Multiple summation inequalities and their application to stability analysis of discrete-time delay systems

Stability analysis of discrete-time systems with arbitrary delay kernels based on kernel-related summation inequality and model transformation

This paper addresses stability analysis and controller design for discrete-time positive systems with multiple delays. The control is under positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. The contribution lies in two aspects. First, a set of necessary and sufficient stability conditions is established for delayed discrete-time positive systems. Second, several necessary and sufficient conditions are presented for the existence of controllers under the positivity constraint. All the controllers are explicitly constructed if existent, and all the results are formulated as linear programming or linear matrix inequality problems, hence easy to be verified.

In this paper, stability analysis of deterministic and stochastic nonlinear discrete-time dynamical systems is considered. It is shown that some difficulties, such as independency or identical distribution of random variable sequence, that may arise in using Lyapunov's and LaSalle's approaches for stability analysis of discrete-time stochastic systems may be overcome by means of fixed point theory. Furthermore, suitable definitions of stability are provided. Ultimately, numerical examples are given in order to show the results.

In this note, we deal with the stability and stabilization problems for nonlinear quadratic discrete-time periodic systems. By using the quadratic Lyapunov function, and a so called periodic invariant set, delay-independent sufficient conditions for local stability and local stabilization for nonlinear quadratic discrete-time periodic systems are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

Stability analysis of nonsmooth systems, using Lyapunov direct method, is a difficult task. An alternative way of studying the stability in such systems is to employ the concept of Lyapunov exponents. However, two difficulties in the calculation of Lyapunov exponents, i.e. numerical artifacts and low computational efficiency, often prohibit the application of this otherwise powerful method. In this paper, a case study of the stability analysis of a switching contact task control system using the concept of Lyapunov exponents is presented. The goal is to carry out a numerical investigation of the method of nonstandard finite difference discretization, for constructing discrete models of differential equations that describe the Lyapunov exponents for such a nonsmooth system. It is shown that, as compared with the standard fourth-order Runge-Kutta method, the nonstandard finite difference discretization scheme provides numerically stable results with a larger critical integration step-size and less computation time. Therefore, from numerical stability and computational efficiency viewpoints, the nonstandard finite difference discretization method is meritorious and should be given consideration when stability analysis of systems using Lyapunov exponents is of concern.

This paper establishes a framework for stability analysis and memoryless state-feedback control synthesis of linear discrete-time systems with time varying state delay. The underlying idea is converting the considered system into an equivalent constrained switched system, leading to sufficient and necessary condition for stability analysis. Based on this switched system modeling, available information about the time-varying delay such as known bounds of variation rate can be incorporated into the analysis in a natural way. Moreover, sufficient conditions for memoryless state-feedback control design are obtained in a similar style. The analysis and synthesis conditions are given in terms of solvability of a set of linear matrix inequalities (LMIs). Numerical examples are included for illustration.

We study the stability analysis and control synthesis of uncertain discrete-time two-dimensional (2D) systems. The mathematical model of the discrete-time 2D system is established upon the well-known Roesser model, and the uncertainty phenomenon, which appears typically in practical environments, is modeled by a convex bounded (polytope type) uncertain domain. The stability analysis and control synthesis of uncertain discrete-time 2D systems are then developed by applying the Lyapunov stability theory. In the processes of stability analysis and control synthesis, the obtained stability/stabilzaition conditions become less conservative by applying some novel relaxed techniques. Moreover, the obtained results are formulated in the form of linear matrix inequalities, which can be easily solved via standard numerical software. Finally, numerical examples are given to demonstrate the effectiveness of the obtained results.

Stability Analysis of Takagi-Sugeno Fuzzy Systems With Linear State-Space Submodels

This paper studies the exponential stability of a class of discrete-time piecewise-linear systems (DPLS). Some basic properties of the proposed DPLS are established, which enables the generating function approach to be used for the system stability analysis. By introducing the generating functions of DPLS and showing their properties, a sufficient and necessary condition for the exponential stability of DPLS is derived. Furthermore, the maximum exponential growth rate of system trajectories can be obtained exactly by computing the radii of convergence of the generating functions. The algorithm for computing the generating functions is developed and two examples are given to demonstrate the proposed approach.

The stability analysis of time-delay systems is a tough task even in the linear case, especially when time-varying delays are considered. In recent years, for the class of positive delay systems, a number of insightful works appeared in the literature, highlighting their peculiar stability properties, which lead to simple and easy to check necessary and sufficient conditions for the delay-independent stability. This fact poses the question whether some results that hold for positive systems can be exported to arbitrary systems. In some recent papers we gave an answer to this question by extending to the time-delay setting the Internally Positive Representation technique, a method that allows to systematically construct positive representations of arbitrary systems. In this work, after a survey on recent stability results for positive delay systems, an overview of the Internally Positive Representation technique and of its implications on the stability analysis of arbitrary systems is reported. Three classes of linear systems are considered: delay differential, delay difference, and coupled delay differential-difference systems.

This paper is concerned with stability criterion of discrete-time descriptor systems in terms of LMIs. Though there have been proposed several LMIs for stability of discrete-time descriptor systems, their use is seemingly limited in comparison with LMIs for continuous-time descriptor systems. A remedy of those existing results is that they are not able to be linearized with respect to the coefficient matrices of the descriptor equation. This makes it difficult to apply them to synthesis problems, while LMIs for analysis of state-space systems or those for continuous-time descriptor systems enjoy change-of-variable methods to obtain LMIs for synthesis. In this paper, a new LMI is presented for admissibility of discrete-time descriptor systems that has such a useful feature of affineness of coefficient matrices in the inequality. After showing the main result with the new LMI, we apply it to several problems on discrete-time descriptor systems, namely, state-feedback synthesis, stability analysis of δ-operator systems and D-stability problems generalized to descriptor systems, and a generalization of quadratic stabilization. Some of these are illustrated via numerical examples.

Improved looped-functional approach for dwell-time-dependent stability analysis of impulsive systems

This paper studies stability analysis of linear time-invariant time-delay feedback systems consisting of a finite-dimensional system and a pure delay. The continuous-time state transition is described by the monodromy operator in discrete-time through the lifting technique, and a stability condition in terms of its spectral radius is first restated in terms of the norm of a power of the operator. Quasi-finite-rank approximation is then applied on the power of the operator, and by evaluating the effect of the approximation error on the norm, a new necessary and sufficient stability condition is given finally.

Improved delay-dependent stability criteria for neutral systems with mixed interval time-varying delays and nonlinear disturbances

Performance and stability analysis of discontinuous PWA systems by piecing together PWQ functions