Doy82]
J. Doyle. Analysis of feedback systems with structured uncertainties. IEE Proc., 129-D(6):242{250, November 1982.
[Gar81]
J. B. Garnett. Bounded Analytic Functions. Academic Press, 1981.
[KGP87] P. P. Khargonekar, T. T. Georgiou, and A. M. Pascoal. On the robust stabilization of time-invariant plants with unstructured uncertainty. IEEE Trans. Aut. Control, AC-32:201{207, 1987. [Koo80]
P. Koosis. Introduction to Hp spaces. Cambridge University Press, 1980.
[Lev96]
W. S. Levine, editor. The Control Handbook. CRC Press, Boca Raton, Florida, 1996.
[PD93]
A. Packard and J. Doyle. The complex structured singular value. Automatica, 29(1):71{109, 1993.
[PP93]
A. Packard and P. Pandey. Continuity properties of the real/complex structured singular value. IEEE Trans. Aut. Control, AC-38:415{428, 1993.
[Rud74]
W. Rudin. Real and Complex Analysis. McGraw-Hill, New York, 1974.
[Saf82]
M. G. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc., 129-D:251{256, November 1982.
[TF95]
A.L. Tits and M.K.H. Fan. On the Small- Theorem. Automatica, 31(8):1199{ 1201, 1995.
[Tit95]
A.L. Tits. The small theorem without real-rational assumption. In Proceedings of the American Control Conference, pages 2870{2872, 1995.
[wJCD98] K. Zhou with J. C. Doyle. Essentials of Robust Control. Prentice Hall, New Jersey, 1998. [wJDG96] K. Zhou with J. Doyle and K. Glover. Robust and Optimal Control. Prentice Hall, 1996.
26
Appendix A: Some results from complex analysis Lemma 6 Let
w : @ D ! R+ be a continuously di erentiable function, with log w being integrable over @ D . De ne the function : D c ! C by
Z 2 + e?j d: 1 log w(e?j ) zz ? (z) = 2 e?j 0
Let (z) = e
(z ) .
(40)
Then:
2 H1(D c ), and admits a continuous extension on D c . j (rej )j converges to w(ej ) as r # 1. Moreover:
If w is conjugate-symmetric, i.e., w(z ) = w(z) on @ D , then is real on the real axis. If w is bounded away from zero over @ D , ?1 2 H1(D c ).
Proof:
This lemma represents an application of the standard Poisson integral formula (see for example, [Gar81, Ch. I, x3]) to construct a function , analytic in D c , such that Re (rej ) ! log w as r # 1. Continuity of (and consequently ) over D c results from the fact that Im (rej ) ! w~ as r # 1, where w~ : @ D ! R is a continuous function (this is guaranteed by the continuity of the derivative of w; see for example, [Koo80, pp. 25{26]). Next, if w is conjugate-symmetric, it is easily veri ed that (z) = (z), whenever z = z , so that , and hence are real-valued on the real axis. Finally, when w is bounded away from zero over @ D , ?1 is analytic in D c , and bounded over D c , so that ?1 2 H1(D c ). We next state a \nontriviality" condition for the set of uncertainties.
Lemma 7 Suppose that w : @ D ! R+ is lower semicontinuous. Let
8 9 c < extension on D = Bw = : 2 H1(D c ) : has a continuous (z) (z) w2I for all z 2 @ D ; ;
and
Bw R = f 2 Bw : is real on the real axisg :
Then the following conditions are equivalent: 1. Bw contains an element besides 0.
27
2. Bw R contains an element besides 0. 3. min(0; log w) is integrable over @ D . Moreover, when any of these conditions hold, there exists an element of the form I in Bw R, with : C ! C , satisfying for z 2 @ D , j (z)j > 0 whenever w(z) > 0.
Proof: Clearly, (2) implies (1). We now show that (1) implies (3). Suppose some nonzero 2 Bw . Let be some nonzero entry of . Then, log jj is integrable over @ D (see for
example, [Gar81, Ch. II, Thm 4.1]). Since w is measurable (being lower semicontinuous) and since w(z) j(z)j for z 2 @ D , min(0; log w) is integrable over @ D as well. Next, we show that (3) implies (2). Suppose that min(0; log w) is integrable over @ D . De ne w^ by w^(z) = min(w(z); w(z )). Then, min(1; w^) is lower semicontinuous and logintegrable over @ D , so that it is is a limit from below of continuous functions which are logintegrable over @ D . Let g : @ D ! R+ be a continuous function, that is conjugate-symmetric, bounded above by min(1; w^), with g being log-integrable. Then g can be uniformly approximated from below by a continuously di erentiable log-integrable function h : @ D ! R+ , also conjugate-symmetric. Then, a function 2 H1(D c ) can be constructed using Lemma 6 with a continuous extension over D c , real on the real axis, and with j (z)j = h(z) for all z 2 @ D . Clearly, I 2 Bw R. Finally, for every z such that w(z) > 0, the functions g and h in the above construction can be chosen to satisfy g(z) > 0 and h(z) > 0 respectively, and the last statement of the lemma follows.
Appendix B. Proof of Lemma 2 The following two lemmas will be used. The rst lemma is a direct consequence of a result of Packard and Pandey [PP93, Lemma 5.1]. It generalizes Lemma 1 of [TF95].
Lemma 8 Let M 2 C nn be such that (M ) > 0 and is discontinuous at M . Then there exists a real matrix ? 2 ? such that (?) = 1=(M ) and det(I + ?M ) = 0. Proof: Let Mrr 2 C nr nr be the top left submatrix of M . From Lemma 5.1 in [PP93], it
follows that discontinuity of at M implies that ?r (Mrr) = (M ), where ?r (Mrr) denotes the structured singular value of Mrr with respect to block structure ?r. Since (M ) > 0, 28
there exists ?r 2 ?r with (?r) = 1=(M ) such that det(I +?rMrr) = 0. It is readily checked that the matrix ? 2 ? de ned by ? = diagf?r; 0nc g satis es the required properties.
Lemma 9 Let M 2 C nn and let i 1, i = 1; : : : ; n. Then, given any block structure ?, (diag( i )M ) (M ):
Proof: If (M ) = 0, the result holds trivially. Thus, suppose (M ) > 0, and let ? 2 ?, with (?) = 1=(M ) satisfy det(I + ?M ) = 0. Let ?0 = ?diag( i)?1 . Since i 1 for all i it follows that (?0) (?). Moreover, det(I + ?0 diag( i)M ) = det(I + ?M ) = 0: The claim follows. Proof of Lemma 2 Let z 0 2 @ D such that (W (z 0 )P (z 0 ))
p3 1 : 1+
There are two cases: either () is continuous at W (z0 )P (z0) or it is discontinuous at that point. First suppose it is discontinuous. In that case, it follows from Lemma 8 that there is p a real ?0 2 ?, with (?0) 3 1 + such that det(I + ?0W (z0 )P (z0)) = 0 i.e., (25) holds with ?^ = 1+1 ?0 2 B?, real, and z^ = z0 . Suppose now that () is continuous at W (z0 )P (z0). If z0 62 f?1; 1g, then the claim follows directly from the de nition of , with z^ = z 0 . Thus suppose that z 0 = 1 (the proof is similar when z 0 = ?1), i.e., 1 : (W (1)P (1)) p3 1+ By continuity, there exists 0 2 (0; ) such that 1 (W (1)P (ej )) 8 2 [0; 0 ); 2 = 3 (1 + ) implying, since is positively homogeneous, that ! 1 1 j 8 2 [0; 0): W (1)P (e ) p3 (1 + ) 1+ 29
Now, since the (diagonal) entries wi of W are lower semicontinuous, there exists ^ 2 (0; 0) such that, for all i, 1 w (1): wi (ej^) p3 1+ i It follows from Lemma 9 that 1 : (W (ej^)P (ej^)) (1 + ) The claim follows readily, with z^ = ej^.
Appendix C. Proof of Lemma 3 Express ? as
? = diag( 1Ir1 ; : : : ; RIr ; 1Is1 : : : : ; S Is ; ?1; : : : ; ?F ); R
S
and let k k denote the Euclidean norm in C n . Let u 2 C n , with kuk = 1, be such that (I + M ?)u = 0: Express u as [^uT ; uT1 ; : : : ; uTF ]T , with u^ 2 C n
r
Ps
+
i
and ui 2 C f , and let i
?0 = diag( 1Ir1 ; : : : ; RIr ; 1Is1 : : : : ; S Is ; ?01; : : : ; ?0F ) R
with, for i = 1; : : : ; F ,
S
8 < ?iuiui =kuik2 if kuik 6= 0; 0 ?i = : 0 otherwise:
Then clearly (?0 ) = (?) and, for i = 1; : : : ; F , rank(?0i) 1. Moreover (I + M ?0 )u = u + M ?u = (I + M ?)u = 0; and thus det(I + M ?0 ) = 0, proving our claim.
Appendix D. Proof of Lemma 5 For i = 1; : : : ; S , let i be an all-pass, stable transfer function de ned by 1 + i z i (z ) = ai z + i 30
where ai 2 [?1; 1] and i 2 (?1; 1] are such that i (^z) = i. It is readily checked that, since z^ 62 f?1; 1g, this is always possible. Next, for i 2 f1; : : : ; F g, let i viwi (i 2 [0; 1]) be the singular value decomposition of ?i and let vij and wij be the j th entries of vi and wi, respectively. Let xi = [x1i ; : : : ; xfi ]T and yi = [yi1; : : : ; yif ]T be real-rational vector functions, with xi analytic and bounded in D c and yi analytic and bounded in the interior of D , de ned by z + ij 1 + ij z ; yij (z ) = bij xji (z ) = aij z+ 1+ z i
i
ij
ij
where aij ; bij 2 [?1; 1], ij ; ij 2 (?1; 1] are such that xji (^z ) = vij ;
yij (^z ) = wij ;
j = 1 : : : ; fi
Let i(z) = i xi(z)yi (1=z)T . Finally, let (z) = diag( 1Ir1 ; : : : ; RIr ; 1(z)Is1 ; : : : ; S (z)Is ; 1(z); : : : ; F (z)): S
R
Clearly, 2 BRR and (^z ) = ?.
31
List of Figures 1 2 3
The P { interconnection modeling an uncertain system. . . . . . . . . . . . 2 ~ is. . . . . 7 The P~ {~ interconnection. I (P; ) is stable if and only if I (P~ ; ) When W is unbounded, uncertainties of arbitrarily large size can leave the system stable while rendering the H1 norm of the transfer function from u to e arbitrarily large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
32
