o delay via reduction to a one-block problem

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erroneously predicts the existence of limit cycles. We have shown by example how one can use the Tsypkin locus analysis to quantify how “nearly tangent” the graphs of the transfer function and the negative reciprocal of the describing function must be in order for the describing function technique to erroneously predict limit cycles. REFERENCES

Fig. 1.

[1] J. E. Gibson, Nonlinear Automatic Control. New York: McGraw-Hill, 1963. [2] B. Hamel, Contribution á L’ètude Mathimatique des Systèmes de Règale Par Tout-On-Rien C.E.M.V. Paris, France: Service Technique Aéronautique, 1949, vol. 17. [3] A. I. Mees and A. R. Bergen, “Describing functions revisited,” IEEE Trans. Automat. Contr., vol. AC-2, pp. 473–478, Apr. 1975. [4] C. L. Phillip and R. D. Harbor, Feedback Control Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. [5] J. E. Slotine and W. Li, Applied Nonlinear Control. Upper Saddle River, NJ: Prentice-Hall, 1991. [6] Y. Z. Tsypkin, Relay Control Systems. Cambridge, MA: Cambridge Univ. Press, 1984.

Control of Systems With I/O Delay Via Reduction to a One-Block Problem Gjerrit Meinsma, Leonid Mirkin, and Qing-Chang Zhong

Abstract—In this note, the standard (four-block) control problem for systems with a single delay in the feedback loop is studied. A simple procedure of the reduction of the problem to an equivalent one-block problem having particularly simple structure is proposed. The one-block problem is then solved by the -spectral factorization approach, resulting in the so-called dead-time compensator (DTC) form of the controller. The advantages of the proposed procedure are its simplicity, intuitively clear derivacontroller, and extensibility to the multion of the DTC form of the tiple delay case. Index Terms—Dead-time compensation, factorization, time-delay systems.

optimization,

-spectral

I. INTRODUCTION AND PROBLEM FORMULATION Consider the dead-time (DT) system in Fig. 1, where P (s) is a finitedimensional generalized plant with the transfer matrix

P (s) =

A

B1

B2

C1

0

D12

C2

D21

0

Manuscript received April 4, 2002. Recommended by Associate Editor K. Gu. This work was supported by the Israel Science Foundation under Grants 384/99 and 106/01, the Fund for the Promotion of Research at the Technion, and the EPSRC under Grant GR/N38190/1. G. Meinsma is with the Faculty of Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands (e-mail: g.meinsma@math. utwente.nl). L. Mirkin is with the Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]). Q.-C. Zhong is with the Department of Electrical and Electronic Engineering, Imperial College, London SW7 2BT, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2002.804472

The “standard problem” for DT systems.

e0sh is the loop delay with the dead-time h > 0, and Kh (s) is a proper part of the controller to be designed. The problem to be studied in this note is formulated as follows. OPh : Given the plant P (s) and the dead time h, determine whether there exists a proper Kh (s), which internally stabilizes the system in Fig. 1 and guarantees

F`

P; e0sh Kh

1


0) OPh is solvable only if so is OP0 . Therefore, combining the parameterization of all solutions to OP0 with the transformation in Fig. 2 the following result can formulated. Lemma 1: OPh is solvable iff so is its delay-free counterpart OP0 1 0sh Kh 1 < . and, in addition, Cr G0 0 ;e Lemma 1 actually implies that OPh can be converted to the following equivalent problem. OPeq : Given the bistable system G0 (s) with the state–space realization (3) and the dead time h, determine whether there exists a proper Kh (s), which guarantees

Cr

1 0sh G0 Kh 0 ;e

1


h iff 622 (t) is nonsingular for all t 2 [0; h].

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the construction of W and the proof that the aforementioned factorization does yield the solution to OPeq . To circumvent this obstacle, the approach of [8] is used. The idea is to exploit the special structure of  and use it to remove the infinite-dimensional part from the factorization. To this end, note that the infinite-dimensional part of  only enters the off-diagonal blocks

5

Fig. 3.

All admissible controllers for OP .

We are now in the position to formulate our main result. Theorem 1: OPeq is solvable iff > h . In that case, all solutions Kh to the OPeq are given by

Kh

= Cr 1I I0

Gh ; Qh

(8)

(see Fig. 3), where the equation shown at the bottom of the page holds, and Qh must satisfy kQh k1 < , but otherwise arbitrary. Having this result, the solution to OPh can now be formulated as follows. Corollary 1: OPh is solvable iff OP0 is solvable, and > h . In that case, all solutions Kh to the OPh are given by (8). Remark 3: The formulas for t and could be further cleaned up as shown in [10]. The reader could also find there the more conventional linear fractional transformation (LFT) form of parametrization (8). The rest of this section is devoted to the proof of Theorem 1. In Section III-B, the main ideas of the proof are outlined, then, in Section III-C, some technical machinery to be used in the sequel is introduced. In Section III-D, we derive the necessary conditions for the solvability of OPeq . Section III-E is devoted to the construction of and Gh , and, finally, in Section III-F, we prove the validity of the formulae.

6( )

1

1

5

5 =

= 0

0

5 =: G J G = WJ W where

G

=: G00 1

e0sh I

0 0I

01  5 = 511 0 512 5225225R21 + R 522 R

=

Gh

=

AF

+ ZB2 B20 X

0XB2 B20 X 0C2 AF

C2

R

522 522

: 01 21 . This R is finite dimensional if we choose for R e0sh 22 to be the stable FIR system

= 1+

1

5 5

e0sh

52201521

5

(incidentally, 22 is invertible because of the structure of G0 ). In Section III-E, we show that this choice yields the of Theorem 1 and that Gh defined in Theorem 1 equals Gh W 01 where W is a finite dimensional J -spectral factor of . Typically, it is the existence of such Gh W 01 that forms the bottleneck of the proof. Here, however, we bypass this difficulty by first showing that must exceed h if OPeq is to have a solution, see Section III-D. Therefore, > h and this guarantees invertibility of 22 and, hence, existence of Gh (see Theorem 1). With Gh known to exist the rest of the proof follows fairly standard arguments. Continuity is used to show that G W01 is J lossless. Finally, as in the finite-dimensional case, all solutions Kh are shown to have the form Kh Cr W01; Qh Cr I I W 01; Qh Cr I I Gh ; Qh with kQk1 < .

5

=

1

=

6

and G W01 is J -lossless; see [14] and [16] for the definitions. If G were finite dimensional, then the existence of such a W would be necessary and sufficient for the solvability of OPeq and the set of all solutions would be parameterized by Kh Cr W01; Qh with kQh k1 < . Yet G is infinite dimensional. This complicates both

1 = h

:

5ij are the subblocks of the (finite-dimensional) transfer matrix 5 = G00 1  J G00 1 . Also, note that the J -spectral factorization of 5 can be reduced to that of  5 =: I0 1I 5 1I I0 (9) provided 1 2 H 1 . Indeed, one can see that W is a bistable J -spectral factor of 5 iff I 0 : W = W 1 I is a bistable J -spectral factor of 5 . The idea then is to choose 1 so as to make 5 finite dimensional. It is easy to verify that

1 = h

In the proof of Theorem 1, we use the J -spectral factorization ap: I 0 2 I . We are looking for a bistable W so that

512 522

esh

Here, :

B. Proof Outline proach. Let J

511

e0sh 521

0B20 X 0 622 0 1 Y Z 0 6021

1

0

=

0 0 1 ZB2

ZY C2 C2 Y Z 0 0 0 0AF 0 XB2B2 Z 0XB2 0 1 C2 Y Z 0 0 0 1 0 Z + 612 622 X B2 600 22 ZY C2 I 0 0 I

=

1

0

=

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C. Preliminary: S -Transformations

D. Necessary Solvability Conditions

Throughout this section, we will extensively use the “Schur complementation” transformations Su O and S` O , which are defined for a 2 2 2 block operator O as follows:

( )

Su (O) =: S` (O) =:

( )

01 O11 0O1101 O12 0 1 01 O12 O21 O11 O22 0 O21 O11 0 1 01 O11 0 O12 O22 O21 O12 O22 : 0 1 0 1 0O22 O21 O22

=

1 O 2

()

1 2 1 2

= Su (O) = S` (O)

1 2 1 2

A C1 C2

B1 I

0

[0 ]

terval ; h (the former system is actually open-loop in this interval). This, in fact, proves that OPeq is solvable only if

0

Su (8(s)) =

0

1I

Cr G00 1 ; 0 =

S` (8(s)) =

B2

0

(10a)

1

11 I

O

I

12

0

I

= S` (O) + 010 2 101

^ =: 0

=:

0

:

)

[ B51

0 ] = J^ [ C51

I

B52

.

(11a)

S` (5) =

0 C52

1=

A5

+ 1 B52 C52 C51 1

C52

]

(13)

1 = h

5 5 (5)

B51 I

0

6(t) = e(A + B C A5 + 02 B52 C52 e0sh

(by (5), one can see that

(11b)

:

6

: 01 21 we use the Now, for the construction of  h e0sh 22 0 1 fact that 0 22 21 is the lower left block of S` . Therefore, consider [using (10b)]

0 02 C52

0 1 B52

0

0 1 I )t ). Then B51

0

and this coincides with the realization in Theorem 1. Moreover

(12)

R

: =1 + e0sh 52201521 1 = A5 + 1 B52 C52

0

This relation will be used in Section III-E.

5=

5=(

0I 0

(10b)

(see also [14, Ch.4], where similar transformations were introduced). Another advantage of looking at the S -transformation of O instead of at O itself is that

0

ZY C20

B20 X

By Lemma 2, we can safely assume hereafter that > h . In this section, the previous assumption is required to ensure that 22 is in1 vertible. Consider G001  J G0 0 , which has the state–space realization [recall (5)] shown in the quation at the bottom of the page. Note that by construction

5 5

I B1 B2 I 0 : 0 I

Su (Su (O)) =S` (S` (O)) = O Su (S`(O)) =S` (Su (O)) = O01

I

+ ZB2 B20 X

E. Factorization of G  J G

where J

The signal swapping interpretation also implies the following relations when corresponding transformations exist:

S`

AF

0



0

=

( )

[0 ]

B2

B1 I

L [0;h] :

1 To find the state–space realization of Cr G0 0 ; , note that it is0equal 0 1 to the (1,2) subblock of S` G0 . Yet, according to (11), S` G0 1 Su G0 . This, together with (10a), yields

then the straightforward flow-tracing yields

A 0 B1 C1 0C1 C2 A 0 B2 C2 C1 0C2

Cr G00 1 ; 0

Thus, we proved the following result. Lemma 2: OPeq is solvable only if > h , where h is the 1 L2 ; h induced norm of Cr G0 0 ; given by(7).

(provided the mappings are well defined). The aforementioned relations prompt an elegant way to perform S -transformations for systems given by their state–space realizations. Indeed, if

8(s) =

We start with finding the necessary condition for the solvability of

OPeq . To this end, note that given any proper Kh , the responses of Cr G00 1 ; e0sh Kh , and Cr G00 1 ; 0 to any input coincide in the in-

>

In the sequel, we call these transformations the upper and lower S -transformations, respectively. It is clear that the upper (lower) S -transformation is well defined iff the upper left (lower right) subblock of O is nonsingular. S -transformations can be thought of as the “swapping” of parts of the inputs and outputs, namely

1 2

1893

ZAL Z 01 0 2 0

C2 C2 0 XB2 B20 X 0Z 00 A0L Z 0 B20 X B20 Z 0 2 C2 Y Z 0

C2 A5 B51 B52 C51 I 0 : 0 0 2 I C52

ZB2 0XB2 I

0

C52

ZY C20 0 2 C20

0

0 2 I

601 B51 0

:

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5

Next, to obtain a realization of as defined in (9), we use (12) and then combine the various realizations

1 5  I 0 0 I 1 I  0 1 =S` (5 ) + 01 0 sh 0sh = e 0 I I0 S` (5) e 0 I I0 + A5 + 1 B52 C52 601 B51 = C51 6 I 1 C52 0

I

S` (5 ) =S`

and M

611 0 612 62201621 = 600 22 . Then, the natural choice for M is = 062201621 , since this yields A = 6022 AF 600 22

which is Hurwitz, so that W is bistable (anothor way to find the required M is to use the J -spectral factorization of generic para-Hermitian transfer matrices [17]). Straightforward algebra yields then that W 01 Gh with Gh as in Theorem 1.

0 1 01 0

=

0 1 B52

F. Necessity and Sufficiency

0

:

0 1 I

By construction, we have that < for Qh defined as

kQh kL

Qh

This, together with (11a) and (10b), yields

601 B51 B52 6 I 0 5 = 0 0 2 I and the “B ” and “C ” matrices of 5 satisfy 0 C52 0 ] [ 601 B51 B52 ] = J^ [ 60 C51 (14) which follows from (13) and the fact that 6 is symplectic and, thus, 6J^60 = J^. To J -factorize 5 , let M = M 0 be any matrix satisfying the folA5 C51 C52

lowing Riccati equation:

[ 0M

I

] 601

A5 0 B51 C51

+ 12 B52 C52 6

I M

=0

(we construct one such M as follows). Then, taking into account (14), one can verify that stable

ZAL Z 01 W

[ I 0] 601 B51

I M 0 1 C52 MI

=

C51

6

I

[ I 0] B52 0

0

I

5 = W  J W . Moreover, the “A” matrix of W 01 , say

2

I M

=[ I 0]

0]

601 B51 C51 6 0

=[ I 0] 601

1 B52 C52

2

601 B51 C51 6 + 12 B52 C52 1 A5 0 B51 C51 + B52 C52 6

A5 0

I M I M

2 0 0 0 1 =[ I 0] 601 AF Z Y C2 C2 Y 0 0 B2 B2 Z 0 0A

2

F

I M

6

+ (1= 2 )B52 C52 commutes with 6

where (5) and the fact that A5 were used. Since is symplectic

6

601 =

6022 0 0621

0 0612 0 611

1 G0 h

I

01

0

I

L

< iff

; Kh :

Now, this Qh is proper if Kh is proper, yet the set of proper operators in L1 is, in fact, H 1 [18] (see also [19, A6.26.c, A6.27]). So, if Kh solves OPeq , then necessarily kQh k1 < . This condition on Qh is also sufficient, as we shall now see. The concept to note is that

e0sh I

I 0 1 I Gh is not only stable and J -unitary (i.e., 2(h) J 2(h) = J ), but in fact J lossless (meaning that, in addition, 222 (h) is bistable). Indeed, from 2(h) J 2(h) = J it follows that 222 (h) 222 (h)  I , and as our 2(t) (which by Lemma 2 exists for all > h ) is stable and continuous as a function of t 2 [0; h], and 222 (0) = I it follows that 222 (h) is bistable. It is well known that for J -lossless 2(h), we have that Q = Cr (2(h); Qh ) is stable for any kQh k1 < , see, e.g., [8, I 0 Th. 6.2]. Also, Kh = Cr 1 I Gh ; Qh is proper for any stable

2(h) =: G00 1

0

0

I

Qh .

IV. CONCLUDING REMARKS

does satisfy A , equals

A =ZAL Z 01 0 [ I

=: Cr

Cr G00 1 ; e0shKh

In this note, two “competing” approaches to H 1 control for systems with a single delay have been put together, and the result is probably the simplest solution to date to the problem. Instrumental is the idea to reduce the problem to a one-block problem with a simple structure. In fact, this idea has been put to use to solve the case where there are multiple delays. This will be reported elsewhere. REFERENCES [1] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Upper Saddle River, NJ: Prentice-Hall, 1995. [2] C. Foias, H. Özbay, and A. Tannenbaum, Robust Control of Infinite Dimensional Systems: Frequency Domain Methods. London, U.K.: Springer-Verlag, 1996, vol. 209, Lecture Notes in Control and Information Sciences. [3] H. Dym, T. Georgiou, and M. C. Smith, “Explicit formulas for optimally robust controllers for delay systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 656–669, Apr. 1995. [4] T. Bas¸ar and P. Bernhard, -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, 2nd ed. Boston, MA: Birkhäuser, 1995. [5] A. Kojima and S. Ishijima, “Robust controller design for delay systems in the gap-metric,” IEEE Trans. Automat. Contr., vol. 40, pp. 370–374, Feb. 1995. [6] K. M. Nagpal and R. Ravi, “ control and estimation problems with delayed measurements: State-space solutions,” SIAM J. Control Optim., vol. 35, no. 4, pp. 1217–1243, 1997. [7] G. Tadmor, “Robust control in the gap: A state space solution in the presence of a single input delay,” IEEE Trans. Automat. Contr., vol. 42, pp. 1330–1335, Sept. 1997. [8] G. Meinsma and H. Zwart, “On control for dead-time systems,” IEEE Trans. Automat. Contr., vol. 45, pp. 272–285, Feb. 2000. [9] G. Tadmor, “The standard problem in systems with a single input delay,” IEEE Trans. Automat. Contr., vol. 45, pp. 382–397, Mar. 2000.

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H

H

H

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[10] L. Mirkin, “On the extraction of dead-time controllers from delay-free parametrizations,” in Proc. LTDS’2000, Ancona, Italy, 2000, pp. 157–162. control of system with I/O delay: A [11] L. Mirkin and G. Tadmor, “ review of some problem-oriented methods,” IMA J. Math. Control Inform., vol. 19, pp. 185–199, 2002. control problem for [12] G. Meinsma and H. Zwart, “The standard dead-time systems,” in Proc. MTNS’98 Symp., Padova, Italy, 1998, pp. 317–320. [13] I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, “The band method for positive and contractive extension problems,” J. Operator Theory, vol. 22, pp. 109–155, 1989. [14] H. Kimura, Chain-Scattering Approach to Control. Boston, MA: Birkhäuser, 1996. [15] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. London, U.K.: Springer-Verlag, 1995. controller synthesis by -lossless coprime factoriza[16] M. Green, “ tion,” SIAM J. Control Optim., vol. 30, no. 3, pp. 522–547, 1992. [17] Q.-C. Zhong, “Frequency-domain solution to delay-type Nehari problem using -spectral factorization,” Automatica, 2002, to be published. [18] E. G. F. Thomas, “Vector-valued integration with applications to the space,” J. Math. Control Inform., vol. 14, pp. operator valued 109–136, 1997. [19] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. New York: Springer-Verlag, 1995.

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H

H

J

H

J

H

BIBO Stability of Linear Switching Systems György Michaletzky and László Gerencsér

=

+

Abstract—In this note, we show that for linear switching systems of the form , where the matrices are chosen arbitrarily from a given set of matrices, bounded-input–bounded-output stability implies uniform exponential stability. Index Terms—Bounded-input–bounded-output (BIBO) stability, switching systems, uniform exponential stability.

I. INTRODUCTION Suppose that we are given a set of real p 2 p matrices A = fA j 2 0g, where 0 is an arbitrary set of indexes. Consider the switching linear system

x(n + 1) = A(n) x(n) + u(n + 1)

(1.1)

where x(0) 2 p ,  : [0; 1; . . .) ! 0 is an arbitrary switching signal, while u(1); u(2); . . . ; 2 p is an input signal. The problem of finding conditions for various kinds of stability of the system above has reManuscript received May 24, 2001; revised April 24, 2002. Recommended by Associate Editor J. M. A. Scherpen. This work was supported in part by the Swedish Research Council for Engineering Sciences (TFR), the Göran Gustafsson Foundation, the National Research Foundation of Hungary (OTKA) under Grants T 015668, T16665, and T032932, and the Research Development Grant of the Ministry of Education, Hungary (FKFP) 0194/1997. G. Michaletzky is with the Department of Probability Theory and Statistics, Eötvös Loránd University, Budapest 1117, Hungary, and also with the Computer and Automation Institute of the Hungarian Academy of Sciences. L. Gerencsér is with the Computer and Automation Institute of the Hungarian Academy of Sciences, Budapest 1111, Hungary. Digital Object Identifier 10.1109/TAC.2002.804470

1895

ceived a fair amount of attention recently, both for discrete-time and continuous-time systems. A number of papers focused on finding appropriate switching strategy in order to stabilize the system; see, e.g., [9], [10], and [15]. Another line of research is to characterize systems which are asymptotically stable for any arbitrary switching signal  ; see, e.g., [5], [7], [8], [12], and [13]. In [1] and [6], Lie-algebraic conditions are given implying the existence of a common quadratic Lyapunov-function. Dayawansa and Martin [3] proved that for compact linear polysystems uniform asymptotic stability implies the existence of a common Lyapunov function, which is not necessarily quadratic. Their proof is carried out for continuous-time systems, but without too much effort it works for discrete-time systems, as well. Stability of switching systems is an important technical issue in recursive identification and stochastic adaptive control. In the context of recursive identification, the estimated inverse system is time varying and the set of inverse systems should be restricted so that the stability of the time varying system is ensured. Assuming that the inverse plants can be described by finite-dimensional systems with fixed dimension the problem reduces to finding conditions under which the response of a time-varying linear system exited by a signal which is bounded in a certain stochastic sense, stays bounded in a certain stochastic sense. Such a condition is given in [4, Condition 4.2], where the existence of a joint quadratic Lyapunov function is assumed. In the light of the results of the work of [3], this condition is obviously too restrictive. The question arises as to what extent can [4, Condition 4.2] be relaxed. Similar problems arise in stochastic adaptive control, where the controller is time-varying, and the set of controllers should be restricted so that the stability of the time-varying closed-loop system is ensured for any choice of the controller. In this note, the following closely related result will be proved: assume that for any arbitrary switching path  the linear time-varying system (1.1) is bounded-input–bounded-output (BIBO) stable in the sense that for any bounded input sequence u(j ), j  1, the output sequence x(j ), j  0 is bounded, as well. Then the homogeneous linear switching-system is exponentially stable in a sense to be defined later. II. BIBO STABILITY p Denote by the set of infinite sequences of p-dimensional vectors, and by lp the set of norm-bounded infinite sequences of p-dimensional vectors. For any switching path  : [0; 1; . . .) 0, let us denote by T the linear operator for which

1

1

!

T (x(0); u(1); u(2); . . .) = (x(0); x(1); x(2); . . .) where the sequence x(j ), j

(2.1)

 0 is generated by

x(n + 1) = A(n) x(n) + u(n + 1) from the initial vector x(0). Obviously p ! p : T : 1 1

Note that introducing the initial condition x(01) = 0 we obtain that x(0) can be considered as u(0) = x(0). Using this, T can be written shortly as

T u = x where u

= (u(0); u(1); . . .), x = (x(0); x(1); . . .).

0018-9286/02$17.00 © 2002 IEEE