Stability Of Discrete-time Systems Research Articles

Comparison of approximation methods of positive stable continuous-time linear systems by positive stable discrete-time systems

Abstract The positive asymptotically stable continuous-time linear systems are approximated by corresponding asymptotically stable discrete-time linear systems. Two methods of the approximation are presented and the comparison of the methods is addressed. The considerations are illustrated by three numerical examples and an example of positive electrical circuit.

Stabilization of Discrete-Time Systems With Multiple Actuator Delays and Saturations

This paper studies the problems of global and semi-global stabilization of discrete-time linear systems with multiple input saturations and arbitrarily large bounded delays. By developing the methodology of truncated predictor feedback (TPF), state feedback control laws using only the current states of the systems are constructed to solve these problems. The feedback gains are dependent on the delay information of the open-loop system and thus are referred to as delay-dependent feedback. A method for determining the exact condition such that the resulting closed-loop system is asymptotically stable is also presented. Moreover, if the delays in the system are time-varying or even unknown, a modified delay-independent TPF is also established to solve the concerned problems. Numerical example has been worked out to illustrate the effectiveness of the proposed approaches.

Robust Output Feedback Stabilization of a Field-Sensed Magnetic Suspension System

The magnetic suspension system (MSS) is very important in many engineering applications. This paper proposes the dynamic output feedback control of a field-sensed MSS (FSMSS). Subsequently, the mathematical model of the MSS is described by discrete-time systems. Ideally, the coefficients of a nominal polynomial can precisely determine the Schur stability. But in reality, the coefficients may contain uncertainties due to reasons such as computational errors. Therefore, there is a need to address the problem of robust stability for discrete-time systems. In this paper, the size of allowable perturbation in polynomial coefficient space was estimated for the output feedback control of the MSS. Theℓ∞-norm and a lower bound for the size of the Schur stability hypercube are provided in this paper.

Stability and stabilization for discrete-time systems with time-varying delays via augmented Lyapunov–Krasovskii functional

This paper is concerned with the stability and stabilization problems for discrete-time systems with interval time-varying delays. By construction of an augmented Lyapunov–Krasovskii functional and utilization of zero equalities, improved delay-dependent criteria for asymptotic stability of the systems are derived in terms of linear matrix inequalities (LMIs). Based on the proposed stability criteria, a sufficient condition for designing feedback gains of time-delayed controllers which guarantee the stability of the concerned system is presented. Through three numerical examples, the effectiveness to enhance the feasible region of the proposed criteria is demonstrated.

On stability of discrete-time systems under nonlinear time-varying perturbations

We give some explicit stability bounds for discrete-time systems subjected to time-varying and nonlinear perturbations. The obtained results are extensions of some well-known results in (Hinrichsen and Son in Int. J. Robust Nonlinear Control 8:1169-1188, 1998; Shafai et al. in IEEE Trans. Autom. Control 42:265-270, 1997) to nonlinear time-varying perturbations. Two examples are given to illustrate the obtained results. Finally, we present an Aizerman-type conjecture for discrete-time systems and show that this conjecture is valid for positive systems. MSC: 39A30; 93D09

Delay-Dependent Stabilization of Discrete-Time Systems With Time-Varying Delay via Switching Technique

In this paper, we investigate the stabilization problem of time-delay discrete-time systems with occurrence probabilities for time delays. Using an augmenting technique, we convert random time-delay discrete-time systems to delay-free systems with stochastic variables. We propose to design a sequence of delay-dependent controllers. The implemented controller is chosen according to the measured time delay at each sampling instant. When the delay is randomly time-varying, the implemented controller is switched in a designed controller set. An example is provided to illustrate the effectiveness of the stability condition and another example is offered to show the applicability of the proposed design algorithm.

Stability of discrete-time systems via polytopes of reflection vector sets

The stability domain of discrete-time systems is investigated via reflection coeffi- cients of characteristic polynomials of the system. Stable polytopes in the coefficients space of characteristic polynomials are defined starting from the sufficient stability condition in the polynomial reflection coefficients space using different reflection vector sets. The volumes of these stable polytopes are calculated via the triangulation method.

Note on stability of discrete-time time-varying delay systems

This note is concerned with the problem of delay-dependent stability for discrete-time systems with time-varying delay. Two novel delay-dependent stability conditions are established by bounding the forward difference of the Lyapunov functional based on the reciprocally convex approach and a scale inequality, respectively. These criteria improve the existing ones with smaller computation burden and less conservatism, which is verified by both theoretical proof and a numerical example.

Finite-time stability of discrete-time systems with time-varying delay

Finite-time stability can be used in all applications where large values of the state are not acceptable. In this paper, finite-time stability problem for a class of linear time-varying delay systems is studied. Based on Lyapunov-like functions method and using an appropriate model transformation of the original system, the sufficient delay-dependent finite-time stability conditions are derived. The criteria are presented in the form of LMIs, which are dependent on the minimum and maximum delay bounds. The numerical examples are presented to illustrate the applicability of the developed results.

New criterion for stability of discrete-time systems joined with a saturation operator on the state-space

A new criterion for the global asymptotic stability of discrete-time systems in a state-space realization using saturation arithmetic and with the coefficient matrix A fulfilling a certain condition is presented. The criterion possesses a considerable degree of freedom (free parameters) for testing the stability condition. The criterion is compared with some earlier criteria.

Parametric Lyapunov Equation Approach to Stabilization of Discrete-Time Systems With Input Delay and Saturation

This paper studies the problem of stabilization of discrete-time linear systems with input delay and saturation nonlinearity. By exploring some further intricate properties of the recently developed parametric Lyapunov equation-based low-gain feedback design approach, solutions are proposed to solve the problems by both state feedback and output feedback. This new approach is not only simpler than the existing method that is based on the eigenstructure assignment technique, but also provides explicit conditions on the low-gain parameter to guarantee the stability of the closed-loop system. Moreover, it is possible by adjusting the low-gain parameter online to achieve global results when the system is subject to both input saturation and time-delay. Also, the delay in the input is allowed to be time-varying in some cases.

Stability analysis and output‐feedback stabilization of discrete‐time systems with a time‐varying state delay and nonlinear perturbation

AbstractThis paper aims to derive stability conditions and an output‐feedback stabilization method for discrete‐time systems with a time‐varying state delay and nonlinear perturbation. With a new way of handling the Lyapunov stability criterion, linear matrix inequality conditions are obtained for estimating bounds on delay to ensure the asymptotic stability. Based on the conditions, a synthesis procedure is developed for finding stabilizing output‐feedback gains, which are formulated as direct design variables. Three numerical examples are employed to demonstrate the effectiveness and advantages of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Novel criterion for stability of discrete-time systems in a state-space realization utilizing saturation nonlinearities

A new criterion for the global asymptotic stability of discrete-time systems in a state-space realization utilizing saturation nonlinearities is presented. The form of the criterion is unconventional in that it offers the flexibility of using even an unsymmetric Lyapunov matrix P. A recently reported criterion is shown to be a special case of the present criterion. An example shows the effectiveness of the new criterion.

Strong stability of discrete-time systems

The paper introduces a new notion of stability for internal (state-space) autonomous system descriptions in discrete-time, referred to as strong stability which extends a parallel notion introduced in the continuous-time case. This is a stronger notion of stability compared to alternative definitions (asymptotic, Lyapunov), which prohibits systems described by natural coordinates to have overshooting responses for arbitrary initial conditions in state-space. Three finer notions of strong stability are introduced and necessary and sufficient conditions are established for each one of them. The class of discrete-time systems for which strong and asymptotic stability coincide is characterized and links between the skewness of the eigen-frame and the violation of strong stability property are obtained. Connections between the notions of strong stability in the continuous and discrete-domains are briefly discussed. Finally strong stabilization problems under state and output feedback are studied. The results of the paper are illustrated with a numerical example.

Stability of discrete-time systems joined with a saturation operator on the state-space: Yet another version of Liu–Michel's criterion

A novel criterion for the global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic is presented. The criterion, which is basically a new version of an earlier criterion due to Liu and Michel, is a generalization and improvement over a recently reported criterion. The generalization is due to the fact that the use of even unsymmetric Lyapunov matrix P is permitted. An example illustrating the improvement realized from the present criterion is given.

Sufficient Conditions for Local Asymptotic Stability and Stabilization for Discrete-Time Varying Systems

The purpose of this paper is to establish sufficient conditions for local asymptotic stability and feedback stabilization for discrete-time systems with time depended dynamics. Our main results constitute generalizations of those developed by same authors in a recent paper, published in same journal, for the case of continuous-time systems.

Stability of discrete-time systems joined with a saturation operator on the state–space: Generalized form of Liu-Michel’s criterion

A novel criterion for the global asymptotic stability of discrete-time systems in a state–space realization using saturation arithmetic was previously given by Liu and Michel. In their approach, as in most (or probably all) existing approaches, the matrix P is assumed to be symmetric. In this communique, it is shown that their approach is such as to be extended to allow the use of an unsymmetric matrix P . Taking note of this, a more general form of their criterion is presented. A modified form of the criterion is also presented. Examples show the effectiveness of the generalized approach. To the best of author’s knowledge, this communique is the first report of exploiting an unsymmetric P to obtain a stability condition with reduced conservatism.

Output-Feedback Stabilization of Discrete-Time Systems with Multiple Time-Varying Delays

Output-Feedback Stabilization of Discrete-Time Systems with Multiple Time-Varying Delays

Stabilization of Discrete-time Nonlinear Systems subject to Input Saturations: a New Lyapunov Function Class

AbstractThis paper addresses the problem of stabilization of discrete-time systems including a cone-bounded nonlinearity and a saturating actuator. In the sense of Lyapunov stability, we introduce a new candidate Lyapunov function which takes nonlinearity behavior into account. The local stability criterion is formulated as a set of Bilinear Matrix Inequalities (BMI) conditions. We present an optimization problem in order to guarantee the closed-loop stability aiming the largest basin of attraction, which may be nonconvex, and/or, nonconnected. Furthermore, a simple iterative algorithm is proposed in order to solve our BMI problem. Some numerical examples are presented to highlight the relevance of the new Lyapunov function in regard to the classical quadratic function.

Stability of Discrete-Time Systems Joined With a Saturation Operator on the State-Space $ $

This note studies the stability of discrete-time dynamical systems acted upon by a saturation process in the state-space. A finite procedure is proposed to focus on the asymptotic behavior of systems, which produces linear constraints imposed on Lyapunov-Stein matrix inequalities to be solved. A little linear algebra broadens the scope of stability test from that of the earlier Liu-Michel's criterion.