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

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.


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 closedloop 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, continuoustime singular system in 25 .This approach was further extended to the analysis of the L 2 gain and L ∞ 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: 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.For real symmetric matrices P , P > 0 P ≥ 0 means that matrix P is positive definite semipositive definite .λ max P λ min P denotes the largest smallest eigenvalue of the positive definite matrix P .R n is the n-dimensional real Euclidean space and R m×n is the set of all real m × n matrices.Z represents the sets of all non-negative integers.The superscript "T " represents matrix transposition, and " * " in a matrix is used to represent the term which is induced by symmetry.diag{• • • } stands for a block-diagonal matrix.Sym{A} is the shorthand notation for A A T .For two integers n 1 and n 2 with n 1 ≤ n 2 , we use

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

2.4
If the state feedback is taken as the general form, that is, u k Fx k , 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 x 2 k is in the function sat • .Hence, if B 2 / 0, for given x 1 0 , the solution of x 2 0 is not unique even A 22 is nonsingular.However, if u k FEx k , then system 2.3 is r.s.e. to which implies that, for given x 1 k , the unique solution of x 2 k can be obtained when A 22 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.Note that regularity and causality of system 2.1 with u k 0 ensure that the solution to this system exists and is unique for any given compatible initial value φ s .Definition 2.4.System 2.1 under feedback law 2.2 is said to be exponentially stable with decay rate λ λ > 1 if, for any compatible initial conditions the initial time step, and > 0 is the decay coefficient.
We are interested in the exponential estimates and design for system 2.3 .For any compatible initial condition x 0 φ ∈ C n,d , denote the state trajectory of system 2.1 as x k, x 0 ; then the domain of attraction of the origin is In general, for a given stabilizing state feedback gain F, 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 F and determine a suitable set of initial condition D {φ ∈ C n,d : φ 2 c ≤ δ} ∈ S 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 δ.

2.8
Lemma 2.5.For any appropriately dimensioned matrices R > 0 and N, two positive time-varying integer d k 1 and d k 2 satisfying d k 1 1 ≤ d k 2 ≤ d, and a scalar λ > 0, the following equality holds Proof.See the Appendix.
Lemma 2.6 see 30 .Given a matrix D, let a positive-definite matrix S and a positive scalar η ∈ 0, 1 exist such that then the matrix D satisfies the bound where χ λ max S /λ min S and λ − ln η .
Lemma 2.7.Let 0 < d ≤ d k ≤ d.Consider the following system: where Then, for any compatible initial function φ ∈ C n,d , the solution x k, φ of 2.10 satisfies where r min{λ/d, β}.
Proof.See the Appendix.
For a matrix F ∈ R m×n , denote the jth row of F as f j and define

2.15
Let P ∈ R n×n be a positive-definite matrix and E T PE ≥ 0, the set Ω E T PE, 1 is defined by Also, let V be the set of m × m diagonal matrices whose diagonal elements are either 1 or 0.
There are 2 m elements in V. Suppose that each element of V is labeled as where α i k for i ∈ I 1, 2 m are some variables satisfying α i k ≥ 0 and 2 m i 1 α i k 1.
Lemma 2.9 see 31 .Given matrices X, Y , and Z with appropriate dimensions, and Y is symmetric.Then there exists a scalar ρ > 0, such that ρI Y > 0 and − Sym {X

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 u k 0 with a predefined decay rate.

and S, and matrices
T such that the following inequality holds where

3.2
and R ∈ R n×n is any constant matrix satisfying RE 0 and rank R n − r, then the unforced system 2.1 is regular, causal, and exponentially stable with λ 1/ √ 1 − α.
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 E is regular and rank E r, there exist two nonsingular matrices G 1 and H 1 such that

3.3
Note that RE 0 and rank R n − r, it can be verified that 3.4 From 3.1 , it is easy to obtain Φ 11 < 0. In view of X > 0, Q l > 0, l 1, 2, 3, Z l > 0, Z 2 > 0, d > 0, and d > 0, it can be further obtained that Pre-and postmultiplying 3.5 by H T 1 and H 1 , 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 A 22 is singular, then, there exists a vector η ∈ R n−r and η / 0 such that A 22 η 0. Pre-and postmultiplying 3.7 by η T and η, respectively, result in Then, it is easy to see that 3.7 is a contradiction since Thus, A 22 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 G 2 and H 2 such that According to 3.8 , define 3.9 By using Schur complement on 3.1 , we get where

3.11
Substituting 3.8 and 3.9 into the above inequality, pre-and postmultiplying by diag{H T 2 , H T 2 } and diag{H 2 , H 2 }, respectively, and using Schur complement yields Pre-and postmultiplying this inequality by −A T d22 I and its transpose, respectively, and noting Therefore, according to Lemma 2.6, there exist constants χ λ max Q 122 /λ min Q 122 and Then, the unforced system 2.1 is r.s.e. to the following one:

3.15
Now, choose the following Lyapunov-Krasovskii function: where Then, it follows from 2.1 that

3.20
Using Lemma 2.5 for the last three terms of V 4 ξ k , respectively, and noting d ≤ d k ≤ d, we have Also, it follows from RE 0 that

3.25
Then, substituting 3.19 -3.24 into 3.18 , using 3.25 , and noting 1 − α > 0, Z 1 > 0 and Z 2 > 0, we get where Φ follows the same definition as Φ defined in 3.1 with A, A d , X, Q l , Z v , R, M, N, and Thus, it follows from 3.18 and 3.26 By iterative substitutions, inequality 3.27 yields On the other hand, it follows from the Lyapunov functional 3.16 that where

3.30
Then, combining 3.28 and 3.29 leads to

3.31
Therefore, the difference subsystem of 3.15 is exponentially stable with a decay rate which is not less than e ln 1−α −1 /2 .The remaining task is to show the exponential stability of the algebraic subsystem.Part iii : Exponential stability of the algebraic subsystem.Set f k −A d21 ξ 1 k−d k ; then, it follows from 3.31 that

3.32
Using the second equation in 3.15 , 3.14 and Lemma 2.7, one gets where

3.34
Combining 3.31 and 3.33 yields that

3.35
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 timevarying 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 E I in system 2.1 , then R 0, and we get the exponential stability condition for the standard delay systems x k 1 Ax k A d x k − d k 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 ε 5 .If there exist symmetric matrices X > 0, Q l > 0, l 1, 2, 3, Z v > 0, v 1, 2, and S, and matrices T , F and H such that the following inequalities hold: where h l denotes the lth row of H, where

3.41
Proof.If 3.38 holds, then the ellipsoid Ω E T XE, 1 is included in L H 27 For more details about the ellipsoids and ellipsoid algorithm, we refer the readers to 32-34 .Suppose that x k ∈ L H , ∀k > 0 will be proved later .Hence, by Lemma 2.8, sat FEx k can be expressed as sat FEx k 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 X > 0, Q l > 0, l 1, 2, 3, Z v > 0, v 1, 2, and S, and matrices M, N, and T 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

3.47
This, together with α j k ≥ 0 and 2 m j 1 α j k 1, implies that

3.49
Applying Schur complement to 3.49 leads to where where

3.53
From 3.37 , it follows that ρI S > 0, and then the following inequality holds from Lemma 2.9 3.54 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 x k ∈ L H , ∀k ≥ 0.
On the other hand, by the proof in Theorem 3.1, inequality 3.44 holding implies φ s is included in the ball B δ , ∀s ∈ I −d, 0 , and inequalities 3.37 and 3.38 hold.Then, from the definition of V ξ k , we get This completes the proof.
Remark 3.5.In practice, we may be interested to obtain a ball B δ of initial condition as large as possible.To maximize the ball of initial condition, the following approximate optimization procedure can be used: In the special case that E I, the following corollary directly from Remark 3.3 and Theorem 3.4.Corollary 3.6.Given constants 0 < α < 1, 0 < d < d, ρ > 0, ε 1 > 0, ε 2 > 0, and ε 3 > 0. If there exist matrices X > 0, T , F and H such that the following hold:

3.56
where h l denotes the lth row of H,

3.59
3.56 , 3.58 and Remark 3.7.Scalars ρ, ε f , f 1, 2, . . ., 5, in Theorem 3.4 are tuning parameters which need to be given first.In fact, 3.37 and 3.38 , for fixed α > 0 and 0 < d < d, 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.

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:  For clarity, we give some comments to explain the above simulation results.Firstly, it is easily seen from Table 2 that the smaller α, that is, the smaller decay rate λ, then the larger is the domain B δ 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.

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.
Note that k i < k i−1 then, by iterative substitutions of 2.10 , one gets where k t is a positive integer.
i There exists a finite positive integer k t such that 0 ≤ k k t −1 and −d ≤ k k t < 0. Then it follows from A. A.5 ii k 0 − i−1 j 0 d k j ≥ 0 for all positive integer i with i ≥ 1.Since d k ≥ 0, the sequence {k 0 − i−1 j 0 d k j } ∞ i 1 is monotonically decreased and has a greatest lower bound with ≥ 0; that is lim i → ∞ k 0 − i−1 j 0 d k j .Noting λ > 0, we have lim i → ∞ e −λk i x k i 0 8 .Then, similar to the process to derive A.5 , the results in the lemma holds.
denote the Banach space of family continuous vector valued functions mapping the interval I −d, 0 to R n with the topology of uniform convergence.Denote x k s x k s , ∀s ∈ I −d, 0 .• refers to either the Euclidean vector norm or the induced matrix two-norm.For a function φ ∈ C n,d , its norm is defined as φ c sup s∈I −d,0 φ s .

Figure 1 :Figure 2 :
Figure 1: Simulation results of and as compared 1.25 −k .

Figure 3 : 1 N 1 l k−d k 2
Figure 3: The actuator output under saturated control.
Sign u i k min{1, |u i k |}.Note that the notation of sat • 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 .

Table 1 :
Maximum allowable decay rates λ for different d with d 1.
For various delay upper bound d and by using Theorem 3.1 in this paper, the maximum allowable values of decay rate λ, that guarantees the regularity, causality and exponential stability of the system for given delay lower bound d, are listed in Table1.It can be seen from Table1that the delay upper bound d is related to the decay rate λ, and a smaller λ allows for a larger d. Figure 1 depicts the simulations results of x 1 k and x 2 k as compared to 1.25 −k when d k 2 sin 2.5πk and initial function is φ k −1, 1 T , k −3, −2, . . ., 0. It is clearly observed from Figure 1 that the states x 1 k and x 2 k exponentially converge to zero with a decay rate greater than 1.25.