Stability Of Discrete-time Systems Research Articles

Robust iterative learning control for discrete-time systems with random initial state shifts

This paper presents the sufficient conditions of the stability for discrete-time systems with random initial state shifts when applying iterative learning control. Different sufficient conditions have been developed for systems with different characteristics to ensure boundedness of tracking errors and convergence of the learning process under random initial condition. First, the relationship between the initial shifts of two adjacent iterations is addressed with the help of the transition matrix. And the different analysis models are established for each type of systems: one-dimensional matrix analysis model and 2-D Roesser model. Second, the new and sufficient conditions are established for the proposed learning control law to ensure systems convergence by matrix theory and two-dimensional system theory. Third, solving the linear matrix inequality (LMI) yields the control gains. Finally, digital simulations illustrate the effectiveness and sufficiency of the presented conditions.

LQR and Stabilization for Discrete-Time Systems With Multiplicative Noises and Input Delays

LQR and Stabilization for Discrete-Time Systems With Multiplicative Noises and Input Delays

Finite-Time Stability for Discrete-Time Systems with Time-Varying Delays and Nonlinear Perturbations Using Relaxed Summation Inequality

Finite-Time Stability for Discrete-Time Systems with Time-Varying Delays and Nonlinear Perturbations Using Relaxed Summation Inequality

Discrete-time state delayed systems with saturation arithmetic: overflow oscillation-free realization

ABSTRACT In many smart systems, such as cloud computing, networked control, manufacturing, and telecommunication, the effects of time delays are unavoidable. Saturation nonlinearity is common in many smart systems (e.g. electric systems with limited actuator power supplies, mechanical systems with position and speed constraints, fixed-point digital filters, etc.). Therefore, the study of stability of discrete-time systems (DTSs) with time delay and saturation is of high importance in practice. The global asymptotic stability (GAS) problem of such systems is investigated in this paper. The saturation nonlinearity is constrained by a convex hull with a free matrix whose infinity norm is smaller than or equal to unity. A new Lyapunov-based GAS criterion for DTSs with time-varying delay and state saturation is established. The GAS problem for DTSs with constant delay and saturation is also discussed. The obtained results can be used to ensure the absence of overflow oscillations in the considered system. In comparison to several existing criteria, the approach yields improved stability results. An example is provided to demonstrate the importance of the presented results.

Stabilization of Discrete-Time Systems with Random Delays and Packet Drops: A Predictor-Like Controller

In this paper, we study stabilization of discrete-time systems in which control inputs are transmitted over a random delay and packet drop channel. A predictor-like linear feedback controller is proposed by a reduction technique in which the reduced system has the same dimension as that of the original system. A necessary and sufficient condition to stabilize the original system in the mean-square sense is obtained based on an algebraic Riccati equation. Some necessary and sufficient stabilizability conditions based on the Lyapunov criterion are also presented. Interestingly, it is shown that the stabilization problem considered in this paper is equivalent to the stabilization problem of a particular delay-free system with control-dependent correlated noises. Simulation examples are provided to illustrate the effectiveness and advantages of the proposed stabilizing controller.

New Criterion for the Stability of Discrete-Time Systems with State Saturation and Time-Varying Delay

New Criterion for the Stability of Discrete-Time Systems with State Saturation and Time-Varying Delay

Delayed impulsive stabilisation of discrete-time systems: a periodic event-triggering algorithm

This paper studies the problem of event-triggered impulsive control for discrete-time systems. A novel periodic event-triggering scheme with two tunable parameters is presented to determine the moments of updating impulsive control signals which are called event times. Sufficient conditions are established to guarantee asymptotic stability of the resulting impulsive systems. It is worth mentioning that the event times are different from the impulse times, that is, the control signals are updated at each event time but the actuator performs the impulsive control tasks at a later time due to time delays. The effectiveness of our theoretical result with the proposed scheme is illustrated by three examples.

Delay partitioning approach to the delay-dependent stability of discrete-time systems with anti-windup

ABSTRACT In this digital era, the basis of every smart instrument is discrete signal models e.g. in Networked control systems, Cyber physical systems etc. It has been shown that time-delays are unavoidable during the digital implementation of an engineering system. Therefore, the stabilization of discrete time delayed systems is gaining the high importance [1–10]. Although a lot of literature is found on the stabilization of time delayed systems for a long time using the construction of proper non-negative Lyapunov functional. Recalling some existing results on this issue, the LMI-based stability conditions are obtained by its forward difference negative-definite in direction to claim the less conservative results [15–25]. In order to seek less conservative stability criteria, this paper introduces an anti-windup scheme appended with Wirtinger inequality, reciprocal convex approach and delay partitioning of a discrete-time delayed systems by using Lyapunov Krasovskii functional. To accomplish this task, delay partitioning technique may be utilized to develop improved stability conditions for the considered system. The Wirtinger-based inequality and reciprocal convex approach has been employed to derive less conservative results. On employing the delay partitioning, a novel linear matrix inequality-based criterion is proposed to stabilize such systems. The considered Lyapunov-Krasovskii functional includes the information of intermediate delay to acknowledge the delay information implicitly that ensures the considered system to be regular, impulse free and stable in terms of linear matrix inequalities. The estimation of the attraction basin is to ensure that the state remains inside the level set of a certain Lyapunov function. Numerical simulation verifies that the presented method reduces conservatism than the existing results.

Discrete-Time Control for Double-Integrator Systems With State Constraint and Learning Gain

This brief investigates a barrier Lyapunov function based discrete-time control with Q-learning based gains for double-integrator systems with state constraint. It is found, from the stability proof, that the high conservatism of analysis for the stabilization of discrete-time system with state constraint is revealed, where the explicit selection of constant high gain is challenging. To address this problem, the fuzzy Q-learning algorithm is employed to search for the nearly optimal control gains for both fast response and low steady-state error in the long view of performance consideration. The numerical and experimental results verify the effectiveness of the proposed method, and varying gains based on fuzzy approximation Q-learning can aid to reduce the steady-state error while fast response to the reference motion trajectories.

Exponential stability of discrete-time systems with slowly time-varying delays

Slowly time-varying delays are seldom, but do need to be, considered in the context of discrete-time systems. This paper addresses the exponential stability issue of discrete-time systems with slowly time-varying delays. The basic idea is to transform, by utilizing the switching transformation approach, the original system with slowly time-varying delays into an equivalent switched system with special switching signal. Different types of delays correspond to different types of switching signals, and the stability issue of the original system is converted into that of a switched system. It is the first time that the method of switched homogeneous polynomial Lyapunov function is applied to general delayed systems. Some sufficient exponential stability conditions for the original system are proposed in several situations. It is numerically shown that the conservativeness of the proposed conditions reduces as the degree of the switched homogeneous polynomial Lyapunov function increases.

Stability of Discrete-Time Systems Under Restricted Switching via Logic Dynamical Generator and STP-Based Mergence of Hybrid States

In this article, we propose a novel technique for the stability analysis of discrete-time switched systems under constrained switching based on the semitensor product (STP) of matrices and vector-representation of logic. We assume that the admissible switching signals are generated by a logic dynamical system, referred to as the logic dynamical generator (LDG) of the admissible switching. We demonstrate that switching with minimum dwell-time and restricted successors represent two special cases of this type, by constructing their respective LDGs. First, we transfer the LDG into algebraic form, based on the vector-representation of logic. Next, we merge the logic state of the LDG and the continuous state of the original switched system using the STP and derive the dynamics of the merged state; a switched system without constraints on switching. The stability of the switched system under constrained switching is equivalent to the stability of the merged system under arbitrary switching. Based on this, sufficient conditions for the asymptotical stability of nonlinear switched systems with analytic subsystems under constrained switching are obtained. Specifically, in the case of all subsystems being linear, necessary and sufficient conditions for asymptotical stability under constrained switching are proposed.

Continued Fraction Expansions of Stable Discrete-Time Systems of Difference Equations

We provide a systematic procedure for generating the coefficients of the continued fraction expansion of the test function associated with the characteristic polynomial of a stable system of difference equations. We illustrate the feasibility of the procedure, and we provide an application on the stability of two-dimensional digital filters.

Global exponential stability of discrete-time Hopfield neural network models with unbounded delays

In this paper, a general setting is presented to study the exponential stability of discrete-time systems with bounded or unbounded delays. Based on the M-matrix theory, we establish sufficient conditions to ensure the global exponential stability of the zero equilibrium of low-order, and high-order, discrete-time Hopfield neural network models with unbounded delays and delay in the leakage terms. A comparison of the literature shows that our results generalize and improve some in recent publications.

A delay-variation-dependent stability criterion for discrete-time systems via a bivariate quadratic function negative-determination lemma

This article concerns with stability analysis of discrete linear systems with time-varying delays. Firstly, we extend a quadratic function negative-determination lemma for a single variable to the bivariate case. Secondly, we construct a novel Lyapunov-Krasovskii functional (LKF) with a quadratically delay-dependent matrix to investigate the stability of discrete-time systems with time-varying delays. Based on the proposed lemma, a new delay-variation-dependent stability criterion is derived. Finally, numerical examples are given to illustrate the theoretical result and the proposed criterion is shown to be less conservative than some previous ones.

Stable learning laws design for long short-term memory identifier for uncertain discrete systems via control Lyapunov functions

This study introduces a method for designing stable learning laws of Long Short-Term Memory (LSTM) networks working as a non-parametric identifier of nonlinear systems with uncertain models. The strategy applies the concept of stability for discrete-time systems in the sense of Lyapunov to prove that origin is a practical stable equilibrium point for the identification error. The laws consider a general class of sigmoidal functions placed at the different gates of a LSTM structure (long and short memory). The design of the learning laws uses a matrix inequality framework to obtain the rate gains associated with the evolution of the weights. Numerical results show the designed learning laws for the non-parametric identifier based on a LSTM approximation tested on two classes of nonlinear systems: the first one describes the ozone-based degradation of organic contaminants, and the second one represents the dynamics of a Van Der Poll oscillator. The LSTM identifier is compared against a classical Lyapunov-based recurrent neural network. This comparison demonstrates how the proposed algorithm approximates the trajectories of both systems with a smaller mean squared error, which serves as an indicator of the benefits obtained with these new learning laws.

Symmetric Properties of Routh–Hurwitz and Schur–Cohn Stability Criteria

It is often noticed in the literature that some key results on the stability of discrete-time systems of difference equations are obtained from their corresponding results on the stability of continuous-time systems of differential equations using suitable conformal mappings or bilinear transformations. Such observations lead to the search for a unified approach to the study of root distribution for real and complex polynomials, with respect to the left-half plane for continuous-time systems (Routh–Hurwitz stability) and with respect to the unit disc for discrete-time systems (Schur–Cohn stability). This paper is a further contribution toward this objective. We present, in a systematic way, the similarities, and yet, the differences between these two types of stability, and we highlight the symmetry that exists between them. We also illustrate how results on the stability of continuous-time systems are conveyed to the stability of discrete-time systems through the proposed techniques. It should be mentioned that the results on Schur–Cohn stability are known to be harder to obtain than Routh–Hurwitz stability ones, giving more credibility to the proposed approach.

Periodic Averaging of Discrete-Time Systems: A Time-Delay Approach

This article is concerned with the stability of discrete-time systems with fast-varying coefficients that may be uncertain. Recently, a constructive time-delay approach to averaging was proposed for continuous-time systems. In the present article, we develop, for the first time, this approach to discrete-time case. We first transform the system to a time-delay system with the delay being the period of averaging, which can be regarded as a perturbation of the classical averaged system. The stability of the original system can be guaranteed by the resulting time-delay system. Then under assumption of the classical averaged system being exponentially stable, we derive sufficient stability conditions for the resulting time-delay system, and find a quantitative upper bound on the small parameter that ensures the exponential stability. Moreover, we extend our method to input-to-state stability (ISS) analysis of the perturbed systems. Finally, we apply the approach to the practical stability of discrete-time switched affine systems, where an explicit ultimate bound in terms of the switching period is presented. Two numerical examples are given to illustrate the efficiency of results.

New results on stability of discrete-time systems with infinite delays

This paper is concerned with problems of stability of discrete-time systems with infinite delays. Several new theorems for stability analysis of the concerned delayed systems are developed. Specifically, uniform stability, uniform asymptotic stability, global uniform asymptotic stability, exponential stability and global exponential stability are included. Our results include some existing works on bounded delays as special cases. Furthermore, an example is furnished to demonstrate the effectiveness of the proposed theorems.

Some novel necessary and sufficient conditions of exponential stability for discrete-time systems with multiple delays: A Lyapunov matrix approach

This paper aims at establishing necessary and sufficient conditions of exponential stability for linear discrete-time systems with multiple delays. Firstly, we introduce a new concept—Lyapunov matrix, and investigate its properties, existence and uniqueness by: (i) characterizing the solution of a boundary value problem of matrix difference equations; and (ii) constructing complete type Lyapunov–Krasovskii functionals with pre-specified forward difference. Secondly, a new constructive analysis methodology, named Lyapunov matrix approach, is proposed to establish necessary and sufficient exponential stability conditions for discrete-time systems with multiple delays. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results. It is worth emphasizing that, from a view of computation, the Lyapunov matrix approach proposed here is concerned with three key steps: (i) solve a systems of linear equations; (ii) check whether a constant matrix is of full-column-rank, and (iii) judge whether a constant matrix is positive definite. All of these can be easily realized by using the tool software—MATLAB.

Output-feedback anti-disturbance predictor-based control for discrete-time systems with time-varying input delays

This paper investigates the robust stabilization of discrete-time systems with time-varying input delays and model uncertainties by predictor-based anti-disturbance output-feedback control strategies. Here, a novel predictor-feedback control combined with an extended state observer is proposed. The objective is to counteract the negative effects of input delays while actively rejecting disturbance signals typically encountered in engineering practice, such as steps or harmonics. Differently from previous approaches, unknown but bounded time-varying delays are taken into consideration. Moreover, the complexity of the algorithm for control synthesis is notably reduced. Finally, an illustrative example from the literature is provided to show that better robust performance can be achieved with the proposed method.