Linear Discrete Time-delay Systems Research Articles

Regulator Problem for Linear Discrete-Time Delay Systems with Symmetrical Constraints

Regulator Problem for Linear Discrete-Time Delay Systems with Symmetrical Constraints

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

White noise estimation for linear discrete fractional order system

<abstract> <p>The process white noise (PWN) and observation white noise (OWN) estimation problem for linear discrete fractional order systems (LDFOS) is addressed in this study. By using the Grünwald-Letnikov (G-L) operator as a definition of the discrete fractional calculus (DFC), LDFOS is transformed into a class of linear discrete time-delay systems. However, it is different from the general time-delay system, in which the time-delay part is the cumulative sum from time 0 to the previous time. Based on the orthogonal projection theorem, a suboptimal one-step predictor of LDFOS is designed. Due to the existence of cumulative sum time-delay in system, the Riccati equation has one more cumulative sum state error variance term, which is different from the classical Kalman filter (KF). Moreover, using innovation analysis technology, the filtering and fixed-lag smoothing estimators of PWN and OWN in the form of noise orthogonal projection gain matrices are derived. Finally, two simulation examples are given to verify the effectiveness of PWN and OWN estimators.</p> </abstract>

A new robust discrete-time sliding mode control design for systems with time-varying delays on state and input and unknown unmatched parameter uncertainties

This paper proposes a new robust Discrete-Time Sliding Mode Control (dt-smc) scheme for linear discrete-time delay system. The delays, affecting the state and the input, are assumed to be known, time-varying and bounded. The considered system is also subject to time-varying Unknown But norm-Bounded (ubb) unmatched uncertainties, that act on the state and the delayed state vectors. The designed controller is based on a novel sliding surface. Using a new Lyapunov functional candidate and by introducing slack variables, a less conservative delay-dependent sufficient condition ensuring the asymptotic stability of the closed loop system is derived in terms of Linear Matrix Inequalities (lmis). With respect to that condition a new dt-smc law is synthesized. The proposed control algorithm is applied to an Autonomous Underwater Vehicle (auv). The simulation results demonstrate the effectiveness of the proposed controller.

Observer-Based PID Security Control for Discrete Time-Delay Systems Under Cyber-Attacks

This article deals with the observer-based proportional-integral-derivative (PID) security control problem for a kind of linear discrete time-delay systems subject to cyber-attacks. The cyber-attacks, which include both denial-of-service and deception attacks, are allowed to be randomly occurring as regulated by two sequences of Bernoulli distributed random variables with certain probabilities. A novel observer-based PID controller is proposed such that the closed-loop system achieves the desired security level and the quadratic cost criterion (QCC) has an upper bound. Sufficient conditions are derived under which the exponentially mean-square input-to-state stability is guaranteed and the desired security level is then achieved. Subsequently, an upper bound of the QCC is obtained and the explicit expression of the desired PID controller is also parameterized. Finally, the validity of the developed design approach is verified via an illustrative example.

Guaranteed Cost Preview and Repetitive Control for Uncertain Linear Discrete Time-Delay Systems

Guaranteed Cost Preview and Repetitive Control for Uncertain Linear Discrete Time-Delay Systems

On Stability Analysis of Discrete-Time Systems With Constrained Time-Delays via Nonlinear Halanay-Type Inequality

This letter aims to study the problem of stability for discrete-time fully nonlinear time-delay systems with constrained time-varying delays. A delays digraph is used to model the topology of the delay signals. By exploiting the Halanay technique and suitable Lyapunov functions, some sufficient conditions for the global asymptotic stability, uniform global asymptotic stability and global exponential stability are established. A matrix inequality is derived, and it is employed to prove the global exponential stability of linear discrete-time delay systems with delay signals obeying to a delays digraph. Finally, examples are given to illustrate the results.

Non-monotonic Lyapunov-Krasovskii functional approach to stability analysis and stabilization of discrete time-delay systems

In this paper, a novel non-monotonic Lyapunov-Krasovskii functional approach is proposed to deal with the stability analysis and stabilization problem of linear discrete time-delay systems. This technique is utilized to relax the monotonic requirement of the Lyapunov-Krasovskii theorem. In this regard, the Lyapunov-Krasovskii functional is allowed to increase in a few steps, while being forced to be overall decreasing. As a result, it relays on a larger class of Lyapunov-Krasovskii functionals to provide stability of a state-delay system. To this end, using the non-monotonic Lyapunov-Krasovskii theorem, new sufficient conditions are derived regarding linear matrix inequalities ( LMIs ) to study the global asymptotic stability of state-delay systems. Moreover, new stabilization conditions are also proposed for time-delay systems in this article. Both simulation and experimental results on a pH neutralizing process are provided to demonstrate the efficacy of the proposed method.

Laguerre-domain Modeling and Identification of Linear Discrete-time Delay Systems

A closed-form Laguerre-domain representation of discrete linear time-invariant systems with constant input time delay is derived. It is shown to be useful in a l2 → l2 system identification setup (with l2 denoting square-summables signals) often arising in biomedical applications, where the experimental protocol does not allow for persistent excitation of the system dynamics. The utility of the proposed system representation is demonstrated on a problem of drug kinetics estimation from clinical data.

Interval estimation for linear discrete-time delay systems

This paper deals with the problem of interval estimation for linear discrete-time systems with a constant time delay. First, an interval observer is designed based on cooperativity and Lyapunov-Krasovskii stability analysis. Second, a zonotope-based interval estimation, which is independent of cooperativity constraint, is proposed. It integrates robust observer design, based on multiple feedbacks, with reachability analysis via zonotopes. In order to enhance the accuracy of interval estimation, an H∞ technique is introduced into observer design to reduce the effects of disturbances and noises. Finally, simulation results are given to illustrate the efficiency of the proposed method.

Finite frequency range iterative learning fault-tolerant control for discrete time-delay uncertain systems with actuator faults

The subject area considered is discrete linear time delay systems operating repetitively on a finite time interval with actuator faults, where the system resets at the end of each operation. Regulation of the dynamics is by iterative learning control and performance goals imposed over finite frequency intervals for the case of uncertainty in the dynamic model. To derive the results, the generalized Kalman–Yakubovich–Popov lemma is used. A simulation based case study is also given to demonstrate the applicability of the new results.

Necessary exponential stability conditions for linear discrete time-delay systems and application

ABSTRACT Necessary conditions for the exponential stability of the linear discrete time-delay systems are presented by employing the so-called Lyapunov–Krasovskii functional approach. These conditions not only provide a new tool for stability analysis of the linear discrete time-delay system by characterising instability domains, but also extend the existing results of the linear discrete time-delay system. Simultaneously, we investigate several crucial properties that connect the Lyapunov matrix and the fundamental matrix of the system. Finally, the robust stability analysis of the linear discrete time-delay systems with norm-bounded uncertainties is presented. Numerical examples illustrate the validity of the obtained results.

Improved Stability Criteria for Discrete-time Delay Systems via Novel Summation Inequalities

This paper is concerned with the stability analysis of linear discrete time-delay systems. New discrete inequalities for single summation and double summation are presented to estimate summation terms in the forward difference of Lyapunov-Krasovskii functional (LKF), which are more general than some commonly used summation inequalities. Through the construction of an augmented LKF, improved delay-dependent stability criteria for discrete time-delay systems are established. Based on this, a time-delayed controller is derived for linear discrete time-delay systems. Finally, the advantages of the proposed criteria are revealed from the solutions of the numerical examples.

JSH transformation and its application to stability analysis for linear discrete time-delay systems

This paper is concerned with stability analysis of linear discrete time-delay systems. Firstly, by JSH transformation, we developed a new necessary and sufficient condition for the asymptotic delay-dependent stability. Secondly, in order to make the designed results of this work more applicable, an unified matrix polynomial approach and numerical radius are used to get a sufficient delay-independent asymptotic stability criterion. The effectiveness and the conservatism reduction of the proposed asymptotic stability results are shown by means of three numerical examples.

On Finite frequency H∞ performance for discrete linear time delay systems

ABSTRACTThis paper is concerned with the finite frequency H∞ performance of discrete linear systems with time delay. The disturbance of the system is considered to be restricted in low/middle/high frequency. A novel summation inequality is proposed by including more free weighed matrices. Then by using generalised Kalman–Yakubovich–Popov Lemma and a Lyapunov–Krasovskii functional like method, a new bounded real lemma is presented for the finite frequency H∞ performance analysis. The advantages and reduced conservatism of the new criterion are illustrated by a classical example from the literature.

Finite-time stability for a linear discrete-time delay systems by using discrete convolution: An LMI approach

The problem of finite-time stability for a class of linear discrete-time systems with state delay is studied. The Lyapunov-Krasovskii-like functional which is based on discrete convolutions of delayed state vector and time-dependent discrete vector functions is used. In order to obtain much less conservative results, new sufficient condition is derived in form of linear matrix inequalities. Numerical example is given to demonstrate the effectiveness of the proposed stability criterion. It was shown that the obtained results are less conservative than some existing ones in the literature. A computer simulation was performed for the analysis of the dynamical behaviour of this system.

Output regulation problem for discrete-time linear time-delay systems by output feedback control

In this paper, we study the output regulation problem of discrete linear time-delay systems by output feedback control. We have established some results parallel to those for the output regulation problem of continuous linear time-delay systems.

Finite-Time Stability and H ∞ Control of Linear Discrete-Time Delay Systems with Norm-Bounded Disturbances

This paper deals with the finite-time stability and H∞ control of linear discrete-time delay systems. The system under consideration is subject to interval time-varying delay and norm-bounded disturbances. Linear matrix inequality approach is used to solve the finite-time stability problem. First, new sufficient conditions are established for robust finite-time stability of the linear discrete-time delay system with norm-bounded disturbances, then the state feedback controller is designed to robustly finite-time stabilize the system and guarantee an adequate level of system performance. The delay-dependent sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed results.

Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems

This paper is concerned with stability of linear discrete time-delay systems. Note that a tighter estimation on a finite-sum term appearing in the forward difference of some Lyapunov functional leads to a less conservative delay-dependent stability criterion. By using Abel lemma, a novel finite-sum inequality is established, which can provide a tighter estimation than the ones in the literature for the finite-sum term. Applying this Abel lemma-based finite-sum inequality, a stability criterion for linear discrete time-delay systems is derived. It is shown through numerical examples that the stability criterion can provide a larger admissible maximum upper bound than stability criteria using a Jensen-type inequality approach and a free-weighting matrix approach.

Statistical χ2 Testing Based Fault Detection for Linear Discrete Time-delay Systems

This paper is concerned with statistical χ2 testing based fault detection (FD) for a class of linear discrete time-varying (LDTV) stochastic systems with delayed state. Different from the traditional residual based FD, we propose to construct the evaluation function by directly using measurement observations. Then an equivalent solution can be given in terms of Riccati recursion by utilizing projection and innovation analysis technique. Moreover, the fault free case evaluation function is with central χ2 distribution and the heavy computational burden is reduced. Furthermore, strategies of χ2 statistic testing on evaluation function are also discussed. Finally, a numerical example is given to illustrate the proposed method.