Stability Analysis Of Discrete-time Systems Research Articles

Stability analysis of discrete-time delayed systems via matrix-injection-based transformation method for bivariate quadratic functions

In this paper, the stability analysis of discrete-time systems with a time-varying delay is studied. First, a selectable matrix-injection-based transformation lemma is proposed to guarantee the negative (or positive) definiteness of bivariate quadratic function. This lemma allows a selection of conditions that combine different delay sets, which is more relaxed than that in most previous studies. Then, an improved Lyapunov–Krasovskii functional (LKF) with delay-squared-product summation is given, which introduces extra cross-terms about delay-variation square for stability analysis. Considering the role of injection matrices in the proposed lemma, together with two types of allowable delay sets, four stability criteria are established. Finally, the less conservatism of the proposed criteria is verified by two numerical examples, which demonstrate the superiority of the presented methods.

Delay-variation-dependent summation inequality and its application to stability analysis of discrete-time systems with time-varying delay

This work tends to research the stability of the discrete-time systems with a time-varying delay based on the Lyapunov-Krasovskii functional (LKF) method. In order to acquire a less conservative stability criterion, some techniques in this work are refined and taken into use. Firstly, to reduce the conservatism generated when estimating the forward difference of the LKF, a newly delay-variation-dependent summation inequality is constructed, which includes the existing free-matrix-based and Bessel function-based summation inequalities, using delay-variation-product relaxed matrices to provide more freedom for the estimation results. Secondly, to further show the influence of the introduced variation information in the time-varying delay, we give another selection of the allowable delay set for the delayed discrete-time systems. Thirdly, by taking advantages of the above method and by constructing a delay-product-type LKF, using extended free-weighting-matrices zero equations, and considering different allowable delay sets, two improved linear matrix inequality (LMI)-based delay-variation-dependent criteria for delayed discrete-time systems are formulated. Some classical numerical instances are presented to explain the effectiveness of these proposed stability criteria.

Stability analysis of discrete-time systems with time-varying delay via a delay-dependent matrix-separation-based inequality

This paper is concerned with the stability analysis of discrete-time systems with a time-varying delay. Different from the recent related works that reduce the conservatism by increasing the complexity of Lyapunov functions, this paper aims at conservatism-reduction by using relatively simple Lyapunov functions with an augmented double summation term. The key point for achieving this task is that a delay-dependent matrix-separation-based inequality is established to estimate the augmented-type summation term arising in the forward difference of Lyapunov functions. The proposed method provides a general form of inequalities, which realizes the effective reduction of estimation gap, and also introduces several tractable delay-dependent matrices, which achieves the full use of available delay-related information. As a result, for two types of time-varying delays, the usage of the proposed inequality under different separations leads to two less conservative and low complex stability criteria, whose advantages are demonstrated via three examples.

Bessel-type inequality in semi-inner-product spaces and its application to stability analysis of discrete-time systems with distributed delays

In this paper, a Bessel-type inequality in semi-inner produce (s.i.p.) spaces is presented to deal with the problem of stability analysis for a class of discrete-time systems with distributed delays. The main purpose of this paper is to obtain less conservative stability conditions for the investigated systems. For this purpose, a novel inequality (which is referred to as Bessel-type inequality in s.i.p. spaces) is constructed by combining the Bessel inequality and a set of polynomials which is pairwise orthogonal in a s.i.p. space. The proposed inequality is more general than the conventional Jensen-type inequality, which is, to the best of our knowledge, the only inequality to deal with the problem of stability analysis for discrete-time systems with distributed delays up to now. Based on the proposed inequality and the Lyapunov-Krasovskii stability theory, a stability condition is presented. Finally, a numerical example is given to indicate that the proposed method is able to effectively reduce the conservatism.

Equivalent description and stability analysis for discrete-time systems with uniformly distributed uncertainty

We are concerned with discrete-time systems with uncertainty, which is assumed to be uniformly distributed over the unit interval [0,1]. To describe this uncertainty, the space {0,1}N, which is composed of all the infinite 0-1 sequences and endowed with the Haar measure, actually constitutes the underlying probability space. Therefore, it is shown that for stability analysis, such an uncertain system can be transformed into a switching system with switching sequences having maximal entropy. A stability criterion then is derived from this transformation. An illustrative example is included to show the effectiveness of the theoretical results.

Delay-Dependent Positivity and Stability Analysis of Discrete-Time Systems With Delay

In this letter, delay-dependent positivity and stability conditions, which are crucially different from existing delay-independent ones, are derived for discrete-time systems with time-varying delay. By utilizing a special property called non-oscillatory behavior of solutions of scalar difference equations with delays, the proposed conditions are formulated in terms of linear programming settings. The efficiency of the obtained results is illustrated by a numerical example with simulations.

Stability Analysis of Discrete-time System with Slowly Time-varying Delays

This paper addresses the stability issue of discrete-time systems with slowly time-varying delays. The basic idea is to convert the stability of the original system into an equivalent switched system, that is, each subsystem of the switched system corresponds to a constant delay of the original system in the entire delay interval via a switching transform approach. Moreover, the method of switched homogeneous polynomial Lyapunov function of higher order is proposed. By this approach, the results proposed are less conservative than the reported results. Finally, an example is provided to illustrate the advantages of the proposed stability results.

Homogeneous Discrete-Time Approximation

In this paper we study some stability properties of discrete-time systems whose transition map can be approximated by a discrete-time homogeneous transition map. This allows us to identify qualitative stability properties of discrete-time systems by only knowing the discrete-time homogeneity degree of its approximation. We show how these results can be applied to the stability analysis of discrete-time systems obtained by means of the explicit and implicit Euler discretization methods.

Stability analysis of discrete-time switched systems with time-varying delays via a new summation inequality

This paper proposes a new summation inequality, which improves the conservatism in the stability analysis for discrete-time systems with time-varying delay. In order to show the effectiveness of the proposed inequality, which provides general lower bound of the summation quadratic term of the form ∑s=abxT(s)Mx(s), a delay-dependent stability criterion for such systems is derived within the framework of linear matrix inequalities (LMIs). Going one step forward, the proposed inequality is applied to a stability problem in discrete-time switched systems with time-varying delays. The advantages of employing the proposed summation inequality are illustrated via three numerical examples.

Stability analysis of discrete-time systems with time-varying delays: generalized zero equalities approach

Summary This paper suggests a generalized zero equality lemma for summations, which leads to making a new Lyapunov–Krasovskii functional with more state terms in the summands and thus applying various zero equalities for deriving stability criteria of discrete-time systems with interval time-varying delays. Also, using a discrete-time counter part of Wirtinger-based integral inequality, Jensen inequality, and a lower bound lemma for reciprocal convexity, the forward difference of the Lyapunov–Krasovskii functional is bounded by the combinations of various state terms including not only summation terms but also their interval-normalized versions, which contributes to making the criteria less conservative. Numerical examples show the improved performance of the criteria in terms of maximum delay bounds. Copyright © 2016 John Wiley & Sons, Ltd.

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.

Further results on stability analysis of discrete-time systems with time-varying delays via the use of novel convex combination coefficients

This paper focuses on deriving an improved stability criterion for discrete time-delay systems via a Lyapunov–Krasovskii functional approach that takes advantage of triple summation terms. To this end, novel convex combination coefficients coupled with second-order time-varying delay terms are introduced, and the corresponding reciprocally convex approach is established with the use of the minimal number of slack matrix variables. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed method, with particular regard to robustness and H∞ performance.

Efficient Approach to Stability Analysis of Discrete-Time Systems with Time-Varying Delay

This paper aims at deriving an efficient criterion for the robust stability analysis of discrete-time systems with time-varying delay. In the derivation, to obtain a larger stability region under the requirement of less computational complexity, this paper proposes a valuable method capable of establishing a less conservative stability criterion without using the free-weighting approach and an extremely augmented state. In parallel, the stabilization problem of systems with time-delayed control input is addressed in connection with the derived stability criterion.

Improved delay-partitioning approach to robust stability analysis for discrete-time systems with time-varying delays and randomly occurring parameter uncertainties

Summary This paper is concerned with the robust stability problem for uncertain discrete-time systems with interval time-varying delays and randomly occurring parameter uncertainties. By construction of a suitable Lyapunov–Krasovskii functional and utilization of new zero equalities with delay-partitioning approach, improved delay-dependent criteria for the robust stability of the systems are derived in terms of linear matrix inequalities for guaranteeing the asymptotic stability of the concerned systems. The effectiveness and reduction of conservatism of the derived results are demonstrated by three numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.

Robust stability analysis of discrete-time systems with parametric and switching uncertainties

Robust stability analysis is investigated for discrete-time linear systems with rational dependency with respect to polytopic type uncertainties. Two type of uncertainties are considered: constant parametric uncertainties and time-varying switching uncertainties. Results are in LMI formalism and proofs involve parameter-dependent, quadratic in the state, Lyapunov functions. The new proposed conditions are shown to extend and merge two important existing results. Conservatism reduction is tackled via a model augmentation technique. Numerical complexity is contained by exploiting the structure of the models with respect to the uncertainties.

Further Input-to-State Stability Subtleties for Discrete-Time Systems

This technical note considers input-to-state stability analysis of discrete-time systems using continuous Lyapunov functions. The main result reveals a relation between existence of a continuous Lyapunov function and inherent input-to-state stability on compact sets with respect to both inner and outer perturbations. If the Lyapunov function is K∞-continuous, the result applies to unbounded sets as well.

Stability Analysis of Discrete-Time Systems With Quantized Feedback and Measurements

This paper considers a quantized system with finite-level quantized input computed from quantized measurements (QIQM). The problem of globally asymptotic stability of QIQM system is transferred to the one of an equivalent system depending on a multiplier which is nonnegative and bounded. It is the focus of this paper to discuss the nonnegativity and boundness of the multiplier. A sufficient condition is given for the globally asymptotic stability of QIQM system. Note the main method used here is similar to the one used by Richter and Misawa, in which the uniform quantizer is employed. This paper is the extension of that work to the logarithmic quantization which is more advantage than the uniform one. A numerical simulation is presented at last to show the effectiveness of the main results and the advantage of logarithmic quantization compared to uniform quantization.

Relaxed inequality approach to robust ℋ∞ stability analysis of discrete-time systems with time-varying delay

This study aims at deriving a less conservative delay-dependent criterion for the robust H ∞ stability analysis of discrete-time systems with interval time-varying state delay. To this end, an appropriate Lyapunov-Krasovskii functional containing triple summation terms is established and a relaxed inequality approach is proposed to address the induced double summation inequality and to exploits novel relaxation variables for null sum terms. Three numerical examples are provided to illustrate the effectiveness of the derived stability criteria.

Regional asymptotic stability analysis for discrete-time delayed systems with saturation nonlinearity

This paper presents a novel method for asymptotic stability analysis of discrete-time systems with state delay and saturation nonlinearity. Based on Lyapunov functional and LMI (linear matrix inequality) framework, new stability criteria are derived in terms of LMIs by using some properties of the saturation nonlinearity. The criteria can be applied to the global and regional stability. Numerical examples are given to verify the theoretical result of the proposed method.

Delay-dependent stability analysis for discrete-time systems with time varying state delay

The stability of discrete systems with time-varying delay is considered. Some sufficient delaydependent stability conditions are derived using an appropriate model transformation of the original system. The criteria are presented in the form of LMI, which are dependent on the minimum and maximum delay bounds. It is shown that the stability criteria are approximately the same conservative as the existing ones, but have much simpler mathematical form. The numerical example is presented to illustrate the applicability of the developed results.