Delay-dependent stabilization of a class of time-delay nonlinear systems: LMI approach

*Correspondence: nadhemechi_fsg@yahoo.fr 1Department of Mathematics, Faculty of Sciences of Gafsa, Gafsa University, Zarroug, Gafsa, 2112, Tunisia Full list of author information is available at the end of the article Abstract This paper deals with the state and output feedback stabilization problems for a family of nonlinear time-delay systems satisfying some relaxed triangular-type condition. A new delay-dependent stabilization condition using a controller is formulated in terms of linear matrix inequalities (LMIs). Based on the Lyapunov-Krasovskii functionals, global asymptotical stability of the closed-loop systems is achieved. Finally, simulation results are shown to illustrate the feasibility of the proposed strategy.


Introduction
Time delays are important components of many dynamical systems that describe coupling or interconnection between dynamics, propagation, or transport phenomena in shared environments, economic models [, ], biological systems [], and in competition in population dynamics []. In the literature, there are two categories of criteria: delay-dependent [, ] and delay-independent criteria. For criteria-dependent delay gives information on the length of delays, this model is used more frequently than delay-independent ones. For delay-dependent criteria, see [-] and the references therein. For a delay-free system, under control laws with high gain observers, asymptotic stability has been achieved by [-]. The analysis of nonlinear systems with time delays is typically more difficult than systems without time delays [, ]. The study of the stabilization of the system with delay was the object of much research (see for example [-] and the references therein). In [], using the Lyapunov-Krasovskii functional approach and the linear matrix inequality (LMI)-based design method, the stability problem for time-delay systems is discussed. In [], a principle of separation has been established in a class of systems, inspired by [], which covers the class of systems considered by []. Under an output feedback controller the global asymptotic stability is obtained. In [] and [], under linear growth conditions, global stabilization by state feedback and output feedback has been studied for a class of nonlinear time-delay systems. In this regard, [ , proposed conditions were expressed in terms of linear matrix inequalities, and used to examine the exponential stability problem for delayed generalized neural networks. The problem of robust finite-time stabilization and guaranteed cost control for neural networks with varying interval time delay has been achieved by []. By constructing a set of improved Lyapunov-Krasovskii functionals, a design of memoryless state feedback guaranteed cost controllers has been presented for the system in terms of linear matrix inequalities.
Under constructing a set of improved Lyapunov-Krasovskii functionals and a Newton-Leibniz formula, the conditions for the exponential stability of the systems have been established in terms of LMIs.
[] proposed delay-dependent conditions for the exponential stability of linear systems with non-differentiable varying interval time delay.
Under delay-dependent conditions, global exponential stability of a class of nonlinear time-delay systems has been achieved by []. The condition on the nonlinearity to cover the time-delay systems, given by [], is a generalization of conditions considered by [, , , ]. Moreover, the generalized conditions cover the systems given by [, ] for a class of nonlinear delay-free systems.
In this paper, we investigate the problem of output feedback stabilization of a class of nonlinear time-delay systems, which cover the systems considered by []. Motivated by [] and [], we use appropriate Lyapunov-Krasovskii functionals to establish global asymptotical stability of the closed-loop systems. Then they are used to obtain a new state and input delay-dependent criterion that ensures the stability of the closed-loop system with a state feedback controller. The rest of this paper is organized as follows. In the next section, some preliminary results are summarized and the system description is given. Our main results are stated in Section . First, a parameter-dependent linear state and output feedback controllers are synthesized to ensure global asymptotical stability of the nonlinear time-delay system. Finally, an illustrative example is discussed to demonstrate the effectiveness of the obtained results.

System description and preliminary
Consider a time-delay system of the form where τ >  denotes the time delay, and ϕ ∈ C is the initial function where C denotes the Banach space of continuous functions mapping the interval [-τ , ] → R n equipped with the supremum-norm with being the Euclidean norm. The map f : R n × R n is smooth and satisfies f (, ) = . The function segment x t is defined by For ϕ ∈ C, we denote by x(t, ϕ) or in short x(t) the solution of () that satisfies x  = ϕ. The segment of this solution is denoted by x t (ϕ) or in short x t .

Definition  The zero solution of () is called
• stable, if for any ε >  there exists δ >  such that • asymptotically stable, if it is stable and attractive, • globally asymptotically stable, if it is stable and δ can be chosen arbitrarily large for sufficiently large ε, and () is satisfied for all σ > .
Sufficient conditions for stability of a functional differential equation are provided by the theory of Lyapunov-Krasovskii functionals [], a generalization of the classical Lyapunov theory of ordinary differential equations []. Let us recall here that a function α : R + → R + is of class K if it is continuous and increasing and α() = , and of class K ∞ if it is of class K and it is unbounded. The following theorem provides sufficient Lyapunov-Krasovskii conditions for global asymptotic stability of the zero solution of system () (see []).
Theorem  Assume that there exists a locally Lipschitz functional V : C → R + , functions α  and α  of class K ∞ , and function α  of class K, such that

Then the zero solution of system () is globally asymptotically stable.
Notation  Throughout the paper, the time argument is omitted and the delayed state vector x(tτ ) is denoted by x τ . A T means the transpose of A. λ max (A) and λ min (A) denote the maximal and minimal eigenvalue of a matrix A, respectively. P >  means that the matrix P is symmetric positive definite. I is an appropriately dimensioned identity matrix, and diag[· · · ] denotes a block-diagonal matrix.
Lemma  See the Schur complement from []. Let M, P, Q be the given matrices such that Q > . Then Lemma  For any vector a, b ∈ R n and scalar ε > , we have In this paper, we consider the time-delay nonlinear system , where x ∈ R n is the state vector, u ∈ R is the input of the system, y ∈ R is the measured output, and τ is a positive known scalar that denotes the time delay affecting the state variables. The matrices A, B and C are given by The mappings f i : R n × R n × R → R, i = , . . . , n, are smooth and satisfy the following assumption.

Global stabilization by state feedback
This section presents the delay-dependent stabilization conditions obtained by means of the LMI method. The state feedback controller is given by where K(ε) = [ k  ε n , . . . , k n ε ] and K = [k  , . . . , k n ] such that A K := A + BK is Hurwitz.
Theorem  Suppose that Assumption  is satisfied. Then there exist symmetric positive definite matrices S, Q, Z and there exists a positive constant ε such that the following LMIs hold: Then the closed loop time-delay system ()-() is asymptotically stable for any time delay Proof The closed loop system is given bẏ

Using the fact that
Let us choose a Lyapunov-Krasovskii functional candidate as follows: where Since S is symmetric positive definite, for all χ ∈ R n , This implies that, on the one hand, and on the other hand, So by Assumption  we get Using Lemma  we deduce thaṫ Using Lemma  and (), the time derivative of W  iṡ Hence, we havė Then, using the Lyapunov-Krasovskii stability Theorem  and the Schur complement Lemma , we conclude that the closed loop time-delay system ()-() is asymptotically stable if () and () hold.

Global stabilization by output feedback
In [], under Assumption , if the conditions does not depend on the delay τ , global exponential stability by the dynamic output feedback control is achieved. In this subsection, we study the problem of global asymptotic stability by output feedback control under Assumption  and delay-dependent conditions. The following system is proposed: where L(ε) = [ l  ε , . . . , l n ε n ] T and L = [l  , . . . , l n ] T such that A L := A + LC is Hurwitz. The output feedback controller is given by

Then the closed loop time-delay system ()-() is asymptotically stable for any time delay
τ satisfying  ≤ τ ≤τ .

Using the fact that
Let us choose a Lyapunov-Krasovskii functional candidate as follows: where where W is given by (). Using () and (), we geṫ Finally, we select α such that

Numerical example
To check the effectiveness of the result, consider the following system: x  = u. () Following the notation used throughout the paper, let f  (x, x τ , u) =   x  sin x  cos u +   x  (tτ ) cos u, and f  (x, x τ , u) = f  (x, x τ , u) = . Since f  depends on x  and x τ  , the output feedback scheme in [, ] is not applicable. It is easy to check that system () satisfies Assumption  with γ  (ε) = γ  (ε) =  ε . Select K = [- - -] and L = [- - -] T . A K and A L are Hurwitz. Applying Theorem  and the MATLAB LMI Control Toolbox, we find that conditions () and () are given,

Conclusion
In this paper, we are concerned with the problem of global asymptotic stability for a certain class of nonlinear time-delay systems, written in triangular form, satisfying a linear growth condition [, , , ]. In [], the authors derived delay-independent conditions to ensure global exponential stability of the closed-loop systems. Using the Lyapunov-Krasovskii functionals given by () and (), we have derived delay-dependent conditions, using a controller that is formulated in terms of LMIs, to ensure global asymptotical stability of the resulting closed-loop systems. The obtained result extends for global exponential stability.