Stability Of Discrete-time Systems Research Articles

Robust finite-time stability of discrete time systems with interval time-varying delay and nonlinear perturbations

The problem of finite-time stability (FTS) for discrete-time systems with interval time-varying delay, nonlinear perturbations and parameter uncertainties is considered in this paper. In order to obtain less conservative stability criteria, a finite sum inequality with delayed states is proposed. Some sufficient conditions of FTS are derived in the form of the linear matrix inequalities (LMIs) by using Lyapunov–Krasovskii-like functional (LKLF) with power function and single/double summation terms. More precisely estimations of the upper bound of the initial value of LKLF and the lower bound of LKLF are proposed. As special cases, the FTS of nominal discrete-time systems with constant or time-varying delay is considered. The numerical examples are presented to illustrate the effectiveness of the results and their improvement over the existing literature.

Further results on delay-dependent stability criteria of discrete systems with an interval time-varying delay

This paper deals with stability of discrete-time systems with an interval-like time-varying delay. By constructing a novel augmented Lyapunov functional and using an improved finite-sum inequality to deal with some sum-terms appearing in the forward difference of the Lyapunov functional, a less conservative stability criterion is obtained for the system under study if compared with some existing methods. Moreover, as a special case, the stability of discrete-time systems with a constant time delay is also investigated. Three numerical examples show that the derived stability criteria are less conservative and require relatively small number of decision variables.

Auxiliary function-based summation inequalities and their applications to discrete-time systems

Auxiliary function-based summation inequalities are addressed in this technical note. By constructing appropriate auxiliary functions, several new summation inequalities are obtained. A novel sufficient criterion for asymptotic stability of discrete-time systems with time-varying delay is obtained in terms of linear matrix inequalities. The advantage of the proposed method is demonstrated by two classical examples from the literature.

Stabilization of discrete time systems by reflection coefficients

Stabilization of discrete time systems by reflection coefficients

Augmented Lyapunov-Krasovskii Functional Approach to Stability of Discrete Systems With Time-Varying Delays

This paper investigates the problem of delay-dependent stability for discrete-time systems with time-varying delays. A novel augmented Lyapunov-Krasovskii functional is proposed in deriving stability criteria in which the feasible region is enhanced. Also, an improved summation inequality is developed and applied to find the lower bound of summation inequalities. Via two numerical examples, improved results will be shown by comparing with maximum delay bounds.

Analysis and comparison of the stability of discrete-time and continuous-time linear systems

Abstract The asymptotic stability of discrete-time and continuous-time linear systems described by the equations xi+1 = Ākxi and x(t) = Akx(t) for k being integers and rational numbers is addressed. Necessary and sufficient conditions for the asymptotic stability of the systems are established. It is shown that: 1) the asymptotic stability of discrete-time systems depends only on the modules of the eigenvalues of matrix Āk and of the continuous-time systems depends only on phases of the eigenvalues of the matrix Ak, 2) the discrete-time systems are asymptotically stable for all admissible values of the discretization step if and only if the continuous-time systems are asymptotically stable, 3) the upper bound of the discretization step depends on the eigenvalues of the matrix A.

Stability analysis of discrete-time systems with variable delays via some new summation inequalities

This paper proposes an improved stability condition of discrete-time systems with variable delays. Based on some mathematical techniques, a series of new summation inequalities are obtained. These new inequalities are less conservative than the Jensen inequality. Based on these new summation inequalities and the reciprocally convex combination inequality, a novel sufficient criterion on asymptotical stability of discrete-time systems with variable delays is obtained by constructing a new Lyapunov-Krasovskii functional. The advantage of the proposed inequality in this paper is demonstrated by a classical example from the literature.

Discrete-time systems with random switches: From systems stability to networks synchronization.

In this article, we develop some approaches, which enable us to more accurately and analytically identify the essential patterns that guarantee the almost sure stability of discrete-time systems with random switches. We allow for the case that the elements in the switching connection matrix even obey some unbounded and continuous-valued distributions. In addition to the almost sure stability, we further investigate the almost sure synchronization in complex dynamical networks consisting of randomly connected nodes. Numerical examples illustrate that a chaotic dynamics in the synchronization manifold is preserved when statistical parameters enter some almost sure synchronization region established by the developed approach. Moreover, some delicate configurations are considered on probability space for ensuring synchronization in networks whose nodes are described by nonlinear maps. Both theoretical and numerical results on synchronization are presented by setting only a few random connections in each switch duration. More interestingly, we analytically find it possible to achieve almost sure synchronization in the randomly switching complex networks even with very large population sizes, which cannot be easily realized in non-switching but deterministically connected networks.

Stabilization of Discrete-time Systems with Multiplicative Noise and Multiple Delays in the Control Variable

This paper focuses on the stabilization problem of discrete-time systems with multiplicative noise and multiple delays in the control variable. First, a transformation is constructed to convert the original system into a delay-free system with its input matrix containing future noise. Next, the finite-horizon linear quadratic regulation problem for the delay-free system subject to a constraint on the control is investigated. We provide the necessary and sufficient condition for the problem having a unique solution and the optimal feedback control using Riccati-type difference equations. Finally, we show that the original system can be stabilized in the mean-square sense if and only if the set of solutions to the Riccati-type difference equations is convergent. This further yields the fact that the system under consideration can be stabilized if and only if the algebraic Riccati-type equations have a unique solution such that a specific matrix is positive definite.

Linear Quadratic Regulation and Stabilization of Discrete-Time Systems With Delay and Multiplicative Noise

This paper is concerned with the long-standing problems of linear quadratic regulation (LQR) control and stabilization for a class of discrete-time stochastic systems involving multiplicative noises and input delay. These fundamental problems have attracted resurgent interests due to development of networked control systems. An explicit analytical expression is given for the optimal LQR controller. More specifically, the optimal LQR controller is shown to be a linear function of the conditional expectation of the state, with the feedback gain based on a Riccati-ZXL difference equation. It is also shown that the system is stabilizable in the mean-square sense if and only if an algebraic Riccati-ZXL equation has a particular solution. These results are based on a new technical tool, which establishes a non-homogeneous relationship between the state and the costate of this class of systems, and the introduction of a new Lyapunov function for the finite-horizon optimal control design.

A Less Conservative LMI Condition for Stability of Discrete-Time Systems With Slope-Restricted Nonlinearities

Many conditions have been found for the absolute stability of discrete-time Lur'e systems in the literature. It is advantageous to find convex searches via LMIs where possible. In this technical note, we construct two less conservative LMI conditions for discrete-time systems with slope-restricted nonlinearities. The first condition is derived via Lyapunov theory while the second is derived via the theory of integral quadratic constraints (IQCs) and noncausal Zames-Falb multipliers. Both conditions are related to the Jury-Lee criterion most appropriate for systems with such nonlinearities, and the second generalizes it. Numerical examples demonstrate a significant reduction in conservatism over competing approaches.

Augmented observer approach for high-impedance haptic system with time delay

Among others measures of performance, a haptic device may be characterized by its behavior in the implementation of a virtual wall in terms of the biggest value of the virtual stiffness leaving stable the device. This value depends on many factors including time delays and the virtual world implementation. The paper proposes a new architecture for this latter one based on the use of an augmented state observer which contributes to improve without additional costs the performance of a haptic device with respect to the standard implementation based on the use of the backward finite difference method. In order to compute the stability boundaries of the device in terms of virtual wall parameters, a new LMI criterion for the stability of discrete-time systems with time-varying delay is given. Some numerical simulation results are included to prove the effectiveness of the proposed approach.

Stability of Discrete-Time Systems With Time-Varying Delays via a Novel Summation Inequality

This technical note is concerned with the stability analysis of discrete linear systems with time-varying delays. The novelty of the technical note comes from the consideration of a new inequality which is less conservative than the celebrated Jensen inequality employed in the context of discrete-time delay systems. This inequality is a discrete-time counterpart of the Wirtinger-based integral inequality that was recently employed for the improved analysis of continuous-tine systems with delays. However, differently from the continuous-time case, the proof of the new inequality is not based on the Wirtinger inequality. The method is also combined with an efficient representation of the improved reciprocally convex combination inequality in order to reduce the conservatism induced by the LMIs optimization setup. The effectiveness of the proposed result is illustrated by some classical examples from the literature.

Robust output feedback stabilization for discrete-time systems with time-varying input delay

In this paper, a robust output feedback stabilization method is proposed for discrete-time systems subject to time-varying input delay. By introducing an augmented state, the robust stabilization problem is converted into that of a delay-free system. Using Artstein's reduction method and the scaled-bounded real lemma, sufficient conditions for robust stability are established via matrix inequalities, by which the output feedback controller can be derived. A tuning parameter is introduced to efficiently solve these matrix inequalities, therefore facilitating the improvement of control performance. An illustrative example is used to show the effectiveness and merit of the proposed method.

An LMI Based Criterion for Global Asymptotic Stability of Discrete-Time State-Delayed Systems with Saturation Nonlinearities.

A linear matrix inequality (LMI) based criterion for the global asymptotic stability of discrete-time systems with multiple state-delays employing saturation nonlinearities is presented. Numerical examples highlighting the effectiveness of the proposed criterion are given.

Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach

This paper addresses the problem of establishing robust asymptotical stability of discrete-time linear systems polynomially affected by time-varying uncertainty confined into a polytope. A linear matrix inequality (LMI) condition for establishing robust asymptotical stability is proposed by introducing a novel approach for establishing the existence of a common homogeneous polynomial Lyapunov function (HPLF). This approach consists, firstly, of introducing a Gram matrix built with respect to the state and parametrized by an arbitrary vector function of the uncertainty, and secondly, of requiring that a transformation of the introduced Gram matrix is a sum of squares (SOS) of matrix polynomials. The approach, hence, is referred to as a Gram-SOS approach. It is shown that the proposed LMI condition is sufficient for any degree of the HPLF candidate, that includes quadratic robust stability as a special case, and that is also necessary for a sufficiently large degree of the HPLF candidate. Numerical examples also show that the proposed LMI condition can outperform alternative ones in terms of conservatism and computational burden.

Stability of Discrete-Time Systems with Time-Varying Delay: Delay Decomposition Approach

This article deals with the problem of obtaining delay-dependent stability conditions for a class of discrete-time systems with interval time-varying delay. Using the decomposition the delay interval into two unequal subintervals by tuning parameter α, a new interval delay-dependent Lyapunov-Krasovskii functional is constructed to derive novel delay-dependent stability conditions which are expressed in terms of linear matrix inequalities. This leads to reduction of conservatism in terms of the upper bounds of the maximum time-delay. The numerical examples show that the obtained result is less conservative than some existing ones in the literature.

New results on state bounding for discrete‐time systems with interval time‐varying delay and bounded disturbance inputs

This study considers the problem of state bounding for a class of discrete-time systems with interval time-varying delay and bounded disturbance inputs. By using an improved Lyapunov–Krasovskii functional combining with the delay-decomposition technique and the reciprocally convex approach, the authors first derive new delay-dependent conditions in terms of matrix inequalities to guarantee the existence of a ball such that, for any initial condition, the state trajectory of the system is either bounded within that ball or converges exponentially within it. On the basis of these new conditions, the authors then derive an improved ellipsoid reachable set bounding and a new result on exponential stability of discrete-time systems with interval time-varying delay. Numerical examples are presented to show the effectiveness of the obtained results and improvement over existing results.

Quantized stabilization of discrete-time systems in a networked environment

This paper is concerned with the problem of output feedback stabilization for a class of discrete-time systems with sector nonlinearities and imperfect measurements. A unified control law model is proposed to take the network-induced delay, random packet dropout and measurement quantization into consideration simultaneously. By choosing appropriate Lyapunov functional, a new stability condition, which is dependent on multiple network status, is established for the resulting closed-loop system. Based on the result, a design criterion for the static output feedback controller is formulated in the form of nonconvex matrix inequalities, and the cone complementary linearization (CCL) procedure is exploited to solve the nonconvex feasibility problem. Incidentally, a less conservative synthesis method is also developed for the state feedback stabilization purpose. Finally, two illustrative examples are provided to illustrate the effectiveness and applicability of the proposed design method.

Approximation of fractional positive stable continuous-time linear systems by fractional positive stable discrete-time systems

Abstract Fractional positive asymptotically stable continuous-time linear systems are approximated by fractional positive asymptotically stable discrete-time systems using a linear Padé-type approximation. It is shown that the approximation preserves the positivity and asymptotic stability of the systems. An optional system approximation is also discussed.