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Tradeoffs in linear time-varying systems: an analogue of Bode's sensitivity integral
Automatica 37 (2001) 1541}1550
Tradeo!s in linear time-varying systems: an analogue of Bode's sensitivity integral夽 Pablo A. Iglesias* Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA Received 30 September 1999; revised 23 February 2001; received in "nal form 9 March 2001
Bode+s integral characterizes some of the tradeows imposed by the open-loop dynamics of a linear, time-invariant system. Here we present an analogous time-domain characterization applicable to time-varying systems. Abstract A new time-domain interpretation of Bode's integral is presented. This allows for a generalization to the class of time-varying systems which possess an exponential dichotomy. It is shown that the sensitivity function is constrained, on average, by the Lyapunov exponents of the open-loop system which take the place of the magnitude of the open-loop poles. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Sensitivity; Entropy; Time-varying systems; Lyapunov exponents; Discrete-time systems
1. Introduction The open loop dynamics of a system impose limitations on what is achievable with a feedback control system. Mathematical characterizations of some of these constraints for linear time-invariant systems have been known for a long time. One of the earliest is due to Bode (1945), who showed that if the open-loop system ¸(z) is stable, the sensitivity function S(z)"1/(1#¸(z)) must satisfy
L
ln S(e S) d"0.
(1.1)
It should be noted that Bode's original result was for continuous-time systems, whereas we have stated the discrete-time analogue, whose derivation can be found in Chen and Nett (1995), Middleton (1991), Sung and Hara (1988). One signi"cant di!erence between the two settings is that in continuous-time, a two-pole roll-o! in the open-loop transfer function is necessary. The 夽
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor L. Qiu under the direction of Editor Roberto Tempo. * Tel.: #1-410-516-6026; fax: #1-410-516-5566. E-mail address: [email protected] (P. A. Iglesias).
corresponding result for discrete-time systems only requires that the open-loop transfer function be strictlyproper. This integral has very important practical implications. For example, in the tracking problem depicted in Fig. 1, where one seeks to keep the error below (1 for reference signals with frequency content 3[0,], the sensitivity function must satisfy ln S(e S)(ln (0, ∀3[0, ]. Since the integral of ln S(e S) is zero, it must follow that ln S(e S)'0, for some ,(, ). In fact, a quick calculation shows that 0)(!ln ))(!) sup ln S(e S) SZ L )(!) lnS so that
S *exp (!ln ) . ! Thus, a smaller or a larger bandwidth will increase the corresponding values of S(e S) in , [0, ]. Bode's result was generalized for arbitrary open loop multivariable systems by Freudenberg and Looze, who
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 1 0 3 - 0
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Fig. 1. Tracking problem.
showed that if ¸(z) has unstable poles p , then the G integral is now equal to:
L
ln det S(e S) d" (!ln p )'0. (1.2) G The appearance of the unstable poles (i.e. p (1) in the G right-hand side of (1.2) serves only to worsen the bandwidth and magnitude tradeo!s mentioned with regard to (1.1). In this paper, we look at a generalization of Bode's integral to time-varying systems. We will "rst show the connection between Bode's integral and the entropy cost function studied in Mustafa and Glover (1990). For linear time-invariant systems, it was shown that the latter is equivalent to a stochastic risk-sensitive cost function. This helps provide a time-domain interpretation to (1.1). Finally, we use the time-varying analogues of the entropy and risk-sensitive cost functions (Peters & Iglesias, 1999) to extend Bode's result to time-varying systems. The rest of the paper is organized as follows. Section 2 provides some necessary preliminary material. These include de"nitions of exponential dichotomy, and inner/outer factorizations of time-varying systems. A timedomain characterization of Bode's integral is given in Section 3, which is done by relating the integral to two cost functions of robust control. The time-domain characterization allows for a generalization to time-varying systems. In Section 4, we show how this generalization can be expressed in terms of the unstable dynamics of the open-loop system. Finally, some concluding remarks are provided in Section 5.
2. Preliminaries The notation used throughout is more or less standard. For a matrix X3"K"L, X denotes the Hermitian transpose. For a matrix function G(z) we use the paraHermitian conjugate: G%(z)"G(1/z ). For a vector x3" L, x " : (xx denotes the usual Euclidean norm. The singular values of X are denoted by (X) and are ordered as (X)* (X)*2* (X). G P These, of course, are the eigenvalues of XX. The norm on X is denoted by X and is equal to (X)": (X). The smallest singular value is also denoted by (¹) and is known as the minimum modulus of X. Note that (¹)"inf¹x: x"1.
The set of all square-summable sequences from 9P"L is denoted by lL . This is a Hilbert space with inner product x, y lL " : x y and induced norm x . \ I I The embedding operator E : ",>KPlK takes , x , and forms x by padding all indices with zeros I I\, k'N. 2.1. State-space representations In the sequel, we will be considering linear time-varying systems admitting a state-space representation % A B x "A x #B w I I . I> I I I I" " : (2.1) % y "C x #D w C D I I I I I I I
We will assume that all the matrix sequences A , etc. I are bounded. With this system, we associate an operator G mapping the input w to output y. This operator has a block matrix representation G with elements GH D if i"j, G " G GH C A A 2A B if i'j G G\ G\ H> H and zero otherwise. If i"j#1, the product A A H H> should be interpreted as the identity.
2.2. Exponential dichotomies and the spectrum of weighted shifts Let k and l be integers, with '0. We say that A is I (1) Uniformly exponentially stable if there exist positive constants and (1 such that A A 2A ( J; I>J\ I>J\ I (2) Uniformly exponentially antistable if there exist positive constants and '1 such that (A A 2A )' J. I N I>J\ I>J\ In both cases, the constants and are independent of k and l. Throughout this paper, stable and antistable refer to uniformly exponentially stable and antistable. In Bode's result, it is important to di!erentiate between the stable poles of ¸(z)*which do not contribute to the right-hand side of (1.2)*and the unstable poles, which do. Time-varying systems which can be decomposed into stable and antistable components are said to possess an exponential dichotomy (Ben-Artzi & Gohberg, 1995; Co!man & SchaK !er, 1967). De5nition 1. Let A be a sequence of n;n matrices I and let P be a bounded sequence of projections in I 1L such that the rank of P is constant. The sequence I P is a dichotomy for A if the commutativity condiI I tion A P "P A is satis"ed for all k, and there exist I I I> I positive constants and '1 such that A A 2A P x' JP x I>J\ I>J\ I I I
(2.2)
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and 1 A A 2A (I!P )x( (I!P )x I>J\ I>J\ I I I
J
(2.3)
for any x. The existence of an exponential dichotomy allows us to de"ne a stability preserving state space transformation (a Lyapunov transformation) that separates the stable and antistable parts of A . I Lemma 2 (Ben-Artzi & Gohberg, 1991). The sequence A in the realization (2.1) admits an exponential dichotI omy iw there exist a bounded sequence of matrices ¹ I with bounded inverses such that
¹
AQ I A ¹\ ¹ B I> I I I> I " 0 C ¹\ D I I I CQ I
0 BQ I AS BS I I , CS D I I
where AQ is stable and AS is antistable. I I
De"ne the weighted shift operator S in the following manner (S x) "A x . I I\ I\ If (S ) denotes the spectrum of the operator, we can characterize the stability conditions as follows: A is I stable i! for (S )L# and antistable i! (S )L"#. These are consistent with the usual time-invariant requirements on the eigenvalues of the `Aa matrix. It can be shown (Ben-Artzi, Gohberg, & Kaashoek, 1993) that the spectrum of the weighted shift will consist of l concentric annuli: J (S )" 3": ) ) , G G G
(2.4)
where 0) ( ( (2( ( (R and J exists a J Lyapunov 1)l)n. Moreover, there transformation so that ¹ A ¹\"diagA,2, AJ, I> I I where AG31P "P , and G
G
(S G )" 3" : ) ) G G for i"1,2, l. Exponential dichotomy is equivalent to the existence of an integer j31,2, l!1 such that, either (1( or (1. H J H> case There is a particular in which each of the annuli is a circle is said to be regular. In this case, positive real values corresponding to the spectrum of S are the dis crete-time Lyapunov exponents. Regularity is hard to
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verify for any particular system, though all time-invariant and periodic systems are regular. In these two cases, the spectral values are the magnitude of the eigenvalues and Floquet characteristic exponents of the system A matrix. For systems involving a #ow with an invariant probability measure, Oseledec\ 's multiplicative ergodic theory states that regularity occurs with probability one; see Arnold (1998). The weighted shifts considered here have received considerable interest recently as a means of extending some classical linear time-invariant system theory results to time-varying systems. For example, a generalization of the PBH test for reachability/stabilizability was introduced in Peters and Iglesias (1999). It is shown that reachability is lost i! there is an almost eigenvector (x 3l ), and eigenvalue ( 3(S )) pair for which L xHS P0. As in the time-invariant case, stabilizability L allows for pairs for which (1. Related results are found in Kamen, Khargonekar, and Poolla (1985) who looked at asymptotic stability, Dullerud and Lall (1999), Iglesias and Peters (1997); Peters and Iglesias (1997) who looked at inducednorm optimal control problems, and (Iglesias, 1999), where it is shown that the `invariant-zero on the unit circlea condition which guarantees the existence of stabilizing solutions to discrete-time algebraic Riccati equations has a natural analogue for Riccati di!erence equations that involves the spectrum of a weighted shift. 2.3. Assumptions on plant We consider the system in Fig. 1, in which the openloop system has a stabilizable and detectable state* space representation
A B " I I . (2.5) * C 0 I It should be noted that this system will include both the controller and plant subsystems. Using (2.5), we have a corresponding state-space representation for the sensitivity operator:
A !B C B I I I I . (2.6) !C I I We will need the following two assumptions. First of all, we would want to ensure that the sensitivity operator S is bounded. This would, of course, be a necessary requirement of any internally stabilizing controller. " 1
Assumption 3. A !B C is stable. I I I Since the sequences A , B and C are bounded, I I I Assumption 3 guarantees that S is bounded.
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We will also need to di!erentiate between the stable and antistable dynamics of . To this purpose, we * assume: Assumption 4. The sequence A admits a rank r I dichotomy with constants and . We should note that the state space matrix sequences A etc. include components from both the open-loop I plant and controller. Note that the `Aa matrix can be represented as
A. 夹 A " I , I 0 A) I where A. and A) are the sequences in the realizations I I of the plant and controller, respectively, and `夹a denotes elements not needed in the analysis. It follows that the spectrum of S is the union of that of S . and S ) and thus contains the unstable portions of both plant and controller dynamics. Remark 5. While Assumption 3 is clearly necessary, it is worth noting that the second assumption is not required in the time-invariant case, where Bode's result allows for open-loop systems having poles on the unit circle. 2.4. Inner/outer factorizations As a step in obtaining Bode's result we will need to compute an inner/outer factorization of the operator S. The existance of such a factorization is a result of Arveson who introduced the concepts of inner and outer operators in nest algebras (Arveson, 1975). In the setting of this paper, we say that F is an inner operator if F is bounded, its matrix representation is lower blocktriangular, and FHF"I. The operator G is outer if it is bounded, lower block-triangular, and its inverse is also bounded and lower block-triangular. For the sensitivity operator arising from (2.6), we can provide state-space representations of its inner and outer factors. Not surprisingly, Riccati di!erence equations play a prominent role. Lemma 6. Consider the Riccati diwerence equation: X "A X (I#B B X )\A . (2.7) I I I> I I I> I There exists a solution to (2.7) with bounded positivesemidexnite X , and stable (I#B B X )\A . MoreI I I I> I over, this solution is unique; and: (i) (ii)
If A is stable then X "0 for all k; I I If A is antistable then inf (X )'0. I I I
(iii)
AQ 0 If A " I admits a dichotomy as in Lemma 2 I 0 AS I 0 0 and inf (XS )'0. then, X " I I I 0 XS I
Proof. That this equation has a stabilizing solution, and that it is unique is shown in Halanay and Ionescu (1994, Theorem 3.2). Item (i) is trivial, as is (iii) once (ii) has been proved. The main technical di$culty is in showing (ii); this is done in the appendix. This Riccati equation is now used to compute the inner/outer factorization of S. Formulae for inner/outer factorizations of general state-space time-varying systems can be found in Dewilde and Van Der Veen (1998), Halanay and Ionescu (1994). These formulae can be applied to the sensitivity operator. De"ne the blockdiagonal operator R " : I#B X B . I I I> I Then, S"S S for G M (I#B B X )\A B R\ I I I> I I I G" (2.8) 1 !R\B X R\ I I I> I inner, and M equals 1 A !B C B I I I I (2.9) (I#B B X )\ (B X A !R C ) R I I I> I I> I I I I outer. We will not be concerned with S , but it should be M worth noting that the `Aa matrix of S\ is M A !B C !B R\[BX A !R C ] I> I I I I I I I I " (I#B B X )\A I I I> I which is stable from Lemma 6. Since this matrix sequence is stable, it follows that S\ is bounded. Moreover, S is M M bounded since its `Aa matrix is stable by Assumption 3. Thus, S! is bounded and hence S is outer. M M
3. A time-domain version of Bode's integral In this section, we outline the connections between Bode's integral and two cost functions that are common in robust control. We "rst consider the entropy cost (Mustafa & Glover, 1990). For a stable function G(z) with norm bound G (, the entropy is de"ned as
L E(G,) " : ! ln det(I!\G%(e S)G(e S)) d. 2 \L Suppose that the sensitivity function has norm bound S ( . This allows one to factor I! \S%(z)S(z)"G%(z)G(z). (3.1)
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Moreover,
3.1. The time-varying case
ln det S(e S)" ln det(I!G(e S)G(e S))#m ln , where m is the matrix dimension of S. The sensitivity integral can be written as
1 2
L
ln det S(e S) d"m ln !E(G,1). \L This shows that Bode's integral can be computed by means of an entropy computation. In Glover and Doyle (1988), it was shown that the entropy cost function is equivalent to the in"nite horizon risk-sensitive cost function. We can use this connection to provide the time-domain characterization of Bode's integral that we seek. Suppose that the input r is white, Gaussian stochastic I process with unit covariance, and that e"Sr. A risksensitive cost function can be de"ned as
,\ (r r !e e ) , I I I I I where E denotes an expectation. We can show, as in Glover and Doyle (1988), that for stable linear timeinvariant discrete-time system S(z) 2 1 ¸(G,, N) " : ln E exp N 2
1 2
L
\L
ln det S(e S) d
1 1 ,\ ln E exp (3.2) " ! lim (r r !e e ) . I I I I 2N 2 ,t I This expression serves as a time-domain characterization of Bode's integral. Remark 7. This characterization of Bode's integral, shows how the frequency domain tradeo!s translate into the time-domain. In particular, if we divide the time line 0,2, N into two regions 0,2, M and M#1,2, N for 0(M(N!1, then the argument of the exponent in (3.2)
1 , (r r !e e ) I I I I 2 I 1 + 1 , (r r !e e ) exp (r r !e e ) . "exp I I I I I I I I 2 2 I I+> Now, if Bode's integral is zero, this means that the terms in (3.2) in the summation with time index k'M cannot decrease without a corresponding increase in those in the horizon k"0,2, M. Under the usual connection between the time and frequency domains: low (high) frequency signals are associated with long (short) time horizons; this new timedomain interpretation agrees with the usual frequency domain version discussed in the introduction. exp
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The relationship between the entropy and risk-sensitive cost functions analogous to that obtained in Glover and Doyle (1988) exists for time-varying systems (Peters & Iglesias, 1999). We now use this connection to extend the time-domain characterization of Bode's integral to time-varying systems. Suppose that the input signal r is a windowed ,
white noise sequence de"ned over the interval k"0,$1,2,$N and let e be the corresponding ,
tracking error. Unlike the time-invariant case, where every signal gives rise to the same cost, in the timevarying systems this set of signals will give rise to a sequence of costs. In the limit, we can consider an average cost function 1 lim sup ! ln E exp((r !e )) (3.3) , , 2(2N#1) , and treat this as the time-varying analogue to Bode's integral. Remark 8. It should be stressed that even though the input sequence r in (3.3) is de"ned over a "nite horizon, ,
this is not true of the tracking error e . Since S is causal, ,
this is de"ned for k*!N. In Section 4.1, we look at the case where the tracking error is only considered over a "nite time. Since r is a Gaussian white-noise sequence, de"ned over a "nite horizon, we can use a standard argument (see, e.g. Glover and Doyle (1988)) to show that (3.3) equals ! ln E exp((r !e )) , , " ln det(SE )H(SE ) (3.4) , , (3.5) " trace ln(EH SHSE ). , , In this latter form of the equation, the matrix logarithm is well de"ned since EH SHSE is a "nite-dimensional , , positive-de"nite matrix (Kenney & Laub, 1998). It will now be shown that this value, as in the timeinvariant case, depends only on the unstable dynamics of the open-loop plant. 4. Constraint on sensitivity due to unstable poles Lemma 9. Suppose that the open loop system satisxes Assumption 3, and that the corresponding sensitivity operator S admits the inner/outer factorization (2.8), (2.9). Then 1 lim sup ln det(EH SHSE ). , , 2N#1 , 1 , "lim sup ln det R I 2N#1 , I\,
(4.1)
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Proof. See the appendix. In the previous result, we showed that the average contribution of the logarithm of the sensitivity operator is tied to the logarithm of R . We now wish to relate this I to the dynamics of the open loop system L. Theorem 10. Suppose that the open loop system satisxes Assumptions 3 and 4. The sensitivity operator S must satisfy 1 lim sup ln det(EH SHSE )*2r ln . , , 2N#1 ,
(4.2)
Proof. See the appendix. Remark 11. Unlike (1.2) where an equality is obtained, Theorem 10 only provides a lower bound on the gain of the sensitivity operator. This lower bound comes from the fact that we only know the smallest rate of exponential growth of the antistable part of ¸. For time-invariant systems, this would be equivalent to replacing every unstable pole in (1.2) with one of a smallest magnitude. We can provide a tighter bound if we know something more about the unstable part of the spectrum of S . For example, if we split the unstable component of A as in I (2.4), then we can replace in (4.2) 1 J lim sup ln det(EH SHSE )*2 r ln , , G 2N#1 G , G *2r ln . Moreover, as shown in Iglesias (2001), if the unstable component is regular, equality is achieved. In this case, the lim sup can be replaced with a limit: J 1 ln det(EH SHSE )"2 r ln , lim , , G G 2N#1 G , where " : " . G G G A second special case where equality is obtained is where the open-loop dynamics are stable, in which case: Corollary 12. Under the assumptions of Theorem 10 and the additional assumption that A is stable, it follows I that 1 lim sup ln det(EH SHSE )"0. , , 2N#1 , Proof. This is a straightforward application of Lemma 9, the de"nition of R and noting that, from Lemma 6, I X "0 for all k. I Remark 13. While we have motivated our results in terms of an analogue of Bode's sensitivity integral, there is an alternative equally valid approach. Let x , be I I
a sequence of random variables and let ¹ be its asso, ciated correlation matrix. If x is stationary, this matrix is I both Toeplitz and Hermitian. It is well known that
L
ln S(e S) d" lim ,> , (4.3) \L , , where S(e S) is the power spectral density, and "det ¹ is the correlation determinant of the process , , x (Papoulis, 1984, p. 478). Moreover, as originally shown by Szego , the limit in (4.3) exists and can be expressed in terms of the poles of the function S(z) (Gohberg, Kaashoek, & van Schagen, 1987). The results presented here are the analogues of these results for non-stationary random variables; see Iglesias (2001). 1 2
4.1. Finite-horizon signals Time-varying systems are usually considered only for "nite-horizon applications. It is worth asking how the results obtained here apply in this case. As was mentioned in the earlier remark, the results obtained so far assumed that the input signal was of "nite duration, but the corresponding error signal was allowed to extend for all time. This means that the operator SE , in (3.4) is in"nitely tall, but has a "nite number of columns. When multiplied by its adjoint, the corresponding matrix is of "nite dimensions. Considering an error signal over a "nite horizon is equivalent to adding a projection operator to SE to form the "nite matrix , S " : EH SE , , , This is a lower block-triangular matrix with identities on the blocks on the diagonal. Hence, its determinant is always one. This means that the time-varying analogue of Bode's integral: ! ln E exp((r !e )) , , " ln det(EH SE )H(EH SE ) , , , , "0 and this is true regardless of the dynamics of L. This last result should not be surprising. Bode's sensitivity integral is always zero unless the loop has unstable dynamics. In "nite-horizon applications, however, every dynamic can be considered as stable as the corresponding signals always remain bounded. Thus, the fact that the analogue of Bode's integral is always zero for systems over a "nite horizon is completely consistent with Bode's result. 4.2. An example We consider a simple example that may help to illustrate some of the results of this paper. We consider
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Fig. 2. Convergence of measure towards a theoretical value. For the size of the tall matrix, a value of n"60 was used.
puting a `talla matrix SI "¹I , where i"!n,2, n G H and j"!N,2, N, for n
a time-varying plant:
1 " I , . 1 0
where " : (2k#1)/(k#1) and the time-invariant I state-feedback controller K"2. State-space representations for the open-loop and sensitivity systems are
1 , " I * 2 0
" 1
!(k#1)\ 1
.
!2
1
The closed-loop system is stable. Let 0(;1, "2!, NM " : W (1!)/ X, and "\,M >(1. It follows that * for k'NM , and 2 * J I I>J I> for any k and l, so that the open-loop dynamics are antistable. Moreover, as the system is one-dimensional, and the weighted shift is clearly invertible, the spectrum of the weighted shift is shown in Shields (1974, Theorem 5) to be the annulus: (S )"z3": (S \ )\)z)(S ), where ( ) ) denotes the spectral radius. This can be computed as in [Halmos (1967), Problem 77]:
I I\ (S )"lim sup "2. L>G I L G Similarly, (S \ )"1/2 and thus the spectrum is the circle of radius 2. We now approximate the successive values of ln det(¹ ), where ¹ " : EH SHSE . This is done by com, , , ,
5. Conclusions By means of the connection between Bode's integral and the entropy cost function, we have provided a timedomain characterization of Bode's sensitivity integral. The traditional frequency domain interpretation is that, if the sensitivity of a closed-loop system is decreased over a particular frequency range*typically the low frequencies*the designer `paysa for this in increased sensitivity outside this frequency range. This interpretation is also valid for the characterization presented here provided one deals with time horizons rather than frequency ranges. Having reinterpreted Bode's sensitivity integral in the time domain, it is possible to consider extensions to systems not admitting frequency domain representations, including linear time-varying systems. In the generalization presented here, the poles of the open loop system are replaced by the Lyapunov exponents of the timevarying di!erence equation. Bode's integral is but one of several known mathematical characterizations of the analytic constraints found in feedback control systems (Chen, 1995, 1998; Freudenberg & Looze, 1985). The corresponding counterparts for time-varying systems are unknown but will be the subject of future research.
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Acknowledgements
Appendix A. Proofs of results
Moreover, from a recursive argument using (A.1), and the fact that X *0 we have that x X x (, and z ( I I I I I for k*0 where z " : B X x . I I I> I> Using z , (A.1) can be rewritten as I x "A x !B z , k*0. I> I I I I In particular, if k"p !1, we have:
A.1. Proof of Lemma 6(ii)
x "A x !B z N N\ N\ N\ N\
We begin with some preliminaries. Since A is antiI stable, there exist constants and with '1 such S S S that
N \ "A 2A x !B z ! A 2A B z . N\ N\ J> J J J Considering the norm of both sides, yields the bound
This work was supported in part by the NSF under Grant ECS-9800057.
A A 2A x' J x I>J\ I>J\ I S S for all x, and all k and l39. Note that this implies that sup A '1, (∀k). I I Similarly, since the solution X is stabilizing, it follows I that the sequence with " : (I#B B X )\A I I I I I> I is stable, i.e., there exist positive constants and with Q Q (1 such that Q 2 x) J x (A.1) I>J\ I>J\ I Q Q for all x, and all k"0,$1,2 and l"1,2,2 . We denote by p the smallest positive integer such that S
NS '2. Similarly, we denote by p the smallest positive S S Q integer such that NQ (1/2. Since (1 and '1 we Q Q Q S know that such integers must exist. Designate p "max(p , p ). Q S To prove the assertion we will use a contradiction argument. Since we know that X *0 for all k, the I statement can only fail if, given an arbitrary , there exists an index k (we will assume k "0 without the loss of generality) and a unit vector x such that 'x X x . We choose an such that
sup A N !1 I I (( sup B . I sup A !1 I I I As mentioned above, sup A '1 so that the inverse I I exists. Moreover, the right-hand side is a positive number. The Riccati di!erence equation can be written as: X " X # X B B X . I I I> I I I> I I I> I Pre- and post-multiplying the k"0 term of this equation by x and x , respectively, yield x X x "x X x #x X B B X x (. Now, de"ne the sequence x " x , I> I I From (A.1), x ) I. I Q Q
k*0.
x *A 2A x !B z N\ N N\ ! A 2A B z J> J J N\ J * N !B z S S N \ ! A 2A B z N\ J> J J J N \ *2! sup B sup A J I I J I I *1.
(A.2) (A.3) (A.4)
In (A.2), we have used the fact that A is antistable and I that x is a unit vector, whereas in (A.3) we have used the norm bound on z and the fact that p was chosen so that I
N '2. Finally, in (A.4), we used the norm bound on . S S However, from (A.1) with k"p we have x ) N (1/2 Q Q N which contradicts (A.4). 䊐 A.2. Proof of Lemma 9 Let ¹ "(SE )H(SE )"(S E )H(S E ). We partition , , , M , M , S according to E : M , 夹 0 0
S " 夹 M 夹
; 0 . (A.5) , V 夹 , Here, ; is a (2N#1)m;(2N#1)m matrix (recall that , m is the dimension of the reference signal), given by ; "(S ) for !N)i, j)N. The other elements of , M G H (A.5) are all in"nite-dimensional operators. According to this partition, trace ln ¹ "ln det ¹ , , "ln det(; ; #VH V ) , , , , "2 ln det ; , # ln det(I#(; )\VH V ;\). , , , ,
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Note that the matrix ; is invertible since it is "nite, dimensional, block-triangular, and all the elements along the diagonal are invertible. In fact, they are R and so: I , 2 ln det ; " ln det R . , I I\, It remains to be seen that 1 , lim ln det (I#(; )\VH V ;\)"0. , , , , 2N#1 , I\, (A.6)
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A.3. Proof of Theorem 10 From Lemma 4 we need to compute ln det R . From I the form of the solution of the Riccati equation, this equals R "I#(BS )XS BS . I I I> I Moreover, we have:
Since VH V is a "nite-dimensional, positive semi-de"nite , , matrix, it has a Cholesky factorization ¹¹. Using this factorization, and the fact that
XS "(AS )XS (I#BS (BS )XS )\AS . (A.8) I I I> I I I> I As we are only concerned with the unstable components of the realization, it will be convenient to drop the superscript `ua for the rest of the proof. From the logarithm of the determinant of (A.8), one obtains
; ; *diag(; ; )*I , , , , we can evaluate
ln det R "2 ln det A #ln det X !ln det X I I I> I from which
ln det (I#;\VH V ; )"ln det (I#¹(; ; )\¹) , , , , , , )ln det (I#¹¹)
, ln det R "ln det(A A 2A ) I , ,\ \, I\, #ln det X !ln det X . (A.9) ,> \, The "rst term in the right-hand side can be written as
"ln det (I#VH V ). (A.7) , , Denote by ,2, the (2N#1)m non-negative ,>K eigenvalues of VH V . , , ,>K ln det(I#VH V )" ln(1# ) , , H H ,>K ) (since '0) H H H , " trace(VH V ) . , , H H H\, We now need to show that this trace is bounded. Because S admits a state-space representation in which the `Aa M matrix is stable, there exist constants *0 and 0)(1 such that (S ) ) I\J. Now, for M I J j"!N,2, N (VH V ) " (S ) (S ) . , , H H M G H M G H GH> Using this, we obtain trace(VH V ) )trace G\HI , , H H GH> m ,>\H " . 1! Therefore, 1 , 0) lim trace(VH V ) , , H H 2N#1 , H\, m (1!,>) "0 ) lim (2N#1)(1!) , as required. 䊐
ln det(A A 2A ) , ,\ \, P "2 ln (A A 2A ). G , ,\ \, G Since (A A 2A )* (A A 2A ) G , ,\ \, P , ,\ \, " (A A 2A ) \, , ,\ * ,> for i"1,2, r; it follows that ln det(A A 2A )*2r ln , ,\ \, P *2r ln #2r(2N#1) ln . Because X is bounded above and below I 1 lim sup (ln det X !ln det X )"0. ,> \, 2N#1 , Similarly, lim sup (2r )/(2N#1)"0. Thus, in (A.9) we , are left with 1 , lim sup ln det R *2r ln I 2N#1 , I\, as claimed.
References Arnold, L. (1998). Random dynamical systems. Berlin: Springer. Arveson, W. (1975). Interpolation problems in nest algebras. Journal of Functional Analysis, 20(3), 208}233.
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Ben-Artzi, A., & Gohberg, I. (1991). Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts. Integral Equations Operator Theory, 14(5), 613}677. Ben-Artzi, A., & Gohberg, I. (1995). Inertia theorems for operator pencils and applications. Integral Equations Operator Theory, 21(3), 270}318. Ben-Artzi, A., Gohberg, I., & Kaashoek, M. A. (1993). A time-varying generalization of the canonical factorization theorem for Toeplitz operators. Indagations Mathematicae (N.S.), 4(4), 385}405. Bode, H. W. (1945). Network analysis and feedback amplixer design. Princeton, NJ: D. Van Nostrand. Chen, J. (1995). Sensitivity integral relations and design trade-o!s in linear multivariable feedback systems. IEEE Transactions on Automatic Control, 40(10), 1700}1716. Chen, J. (1998). Multivariable gain-phase and sensitivity integral relations and design tradeo!s. IEEE Transactions on Automatic Control, 43(3), 373}385. Chen, J., & Nett, C. N. (1995). Sensitivity integrals for multivariable discrete-time systems. Automatica, 31(8), 1113}1124. Co!man, C. V., & SchaK !er, J. J. (1967). Dichotomies for linear di!erence equations. Mathematische Annalen, 172, 139}166. Dewilde, P., & Van Der Veen, A.-J. (1998). Time-varying systems and computations. Boston, MA: Kluwer. Dullerud, G. E., & Lall, S. (1999). A new approach for analysis and synthesis of time-varying systems. IEEE Transactions on Automatic Control, 44(8), 1486}1497. Freudenberg, J. S., & Looze, D. P. (1985). Right half plane poles and zeros and design tradeo!s in feedback systems. IEEE Transactions on Automatic Control, 30(6), 555}565. Glover, K., & Doyle, J. C. (1988). State-space formulae for all stabilizing controllers that satisfy an H -norm bound and relations to risk sensitivity. Systems Control Letters, 11(3), 167}172. Gohberg, I., Kaashoek, M. A., & van Schagen, F. (1987). Szego } Kac}Achiezer formulas in terms of realizations of the symbol. Journal of Functional Analysis, 74(1), 24}51. Halanay, A., & Ionescu, V. (1994). Time-varying discrete linear systems. Basel: BirkhaK user-Verlag. Halmos, P. R. (1967). A Hilbert space problem book. Princeton: D. Van Nostrand Co. Iglesias, P. A. (1999). On the Riccati di!erence equation of optimal control. Linear and Multilinear Algebra, 46(4), 313}326. Iglesias, P. A. (2001). A time-varying analogue of Jensen's formula. Integral Equations Operator Theory, to appear. Iglesias, P. A., & Peters, M. A. (1997). On the induced norms of discrete-time and hybrid time-varying systems. International Journal of Robust Nonlinear Control, 7(9), 811}833.
Kamen, E. W., Khargonekar, P. P., & Poolla, K. R. (1985). A transferfunction approach to linear time-varying discrete-time systems. SIAM Journal on Control and Optimization, 23(4), 550}565. Kenney, C. S., & Laub, A. J. (1998). A Schur}FreH chet algorithm for computing the logarithm and exponential of a matrix. SIAM Journal of Matrix Analysis and Applications, 19(3), 640}663. Middleton, R. H. (1991). Trade-o!s in linear control system design. Automatica, 27(2), 281}292. Mustafa, D., & Glover, K. (1990). Minimum entropy H control. Berlin: Springer. Papoulis, A. (1984). Probability, random variables, and stochastic processes (2nd ed.). New York: McGraw-Hill. Peters, M. A., & Iglesias, P. A. (1997). Minimum entropy control for discrete-time time-varying systems. Automatica Journal of IFAC, 33(4), 591}605. Peters, M. A., & Iglesias, P. A. (1999). The relationship between minimum entropy control and risk-sensitive control for timevarying systems. IEEE Transactions and Automatic Control, 44(5), 1065}1069. Shields, A. L. (1974). Weighted shift operators and analytic function theory. In C. Pearcy (Ed.), Topics in operator theory, Mathematical Surveys, Vol. 13 (pp. 49}128). Providence, RI: American Mathematical Society. Sung, H.-K., & Hara, S. (1988). Properties of sensitivity and complementary sensitivity functions in single-input single-output digital control systems. International Journal of Control, 48(6), 2429}2439.
Pablo A. Iglesias was born in Caracas, Venezuela in 1964. He received the B.A.Sc. degree in Engineering Science from the University of Toronto in 1987, and the Ph.D. degree in Control Engineering from Cambridge University in 1991. Since then he has been on the faculty of the Department of Electrical and Computer Engineering at the Johns Hopkins University, where he currently holds the rank of Professor. He is also a member of the Center for Computational Biology and Medicine. He has had visiting appointments at Lund University. The Weizmann Institute of Science, and the California Institute of Technology. He is also the coauthor of the monograph Minimum Entropy Control for Time-Varying Systems. His current research interests include the control of time-varying systems, and robust control theory, and the use of control theory to study biological signal transduction pathways.
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Tradeo!s in linear time-varying systems: an analogue of Bode's sensitivity integral夽 Pablo A. Iglesias* Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA Received 30 September 1999; revised 23 February 2001; received in "nal form 9 March 2001
Bode+s integral characterizes some of the tradeows imposed by the open-loop dynamics of a linear, time-invariant system. Here we present an analogous time-domain characterization applicable to time-varying systems. Abstract A new time-domain interpretation of Bode's integral is presented. This allows for a generalization to the class of time-varying systems which possess an exponential dichotomy. It is shown that the sensitivity function is constrained, on average, by the Lyapunov exponents of the open-loop system which take the place of the magnitude of the open-loop poles. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Sensitivity; Entropy; Time-varying systems; Lyapunov exponents; Discrete-time systems
1. Introduction The open loop dynamics of a system impose limitations on what is achievable with a feedback control system. Mathematical characterizations of some of these constraints for linear time-invariant systems have been known for a long time. One of the earliest is due to Bode (1945), who showed that if the open-loop system ¸(z) is stable, the sensitivity function S(z)"1/(1#¸(z)) must satisfy
L
ln S(e S) d"0.
(1.1)
It should be noted that Bode's original result was for continuous-time systems, whereas we have stated the discrete-time analogue, whose derivation can be found in Chen and Nett (1995), Middleton (1991), Sung and Hara (1988). One signi"cant di!erence between the two settings is that in continuous-time, a two-pole roll-o! in the open-loop transfer function is necessary. The 夽
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor L. Qiu under the direction of Editor Roberto Tempo. * Tel.: #1-410-516-6026; fax: #1-410-516-5566. E-mail address: [email protected] (P. A. Iglesias).
corresponding result for discrete-time systems only requires that the open-loop transfer function be strictlyproper. This integral has very important practical implications. For example, in the tracking problem depicted in Fig. 1, where one seeks to keep the error below (1 for reference signals with frequency content 3[0,], the sensitivity function must satisfy ln S(e S)(ln (0, ∀3[0, ]. Since the integral of ln S(e S) is zero, it must follow that ln S(e S)'0, for some ,(, ). In fact, a quick calculation shows that 0)(!ln ))(!) sup ln S(e S) SZ L )(!) lnS so that
S *exp (!ln ) . ! Thus, a smaller or a larger bandwidth will increase the corresponding values of S(e S) in , [0, ]. Bode's result was generalized for arbitrary open loop multivariable systems by Freudenberg and Looze, who
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 1 0 3 - 0
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Fig. 1. Tracking problem.
showed that if ¸(z) has unstable poles p , then the G integral is now equal to:
L
ln det S(e S) d" (!ln p )'0. (1.2) G The appearance of the unstable poles (i.e. p (1) in the G right-hand side of (1.2) serves only to worsen the bandwidth and magnitude tradeo!s mentioned with regard to (1.1). In this paper, we look at a generalization of Bode's integral to time-varying systems. We will "rst show the connection between Bode's integral and the entropy cost function studied in Mustafa and Glover (1990). For linear time-invariant systems, it was shown that the latter is equivalent to a stochastic risk-sensitive cost function. This helps provide a time-domain interpretation to (1.1). Finally, we use the time-varying analogues of the entropy and risk-sensitive cost functions (Peters & Iglesias, 1999) to extend Bode's result to time-varying systems. The rest of the paper is organized as follows. Section 2 provides some necessary preliminary material. These include de"nitions of exponential dichotomy, and inner/outer factorizations of time-varying systems. A timedomain characterization of Bode's integral is given in Section 3, which is done by relating the integral to two cost functions of robust control. The time-domain characterization allows for a generalization to time-varying systems. In Section 4, we show how this generalization can be expressed in terms of the unstable dynamics of the open-loop system. Finally, some concluding remarks are provided in Section 5.
2. Preliminaries The notation used throughout is more or less standard. For a matrix X3"K"L, X denotes the Hermitian transpose. For a matrix function G(z) we use the paraHermitian conjugate: G%(z)"G(1/z ). For a vector x3" L, x " : (xx denotes the usual Euclidean norm. The singular values of X are denoted by (X) and are ordered as (X)* (X)*2* (X). G P These, of course, are the eigenvalues of XX. The norm on X is denoted by X and is equal to (X)": (X). The smallest singular value is also denoted by (¹) and is known as the minimum modulus of X. Note that (¹)"inf¹x: x"1.
The set of all square-summable sequences from 9P"L is denoted by lL . This is a Hilbert space with inner product x, y lL " : x y and induced norm x . \ I I The embedding operator E : ",>KPlK takes , x , and forms x by padding all indices with zeros I I\, k'N. 2.1. State-space representations In the sequel, we will be considering linear time-varying systems admitting a state-space representation % A B x "A x #B w I I . I> I I I I" " : (2.1) % y "C x #D w C D I I I I I I I
We will assume that all the matrix sequences A , etc. I are bounded. With this system, we associate an operator G mapping the input w to output y. This operator has a block matrix representation G with elements GH D if i"j, G " G GH C A A 2A B if i'j G G\ G\ H> H and zero otherwise. If i"j#1, the product A A H H> should be interpreted as the identity.
2.2. Exponential dichotomies and the spectrum of weighted shifts Let k and l be integers, with '0. We say that A is I (1) Uniformly exponentially stable if there exist positive constants and (1 such that A A 2A ( J; I>J\ I>J\ I (2) Uniformly exponentially antistable if there exist positive constants and '1 such that (A A 2A )' J. I N I>J\ I>J\ In both cases, the constants and are independent of k and l. Throughout this paper, stable and antistable refer to uniformly exponentially stable and antistable. In Bode's result, it is important to di!erentiate between the stable poles of ¸(z)*which do not contribute to the right-hand side of (1.2)*and the unstable poles, which do. Time-varying systems which can be decomposed into stable and antistable components are said to possess an exponential dichotomy (Ben-Artzi & Gohberg, 1995; Co!man & SchaK !er, 1967). De5nition 1. Let A be a sequence of n;n matrices I and let P be a bounded sequence of projections in I 1L such that the rank of P is constant. The sequence I P is a dichotomy for A if the commutativity condiI I tion A P "P A is satis"ed for all k, and there exist I I I> I positive constants and '1 such that A A 2A P x' JP x I>J\ I>J\ I I I
(2.2)
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and 1 A A 2A (I!P )x( (I!P )x I>J\ I>J\ I I I
J
(2.3)
for any x. The existence of an exponential dichotomy allows us to de"ne a stability preserving state space transformation (a Lyapunov transformation) that separates the stable and antistable parts of A . I Lemma 2 (Ben-Artzi & Gohberg, 1991). The sequence A in the realization (2.1) admits an exponential dichotI omy iw there exist a bounded sequence of matrices ¹ I with bounded inverses such that
¹
AQ I A ¹\ ¹ B I> I I I> I " 0 C ¹\ D I I I CQ I
0 BQ I AS BS I I , CS D I I
where AQ is stable and AS is antistable. I I
De"ne the weighted shift operator S in the following manner (S x) "A x . I I\ I\ If (S ) denotes the spectrum of the operator, we can characterize the stability conditions as follows: A is I stable i! for (S )L# and antistable i! (S )L"#. These are consistent with the usual time-invariant requirements on the eigenvalues of the `Aa matrix. It can be shown (Ben-Artzi, Gohberg, & Kaashoek, 1993) that the spectrum of the weighted shift will consist of l concentric annuli: J (S )" 3": ) ) , G G G
(2.4)
where 0) ( ( (2( ( (R and J exists a J Lyapunov 1)l)n. Moreover, there transformation so that ¹ A ¹\"diagA,2, AJ, I> I I where AG31P "P , and G
G
(S G )" 3" : ) ) G G for i"1,2, l. Exponential dichotomy is equivalent to the existence of an integer j31,2, l!1 such that, either (1( or (1. H J H> case There is a particular in which each of the annuli is a circle is said to be regular. In this case, positive real values corresponding to the spectrum of S are the dis crete-time Lyapunov exponents. Regularity is hard to
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verify for any particular system, though all time-invariant and periodic systems are regular. In these two cases, the spectral values are the magnitude of the eigenvalues and Floquet characteristic exponents of the system A matrix. For systems involving a #ow with an invariant probability measure, Oseledec\ 's multiplicative ergodic theory states that regularity occurs with probability one; see Arnold (1998). The weighted shifts considered here have received considerable interest recently as a means of extending some classical linear time-invariant system theory results to time-varying systems. For example, a generalization of the PBH test for reachability/stabilizability was introduced in Peters and Iglesias (1999). It is shown that reachability is lost i! there is an almost eigenvector (x 3l ), and eigenvalue ( 3(S )) pair for which L xHS P0. As in the time-invariant case, stabilizability L allows for pairs for which (1. Related results are found in Kamen, Khargonekar, and Poolla (1985) who looked at asymptotic stability, Dullerud and Lall (1999), Iglesias and Peters (1997); Peters and Iglesias (1997) who looked at inducednorm optimal control problems, and (Iglesias, 1999), where it is shown that the `invariant-zero on the unit circlea condition which guarantees the existence of stabilizing solutions to discrete-time algebraic Riccati equations has a natural analogue for Riccati di!erence equations that involves the spectrum of a weighted shift. 2.3. Assumptions on plant We consider the system in Fig. 1, in which the openloop system has a stabilizable and detectable state* space representation
A B " I I . (2.5) * C 0 I It should be noted that this system will include both the controller and plant subsystems. Using (2.5), we have a corresponding state-space representation for the sensitivity operator:
A !B C B I I I I . (2.6) !C I I We will need the following two assumptions. First of all, we would want to ensure that the sensitivity operator S is bounded. This would, of course, be a necessary requirement of any internally stabilizing controller. " 1
Assumption 3. A !B C is stable. I I I Since the sequences A , B and C are bounded, I I I Assumption 3 guarantees that S is bounded.
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We will also need to di!erentiate between the stable and antistable dynamics of . To this purpose, we * assume: Assumption 4. The sequence A admits a rank r I dichotomy with constants and . We should note that the state space matrix sequences A etc. include components from both the open-loop I plant and controller. Note that the `Aa matrix can be represented as
A. 夹 A " I , I 0 A) I where A. and A) are the sequences in the realizations I I of the plant and controller, respectively, and `夹a denotes elements not needed in the analysis. It follows that the spectrum of S is the union of that of S . and S ) and thus contains the unstable portions of both plant and controller dynamics. Remark 5. While Assumption 3 is clearly necessary, it is worth noting that the second assumption is not required in the time-invariant case, where Bode's result allows for open-loop systems having poles on the unit circle. 2.4. Inner/outer factorizations As a step in obtaining Bode's result we will need to compute an inner/outer factorization of the operator S. The existance of such a factorization is a result of Arveson who introduced the concepts of inner and outer operators in nest algebras (Arveson, 1975). In the setting of this paper, we say that F is an inner operator if F is bounded, its matrix representation is lower blocktriangular, and FHF"I. The operator G is outer if it is bounded, lower block-triangular, and its inverse is also bounded and lower block-triangular. For the sensitivity operator arising from (2.6), we can provide state-space representations of its inner and outer factors. Not surprisingly, Riccati di!erence equations play a prominent role. Lemma 6. Consider the Riccati diwerence equation: X "A X (I#B B X )\A . (2.7) I I I> I I I> I There exists a solution to (2.7) with bounded positivesemidexnite X , and stable (I#B B X )\A . MoreI I I I> I over, this solution is unique; and: (i) (ii)
If A is stable then X "0 for all k; I I If A is antistable then inf (X )'0. I I I
(iii)
AQ 0 If A " I admits a dichotomy as in Lemma 2 I 0 AS I 0 0 and inf (XS )'0. then, X " I I I 0 XS I
Proof. That this equation has a stabilizing solution, and that it is unique is shown in Halanay and Ionescu (1994, Theorem 3.2). Item (i) is trivial, as is (iii) once (ii) has been proved. The main technical di$culty is in showing (ii); this is done in the appendix. This Riccati equation is now used to compute the inner/outer factorization of S. Formulae for inner/outer factorizations of general state-space time-varying systems can be found in Dewilde and Van Der Veen (1998), Halanay and Ionescu (1994). These formulae can be applied to the sensitivity operator. De"ne the blockdiagonal operator R " : I#B X B . I I I> I Then, S"S S for G M (I#B B X )\A B R\ I I I> I I I G" (2.8) 1 !R\B X R\ I I I> I inner, and M equals 1 A !B C B I I I I (2.9) (I#B B X )\ (B X A !R C ) R I I I> I I> I I I I outer. We will not be concerned with S , but it should be M worth noting that the `Aa matrix of S\ is M A !B C !B R\[BX A !R C ] I> I I I I I I I I " (I#B B X )\A I I I> I which is stable from Lemma 6. Since this matrix sequence is stable, it follows that S\ is bounded. Moreover, S is M M bounded since its `Aa matrix is stable by Assumption 3. Thus, S! is bounded and hence S is outer. M M
3. A time-domain version of Bode's integral In this section, we outline the connections between Bode's integral and two cost functions that are common in robust control. We "rst consider the entropy cost (Mustafa & Glover, 1990). For a stable function G(z) with norm bound G (, the entropy is de"ned as
L E(G,) " : ! ln det(I!\G%(e S)G(e S)) d. 2 \L Suppose that the sensitivity function has norm bound S ( . This allows one to factor I! \S%(z)S(z)"G%(z)G(z). (3.1)
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Moreover,
3.1. The time-varying case
ln det S(e S)" ln det(I!G(e S)G(e S))#m ln , where m is the matrix dimension of S. The sensitivity integral can be written as
1 2
L
ln det S(e S) d"m ln !E(G,1). \L This shows that Bode's integral can be computed by means of an entropy computation. In Glover and Doyle (1988), it was shown that the entropy cost function is equivalent to the in"nite horizon risk-sensitive cost function. We can use this connection to provide the time-domain characterization of Bode's integral that we seek. Suppose that the input r is white, Gaussian stochastic I process with unit covariance, and that e"Sr. A risksensitive cost function can be de"ned as
,\ (r r !e e ) , I I I I I where E denotes an expectation. We can show, as in Glover and Doyle (1988), that for stable linear timeinvariant discrete-time system S(z) 2 1 ¸(G,, N) " : ln E exp N 2
1 2
L
\L
ln det S(e S) d
1 1 ,\ ln E exp (3.2) " ! lim (r r !e e ) . I I I I 2N 2 ,t I This expression serves as a time-domain characterization of Bode's integral. Remark 7. This characterization of Bode's integral, shows how the frequency domain tradeo!s translate into the time-domain. In particular, if we divide the time line 0,2, N into two regions 0,2, M and M#1,2, N for 0(M(N!1, then the argument of the exponent in (3.2)
1 , (r r !e e ) I I I I 2 I 1 + 1 , (r r !e e ) exp (r r !e e ) . "exp I I I I I I I I 2 2 I I+> Now, if Bode's integral is zero, this means that the terms in (3.2) in the summation with time index k'M cannot decrease without a corresponding increase in those in the horizon k"0,2, M. Under the usual connection between the time and frequency domains: low (high) frequency signals are associated with long (short) time horizons; this new timedomain interpretation agrees with the usual frequency domain version discussed in the introduction. exp
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The relationship between the entropy and risk-sensitive cost functions analogous to that obtained in Glover and Doyle (1988) exists for time-varying systems (Peters & Iglesias, 1999). We now use this connection to extend the time-domain characterization of Bode's integral to time-varying systems. Suppose that the input signal r is a windowed ,
white noise sequence de"ned over the interval k"0,$1,2,$N and let e be the corresponding ,
tracking error. Unlike the time-invariant case, where every signal gives rise to the same cost, in the timevarying systems this set of signals will give rise to a sequence of costs. In the limit, we can consider an average cost function 1 lim sup ! ln E exp((r !e )) (3.3) , , 2(2N#1) , and treat this as the time-varying analogue to Bode's integral. Remark 8. It should be stressed that even though the input sequence r in (3.3) is de"ned over a "nite horizon, ,
this is not true of the tracking error e . Since S is causal, ,
this is de"ned for k*!N. In Section 4.1, we look at the case where the tracking error is only considered over a "nite time. Since r is a Gaussian white-noise sequence, de"ned over a "nite horizon, we can use a standard argument (see, e.g. Glover and Doyle (1988)) to show that (3.3) equals ! ln E exp((r !e )) , , " ln det(SE )H(SE ) (3.4) , , (3.5) " trace ln(EH SHSE ). , , In this latter form of the equation, the matrix logarithm is well de"ned since EH SHSE is a "nite-dimensional , , positive-de"nite matrix (Kenney & Laub, 1998). It will now be shown that this value, as in the timeinvariant case, depends only on the unstable dynamics of the open-loop plant. 4. Constraint on sensitivity due to unstable poles Lemma 9. Suppose that the open loop system satisxes Assumption 3, and that the corresponding sensitivity operator S admits the inner/outer factorization (2.8), (2.9). Then 1 lim sup ln det(EH SHSE ). , , 2N#1 , 1 , "lim sup ln det R I 2N#1 , I\,
(4.1)
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Proof. See the appendix. In the previous result, we showed that the average contribution of the logarithm of the sensitivity operator is tied to the logarithm of R . We now wish to relate this I to the dynamics of the open loop system L. Theorem 10. Suppose that the open loop system satisxes Assumptions 3 and 4. The sensitivity operator S must satisfy 1 lim sup ln det(EH SHSE )*2r ln . , , 2N#1 ,
(4.2)
Proof. See the appendix. Remark 11. Unlike (1.2) where an equality is obtained, Theorem 10 only provides a lower bound on the gain of the sensitivity operator. This lower bound comes from the fact that we only know the smallest rate of exponential growth of the antistable part of ¸. For time-invariant systems, this would be equivalent to replacing every unstable pole in (1.2) with one of a smallest magnitude. We can provide a tighter bound if we know something more about the unstable part of the spectrum of S . For example, if we split the unstable component of A as in I (2.4), then we can replace in (4.2) 1 J lim sup ln det(EH SHSE )*2 r ln , , G 2N#1 G , G *2r ln . Moreover, as shown in Iglesias (2001), if the unstable component is regular, equality is achieved. In this case, the lim sup can be replaced with a limit: J 1 ln det(EH SHSE )"2 r ln , lim , , G G 2N#1 G , where " : " . G G G A second special case where equality is obtained is where the open-loop dynamics are stable, in which case: Corollary 12. Under the assumptions of Theorem 10 and the additional assumption that A is stable, it follows I that 1 lim sup ln det(EH SHSE )"0. , , 2N#1 , Proof. This is a straightforward application of Lemma 9, the de"nition of R and noting that, from Lemma 6, I X "0 for all k. I Remark 13. While we have motivated our results in terms of an analogue of Bode's sensitivity integral, there is an alternative equally valid approach. Let x , be I I
a sequence of random variables and let ¹ be its asso, ciated correlation matrix. If x is stationary, this matrix is I both Toeplitz and Hermitian. It is well known that
L
ln S(e S) d" lim ,> , (4.3) \L , , where S(e S) is the power spectral density, and "det ¹ is the correlation determinant of the process , , x (Papoulis, 1984, p. 478). Moreover, as originally shown by Szego , the limit in (4.3) exists and can be expressed in terms of the poles of the function S(z) (Gohberg, Kaashoek, & van Schagen, 1987). The results presented here are the analogues of these results for non-stationary random variables; see Iglesias (2001). 1 2
4.1. Finite-horizon signals Time-varying systems are usually considered only for "nite-horizon applications. It is worth asking how the results obtained here apply in this case. As was mentioned in the earlier remark, the results obtained so far assumed that the input signal was of "nite duration, but the corresponding error signal was allowed to extend for all time. This means that the operator SE , in (3.4) is in"nitely tall, but has a "nite number of columns. When multiplied by its adjoint, the corresponding matrix is of "nite dimensions. Considering an error signal over a "nite horizon is equivalent to adding a projection operator to SE to form the "nite matrix , S " : EH SE , , , This is a lower block-triangular matrix with identities on the blocks on the diagonal. Hence, its determinant is always one. This means that the time-varying analogue of Bode's integral: ! ln E exp((r !e )) , , " ln det(EH SE )H(EH SE ) , , , , "0 and this is true regardless of the dynamics of L. This last result should not be surprising. Bode's sensitivity integral is always zero unless the loop has unstable dynamics. In "nite-horizon applications, however, every dynamic can be considered as stable as the corresponding signals always remain bounded. Thus, the fact that the analogue of Bode's integral is always zero for systems over a "nite horizon is completely consistent with Bode's result. 4.2. An example We consider a simple example that may help to illustrate some of the results of this paper. We consider
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Fig. 2. Convergence of measure towards a theoretical value. For the size of the tall matrix, a value of n"60 was used.
puting a `talla matrix SI "¹I , where i"!n,2, n G H and j"!N,2, N, for n
a time-varying plant:
1 " I , . 1 0
where " : (2k#1)/(k#1) and the time-invariant I state-feedback controller K"2. State-space representations for the open-loop and sensitivity systems are
1 , " I * 2 0
" 1
!(k#1)\ 1
.
!2
1
The closed-loop system is stable. Let 0(;1, "2!, NM " : W (1!)/ X, and "\,M >(1. It follows that * for k'NM , and 2 * J I I>J I> for any k and l, so that the open-loop dynamics are antistable. Moreover, as the system is one-dimensional, and the weighted shift is clearly invertible, the spectrum of the weighted shift is shown in Shields (1974, Theorem 5) to be the annulus: (S )"z3": (S \ )\)z)(S ), where ( ) ) denotes the spectral radius. This can be computed as in [Halmos (1967), Problem 77]:
I I\ (S )"lim sup "2. L>G I L G Similarly, (S \ )"1/2 and thus the spectrum is the circle of radius 2. We now approximate the successive values of ln det(¹ ), where ¹ " : EH SHSE . This is done by com, , , ,
5. Conclusions By means of the connection between Bode's integral and the entropy cost function, we have provided a timedomain characterization of Bode's sensitivity integral. The traditional frequency domain interpretation is that, if the sensitivity of a closed-loop system is decreased over a particular frequency range*typically the low frequencies*the designer `paysa for this in increased sensitivity outside this frequency range. This interpretation is also valid for the characterization presented here provided one deals with time horizons rather than frequency ranges. Having reinterpreted Bode's sensitivity integral in the time domain, it is possible to consider extensions to systems not admitting frequency domain representations, including linear time-varying systems. In the generalization presented here, the poles of the open loop system are replaced by the Lyapunov exponents of the timevarying di!erence equation. Bode's integral is but one of several known mathematical characterizations of the analytic constraints found in feedback control systems (Chen, 1995, 1998; Freudenberg & Looze, 1985). The corresponding counterparts for time-varying systems are unknown but will be the subject of future research.
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Acknowledgements
Appendix A. Proofs of results
Moreover, from a recursive argument using (A.1), and the fact that X *0 we have that x X x (, and z ( I I I I I for k*0 where z " : B X x . I I I> I> Using z , (A.1) can be rewritten as I x "A x !B z , k*0. I> I I I I In particular, if k"p !1, we have:
A.1. Proof of Lemma 6(ii)
x "A x !B z N N\ N\ N\ N\
We begin with some preliminaries. Since A is antiI stable, there exist constants and with '1 such S S S that
N \ "A 2A x !B z ! A 2A B z . N\ N\ J> J J J Considering the norm of both sides, yields the bound
This work was supported in part by the NSF under Grant ECS-9800057.
A A 2A x' J x I>J\ I>J\ I S S for all x, and all k and l39. Note that this implies that sup A '1, (∀k). I I Similarly, since the solution X is stabilizing, it follows I that the sequence with " : (I#B B X )\A I I I I I> I is stable, i.e., there exist positive constants and with Q Q (1 such that Q 2 x) J x (A.1) I>J\ I>J\ I Q Q for all x, and all k"0,$1,2 and l"1,2,2 . We denote by p the smallest positive integer such that S
NS '2. Similarly, we denote by p the smallest positive S S Q integer such that NQ (1/2. Since (1 and '1 we Q Q Q S know that such integers must exist. Designate p "max(p , p ). Q S To prove the assertion we will use a contradiction argument. Since we know that X *0 for all k, the I statement can only fail if, given an arbitrary , there exists an index k (we will assume k "0 without the loss of generality) and a unit vector x such that 'x X x . We choose an such that
sup A N !1 I I (( sup B . I sup A !1 I I I As mentioned above, sup A '1 so that the inverse I I exists. Moreover, the right-hand side is a positive number. The Riccati di!erence equation can be written as: X " X # X B B X . I I I> I I I> I I I> I Pre- and post-multiplying the k"0 term of this equation by x and x , respectively, yield x X x "x X x #x X B B X x (. Now, de"ne the sequence x " x , I> I I From (A.1), x ) I. I Q Q
k*0.
x *A 2A x !B z N\ N N\ ! A 2A B z J> J J N\ J * N !B z S S N \ ! A 2A B z N\ J> J J J N \ *2! sup B sup A J I I J I I *1.
(A.2) (A.3) (A.4)
In (A.2), we have used the fact that A is antistable and I that x is a unit vector, whereas in (A.3) we have used the norm bound on z and the fact that p was chosen so that I
N '2. Finally, in (A.4), we used the norm bound on . S S However, from (A.1) with k"p we have x ) N (1/2 Q Q N which contradicts (A.4). 䊐 A.2. Proof of Lemma 9 Let ¹ "(SE )H(SE )"(S E )H(S E ). We partition , , , M , M , S according to E : M , 夹 0 0
S " 夹 M 夹
; 0 . (A.5) , V 夹 , Here, ; is a (2N#1)m;(2N#1)m matrix (recall that , m is the dimension of the reference signal), given by ; "(S ) for !N)i, j)N. The other elements of , M G H (A.5) are all in"nite-dimensional operators. According to this partition, trace ln ¹ "ln det ¹ , , "ln det(; ; #VH V ) , , , , "2 ln det ; , # ln det(I#(; )\VH V ;\). , , , ,
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Note that the matrix ; is invertible since it is "nite, dimensional, block-triangular, and all the elements along the diagonal are invertible. In fact, they are R and so: I , 2 ln det ; " ln det R . , I I\, It remains to be seen that 1 , lim ln det (I#(; )\VH V ;\)"0. , , , , 2N#1 , I\, (A.6)
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A.3. Proof of Theorem 10 From Lemma 4 we need to compute ln det R . From I the form of the solution of the Riccati equation, this equals R "I#(BS )XS BS . I I I> I Moreover, we have:
Since VH V is a "nite-dimensional, positive semi-de"nite , , matrix, it has a Cholesky factorization ¹¹. Using this factorization, and the fact that
XS "(AS )XS (I#BS (BS )XS )\AS . (A.8) I I I> I I I> I As we are only concerned with the unstable components of the realization, it will be convenient to drop the superscript `ua for the rest of the proof. From the logarithm of the determinant of (A.8), one obtains
; ; *diag(; ; )*I , , , , we can evaluate
ln det R "2 ln det A #ln det X !ln det X I I I> I from which
ln det (I#;\VH V ; )"ln det (I#¹(; ; )\¹) , , , , , , )ln det (I#¹¹)
, ln det R "ln det(A A 2A ) I , ,\ \, I\, #ln det X !ln det X . (A.9) ,> \, The "rst term in the right-hand side can be written as
"ln det (I#VH V ). (A.7) , , Denote by ,2, the (2N#1)m non-negative ,>K eigenvalues of VH V . , , ,>K ln det(I#VH V )" ln(1# ) , , H H ,>K ) (since '0) H H H , " trace(VH V ) . , , H H H\, We now need to show that this trace is bounded. Because S admits a state-space representation in which the `Aa M matrix is stable, there exist constants *0 and 0)(1 such that (S ) ) I\J. Now, for M I J j"!N,2, N (VH V ) " (S ) (S ) . , , H H M G H M G H GH> Using this, we obtain trace(VH V ) )trace G\HI , , H H GH> m ,>\H " . 1! Therefore, 1 , 0) lim trace(VH V ) , , H H 2N#1 , H\, m (1!,>) "0 ) lim (2N#1)(1!) , as required. 䊐
ln det(A A 2A ) , ,\ \, P "2 ln (A A 2A ). G , ,\ \, G Since (A A 2A )* (A A 2A ) G , ,\ \, P , ,\ \, " (A A 2A ) \, , ,\ * ,> for i"1,2, r; it follows that ln det(A A 2A )*2r ln , ,\ \, P *2r ln #2r(2N#1) ln . Because X is bounded above and below I 1 lim sup (ln det X !ln det X )"0. ,> \, 2N#1 , Similarly, lim sup (2r )/(2N#1)"0. Thus, in (A.9) we , are left with 1 , lim sup ln det R *2r ln I 2N#1 , I\, as claimed.
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Pablo A. Iglesias was born in Caracas, Venezuela in 1964. He received the B.A.Sc. degree in Engineering Science from the University of Toronto in 1987, and the Ph.D. degree in Control Engineering from Cambridge University in 1991. Since then he has been on the faculty of the Department of Electrical and Computer Engineering at the Johns Hopkins University, where he currently holds the rank of Professor. He is also a member of the Center for Computational Biology and Medicine. He has had visiting appointments at Lund University. The Weizmann Institute of Science, and the California Institute of Technology. He is also the coauthor of the monograph Minimum Entropy Control for Time-Varying Systems. His current research interests include the control of time-varying systems, and robust control theory, and the use of control theory to study biological signal transduction pathways.
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