Stability Of Discrete-time Systems Research Articles

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.

Stability analysis and output-feedback stabilisation of discrete-time systems with an interval time-varying state delay

This study focuses on the stability of discrete-time systems with a time-varying state delay and uncertainties in system matrices. New linear matrix inequality conditions are derived for determining bounds of the delay size to ensure the asymptotic stability. There is no need to assume that the system is stable when the delay vanishes. Conservativeness of the conditions is reduced as much as possible by introducing the free-weighting matrices in a novel way, and using the least inequality in the derivation process. Then a new static output-feedback stabilisation method is proposed with the gain matrix as a direct design variable. Compared with the existing results, the proposed methods indeed give larger delay bounds for many cases.

Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities

The asymptotic stability problem for discrete-time systems with time-varying delay subject to saturation nonlinearities is addressed in this paper. In terms of linear matrix inequalities (LMIs), a delay-dependent sufficient condition is derived to ensure the asymptotic stability. A numerical example is given to demonstrate the theoretical results.

Qualitative convergence of matrices

In this paper, we study potential convergence of modulus patterns. A modulus pattern Z is convergent if all complex matrices in Q ( Z ) (i.e. all matrices with modulus pattern Z ) are convergent. A modulus pattern is potentially (absolutely) convergent if there exists a (nonnegative) convergent matrix in Q ( Z ) . We also introduce types of potential convergence that correspond to diagonal and D-convergence, studied in [E. Kaszkurewicz, A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhauser, 2000]. Convergent modulus patterns have been completely characterized by Kaszkurewicz and Bhaya [E. Kaszkurewicz, A. Bhaya, Qualitative stability of discrete-time systems, Linear Algebra Appl. 117 (1989) 65–71]. This paper presents some techniques that can be used to establish potential convergence. Potential absolute convergence and potential diagonal convergence are shown to be equivalent, and their complete characterization for n × n modulus patterns is given. Complete characterizations of all introduced types of potential convergence for 2 × 2 modulus patterns are also presented.

Weakly coprime factorization and state-feedback stabilization of discrete-time systems

The LQ-optimal state feedback of a finite-dimensional linear time-invariant system determines a coprime factorization NM −1 of the transfer function. We show that the same is true also for infinite-dimensional systems over arbitrary Hilbert spaces, in the sense that the factorization is weakly coprime, i.e., Nf, $${Mf \in \mathcal {H}^2 \Longrightarrow f \in \mathcal {H}^2}$$ for every function f. The factorization need not be Bezout coprime. We prove that every proper quotient of two bounded holomorphic operator-valued functions can be presented as the quotient of two bounded holomorphic weakly coprime functions. This result was already known for matrix-valued functions with the classical definition gcd(N, M) = I, which we prove equivalent to our definition. We give necessary and sufficient conditions and further results for weak coprimeness and for Bezout coprimeness. We then establish a variant of the inner–outer factorization with the inner factor being “weakly left-invertible”. Most of our results hold also for continuous-time systems and many are new also in the scalar-valued case.

New delay-dependent conditions on robust stability and stabilisation for discrete-time systems with time-delay

This article is concerned with new robust stability conditions and robust stabilisation method for a discrete-time system with time-delay and time-varying structured uncertainties that come into state and input matrices. An improved approach to obtain new robust stability conditions is proposed. Our approach employs a generalised Lyapunov functional combined with the parameterised model transformation method and the generalised free weighting matrix method. These generalisations lead to generalised robust stability conditions that are given in terms of linear matrix inequalities. Moreover, based on new robust stability conditions, a robust stabilisation method for uncertain discrete-time systems with time-delay is given. Numerical examples compare our robust stability conditions with some existing conditions to show the effectiveness of our approach and also illustrate the improvement of our robust stabilisation method.

Stability of discrete-time systems in the simplest critical case

The paper considers nonlinear time-invariant discrete-time systems of arbitrary orders. The right-hand sides of the equations are sums of polynomials of arbitrary degrees and continuous functions which are infinitely small with respect to the highest-degree terms of the polynomials. It is assumed that the characteristic polynomial of the linearized system has one root equal to unity, while its other roots lie inside the unit disk.

Resilient control of non-linear discrete-time state-delay systems

A class of non-linear discrete-time systems with state-delay is considered. We develop an LMI-based analysis and design procedures to check primarily into the robust stability of discrete-time systems with state-delay and bounded non-linearities. Then we address the robust stabilization using nominal and resilient feedback designs. In both cases the trade-off between the size of the controller gains and the bounding factors is illuminated and incorporated into the design formalism. Seeking computational convenience, all the developed results are cast in the format of linear matrix inequalities (LMIs) and several numerical examples are presented throughout the paper to illustrate the feasibility of the theoretical developments.

Equivalence between the Lyapunov–Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems

The stability of discrete-time systems with time varying delay in the state can be analyzed by using a discrete-time extension of the classical Lyapunov–Krasovskii approach. In the networked control systems domain a similar delay stability problem is treated using a switched system transformation approach. The paper aims to establish a relation between the switched system transformation approach and the classical Lyapunov–Krasovskii method. It is shown that using the switched systems transformation is equivalent to using a general delay dependent Lyapunov–Krasovskii functionals. This functional represents the most general form that can be obtained using sums of quadratic terms. Necessary and sufficient LMI conditions for the existence of such functionals are presented.

Absolute Stability of Discrete-Time Systems with Delay

We investigate the stability of nonlinear nonautonomous discrete-time systems with delaying arguments, whose linear part has slowly varying coefficients, and the nonlinear part has linear majorants. Based on the "freezing" technique to discrete-time systems, we derive explicit conditions for the absolute stability of the zero solution of such systems.

Robust stability of discrete-time systems of difference equations (Removed from Archives)

This paper deals with the robust stability of discrete-time systems of difference equations. Given the nominal characteristic polynomial of a certain discrete-time system, we determine the maximum allowable perturbation that such a polynomial can undergo while preserving the Schur stability of the corresponding system.

Input–output-to-state stability for discrete-time systems

This paper presents Lyapunov characterizations of uniform output-to-state stability and uniform input–output-to-state stability (IOSS) (with respect to disturbances) for discrete-time (DT) nonlinear systems. We show the equivalence of the following three properties for DT systems: uniform IOSS, existence of a smooth Lyapunov function for uniform IOSS, and existence of a (state-)norm estimator. This equivalence result is a DT counterpart of Krichman et al. [2001. Input–output-to-state stability. SIAM Journal on Control and Optimization 39, 1874–1928. Theorem 2.4] and a generalization of Jiang and Wang [2002. A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58. Theorem 1.1] and Jiand and Wang [2001. Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869. Theorem 1, 1 ⇔ 4 ].

Equivalence between the lyapunov-krasovskii functional approach for discrete delay systems and the stability conditions for switched systems

The stability of discrete time systems with time varying delay in the state can be analyzed by using a discrete time extension of the classical Lyapunov-Krasovskii approach. In the networked control systems domain a similar delay stability problem is treated using a switched system transformation approach. The paper aims to establish a relation between the switched system transformation approach and the classical Lyapunov-Krasovskii method. It is shown that using the switched systems transformation is equivalent to using a general delay dependent Lyapunov-Krasovskii functional. This functional represents the most general form that can be obtained using sums of quadratic terms. Necessary and sufficient LMI conditions for the existence of such functionals are presented.

Resilient feedback stabilization of discrete-time systems with delays

A class of linear, uncertain discrete-time systems with state delay is considered. We develop an linear matrix inequality (LMI)-based analysis and redesign procedures for improved robust stability of discretetime systems with state delay and bounded nonlinearities. Then, we address the robust stabilization using nominal and resilient feedback designs. In both cases, the trade-off between the size of the controller gains and the bounding factors is illuminated and incorporated into the design formalism. Seeking computational convenience, all the developed results are cast in the format of LMIs and several numerical examples are presented throughout the paper.

Asymptotic characterizations of input–output-to-state stability for discrete-time systems

We establish the equivalence between global detectability and output-to-state stability for difference inclusions with outputs, and we present equivalent asymptotic characterizations of input–output-to-state stability for discrete-time nonlinear systems. These new stability characterizations for discrete-time systems parallel what have been developed for continuous-time systems in Angeli et al. [Uniform global asymptotic stability of differential inclusions, J. Dynamical Control Systems 10 (2004) 391–412] and Angeli et al. [Seperation principles for input–output and integral-input-to-state stability, SIAM J. Control Optim. 43 (2004) 256–276].

New Results on Stability of Discrete-Time Systems With Time-Varying State Delay

This note is concerned with the stability analysis of discrete-time systems with time-varying state delay. By defining new Lyapunov functions and by making use of novel techniques to achieve delay dependence, several new conditions are obtained for the asymptotic stability of these systems. The merit of the proposed conditions lies in their less conservativeness, which is achieved by circumventing the utilization of some bounding inequalities for cross products between two vectors and by paying careful attention to the subtle difference between the terms Sigmam=k-dk k-1(middot) and Sigma m=k-dM k-1(middot), which is largely ignored in the existing literature. These conditions are shown, via several examples, to be much less conservative than some existing result

THE PROBABILITY STABILITY ESTIMATION OF DISCRETE-TIME SYSTEMS WITH RANDOM PARAMETERS

The simple and effective method for determining the probability stability of discrete-time systems consisting of two and more random parameters is presented in this paper. Systems with random parameters are often encountered in industry of plastic materials, rubber industry, chemical industry, etc., therefore this method can be applied in practice. The presented method provides the choice of such values of parameters for which the system has the largest probability stability. The selection of the adequate parameters values enables the system stability and the correctness of system work.

Delay-dependent robust stabilisation of discrete-time systems with time-varying delay

The stability of discrete systems with time-varying delay is considered. New delay-dependent stability criteria are devised, which are dependent on the minimum and maximum delay bounds. An initial analysis leads to a criterion depending on an inequality involving certain matrices that can be freely chosen. By carefully choosing them to reflect the appropriate relationship between states at differing times, a stricter criterion is thereby obtained. Furthermore, new results for delay-dependent robust stabilisation of uncertain systems with time-varying delay are provided on the basis of a linear matrix inequality (LMI) framework. As the conditions obtained for the existence of admissible controllers are not expressed using strict LMI conditions, a cone complementary linearisation procedure is used to find suitable controllers. Finally, the results obtained, including the stability analysis, static output-feedback stabilisation and dynamic output feedback stabilisation are further extended to discrete time-delay systems having uncertain but norm-bounded parameters. Numerical examples demonstrate the validity of the approach proposed.

Necessary and sufficient conditions for robust global asymptotic stabilization of discrete-time systems†

We give Lyapunov-like conditions for non-uniform in time output stabilization of discrete-time systems. Particularly, it is proved that for a discrete-time control system there exists a (continuous) output stabilizing feedback if and only if there exists a (strong) output control Lyapunov function (OCLF). Moreover, strategies for the construction of continuous robust feedback stabilizers are presented.