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Stability and stabilization of linear discrete-time systems with interval delay.

1. Introduction

The problem of stability and stabilization for time-delay systems have been given considerable attention over the past decades. The existing stabilization results for time delay systems can be classified into two type, i.e. delay independent stabilization (see [7, 8,9,10]) and delay-dependent stabilization (see [11,12,13]). The delay- independent stabilization provides a controller which stabilizes a system irrespective of the extent of the delay. On the other hand, the delay-dependent stabilization is concerned with the size of the delay which usually provides an upper bound of the delay capable of ensuring the stability for any delay lower than the upper bound.

Recently, the problem of delay-dependent stability analysis for time-delay systems has received considerable attention, and lots of significant results have been reported (see, for example, Chen et al. [18], He et al. [19], Lin et al. [20], Phat [21], and Xu and Lam [22], and the references therein). Among these references, we note that the delay-dependent stability problem for discrete-time systems with interval-like time-varying delays (i.e., the delay h(k) satisfies 0 < [h.sub.m] [less than or equal to] h(k) [less than or equal to] [h.sub.M]) has been studied by Fridman and Shaked [6], Gao and Chen [2], Gao et al. [3], and Jiang et al. [4], where some LMI-based stability criteria have been presented by constructing appropriate Lyapunov functionals and introducing free-weighting matrices. It should be pointed out that the Lyapunov functionals considered in these references are more restrictive due to the ignorance of the term

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover, the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also ignored in Gao and Chen [2] and Gao et al. [3]. The ignorance of these terms may lead to considerable conservativeness.

As most physical systems occur in continuous time, consequently the theories for stability analysis and controller synthesis are mainly developed for the continuous time. However, it is more feasible that a discrete-time approach is used for the purpose, as the controller is usually digitally implemented. Despite this significance, discrete-time systems with delays (see [6, 14, 15, 16, 17]) have not been paid due attention. It is mainly due to the fact that the delay-difference equations with known delays can be converted into a higher-order smaller-delay system by augmentation approach. However, in systems with great delay extent, this scheme will lead to large-dimensional systems. Futhermore, for systems with unknown delay the augmentation scheme is not applicable.

The present study consists of two parts. In the first part, based on a new Lyapunov functional, an improved delay-dependent stability criterion for discrete-time systems with time-varying delays is presented in terms of LMIs. It is shown that the obtained result is less conservative than those by Ratchagit and Phat [1], Zhang et al. [5], Fridman and Shaked [6], Gao and Chen [2], Gao et al. [3], and Jiang et al. [4]. In the second parts, the stability result proposed in the first part is applied to design a time-delayed controller for a discrete-time linear system. Two examples are providing, respectively, to demonstrate the reduced conservatism of the proposed stability and stabilization criterion.

2. Preliminaries

We collect the following lemmas which be used in the proof for an improved stability criterion in the next section.

Lemma 2.1. (see [23]) The zero solution of difference system is asymptotic stability if there exists a positive definite function V(x(k)): [R.sup.n] [right arrow] [R.sup.+] such that

[DELTA]V(x(k)) = V(x(k + 1)) - V(x(k)) [less than or equal to] - [beta][[parallel]x(k)[parallel].sup.2], [there exist] [beta] > 0,

along the solution of the system. In the case the above condition holds for all x(k) [member of] [V.sub.[delta]] := {x [member of] [R.sup.n]: [parallel]x[parallel] < [delta]}, say one that the zero solution is locally asymtotically stable.

3. Stability Criterion

In this section, we give a novel delay-dependent stability condition for discrete-time systems with interval-like time-varying delays. Now, consider the following system:

x(k + 1) = Ax(k) + Bx(k - h(k)), (3.1)

where x(k) [member of] [R.sup.n] is the state vector, A and B are known constant matrices, and h(k) is a time-varying delay satisfying 0 < h(k) [less than or equal to] h, where h is a positive integer. For the system (3.1), we have the following result.

Theorem 3.1. The discrete time-delay system (3.1) is asymptotically stable for any time delay h(k) [less than or equal to] h, if there exist symmetric positive definite matrices P, Q and R, and positive definite matrices S and W such that S [less than or equal to] R and W [less than or equal to] P satisfying the following LMI:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Proof. Consider the Lyapunov function

V(x(k)) = [x.sup.T](k)Px(k) + [k-1.summation over (i=k-h(k))] (h(k) - k + i)[x.sup.T](i) Qx(i) + [k-1.summation over (i=k-h(k))] [x.sup.T](i)Rx(i),

where P, Q and R being symmetric positive definite of solution (3.2).

Let S and W be positive definite such that S [less than or equal to] R and W [less than or equal to] P. Then, difference of V(x(k)) along trajectory of solution (3.1) is given by [DELTA]V(x(k)) and by the positively definite of Q, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from condition (3.2) that there exists a number [delta] > 0 such that [DELTA]V(x(k)) [less than or equal to] -[delta][[parallel]x(k)[parallel].sup.2], and we have V(x(k)) > 0 by the positively definite of P, Q and R, and Hence, the asymptotic stability of the system (3.1) immediately follows from Lemma 2.1. This completes the proof.

Remark 3.2. Theorem 3.1 gives a sufficient condition for stability criterion for discrete-time system (3.1). These condition are described in terms of certain symmetric matrix over [R.sup.2nx2] inequalities, which the obtained result is less conservative than those condition by Ratchagit and Phat [1] are described in terms of certain diagonal matrix over [R.sup.3nx3n] inequalities, and Zhang et al. [5] are described in terms of certain symmetric matrix over [R.sup.7nx7n] inequalities, and those condition can be realized by using the Schur complement lemma and linear matrix inequality algorithm proposed in Callier and Desoer [24].

4. Time-delayed controller design

The delay-dependent stability condition obtained in the previous section will be applied in this section to design a time delayed state feedback controller for the discrete-time linear system described by

x(k + 1) = Ax(k) + Cu(k), (4.1)

where x(k) [member of] [R.sup.n] is the state, u(k) [member of] [R.sup.m] is the control input, A and C are known constant matrices of appropriate dimensions. Now, consider a time-delayed state feedback control law

u(k) = Fx(k - h(k)), (4.2)

where F is the controller gain to be determined and h(k) is a time-varying delay satisfying 0 < h(k) [less than or equal to] h. Applying the controller (4.2) to the system (4.1) results in the closed-loop system

x(k + 1) = Ax(k) + CFx(k - h(k)). (4.3)

For the system (4.1), we have the following result.

Theorem 4.1. The discrete time-delay system (4.1) is stabilization by a time- delayed state feedback control u(k) = [C.sup.-1]Ux(k - h(k)), where U [member of] [R.sup.nxn] for any time delay h(k) [less than or equal to] h, if there exist symmetric positive definite matrices P, Q and R, and positive definite matrices S and W such that S [less than or equal to] R and W [less than or equal to] P satisfying the following LMI:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

Proof. It is concluded obviously, from condition (3.2) and denote B = CF = C[C.sup.-1]U = U, by Theorem 3.1, the desired result immediately follows.

5. Numerical examples

In this section, two examples are provided to demonstrate the reduced conservatism of the newly proposed stability and stabilization criterion by using the Matlab LMI Control Toolbox.

Example 5.1. Consider a time varying delay h(k) satisfying 0 < [h.sub.m] [less than or equal to] h(k) [less than or equal to] [h.sub.M]. For given [h.sub.m] and by using the Matlab LMI Control Toolbox for A = s [I.sub.n] and B = p [I.sub.n] where s < 0, p > 0 and In is an identity matrix in the n dimension, guarantees that there exist A,B,P,Q,R,S,W and [h.sub.M] satisfying LMI (3.2) in Theorem 3.1.

Step 1: To find B, P, Q, R, S, W and [h.sub.M] for given [h.sub.m] = 6,A = - 0.1234[I.sub.2] and B > 0 where B [member of] [R.sup.2x2], and also seted conditions P, Q,R > 0, 0 < S [less than or equal to] R and 0 < W [less than or equal to] P. By using the Matlab LMI Control Toolbox, we obtain [h.sub.M] = 10, B = 0.9574 x [10.sup.-12][I.sub.2], P = 1.2004[I.sub.2], Q = 0.4499 x [10.sup.- 12][I.sub.2], R = 0.5235[I.sub.2], S = 0.5235[I.sub.2] and W = 1.2004[I.sub.2].

Step 2: To verify the LMI (3.2) in Theorem 3.1. By step 1, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all h = 6, 7,..., 10 such that 6 [less than or equal to] h(k) [less than or equal to] h.

Step 3: To verify the graph of system (3.1) for A = -0.1234[I.sub.2] and B = 0.9574 x [10.sup.-12][I.sub.2] such that 6 [less than or equal to] h(k) [less than or equal to] 10. These graph is shown in Figure 1.

[FIGURE 1 OMITTED]

Therefore, for given A = -0.1234[I.sub.2], B = 0.9574 x [10.sup.-12][I.sub.2] such that 6 [less than or equal to] h(k) [less than or equal to] 10, there exist P = 1.2004[I.sub.2], Q = 0.4499 x [10.sup.-12][I.sub.2], R = 0.5235[I.sub.2], S = 0.5235[I.sub.2] and W = 1.2004[I.sub.2] such that S [less than or equal to] R and W [less than or equal to] P, that guarantees the asymptotic stability of system (3.1). For more examples are presented in Table 1.

Example 5.2. Consider the following system

x(k + 1) = Ax(k) + Cu(k), (5.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and 1 [less than or equal to] h(k) [less than or equal to] 20. The state response of the open-loop system with u(k) = x(k - h(k)) is shown in Figure 2. It is obvious that this system is not stable. Now, by applying Theorem 4.1 and using the Matlab LMI Control Toolbox, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that guarantees the stabilization of system (5.1) by a time-delayed state feedback control

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Using this controller, the state response of the closed-loop system is shown in Figure 3.

6. Conclusions

In this paper, improved delay-dependent conditions for stability and stabilization of discrete-time linear systems with time-varying delays have been presented in terms of LMIs. Numerical examples have been provided to show the effectiveness of the conditions by using Matlab LMI Control Toolbox.

References

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[2] H. Gao and T. Chen, New results on stability of discrete-time systems with time-varying state delay, IEEE Transactions on Automatic Control, 52(2):328-334, 2007.

[3] H. Gao, J. Lam, C. Wang, and Y. Wang, Delay-dependent output-feedback stabilization of discrete-time systems with time-varying state delay, IEEE Proceedings: Control Theory and Applications, 151(6):691-698, 2004.

[4] X. Jiang, Q. L. Han, and X. Yu, Stability criteria for linear discrete-time systems with interval-like time-varying delay, In Proceedings of the American control conference, pp. 2817-2822, 2005.

[5] B. Zhang, S. Xu, and Y. Zou, Improved stability criterion and its applications in delayed controller design for discrete-time systems, Automatica, 44(11):2963 2967, 2008.

[6] E. Fridman and U. Shaked, Stability and guaranteed cost control of uncertain discrete delay systems, International Journal of Control, 78(4):235-246, 2005.

[7] J. Chen, H. A. Latchman, Asymptotic stability independent of delays: simple necessary and sufficient conditions, Proceedings of American Control Conference, Baltimore, USA, pp. 1027-1031, 1994.

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[11] T. Mori, Criteria for asymptotic stability of linear time delay systems, IEEE Transaction on Automatic Control, 30(2):158-161, 1985.

[12] J. Chiasson, A method for computing the interval of delay values for which a differential delay system is stable, IEEE Transaction on Automatic Control, 33(12):11761178, 1988.

[13] R. Manzanilla, L. G. Marmol, and C. J. Vanegas, On the Controllability of a Differential Equation with Delayed and Advanced Arguments, Abstract and Applied Analysis, Vol. 2010, Article ID 307409, 16 pages, doi:10.1155/2010/307409.

[14] T. Mori, N. Fukuma, and M. Kuwahara, Delay-independent stability criteria for discrete-delay systems, IEEE Transaction on Automatic Control, 27(4):964-966, 1982.

[15] V. Kapila, W. M. Haddad, Memoryless Ho controllers for discrete-time systems with time delay, Automatica, 34(9):1141-1144, 1998.

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[17] S. Song, J. Kim, C. Yim, and H. Kim, Ho control of discrete-time linear systems with time varying delays in state, Automatica, 35(9):1587-1591, 1999.

[18] W. H. Chen, Z. H. Guan, and X. Lu, Delay-dependent guaranteed cost control for uncertain discrete-time systems with delay, IEEE Proceedings: Control Theory and Applications, 150(4):412-416, 2003.

[19] Y. He, Q.-G. Wang, C. Lin, and M. Wu, Delay-range-dependent stability for systems with time-varying delay, Automatica, 43(2):371-376, 2007.

[20] C. Lin, Q.-G. Wang, and T. H. Lee, A less conservative robust stability test for linear uncertain time-delay systems, IEEE Transactions on Automatica Control, 51(1):87-91, 2006.

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Pattanapong Tianchai

Faculty of Science, Maejo University, Chiangmai 50290, Thailand.

E-mail: pattana@mju.ac.th
Table 1: Calculated B, P, Q, R, S, W and [h.sub.M] for
given [h.sub.m] = 6,A = -0.1234[I.sub.2] and B>0.

[h.sub.m] [h.sub.M] A < 0, B > 0

6 10 A = -0.1234[I.sub.2]
 B = 0.9574 x
 [10.sup.-12] [I.sub.2]
6 20 A = -0.1234[I.sub.2]
 B = 0.5923 x
 [10.sup.-12] [I.sub.2]
6 30 A = -0.1234[I.sub.2]
 B = 0.1824 x
 [10.sup.-13] [I.sub.2]
6 40 A = -0.1234[I.sub.2]
 B = 0.5549 x
 [10.sup.-11] [I.sub.2]
6 50 A = -0.1234[I.sub.2]
 B = 0.4349 x
 [10.sup.-11] [I.sub.2]

[h.sub.m] P,Q,R > 0

6 P = 1.2004[I.sub.2], Q = 0.4499 x
 [10.sup.-12] [I.sub.2] R = 0.5235
 [I.sub.2]
6 P = 1.2004[I.sub.2], Q = 0.1749 x
 [10.sup.-12] [I.sub.2] R = 0.5235
 [I.sub.2]
6 P = 1.2004[I.sub.2], Q = 0.3935 x
 [10.sup.-14] [I.sub.2] R = 0.5235
 [I.sub.2]
6 P = 1.2004[I.sub.2], Q = 0.8902 x
 [10.sup.-12] [I.sub.2] R = 0.5235
 [I.sub.2]
6 P = 1.2004[I.sub.2], Q = 0.2761 x
 [10.sup.-12] [I.sub.2] R = 0.5235
 [I.sub.2]

[h.sub.m] 0 < S [less than or
 equal to] R, 0 < W
 [less than or equal to] P

6 S = 0.5235 [I.sub.2]
 W = 1.2004 [I.sub.2]

6 S = 0.5235 [I.sub.2]
 W = 1.2004 [I.sub.2]

6 S = 0.5235 [I.sub.2]
 W = 1.2004 [I.sub.2]

6 S = 0.5235 [I.sub.2]
 W = 1.2004 [I.sub.2]

6 S = 0.5235 [I.sub.2]
 W = 1.2004 [I.sub.2]
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Author:Tianchai, Pattanapong
Publication:International Journal of Computational and Applied Mathematics
Date:Nov 1, 2010
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