Finite-Time Control of Discrete-Time Linear Systems

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005

also like to gratefully acknowledge the helpful comments of an anonymous reviewer who suggested Corollary 4.1 and a simplification of the proof of Theorem 2.1.

Finite-Time Control of Discrete-Time Linear Systems

REFERENCES

Abstract—In this note, we consider the finite-time stabilization of discrete-time linear systems subject to disturbances generated by an exosystem. Finite-time stability can be used in all those applications where large values of the state should not be attained, for instance in the presence of saturations. The main result provided in the note is a sufficient condition for finite-time stabilization via state feedback. This result is then used to find some sufficient conditions for the existence of an output feedback controller guaranteeing finite-time stability. All the conditions are then reduced to feasibility problems involving linear matrix inequalities (LMIs). Some numerical examples are presented to illustrate the proposed methodology.

[1] B. D. O. Anderson and S. Vongpanitlered, Network Analysis and Synthesis: A Modern Systems Theory Approach. Upper Saddle River, NJ: Prentice–Hall, 1973. [2] V. Balakrishnan and L. Vandenberghe, “Semidefinite programming duality and linear time-invariant systems,” IEEE Trans. Autom. Control, vol. 48, no. 1, pp. 30–41, Jan. 2003. [3] R. Bhatia, Matrix Analysis. New York: Springer-Verlag, 1997. [4] O. Brune, “The synthesis of a finite two–terminal network whose driving–point impedance is a prescribed function of frequency,” J. Math. Phys., vol. 10, pp. 191–236, 1931. [5] M. K. Çamlibel, J. C. Willems, and M. N. Belur, “On the dissipativity of uncontrollable systems,” in Proc. 42nd IEEE Conf. Decision Control, Maui, HI, 2003, pp. 1645–1650. [6] J. Collado, R. Lozano, and R. Johansson, “On Kalman–Yakubovich–Popov Lemma for stabilizable systems,” IEEE Trans. Autom. Control, vol. 46, no. 7, pp. 1089–1093, Jul. 2001. [7] P. Faurre, M. Clerget, and F. Germain, Opérateurs Rationnels Positifs. Paris, France: Dunod, 1979. [8] A. Ferrante and L. Pandolfi, “On the solvability of the positive real Lemma equations,” Syst. Control Lett., vol. 47, no. 3, pp. 209–217, 2002. [9] T. Iwasaki, S. Hara, and H. Yamauchi, “Structure/control design integration with finite frequency positive real property,” in Proc. Amer. Control Conf., Chicago, IL, 2000, pp. 549–553. [10] T. Kailath, Linear System Theory. Upper Saddle River, NJ: PrenticeHall, 1980. [11] R. E. Kalman, “Lyapunov functions for the problem of Lur’e in automatic control,” in Proc. Nat. Acad. Sci., vol. 49, 1963, pp. 201–205. [12] P. Lancaster and R. Rodman, Algebraic Riccati Equation. Oxford, U.K.: Oxford Science Public, 1995. [13] A. Lindquist and G. Picci, “A geometric approach to modeling and estimation of linear stochastic systems,” J. Math. Syst., Estimat., Control, vol. 1, pp. 241–333, 1991. [14] G. Meinsma, Y. Shrivastava, and M. Fu, “A dual formulation of mixed and on the losslessness of ( ) scaling,” IEEE Trans. Autom. Control, vol. 42, no. 7, pp. 1032–1036, Jul. 1997. [15] A. A. Nudel’man and N. A. Schwartzman, “On the existence of the solutions to certain operatorial inequalities” (in Russian), Sib. Math. Z., vol. 16, pp. 562–571, 1975. [16] Y. Oono and K. Yasuura, “Synthesis of finite passive 2 -terminal networks with prescribed scattering matrices,” Mem. Eng. Faculty, Kyushu Univ., vol. 14, no. 2, pp. 125–177, 1954. [17] L. Pandolfi, “An observation on the positive real Lemma,” J. Math. Anal. Appl., vol. 255, pp. 480–490, 2001. [18] V. M. Popov, “Absolute stability of nonlinear systems of automatic control” (in Russian), Avt. i Telemekh., vol. 22, pp. 961–979, 1961. [19] A. Rantzer, “On the Kalman–Yakubovich–Popov Lemma,” Syst. Control Lett., vol. 28, pp. 7–10, 1996. [20] J. C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation,” IEEE Trans. Autom. Control, vol. AC-16, no. 6, pp. 621–634, Dec. 1971. [21] V. A. Yakubovich, “The frequency theorem in control theory,” Siberian J. Math., vol. 14, pp. 384–419, 1973. [22] D. C. Youla, “On the factorization of rational matrices,” IRE Trans. Inf. Theory, vol. IT-7, pp. 172–189, 1961.

Francesco Amato and Marco Ariola

Index Terms—Finite-time stability, linear systems, output feedback, state feedback.

I. INTRODUCTION When dealing with the stability of a system, a distinction should be made between classical Lyapunov stability and finite-time stability (FTS) (or short-time stability). The concept of Lyapunov asymptotic stability is largely known to the control community; conversely a system is said to be finite-time stable if, once we fix a time-interval, its state does not exceeds some bounds during this time-interval. Often asymptotic stability is enough for practical applications, but there are some cases where large values of the state are not acceptable, for instance in the presence of saturations. In these cases, we need to check that these unacceptable values are not attained by the state; for these purposes FTS could be used. Most of the results in the literature are focused on Lyapunov stability. Some early results on FTS can be found in [1], [2] and [3]; more recently the concept of FTS has been revisited in the light of recent results coming from linear matrix inequalities (LMIs) theory, which has made it possible to find less conservative conditions for guaranteeing FTS and finite time stabilization of uncertain, linear continuous-time systems (see [4]). Conversely, in this note, we deal with discrete-time systems. The first result is a sufficient condition for the existence of a state feedback controller which guarantees the finite-time stabilization of a discretetime linear system subject to disturbances generated by an exosystem. This result is then used to find a possible solution to the static output feedback finite-time stabilization problem. It is finally shown how these conditions can be turned into LMIs based feasibility problems. The note is organized as follows. In Section II, the definition of finite-time stability is recalled and specialized to the discrete-time case, and the problem we want to solve is formally stated. In Section III, the first main result of this note, a sufficient condition for the existence of a state feedback controller guaranteeing finite time stabilization of the closed-loop system, is provided. We also show that our sufficient condition for finite-time stabilization recovers, under stronger assumptions, asymptotic stability. In Section IV, the static output feedback case is discussed. The conditions for finite time stabilization found in the previous sections are then turned into optimization problems involving Manuscript received April 11, 2003; revised May 6, 2004 and December 21, 2004. Recommended by Associate Editor D. E. Miller. F. Amato is with the School of Computer and Biomedical Engineering, Università degli Studi Magna Græcia di Catanzaro, 88100 Catanzaro, Italy. M. Ariola is with the Dipartimento di Informatica e Sistemistica, Università degli Studi di Napoli Federico II, 80125 Napoli, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.847042

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LMIs, and some numerical examples are provided in Section V. Our conclusions are drawn in Section VI. II. PROBLEM STATEMENT AND PRELIMINARIES In this note, we consider the following discrete-time linear system:

x(k + 1) = Ax(k) + Bu(k) + Gw(k) w(k + 1) = F w(k)

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Problem 1: Given (1), find a state feedback controller (2) such that the closed-loop system is finite-time bounded with respect to (x ; w ; ; R; N ). This problem will be solved in Section III. In Section IV, we will make use of these results to tackle the more challenging output-feedback problem. III. STATE FEEDBACK STABILIZATION

(1a) (1b)

Let us first consider the following discrete-time system:

where A 2 n2n , B 2 n2m , G 2 n2r , and F 2 r2r . Given system (1), we consider the static state feedback controller

u(k) = Kx(k)

(2)

where K 2 m2n . The main aim of this note is to find some sufficient conditions which guarantee that the system given by the interconnection of (1) with the controller (2) is bounded over a finite-time interval. The general idea of finite-time stability concerns the boundedness of the state of a system over a finite time interval for given initial conditions; this concept can be formalized through the following definition, which is an extension to discrete-time systems of the one given in [1]. Definition 1 (Finite-Time Stability): The discrete-time linear system

x(k + 1) = Ax(k);

k2

0

is said to be finite-time stable with respect to (x ; ; R; N ), where R is a positive–definite matrix, 0 < x < , and N 2 0 , if

xT (0)Rx(0) x2

) xT (k)Rx(k) < 2

k2

AT P1 A 0 P1 GT P1 A

AT P1 G 0: x 1 0

2

0

Q12

=

Ip

0

0

0

:

If the problem is feasible, then the controller Ko given by the first p columns of LSQ101 renders system (26) finite-time bounded with respect to (x ; w ; ; R; N ). Example 1: Let us consider system

(31a)

x(k + 1) = Ax(k) + Bu(k) + Gw(k) w(k + 1) = F w(k)

(31c)

y (k) = Cx(k)

(32)

Now, starting from Theorem 1, we obtain the following result. LMI Feasibility Problem 1 (From Theorem 1): Given (13) and (x ; w ; ; R; N ), fix a 1 and find matrices Q1 > 0, Q2 > 0 and L, and positive scalars 1 and 2 satisfying the LMIs (14a) and

1 R01 < Q1 < R01

0

(31b)

for some positive numbers 1 and 2 . Inequality (31c) can be converted to an LMI using Schur complements; indeed it is equivalent to

Q11

=

2 n2n is defined as

V. COMPUTATIONAL ISSUES AND NUMERICAL EXAMPLES In this section, we will show how, once we have fixed a value for , the feasibility of the conditions stated in the previous Sections can be turned into LMI-based feasibility problems [8]. To this aim, first of all it is easy to check that condition (14b) is guaranteed by imposing the conditions

C H

=

with

A= F

=

1

3

01 01 0: 8

0 0 :6

0:6

0 :8

B

=

C

01

G=

1

= (2

1

0

0

1

1):

The chosen F matrix has complex eigenvalues of unitary modulus; therefore, we are considering the case of a sinusoidal disturbance. We have considered the following three cases: 1) finite-time stability via state feedback: a) w(0) = 0; b) u(k) = Kx(k); 2) finite-time boundedness via state feedback: a) w(0) 6= 0; b) u(k) = Kx(k); 3) finite-time boundedness via output feedback: a) w(0) 6= 0; b) u(k) = Ko y (k).


In this example, we decided to perform an optimization over using the algorithm sketched below, with the aid of the Matlab LMI Control Toolbox [9]. Step 1) We chose some given fixed values for x , w , R and N . Step 2) We decided an initial value for . Step 3) We solved the specific LMI feasibility problem. Since the parameter can range from 1 to (=x )(2=N ) (this bound is obtained from (31c) letting 1 = 1 and 2 = 0), starting from = 1 we kept increasing until a solution is found or the maximum value for is reached. Step 4) If no solution is found then the initial value for needs to be increased, otherwise can be decreased until its minimum value is found. In this case, we chose x = 1, w = 3, R = I2 , N = 4, and an initial value for equal to 30. Case 1) We solved the LMI Problem 2 and we found that the controller

K

Case 2)

1:54)

solves our problem with = 2:8 and = 1. The value of (see Remark 3) implies that the state boundedness is guaranteed for all N 2 0 and that the closed-loop system is also asymptotically stable. We solved the LMI Problem 1 and found that the controller

K

Case 3)

= (1:02

= (1:00

1:50)

guarantees the desired closed-loop properties with = 16:9 and = 1:36. The solution of the LMI Problem 3 with H = (00:1 10) leads to the controller Ko

Ko

= 0:51

which solves our problem with = 25:5 and = 1:43. Remark 6: Note that in Example 1, we keep x fixed and optimize over . In a similar way one can fix and look for the maximum admissible x guaranteeing the desired closed-loop finite-time property. VI. CONCLUSION In this note, we have considered the finite-time stabilization problem for a discrete-time linear system subject to disturbances generated by an exosystem. The first result of the note is a sufficient condition for finite-time state feedback stabilization; then the output feedback problem has been considered and a further condition guaranteeing the existence of a static output feedback controller has been provided. These conditions have been then turned into optimization problems involving LMI’s. The proposed method has been illustrated through some numerical examples. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewer for suggesting the necessary and sufficient condition (12). REFERENCES [1] P. Dorato, “Short time stability in linear time-varying systems,” in Proc. IRE Int. Convention Record Part 4, 1961, pp. 83–87. [2] L. Weiss and E. F. Infante, “Finite time stability under perturbing forces and on product spaces,” IEEE Trans. Autom. Control, vol. AC-12, pp. 54–59, Jan. 1967.

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[3] H. D’Angelo, Linear Time-Varying Systems: Analysis and Synthesis. Boston, MA: Allyn and Bacon, 1970. [4] F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica, vol. 37, no. 9, pp. 1459–1463, Sep. 2001. [5] F. M. Callier and C. A. Desoer, Linear System Theory. New York: Springer-Verlag, 1991. [6] G. Garcia, J. Bernussou, and D. Arzelier, “Stabilitazion of an uncertain linear dynamic system by state and output feedback: A quadratic stabilizability approach,” Int. J. Control, vol. 64, no. 5, pp. 839–858, 1996. synthesis,” [7] P. Gahinet, “Explicit controller formulas for LMI-based Automatica, vol. 32, no. 7, pp. 1007–1014, 1996. [8] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [9] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: The Mathworks, Inc., 1995.

On Distributed Delay in Linear Control Laws—Part II: Rational Implementations Inspired From the -Operator Qing-Chang Zhong Abstract—This note proposes rational implementations for distributed delay in linear control laws. The main benefit of doing so is the easy implementation of rational transfer functions. The proposed approach was inspired from the -operator. The resulting rational implementation has an elegant structure of chained low-pass filters. The stability of each node can be guaranteed by the choice of the number of the nodes. The sta-norm bility of the closed-loop system can also be guaranteed since the of the implementation error approaches 0 when goes to . Moreover, the steady-state performance of the system is retained without the need to change the control structure. Index Terms— -operator, dead-time compensator, distributed delay, finite-spectrum assignment, implementation error, modified Smith predictor, rational implementation.

I. INTRODUCTION Distributed delays, which are finite-impulse-response (FIR) blocks, often appear as a part of dead-time compensators for processes with dead time, in particular, as a part of the finite-spectrum-assignment (FSA) control law [1]–[3] or in the form of a modified Smith predictor [4], [5] for unstable processes. Distributed delays also appear in H control of (even, stable) dead-time systems [6]–[10] and continuous-time deadbeat control [11]. Due to the requirement of internal stability, such an FIR block has to be, approximately, implemented as a stable block without hidden unstable poles. A common way to do this is to replace the distributed delay by the sum of a series of discrete (often commensurate) delays [1], [4], [5] (other interesting implementations using resetting mechanism can be found in [12] and [13]). There have been some arguments about the possibility of causing instability by doing this, which has attracted a lot of attention from the delay community; see [13]–[22]. It was proposed as an open problem in the survey paper [23]. Very recently, it has been

1

Manuscript received July 18, 2004; revised December 9, 2004. Recommended by Associate Editor Hong Wang. This work was supported by the EPSRC under Grant EP/C005953/1. The author is with the School of Electronics, University of Glamorgan, Pontypridd CF37 1DL, U.K. (email: [email protected]). Digital Object Identifier 10.1109/TAC.2005.847043

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