Stability Criteria for Uncertain Discrete-Time Systems under the Influence of Saturation Nonlinearities and Time-Varying Delay

The problem of global asymptotic stability of a class of uncertain discrete-time systems in the presence of saturation nonlinearities and interval-like time-varying delay in the state is considered. The uncertainties associated with the system parameters are assumed to be deterministic and normbounded. The objective of the paper is to propose stability criteria having considerably smaller numerical complexity. Two new delay-dependent stability criteria are derived by estimating the forward difference of the Lyapunov functional using the concept of reciprocal convexity and method of scale inequality, respectively. The presented criteria are compared with a previously reported criterion. A numerical example is provided to illustrate the effectiveness of the presented criteria.

Physical systems may suffer from parameter uncertainties that arise due to modeling errors, variations in system parameters, or some ignored factors.The existence of parameter uncertainties may result in instability of the designed system [20].
The stability analysis of discrete-time systems involving overflow nonlinearities, parameter uncertainties, and state delays is an important problem.Delay-independent stability criteria for a class of discrete-time state-delayed systems with saturation nonlinearities have been presented in [16,19].A delay-dependent approach for the stability analysis of uncertain discrete-time systems with time-varying delays and quantization/overflow nonlinearities has been proposed in [17].In a recent work [18], a delay-dependent global asymptotic stability criterion for a class of uncertain discretetime state-delayed systems with saturation nonlinearities has been established.

ISRN Applied Mathematics
A major concern of the delay-dependent stability criteria is computational complexity.The objective of this paper is to present global asymptotic stability criteria for uncertain discrete-time systems under the influence of saturation nonlinearities and time-varying delay.In particular, inspired by [23,29,39], we are interested to develop delay-dependent stability criteria which are numerically less complex as compared to [18].
The paper is organized as follows.Section 2 defines the system under consideration and presents a recently reported criterion.In Section 3, we specify the lemmas used and then present our results.A numerical example illustrating the usefulness of the presented results is given in Section 4.
Notations.The notations used throughout this paper are standard.R  denotes the p-dimensional Euclidean space; R × is the set of  ×  real matrices; 0 represents null matrix or null vector of appropriate dimension; I is the identity matrix of appropriate dimension; B  stands for the transpose of the matrix (or vector) B; B > 0 (≥0) means that B is positive definite (semidefinite) symmetric matrix; B < 0 represents that B is negative definite symmetric matrix; ⌊⌉ denotes a function which returns the nearest integer to ; || stands for absolute value of a real number ; the symbol * represents the symmetric terms in a symmetric matrix.

System Description and Existing Criterion
The system under consideration is given by where x() ∈ R  is the system state vector; A, A  ∈ R × are the known constant matrices; ΔA, ΔA  ∈ R × are the unknown matrices representing parametric uncertainties in the state matrices; () ∈ R  is the initial condition at ; and the time-varying delay () is a positive integer which satisfies where ℎ 1 and ℎ 2 are known nonnegative integers representing the lower and upper delay bounds, respectively.The saturation nonlinearities given by are under consideration.The uncertainties are assumed to be of the form [16,17,25,38,40] where H  ∈ R ×  and E  ∈ R   × ( = 0, 1) are known constant matrices and F  ∈ R   ×  ( = 0, 1) is an unknown matrix which satisfies Pertaining to the system given by ( 1)-( 8), the following criterion has been recently reported in [18].

Main Results
Before presenting the main results of the paper, we recall the following lemmas.
Proof.Consider the following Lyapunov functional candidate [23]: where Taking the forward difference of the Lyapunov functional (21) along the solutions of (1) yields
Clearly, in such case C reduces to a positive scalar, namely,  [14]).From (29), it is clear that Δ(x()) < 0, if Ψ 1 < 0. Thus, Ψ 1 < 0 and ( 19) are sufficient conditions for the global asymptotic stability of the system given by ( 1)-( 8).Further, using (6), the condition Ψ 1 < 0 can be rewritten in the following form: where By Lemma 4, ( 34) is equivalent to where  0 > 0. By employing the well-known Schur complement [41], (36) can also be expressed as where From (40), it is clear that Δ(x()) < 0 if ( 19) and (39) hold true.This completes the proof of Corollary 8. Remark 9.It may be mentioned that a criterion for the global asymptotic stability of the system (38) is reported in [29] (see [29,Theorem 1]).It can be verified that Corollary 8 is the same as Theorem 1 of [29].Thus, Theorem 1 of [29] is recovered from the presented approach as a special case.In other words, Theorem 6 may be considered as an extension of the delaydependent criterion given in Theorem 1 of [29] for timevarying delay systems to a model that includes, in addition, parameter uncertainty as well as saturation nonlinearities.
It may be observed that Theorem 6 has a free-weighting matrix S.Although the presence of these free-weighting matrices helps in obtaining reduced conservative results, but they contribute heavily towards the computational complexity.
Next, we present a criterion which does not include any free-weighting matrix.

Conclusions
Two new delay-dependent stability criteria (Theorems 6 and 10) have been proposed for a class of uncertain discrete-time systems with time-varying delay in the presence of saturation nonlinearities.A numerical example has been considered for illustrating the effectiveness of the presented results.As compared to [18], the proposed criteria turn out to be numerically less complex.The presented approach can easily be extended to a class of nonlinear uncertain discrete-time systems with multiple time delays.

Figure 1 :
Figure 1: State response of the system in Example 1.

Figure 2 :
Figure 2: Time-varying delay d(k) used in the simulation.

Figure 4 :
Figure 4: State response of the nominal system.

Table 1 :
Comparison of numerical complexity.