Stability Of Linear Discrete Systems Research Articles

Exponential stability of linear discrete systems with multiple delays by degenerated Lyapunov–Krasovskii functionals

The problem of exponential stability of delayed discrete systems with multiple delays x(n+1)=(I+A)x(n)+∑i=1sBix(n−i),n=0,1,…is studied, where x=(x1x2…xm)T is an unknown vector, m and s are fixed positive integers, A, Bi are square constant matrices and I is a unit matrix. A new degenerated Lyapunov–Krasovskii functional is used to derive sufficient conditions for exponential stability and to derive an exponential estimate of the norm of solutions. Though often used in the study of stability, the assumption that the spectral radius of the matrix of linear terms is less than 1 is not applied here. The criterion derived is illustrated by an example and compared with previously known results.

Exponential Stability of Linear Discrete Systems with Multiple Delays

The paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with multiple delays xk+1=Axk+∑i=1sBixk-mi, k=0,1,…, where s∈N, A and Bi are square matrices, and mi∈N. New criterion for exponential stability is proved by the Lyapunov method. An estimate of the norm of solutions is given as well and relations to the well-known results are discussed.

Exponential Stability of Linear Discrete Systems with Variable Delays via Lyapunov Second Method

The paper investigates the exponential stability of a linear system of difference equations with variable delays xk+1=Axk+∑i=1sBikxk-mik, k=0,1,… , where s∈N, A is a constant square matrix, Bik are square matrices, mik∈N∪0, and mik≤m for an m∈N. New criteria for exponential stability are derived using the method of Lyapunov functions and formulated in terms of the norms of matrices of linear terms and matrices solving an auxiliary Lyapunov equation. An exponential-type estimate of the norm of solutions is given as well. The efficiency of the derived criteria is numerically demonstrated by examples and their relations to the well-known results are discussed.

Discrete inequalities based on multiple auxiliary functions and their applications to stability analysis of time-delay systems

This paper presents new discrete inequalities for single summation and double summation. These inequalities are based on multiple auxiliary functions and include the Jensen discrete inequality and the discrete Wirtinger-based inequality as special cases. An application of these discrete inequalities to analyze stability of linear discrete systems with an interval time-varying delay is studied and a less conservative stability condition is obtained. Three numerical examples are given to show the effectiveness of the obtained stability condition.

Stability and exponential stability of linear discrete systems with constant coefficients and single delay

This paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with single delayx(k+1)=Ax(k)+Bx(k−m),k=0,1,…where A, B are square constant matrices and m∈N. Sufficient conditions for exponential stability are derived using the method of Lyapunov functions and its efficiency is demonstrated by examples.

Exponential stability of linear discrete systems with constant coefficients and single delay

In the paper the exponential stability and exponential estimation of the norm of solutions to a linear system of difference equations with single delay x ( k + 1 ) = A x ( k ) + B x ( k − m ) , k = 0 , 1 , … is studied, where A , B are square constant matrices and m ∈ N . New sufficient conditions for exponential stability are derived using the method of Lyapunov functions. Illustrative examples are given as well.

Discrete Wirtinger-based inequality and its application

In this paper, we derive a new inequality, which encompasses the discrete Jensen inequality. The new inequality is applied to analyze stability of linear discrete systems with an interval time-varying delay and a less conservative stability condition is obtained. Two numerical examples are given to show the effectiveness of the obtained stability condition.

Vibrational stabilization and the Brockett problem

The present paper deals with the exposition of methods for solving the Brockett problem on the stabilization of linear control systems by a nonstationary feedback. The paper consists of two parts. We consider continuous linear control systems in the first part and discrete systems in the second part. In the first part, we consider two approaches to the solution of the Brockett problem. The first approach permits one to obtain low-frequency stabilization, and the second part deals with high-frequency stabilization. Both approaches permit one to derive necessary and sufficient stabilization conditions for two-dimensional (and three-dimensional, for the first approach) linear systems with scalar inputs and outputs. In the second part, we consider an analog of the Brockett problem for discrete linear control systems. Sufficient conditions for low-frequency stabilization of linear discrete systems are obtained with the use of a piecewise constant periodic feedback with sufficiently large period. We obtain necessary and sufficient conditions for the stabilization of two-dimensional discrete systems. In the second part, we also consider the control problem for the spectrum (the pole assignment problem) of the monodromy matrix for discrete systems with a periodic feedback.

FIXED‐ORDER DYNAMIC OPTIMAL CONTROL DESIGN FOR ROBUST STABILITY OF LINEAR DISCRETE SYSTEMS

SUMMARYThis paper presents a fixed‐order dynamic optimal control design method for robust stability of linear discrete systems. The controller determination includes a stability robustness component in addition to the standard quadratic state and control terms in the performance criterion. The control design is based on a parameter optimization technique. The resulting controller involves the solution of three algebraic matrix equations, two of which are discrete‐time Lyapunov matrix equations.

Structural stability of linear discrete systems via the exponential dichotomy

Structural stability of linear discrete systems via the exponential dichotomy

A necessary and sufficient condition for stability of linear discrete systems with parameter-variation

A simple sufficient stability criterion for linear discrete systems obtained previously is proved to be necessary and sufficient for the stability of a class of such systems with parameter-variation.

Further comments on ‘A simple criterion for stability of linear discrete systems’

Some additional comments are offered on the stability condition for linear discrete systems in Dabke (1983). It is pointed out that the condition can be naturally extended to cover time-varying systems.

Comments on ‘ A simple criterion for stability of linear discrete systems ’

Comments on ‘ A simple criterion for stability of linear discrete systems ’

A simple criterion for stability of linear discrete systems

Abstract A simple sufficient condition is presented for confining the poles of a linear discrete system on an arbitrarily specified circular region in the z plane

Proof of a certain conjecture on the stability of linear discrete systems

The proof of a previously published conjecture on the stability of linear discrete systems is obtained using Lyapunov's second stability theorem.

A note on the stability of linear discrete systems and lyapunov method

In a publication by Y. P. Harn and C. T. Chen [1] a proof of a discrete stability test by using the Lyapunov method was given. In this note the main results obtained by the author in this direction in 1965 [6], [7] and the extensions and applications obtained partially with other authors since then and published in different journals [8]-[10] are surveyed. The similarity to the results in [1] is obvious.

Unconditional stabilization of linear discrete systems via adaptive control

This paper deals with the conditions of overall deterministic stability of the adaptive control schemes which may be obtained by associating any usual identification and control methods (e.g. Recursive Minimum Square, Stochastic Approximation, Extended Kalman Filter, for Identification; Model Reference, Pole Placement, Quadratic Optimization, for Control). The deterministic and time invariance hypotheses are introduced here only for the analysis of the above schemes, but not for their synthesis. There are no assumptions of sufficiently exciting input, or any requirement for parameter convergence, nor an hypothesis of bounded input and output signals.

Asymptotic stablity of linear continuous systems containing uncertain parameters†

In a recent paper by the author, the concept of asymptotic stability was extended to systems containing uncertain parameters and a sufficient condition for asymptotic stability of linear discrete systems containing uncertain parameters was developed. In this paper the results are extended and a sufficient condition for asymptotic stability of linear continuous systems containing time-varying uncertain parameters is developed.

Stability theorem for linear discrete systems

A stability theorem is arrived at and is shown to lead to coefficient conditions that serve as useful inspection tests in the determination of the stability of linear discrete systems. Examples are included to support this statement.

A Simplified Stability Criterion for Linear Discrete Systems

In this study a simplified analytic test of stability of linear discrete systems is obtained. This test also yields the necessary and sufficient conditions for a real polynomial in the variable z to have all its roots inside the unit circle in the z plane. The new stability constraints require the evaluation of only half the number of Schur-Cohn determinants [1], [2]. It is shown that for the test of a fourth-order system only a third-order determinant is required and for the fifth-order, one second-order and one fourth-order determinant are required. The test is applied directly in the z plane and yields the minimum number of constraint terms. Stability constraints up to the sixth-order case are obtained and for the nth-order case are formulated. The simplicity of this criterion is similar to that of the Lienard-Chipard criterion [3] for the continuous case which has a decisive advantage over the Routh-Hurwitz criterion [4], [5]. Finally, general conditions on the number of roots inside the unit circle for n even and odd are also presented in this paper.