Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 414373, 27 pages

doi:10.1155/2012/414373

Research Article

Exponential Estimates and Stabilization of Discrete-Time Singular Time-Delay Systems Subject to Actuator Saturation

Jinxing Lin1,2

¹College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

²Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing 210096, China

Received 29 February 2012; Revised 20 June 2012; Accepted 24 July 2012

Academic Editor: Recai Kilic

Copyright © 2012 Jinxing Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with exponential estimates and stabilization of a class of discrete-time singular systems with time-varying state delays and saturating actuators. By constructing a decay-rate-dependent Lyapunov-Krasovskii function and utilizing the slow-fast decomposition technique, an exponential admissibility condition, which not only guarantees the regularity, causality, and exponential stability of the unforced system but also gives the corresponding estimates of decay rate and decay coefficient, is derived in terms of linear matrix inequalities (LMIs). Under the proposed condition, the exponential stabilization problem of discrete-time singular time-delay systems subject actuator saturation is solved by designing a stabilizing state feedback controller and determining an associated set of safe initial conditions, for which the local exponential stability of the saturated closed-loop system is guaranteed. Two numerical examples are provided to illustrate the effectiveness of the proposed results.

1. Introduction

Singular time-delay systems (STDSs) arise naturally in many engineering fields such as electric networks, chemical processes, lossless transmission lines, and so forth [1]. A STDS is a mixture of delay differential equations and delay difference equations; such a complex nature of STDS leads to abundant dynamics, for example, non-strictly proper transcendental equations, irregularity, impulses or non-causality. Therefore, the study of such systems is much more complicated than that for normal state-space time-delay systems. In the past two decades, a great number of stability results on STDSs have been reported in the literature; see, for example, [2–8] and the references therein.

It is noted that many stability results for STDSs are concerned with asymptotic stability. Practically, however, exponential stability is more important because the transient process of a system can be described more clearly once the decay rate is determined [9]. Therefore, in recent years, the study of exponential estimates problem of STDSs has received increasing attention, and a few approaches have been proposed. For example, in [10, 11], the STDS was decomposed into slow (differential) and fast (algebraic) subsystems and the exponential stability of the slow subsystem was proved by using the Lyapunov method. Subsequently, the solutions of the fast subsystem was bounded by an exponential term using a function inequality. However, this approach cannot give an estimate of the convergence rate of the system. To overcome this difficulty, Shu and Lam [12] and Lin et al. [13] adopted the Lyapunov-Krasovskii function method [14, 15] and some improvements have been obtained. In [16, 17], an exponential estimates approach for SSTDs was presented by employing the graph theory to establish an explicit expression of the state variables of fast subsystem in terms of those of slow subsystem and the initial conditions, which allows to prove the exponential stability of the fast subsystem. However, all of the above results are related to continuous-time STDSs. To the best of the authors' knowledge, the problem of exponential estimates of discrete-time STDSs has not been investigated yet. One possible reason is the difficulty in obtaining the estimates for solutions of the corresponding fast subsystem. Therefore, the first aim is to develop effective approach to give the exponential estimates of discrete-time STDSs.

On the other hand, actuator saturation is also an important phenomenon arising in engineering. Saturation nonlinearity not only deteriorates the performance of the closed-loop systems but also is the source of instability. Stabilization of normal state-space systems subject to actuator saturation has therefore attracted much attention from many researchers; see, for example, [18–22], and the references cited therein. Recently, some results for normal state-space systems have been generalized to singular systems. For example, semiglobal stabilization and output regulation of continuous-time singular system subject to input saturation were addressed in [23] by assuming that the open-loop system is semistable and impulse free which allows a state transformation such that the singular system is transformed into a normal system. Also, an algebraic Riccati equation approach to semiglobal stabilization of discrete-time singular linear systems with input saturation was proposed in [24] without any transformation of the original singular system. The invariant set approach developed for state-space system in [18] was extended to general, not necessarily semistable, continuous-time singular system in [25]. This approach was further extended to the analysis of the gain and performance for continuous-time singular systems under actuator saturation [26] and the analysis and design of discrete-time singular systems under actuator saturation [27, 28], respectively. In [17], estimation of domain of attraction for continuous-time STDSs with actuator saturation and the design of static output feedback controller that maximize it were proposed. However, so far, few work exists to address the stabilization problem for discrete-time STDSs subject to actuator saturation, which forms the second object of this paper.

In this paper, we investigate the problems of exponential estimates and stabilization for a class of discrete-time singular systems with time-varying delays and saturating actuators. The main contributions of the paper are twofold:

(1)In terms of linear matrix inequalities (LMIs), an exponential admissibility condition, which not only guarantees the regularity, causality and exponential stability of the unforced system but also gives the corresponding estimates of decay rate and decay coefficient, is derived by constructing a decay-rate-dependent Lyapunov-Krasovskii function and using the slow-fast decomposition. (2)The exponential stabilization problem of STDSs with saturating actuators is solved by designing a stabilizing state feedback controller and determining an associated set of safe initial conditions, for which the local exponential stability of the saturated closed-loop system is guaranteed. The existence criterion of the desired controller is formulated, and an LMI optimization approach is proposed to enlarge the domain of safe initial conditions.

The paper is organized as follows. Problem statement and the preliminaries are given in Section 2. In Section 4, we present the exponential estimates for the STDSs and the solutions to the stabilization problem for the system with saturating actuators. Numerical examples will be given in Section 4 to illustrate the effectiveness of the proposed method. The paper will be concluded in Section 5.

Notation 1. For real symmetric matrices , ) means that matrix is positive definite (semipositive definite). ) denotes the largest (smallest) eigenvalue of the positive definite matrix . is the -dimensional real Euclidean space and is the set of all real matrices. represents the sets of all non-negative integers. The superscript “” represents matrix transposition, and “” in a matrix is used to represent the term which is induced by symmetry. stands for a block-diagonal matrix. is the shorthand notation for . For two integers and with , we use to denote the integer set . Let denote the Banach space of family continuous vector valued functions mapping the interval to with the topology of uniform convergence. Denote , . refers to either the Euclidean vector norm or the induced matrix two-norm. For a function , its norm is defined as .

2. Problem Statement and Preliminaries

2.1. Problem Statement

Consider a class of discrete-time singular system subject to time-varying delay and actuator saturation as follows:

where is the system state, is the control input, and is a compatible vector valued initial function. is a time-varying delay satisfying , where and are constant positive scalars representing the minimum and maximum delays, respectively. The matrix is singular and . , and are known constant matrices. The function is the standard saturation function defined as , where . Note that the notation of is slightly abused to denote scalar values and vector valued saturation functions. Also note that it is without loss of generality to assume unity saturation level [20].

In this paper, we consider the design of a linear state feedback control law of the following form [27]:

where and . The closed-loop system under this feedback is given by

Remark 2.1. The state feedback of the form (2.2) is used to guarantee the uniqueness of the solution of system (2.1). For that, since , there exist two nonsingular matrices such that

If the state feedback is taken as the general form, that is, , by using (2.4), then system (2.3) is restricted system equivalent (r.s.e.) to the following one:

From the second equation of (2.5), it can be seen that is in the function . Hence, if , for given , the solution of is not unique even is nonsingular. However, if , then system (2.3) is r.s.e. to

which implies that, for given , the unique solution of can be obtained when is nonsingular. Nevertheless, it should be pointed that the state feedback (2.2) may be conservative due to its special structure.

To describe the main objective of this paper more precisely, we introduce the following definitions.

Definition 2.2 (see [29]). System (or the pair ) is said to be regular if is not identically zero, and if , then it is further said to be causal.

Definition 2.3 (see [6]). System (2.1) with is said to be regular and causal, if the pair is regular and causal.

Note that regularity and causality of system (2.1) with ensure that the solution to this system exists and is unique for any given compatible initial value .

Definition 2.4. System (2.1) under feedback law (2.2) is said to be exponentially stable with decay rate () if, for any compatible initial conditions , , its solution satisfies for all , where , is the initial time step, and is the decay coefficient.

We are interested in the exponential estimates and design for system (2.3). For any compatible initial condition , denote the state trajectory of system (2.1) as ; then the domain of attraction of the origin is

In general, for a given stabilizing state feedback gain , it is impossible to determine exactly the domain of attraction of the origin with respect to system (2.3). Therefore, the purpose of this paper is to design a state feedback gain and determine a suitable set of initial condition from which the regularity, causality, and exponential stability of the closed-loop system (2.3) is ensured. Also, we are interested in maximizing the size of this set, that is, obtaining the maximal value of .

2.2. Preliminary Results

Define

Lemma 2.5. For any appropriately dimensioned matrices and , two positive time-varying integer and satisfying , and a scalar , the following equality holds

where .

Proof. See the Appendix.

Lemma 2.6 (see [30]). Given a matrix , let a positive-definite matrix and a positive scalar exist such that

then the matrix satisfies the bound

where and .

Lemma 2.7. Let . Consider the following system:

where ,  ,  ,  and  , . If

Then, for any compatible initial function , the solution of (2.10) satisfies

where .

Proof. See the Appendix.

For a matrix , denote the row of as and define

Let be a positive-definite matrix and , the set is defined by

Also, let be the set of diagonal matrices whose diagonal elements are either or . There are elements in . Suppose that each element of is labeled as and denote . Clearly, is also an element of if .

Lemma 2.8 (see [18]). Let be given. If , then can be expressed as

where for are some variables satisfying and .

Lemma 2.9 (see [31]). Given matrices , , and with appropriate dimensions, and is symmetric. Then there exists a scalar , such that and .

3. Main Results

In this section, we will first present a delay-dependent LMI condition which guarantees the regularity, causality and exponential stability of the unforced system (2.1) (i.e., with ) with a predefined decay rate.

Theorem 3.1. Given constants and . If there exist symmetric matrices , , , , and , and matrices , and such that the following inequality holds

where

and is any constant matrix satisfying and , then the unforced system (2.1) is regular, causal, and exponentially stable with .

Proof. The proof is divided into three parts: (i) To show the regularity and causality; (ii) to show the exponential stability of the difference subsystem; (iii) to show the exponential stability of the algebraic subsystem.

Part (i): Regularity and causality. Since is regular and , there exist two nonsingular matrices and such that

Note that and , it can be verified that , and , that is,

From (3.1), it is easy to obtain . In view of ,  ,  ,  ,  ,  ,  and , it can be further obtained that

Pre- and postmultiplying (3.5) by and , respectively, and using (3.3) and (3.4), it is obtained that

where represents matrices that are not relevant in the following discussion. Thus,

Now, we assume that the matrix is singular, then, there exists a vector and such that . Pre- and postmultiplying (3.7) by and , respectively, result in . Then, it is easy to see that (3.7) is a contradiction since . Thus, is nonsingular, which implies that the unforced system (2.1) is regular and causal by Definition 2 and Theorem 1 in [4].

Part (ii): Exponential stability of the difference subsystem. From [29], the regularity and causality of the unforced system (2.1) imply that there exist two nonsingular matrices and such that

According to (3.8), define

By using Schur complement on (3.1), we get

where

Substituting (3.8) and (3.9) into the above inequality, pre- and postmultiplying by and , respectively, and using Schur complement yields

Pre- and postmultiplying this inequality by and its transpose, respectively, and noting , we have

Therefore, according to Lemma 2.6, there exist constants and such that

Let , where and . Then, the unforced system (2.1) is r.s.e. to the following one:

Now, choose the following Lyapunov-Krasovskii function:

where

with , and . Define

Then, it follows from (2.1) that

Let

Using Lemma 2.5 for the last three terms of , respectively, and noting , we have

Note that provides

Also, it follows from that

Then, substituting (3.19)–(3.24) into (3.18), using (3.25), and noting , and , we get

where follows the same definition as defined in (3.1) with , , , , , , , , and instead of , , , , , , , , and . Performing a congruence transformation on (3.1) by , and then using the Schur complement implies . Thus, it follows from (3.18) and (3.26) that , which leads to

By iterative substitutions, inequality (3.27) yields

On the other hand, it follows from the Lyapunov functional (3.16) that

where

Then, combining (3.28) and (3.29) leads to

Therefore, the difference subsystem of (3.15) is exponentially stable with a decay rate which is not less than . The remaining task is to show the exponential stability of the algebraic subsystem.

Part (iii): Exponential stability of the algebraic subsystem. Set ; then, it follows from (3.31) that

Using the second equation in (3.15), (3.14) and Lemma 2.7, one gets

where

Combining (3.31) and (3.33) yields that

Thus, we have

The proof is completed.

Remark 3.2. Theorem 3.1 is obtained by applying a Lyapunov-Krasovskii function method to both the difference and algebraic subsystems of a discrete-time singular system with time-varying delay. Such a method in dealing with the algebraic subsystem of the discrete-time singular delay system has not been reported in the literature.

Remark 3.3. When in system (2.1), then , and we get the exponential stability condition for the standard delay systems from (3.1).

Based on the result of Theorem 3.1, we now present the existence conditions of a stabilizing state feedback controller for system (2.1) and the corresponding set of initial condition.

Theorem 3.4. Given constants , , , , , , , and . If there exist symmetric matrices , , , , , and , and matrices , , , and such that the following inequalities hold:

where denotes the row of ,

, , , , are defined in (3.1), and is any constant matrix satisfying with , then system (2.3) is regular, causal, and locally exponentially stable with for any compatible initial condition in the ball

where

Proof. If (3.38) holds, then the ellipsoid is included in [27] (For more details about the ellipsoids and ellipsoid algorithm, we refer the readers to [32–34]). Suppose that , (will be proved later). Hence, by Lemma 2.8, can be expressed as

and it follows that

Choose a Lyapunov function as in (3.16), and then, by Theorem 3.1, system (3.43) is regular, causal and locally exponentially stable if there exist symmetric matrices , , , , , and , and matrices , , and defined in (3.1) such that

where

Now, provided that inequalities (3.37) hold, then by using Schur complement and Lemma 2.9, one has

This, together with and , implies that

that is,

Applying Schur complement to (3.49) leads to

where

Rewrite

where

From (3.37), it follows that , and then the following inequality holds from Lemma 2.9

which, together with (3.50) and (3.52), implies that

Applying Schur complement, it results that inequality (3.55) is equivalent to inequality (3.44). Therefore, if inequalities (3.37) hold, then inequality (3.44) holds, that is, system (3.43) is regular, causal, and exponentially stable, provided that , .

On the other hand, by the proof in Theorem 3.1, inequality (3.44) holding implies . Assume the initial condition is included in the ball , , and inequalities (3.37) and (3.38) hold. Then, from the definition of , we get , that is, . Since inequalities (3.38) are satisfied, it follows that . Furthermore, because , we can conclude that , which means that , . This completes the proof.

Remark 3.5. In practice, we may be interested to obtain a ball of initial condition as large as possible. To maximize the ball of initial condition, the following approximate optimization procedure can be used:

where .

In the special case that , the following corollary directly from Remark 3.3 and Theorem 3.4.

Corollary 3.6. Given constants , , , , , and . If there exist matrices , , , , , , , , and such that the following hold:

where denotes the th row of ,

and and are defined in (3.1), then, under the state feedback , system (2.3) with is locally exponentially stable with for any compatible initial condition in the ball

where

Also, as Remark 3.5, the corresponding approximate optimization problem of Corollary 3.6 becomes

where .

Remark 3.7. Scalars , , , in Theorem 3.4 are tuning parameters which need to be given first. In fact, (3.37) and (3.38), for fixed and , are bilinear matrix inequalities (BMIs) regarding to these tuning parameters. If one can accept more computational burden, the optimal values of these parameters can be obtained by applying some global optimization algorithms [35] to solve the BMIs.

4. Numerical Examples

In this section, two examples are given to show the effectiveness of the proposed method.

Example 4.1. Consider the discrete-time singular time-delay system (2.1) without input and with the following parameters:

Let . For various delay upper bound and by using Theorem 3.1 in this paper, the maximum allowable values of decay rate , that guarantees the regularity, causality and exponential stability of the system for given delay lower bound , are listed in Table 1. It can be seen from Table 1 that the delay upper bound is related to the decay rate , and a smaller allows for a larger . Figure 1 depicts the simulations results of and as compared to when and initial function is , . It is clearly observed from Figure 1 that the states and exponentially converge to zero with a decay rate greater than .

Table 1: Maximum allowable decay rates for different with d   .

1.2642
1.1501
1.0936
1.0613
1.0408
1.0101
1.0049

Figure 1: Simulation results of and as compared .

Example 4.2. Consider the discrete-time singular time-delay system (2.1) with the following parameters:

The time-delay is . Our purpose is to design the stabilizing controllers for different decay rate and estimate the domain of attraction for the above system.

Let , , , , , , and . For various , solving by using the mincx problem in LMI Toolbox [36], the maximal values of , denoted , and the corresponding controller gains are listed in Table 2. Especially, when , the feasible solutions to are

Furthermore, applying the obtained controller and giving a possible time-varying delay , we obtain the state response of the resulting closed-loop system and the corresponding actuator output as shown in Figures 2 and 3 for the given initial function , .

Table 2: and for different values of .

0.1 (1.0541)
2.1025
[1.3690 0]

0.2 (1.1180)
2.0932
[1.3206 0]

0.3 (1.1952)
2.0318
[1.2124 0]

Figure 2: State response of the closed-loop system under saturated control.

Figure 3: The actuator output under saturated control.

For clarity, we give some comments to explain the above simulation results. Firstly, it is easily seen from Table 2 that the smaller , that is, the smaller decay rate , then the larger is the domain for which we guarantee the exponential stability of the saturated system. It shows the tradeoff between the size of the ball of the admissible initial condition and the desired decay rate. Secondly, from the curves in Figures 2 and 3, one can see that despite the partly unknown transition probabilities, either the designed state-feedback or the designed output-feedback controllers are feasible and effective ensuring that the resulting closed-loop systems are stable.

It should be pointed out that the results in [27] cannot be applied to this system, because the system considered here contains state delay.

5. Conclusions

The problems of exponential estimates and stabilization for a class of discrete-time singular systems with time-varying state delays and saturating actuators have been investigated in this paper. An exponential admissibility condition has been developed for the system. Also, a saturated state feedback controller and a domain of safe initial conditions have been determined by using LMI optimization-based approach. The effectiveness of the results has been illustrated by numerical examples. Note that in the present work, a quadratic Lyapunov-Krasovskii function was used for the saturated closed-loop system modeled by differential inclusion, which may lead to conservatism. Applying the saturation-dependent Lyapunov function approach proposed in [20] to capture the real-time information on the severity of saturation and thus leading to less conservative results will be interesting topics for further research.

Appendix

Proof of Lemma 2.5. For given and defined in (2.8), the following equation holds

Rearranging the above equation yields (2.9). This completes the proof.

Proof of Lemma 2.7. Case  1 . Clearly, , and then .

Case  2 . Inspired by the idea in [16], to model the dependency of on past instances, define

Note that then, by iterative substitutions of (2.10), one gets

where is a positive integer.

(i) There exists a finite positive integer such that and . Then it follows from (A.2) that

By (A.3), (A.4), (2.11) and , the following is true:

(ii)   for all positive integer with . Since , the sequence is monotonically decreased and has a greatest lower bound with ; that is . Noting , we have [8]. Then, similar to the process to derive (A.5), the results in the lemma holds.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant 60904020, Open Fund of Key Laboratory of Measurement and Control of CSE (no. MCCSE2012A06), Ministry of Education, China, Southeast University, and the Scientific Research Foundation of Nanjing University of Posts and Telecommunications (NY210080).

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