Sufficient Conditions For Exponential Stability Research Articles

Aperiodic sampled‐data stabilization of switched singular systems with asynchronous switching: A hybrid control approach

AbstractThis article investigates the sampled‐data stabilization problem for a class of switched singular systems with aperiodic sampling. The asynchronous switching between the plant and the switched controller caused by sampling is considered. Unlike normal systems, the conventional zero‐order hold type sampled‐data control law may be inapplicable to singular systems. This is because under that control law, the algebraic equations of singular systems may have no solutions at sampling instants. To cope with this problem, a new hybrid control law composed of a mode‐dependent impulsive controller and a mode‐dependent sampled‐data state feedback controller is devised. The impulsive controller adjusts the values of differential substate at sampling instants so that the algebraic equations are solvable. Then, by constructing a novel piecewise sampling‐time‐dependent Lyapunov function and using the average dwell time method as well as convex technique, a sampling‐range‐dependent sufficient condition for exponential stability of the resulting closed‐loop system is established. Moreover, an optimization‐based method is proposed to solve the hybrid controller gains. Simulation results on two examples including a mechanical arm system show the effectiveness of the proposed control method.

Parameterization to fault detection filtering for Itô stochastic T‐S fuzzy systems

AbstractThis article focuses on the fault detection filtering problem for Itô stochastic Takagi–Sugeno (T‐S) fuzzy systems. In terms of line integral Lyapunov function, a sufficient condition of exponential stability in mean square and extended dissipativity for the systems under consideration is established which has less conservatism than the one based on quadratic Lyapunov function. The obtained sufficient condition is nonlinear, which makes the synthesis of fault detection filter being difficult and challenging. By choosing general matrix variables and constructing the appropriate orthogonal complement matrices, the nonlinear sufficient condition can be converted into linear one ingeniously via utilizing Finsler's lemma twice. Thus, the fault detection filter can be developed by virtue of parameterization. It should be pointed out that our filter determined by the parameterization approach includes the one obtained by the equivalent transformation method as a special case. Finally, we demonstrate the feasibility and superiority of our proposed approach through three numerical examples.

Asynchronous control of switched systems with time-delay based on mode-dependent average dwell time

Switched system is a kind of important hybrid system, which is usually composed of finite subsystems and an organization switching rule. In the study of switched system, it is usually default that the subsystem and the sub-controller are switched synchronously. However, in practice, the switching of sub-controller often lags behind the subsystem, resulting in asynchronous switching between subsystem and sub-controller, which affects the stability of the system. To solve this problem, this paper discusses the asynchronous control problem of a class of time-varying delay switched systems based on mode-dependent average dwell time (MDADT). Firstly, by constructing a dynamic output feedback controller and using piecewise Lyapunov function, the sufficient conditions for exponential stability and H∞ performance of the closed-loop switching system are obtained. At the same time, the design algorithm of dynamic output feedback controller is given. Finally, a simulation of a switched system with two subsystems under ADT switching signal and MDADT switching signal is given, and the effectiveness of the proposed method is verified by comparison.

CBF‐guided iterative learning control performance optimization for switched systems

AbstractThis article studies the control barrier function (CBF) guided tracking performance optimization for switched systems under iterative learning control (ILC). CBF is an effective method for handling control system state safety. If the safe set is designed as expected performance, and the unsafe set is not expected, then this may be a good choice for optimizing control system performance. We propose a tracking performance optimization method based on CBF to speed up the convergence of the tracking error. A safe set is used to describe the expected tracking performance, by utilizing CBF, safety is further transformed into a set of inequality conditions that constrain ILC controller gains, which ensures that the tracking error is always within the safe set. In addition, an event‐triggering mechanism (ETM) is considered to reduce unnecessary data transmission, and sufficient conditions for exponential stability of the tracking error of switched systems and ILC controllers design are obtained. Finally, a simulation example is given to verify the feasibility of the proposed method.

Event‐triggered control of switched linear systems with state‐dependent switching law and unstabilizable modes

AbstractThis paper studies the event‐triggered control of switched linear system with state‐dependent switching law. A dynamic term is introduced to the switching law to acclimatize the dynamic mode‐dependent event‐triggered controller. Besides, neither the switching law nor the controller needs continuous detection of the system states. Because of the event‐triggered and sampling behavior of the controller, asynchronous between the controller and the system modes are encountered. A new multiple Lyapunov functional is proposed to appropriately characterize the synchronous and asynchronous working modes of the system, and a sufficient exponential stability condition is obtained, which is applicable for the cases when some modes or all modes of the switched system are unstabilizable. Moreover, the event‐triggered control and switching law parameters are jointly designed using linear matrix inequalities. Numerical and comparison studies are offered to provide the validity and superiority of the result.

The Necessary and Sufficient Conditions of Exponential Stability for Heat and Wave Equations with Memory

The Necessary and Sufficient Conditions of Exponential Stability for Heat and Wave Equations with Memory

Stabilization of genetic regulatory networks with time‐varying delays via event‐triggered impulsive control

This paper considers the exponential stabilization of genetic regulatory networks (GRNs) with time‐varying delays by means of event‐triggered impulsive control (ETIC). Firstly, an ETIC strategy is designed, and impulse instants are determined by the state of the system. Then, some sufficient conditions for exponential stability of the GRNs with time‐varying delays under the proposed ETIC are established by Lyapunov functions together with Razumikhin technique. Also, it's shown that Zeno behavior can be eliminated in the designed strategy. Finally, two numerical examples are given to illustrate the effectiveness of the presented results.

Boundary feedback stabilization of an unstable cascaded heat–heat system with different reaction coefficients

The present paper addresses a boundary stabilization problem for a cascaded heat–heat system with different reaction coefficients. A sufficient and necessary condition about two reaction coefficients of the cascaded heat–heat model is found to guarantee the controllability of the coupled system. Based on modal decomposition method, a feedback control is proposed to stabilize the coupled system. Sufficient conditions for exponential stability and well-posedness of the closed-loop systems are derived via the Lyapunov method. The performance is validated by the numerical simulation, which illustrates the efficiency of the proposed control scheme.

Exponential stability of linear systems under a class of Desch–Schappacher perturbations

In this paper, we investigate the uniform exponential stability of the system where the unbounded operator is the infinitesimal generator of a linear semigroup of contractions in a Hilbert space and is a Desch–Schappacher operator. Then we give sufficient conditions for exponential stability of the above system. The obtained stability result is then applied to prove the uniform exponential stabilization of some bilinear partial differential equations.

Stability of Delayed Reaction-Diffusion Neural-Network Models With Hybrid Impulses via Vector Lyapunov Function.

This article focuses on stability analysis of delayed reaction-diffusion neural-network models with hybrid impulses based on the vector Lyapunov function. First, several properties of a vector Halanay-type inequality are given to be the key ingredient for the stability analysis. Then, the Krasovskii-type theorems are established for sufficient conditions of exponential stability, which removes the common threshold of impulses in each neuron subsystem at every impulse time. It shows that the stability of neural networks can be retained with hybrid impulses involved in neural networks, and the synchronization of neural networks can be achieved by designing an impulsive controller, which allows the existence of impulsive perturbation in some nodes and time. Finally, the effectiveness of theoretical results is verified by numerical examples with a successful application to image encryption.

Positivity and decentralized [formula omitted] control of nonlinear interconnected switched positive systems under MDDT constraint

The article is focused on the problems of positivity and decentralized L1 control for nonlinear interconnected switched positive systems(NISPSs) under mode-dependent dwell time(MDDT) constraint, which includes mode-dependent constant dwell time(MDCDT), mode-dependent minimum dwell time(MDMDT) and mode-dependent ranged dwell time(MDRDT) constraints. Both nonlinear subsystems and nonlinear interconnections are included in the considered NISPSs. First, a sufficient and necessary condition of positivity is presented for NISPSs. Next, by applying a novel discretized linear copositive Lyapunov function method, a sufficient condition of exponential stability with L1-gain performance is provided for NISPSs under MDCDT, MDMDT and MDRDT constraints, respectively. Further, by applying the matrix decomposition technology to the decentralized controller gain matrix, an effective mode-dependent piece-wise design scheme of decentralized L1 controller in the solvable linear programming(LP) form is proposed for NISPSs under MDCDT, MDMDT and MDRDT constraints, respectively. The proposed decentralized L1 controller design scheme for NISPSs under MDDT constraint can be degenerated into the one in mode-independent constraint case accordingly. At last, four examples with comparisons are given to illustrate the importance and effectiveness of the obtained results.

Exponential stability of time‐delay systems with flexible delayed impulse

AbstractThe problem of exponential stability for time‐delay systems with flexible delayed impulse control is investigated. Unlike the conventional Razumikhin‐type inequality, variable parameter based on the flexible impulsive gain is introduced. Sufficient conditions for exponential stability are developed for a class of time‐delay systems, for which the size of delay is not limited by impulsive intervals. A flexible impulse control scheme is developed by utilizing the flexible impulsive gain and time‐varying delays. When the system is disturbed by the impulse interferences, both the impulse gain and impulse delay can be adjusted accordingly to make the system exponential stable. The effectiveness of the present results is illustrated by two numerical examples and an application.

Asynchronous Switching Control of Discrete Time Delay Linear Switched Systems Based on MDADT

Ideally, switching between subsystems and controllers occurs synchronously. In other words, whenever a subsystem requires switching, its corresponding sub-controller will be promptly activated. However, in reality, due to network delays, system detection, etc., the activation of candidate controllers frequently lags, which causes issues with asynchronous switching between controllers and subsystems. This asynchronous switching problem may affect system performance and even make the system unstable because the state between the subsystem and the controller may be inconsistent, resulting in the controller not being able to control the subsystem correctly. To keep the system stable while using asynchronous switching, this work suggests an asynchronous control technique for a class of discrete linear switching systems with time delay based on the mode-dependent average dwell time (MDADT). First, we construct a state feedback controller and establish a closed-loop system. In the asynchronous and synchronous intervals of subsystems and controllers, different Lyapunov functions are selected, and sufficient conditions for exponential stability and the H∞ performance of the closed-loop system under asynchronous switching are obtained. In addition, using the MDADT switching strategy, the relevant parameters of each subsystem are designed and the corresponding state–feedback controller gain matrix can be obtained. Finally, a switching system with three subsystems is shown. The approach is confirmed by simulating it using the average dwell time (ADT) switching strategy and the MDADT switching strategy separately.

Stability of Cayley dynamic systems with impulsive effects

Linear dynamic systems with impulsive effects are considered. For such a system we define a new impulsive exponential matrix. Necessary and sufficient conditions for exponential stability and boundedness have been established. The fundamental tool is an impulsive exponential matrix for exponential stability.

Stability Analysis of Second-Order Linear PDEs on Time Scales

This paper presents the stability analysis of a class of second-order linear partial differential equations (PDEs) on time scales with diffusion operator and first-order partial derivative. According to the time scale theory, the Lyapunov functional method, and some inequality techniques, sufficient conditions for exponential stability are strictly obtained, and the results are generalized for that where both the discrete-time and continuous-time cases are considered jointly. In addition, the theoretical results are applied to exponential synchronization of reaction-diffusion neural networks (RDNNs). Simulation examples are given to verify the feasibility of our results.

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 nonlinear switched systems with hybrid delayed impulses

AbstractThis article addresses the exponential stability problem of switched systems with hybrid delayed impulses that depend on both current states and historical states. Some sufficient conditions for exponential stability based on Lyapunov functions are obtained. In particular, the potential stabilizing and destabilizing effects of time delays in impulses are investigated. It is shown that when continuous dynamics are stable, the primal stabilizing impulses may destabilize the system if there exist delays in the impulses. Interestingly, under certain conditions, such delayed impulses cannot destroy the stability of the whole system for all impulsive time sequences with arbitrarily bounded impulsive intervals, and the sizes of the delays in the impulses can theoretically be arbitrarily finite. Finally, to illustrate the validity of the derived results, some examples and simulations are provided.

Subpredictor-Based Fuzzy Control Design for a Reaction–Diffusion Equation

This article addresses the stabilization of the nonlinear reaction–diffusion equation with an unknown but bounded delay. Based on Takagi–Sugeno (T–S) fuzzy rules, a T–S fuzzy partial differential equation (PDE) model is constructed to describe the nonlinear distributed parameter system. More importantly, we contribute to the large delay compensation and the future state prediction by introducing a chain of subpredictors for the T–S fuzzy PDE model, and the convergence of observation errors is guaranteed as well. Different from the conventional approaches, the proposed observer can compensate a larger delay. To stabilize the system, for the first time, a new subpredictor-based T–S fuzzy control law is suggested by using the proposed observer, which is composed of elementary observers connected in series. The implementation of the subpredictor feedback is simpler than the classical methods that only one of the elementary observers is employed. Via Lyapunov–Krasovskii approach, sufficient exponential stability conditions are established in the framework of linear matrix inequalities. A numerical example is provided to illustrate the effectiveness of theoretical results.

A generalized proportional Caputo fractional model of multi-agent linear dynamic systems via impulsive control protocol

This paper deals with multi-agent systems that, due to using the generalized proportional Caputo fractional derivative, possess memories. The information exchange between agents does not occur continuously but only at fixed given update times, and the lower limit of the fractional derivative changes according to the update times. Two types of multi-agent systems are studied, namely systems without a leader and systems with a leader. For a generalized proportional Caputo fractional model of multi-agent linear dynamic systems, sufficient conditions for exponential stability via impulsive control are obtained. In the case of the presence of a leader in the multi-agent system, we derive sufficient conditions for the leader-following consensus via impulsive control based on the leader’s influence. Simulation results are provided to verify the essential role of the generalized proportional Caputo fractional derivative and impulsive control in realizing the consensus of multi-agent systems.

Stabilization of networked periodic piecewise linear systems under asynchronous switching with packet dropouts

In this paper, the networked stabilization of discrete-time periodic piecewise linear systems under transmission package dropouts is investigated. The transmission package dropouts result in the loss of control input and the asynchronous switching between the subsystems and the associated controllers. Before studying the networked control, the sufficient conditions of exponential stability and stabilization of discrete-time periodic piecewise linear systems are proposed via the constructed dwell-time dependent Lyapunov function with time-varying Lyapunov matrix at first. Then to tackle the bounded time-varying packet dropouts issue of switching signal in the networked control, a continuous unified time-varying Lyapunov function is employed for both the synchronous and asynchronous subintervals of subsystems, the corresponding stabilization conditions are developed. The state-feedback stabilizing controller can be directly designed by solving linear matrix inequalities (LMIs) instead of iterative optimization used in continuous-time periodic piecewise linear systems. The effectiveness of the obtained theoretical results is illustrated by numerical examples.