Linear Impulsive Systems Research Articles

Local and nonlocal problems for fractional impulsive evolution systems with order $\nu\in(1,2)$

In this paper, we study the existence and uniqueness of $PC$-mild solutions for fractional impulsive evolution systems with Caputo derivative. First, we give a compact result of the solution operators of linear fractional evolution system while the cosine family is compact. Second, the representation of $PC$-mild solutions of linear fractional impulsive systems with initial value conditions and nonlocal initial value conditions are given by using the Laplace transform method. Third, several sufficient conditions for judging the existence and uniqueness of solutions of our problem are established via some fixed point theorems and operator semigroup theory. Finally, an example is given to illustrate the results of our paper.

Reinforcement learning for optimal control of linear impulsive systems with periodic impulses

This paper focuses on the finite- and infinite-horizon optimal control problems of linear impulsive systems with periodic impulses under the quadratic performance index. Necessary and sufficient conditions for the optimal impulsive system are derived in terms of hybrid Riccati equations by utilizing the variational method and the collocation method combined with a time-varying Lyapunov function. Different from the existing model-based impulsive control schemes, three reinforcement learning (RL)-based algorithms are proposed to solve the optimal impulsive controller and hybrid controller for the impulsive system without the exact knowledge of the system dynamics. The asymptotical stability of the impulsive control system and the convergence of the RL-based algorithms are proved rigorously. Finally, a numerical simulation illustrates the effectiveness of the proposed control methods.

Model matching problems for impulsive linear systems with polytopic uncertainties

This work deals with the problem of designing a feedback compensator that forces the output of a linear system with abrupt discontinuities in the state evolution and polytopic uncertainties to match that of a given model with the same features. First, the case in which the system and the model are initialized at zero and output matching is required to be exact is considered. Then, the case in which, for arbitrary initialization, output matching is required to be asymptotic for sufficiently slow sequences of the time instants wherein the state exhibits abrupt discontinuities is studied. In addition, on the assumption that the model is stable for sufficiently slow jump time sequences, also the further requirement that asymptotic output matching be achieved with stability of the compensated system is investigated. Constructive, directly checkable, solvability conditions for the problems addressed are derived by leveraging on appropriate structural notions and geometric tools. Algorithmic procedures for the synthesis of the compensators, when the solvability conditions are met, are devised. Some illustrative examples conclude the work.

Sliding Mode Control with Time-Varying Sliding Surface for Linear Uncertain Impulsive Systems

Sliding Mode Control with Time-Varying Sliding Surface for Linear Uncertain Impulsive Systems

ASYMPTOTIC EXPANSION OF THE SOLUTION FOR SINGULARPERTURBED LINEAR IMPULSIVE SYSTEMS

In this study, a singularly perturbed linear impulsive system with singularly perturbed impulses is considered. Many books discuss different types of singular perturbation problems. In the present work, an impulse system is considered in which a small parameter is introduced into the impulse equation. This is the main novelty of our study, since other works [25] have only considered a small parameter in the differential equation. A necessary condition is also established to prevent the impulse function from bloating as the parameter approaches zero. As a result, the notion of singularity for discontinuous dynamics is greatly extended. An asymptotic expansion of the solution of a singularly perturbed initial problem with an arbitrary degree of accuracy for a small parameter is constructed. A theorem for estimating the residual term of the asymptotic expansion is formulated, which estimates the difference between the exact solution and its approximation. The results extend those of [32], which formulates an analogue of Tikhonov’s limit transition theorem. The theoretical results are confirmed by a modelling example.

Leader-Following Consensus of Linear Multiagent Systems With Aperiodically Sampled Outputs: A Distributed Impulsive-Observer-Based Approach.

This article studies the leader-following consensus problem for a class of linear multiagent systems over a directed graph with aperiodically sampled outputs. First, a novel distributed impulsive-observer-based consensus protocol is designed. This protocol requires only the output measurements at sporadic time instants for the observer and control gain design. Second, by using time-varying Lyapunov function techniques, sufficient conditions for exponential stability of a class of linear impulsive systems are established; subsequently, these stability results are applied for the distributed impulsive observer design. Different from the existing related works, the impulsive observer gain designed in this work is decoupled from the graph properties. As a result, once the impulsive observer gain is designed for one network topology, it can be directly used for other network topologies, as long as the graph properties and the dynamics of local agents satisfy certain conditions. Furthermore, the resilience of the designed protocol is tested under denial of service (DoS) attacks. It is shown that the protocol is robust with respect to low-frequency DoS attacks occurring in the observer communication network. Finally, two examples illustrating the validity and effectiveness of the proposed protocol are included.

Stability and guaranteed cost control for linear impulsive systems with random impulses: Application to stochastic event‐triggered control

SummaryThis paper proposes a systematic framework for stability and guaranteed cost control of linear impulsive systems subject to random time‐ and event‐triggering impulses. Moment stability criteria and design methods of time‐ and event‐triggered controllers with guaranteed quadratic performance for the impulsive system are established. It shows that the proposed event‐triggered controller in the sense that it guarantees at least the same quadratic performance function bound as the designed time‐triggered guaranteed cost controller, however, with a larger, or at most equal, average inter‐transmission time. Based on the proposed framework of random‐impulses systems, a stochastic dynamic event‐triggered control strategy is proposed to solve sampled‐data control systems in the presents of sporadic measurements or stochastic sampling. Numerical simulations illustrate the effectiveness of the proposed theoretical results.

A Study on Existence and Controllability of Conformable Impulsive Equations

We study the existence/uniqueness of conformable fractional type impulsive nonlinear systems as well as the controllability of linear/semilinear conformable fractional type impulsive controlled systems. Using the conformable fractional derivative approach, we introduce the conformable controllability operator and the conformable controllability Gramian matrix in order to obtain the necessary and sufficient conditions for the complete controllability of linear impulsive conformable systems. We present a set of sufficient conditions for the controllability of the conformable semilinear impulsive systems.

IQC based analysis and estimator design for discrete-time systems affected by impulsive uncertainties

We propose novel quadratic performance tests for linear discrete-time impulsive systems based on viewing these systems as feedback interconnections of some non-impulsive linear system with an impulsive operator. In order to systematically analyze such interconnections, we employ the framework of integral quadratic constraints and propose novel constraints of this kind for capturing the behavior of the involved impulsive operator. As a major benefit, the modularity of this framework permits seamless extensions to interconnections affected by heterogeneous uncertainties in a straightforward fashion. This contrasts with alternative approaches which are based on capturing the system’s impulsive behavior by means of a clock. Building upon the developed analysis criteria, we characterize the existence of non-impulsive estimators for such impulsive interconnections in a lossless fashion and in terms of linear matrix inequalities. Finally, our approach is illustrated by means of several numerical examples.

Controllability Criteria on Discrete-Time Impulsive Hybrid Systems With Input Delay

Due to the importance of the hybrid systems, the controllability for linear discrete-time impulsive hybrid systems with input delay (DIHSID) is investigated by resorting to the geometric and algebraic analytical methods in this article. First, the null reachability and controllability geometric conditions are studied. Specifically, the null reachable set and controllable set for every impulsive and switching sequence are obtained by the properties of the invariant subspace. Besides, a new subspace sequence is constructed to analyze the null reachability and controllability of DIHSID. Second, the complete controllability of DIHSID is investigated by algebraic methods. In the form of Gramian matrices, several sufficient complete controllability conditions for DIHSID are established without assuming the nonsingularity of all impulsive matrices. Furthermore, a less conservative complete controllability criterion that is necessary and sufficient is developed by introducing a row matrix of some Gramian matrices. Finally, two illustrating examples show the effectiveness of the developed controllability theory.

Relative Controllability for a Class of Linear Impulsive Systems

Hybrid systems are systems that involve continuous and discrete event dynamical behaviors. A impulsive system is a special hybrid system. The continuous dynamics of impulsive systems are usually described by ordinary differential equations and the discrete event dynamics with instantaneously rapid jumps are described by switching laws. Various complex dynamical phenomena that can be modeled by impulsive systems arise in many areas of modern science and technology such as economics, physics, chemistry, biology, information science, radiotherapy, acupuncture, robotics, neural networks, automatic control, artificial intelligence, space technology, and telecommunications, etc. In modern control theory, controllability is one of the most important dynamical properties of considered impulsive systems, therefore, the controllability problem is regarded as one of the fundamental issues of impulsive systems. The basic questions for controllability of impulsive systems as well as for the ordinary systems without impulses and with control function are to obtain useful criteria that allow us to identify whether given dynamic systems are controllable or not. Up to now there have been being many investigation results for controllability of different kinds of impulsive systems with respect to the terminal state constraints of a point type. The purpose of this paper is to study relative controllability with respect to the terminal state constraint of a general type for a class of linear time-varying impulsive systems. In this paper, several types of criteria for relative controllability of such systems are established by a algebraic method, that is, specially speaking, by the matrix rank method. Some corresponding necessary and sufficient conditions for controllability of linear time-invariant impulsive systems are also obtained more compactly. Meanwhile, for given impulsive systems some equivalent relationships between different kinds of controllability are investigated and our criteria for relative controllability are compared with the existing results. A simple example is given to illustrate the utility of our criteria.

Hybrid output regulation for linear impulsive systems with aperiodic jumps: A discrete-time feedback controller design approach

This paper is concerned with the problem of hybrid output regulation for a class of linear impulsive systems with aperiodic jumps. Firstly, by leveraging time-dependent Lyapunov function technique and impulsive control theory, sufficient conditions for achieving output regulation are obtained in state feedback case. Then, the results are extended to error feedback case by constructing an impulsive observer. In this framework, two novel hybrid controllers are designed. Such controllers only need the discrete-time system state or error signal for feedback. The complete procedures for controller designs are also presented. Finally, two illustrative examples, including a numerical example and an LC circuit, are given to show the validity and applicability of the proposed control laws.

ON THE STABILITY AND MEAN SQUARE STABILIZATION OF A CLASS OF LINEAR STOCHASTIC SYSTEMS CONTROLLED BY IMPULSES

This paper is devoted to the analysis of the mean-square stability of a class of linear time-varying impulsive Ito-type stochastic systems. We succeed to obtain necessary and sufficient stability conditions in the general time-varying case. This result is obtained thanks to the theory of positive operators on Hilbert spaces. The problem of state­feedback stabilization is treated as well.

Exponential and Hyers-Ulam stability of impulsive linear system of first order

Exponential and Hyers-Ulam stability of impulsive linear system of first order

Moment observability and output feedback stabilisation for linear stochastic impulsive systems

ABSTRACT This paper is concerned with the observability of linear stochastic impulsive systems with multiplicative noise. The concept of 2pth moment observability is first introduced. Then, to explore the moment observability, a deterministic expanded system is established for the linear stochastic impulsive systems by the generalised -representation technique. It shows that the linear stochastic impulsive system is 2pth moment observability if and only if the expanded system is completely observable. Subsequently, a series of algebraic criteria and geometric criteria for 2pth moment observability of linear stochastic impulsive systems are established by analysing the solution of the expanded system. The impulsive observer-based stabilisation problem for linear stochastic systems is further studied based on these observability criteria. Finally, two numerical examples illustrate the proposed theoretical results.

Sliding Mode Control for Linear Impulsive Systems With Matched Disturbances

This article develops the sliding mode control (SMC) strategy for linear impulsive systems with matched disturbances. Such systems describe dynamical systems that evolve according to continuous process most of the time with matched disturbances, but instantaneously exhibit discontinuities, i.e., impulsive disturbances. A suitable sliding surface function is proposed and the resulting sliding mode dynamics are a class of linear impulsive systems. Some sufficient conditions coupled with impulse time sequences are proposed to guarantee the reachability of the predesigned sliding surface and the stability of the reduced-order impulsive systems under sliding motion, respectively. We show how restrictions on the structure of the systems and impulses should be imposed to ensure the feasibility of the proposed SMC strategy in comparison with general SMC in continuous space. To show the effectiveness of the proposed approach, two simulation examples are presented at the end.

Controllability Results for a Class of Piecewise Nonlinear Impulsive Fractional Dynamic Systems

We study the total controllability for a new class of piecewise nonlinear Langevin fractional dynamic equations with non-instantaneous impulses. New necessary and sufficient conditions are presented for the total controllability of the corresponding linear impulsive systems. Also, the nonlinear case is investigated and controllability criteria are established. We transform the controllability problem into the existence of a fixed point task by defining a nonlinear operator and a proper admissible piecewise control function on a Banach space. Fractional calculus, Mittag-Leffler functions, Gramian type matrices, and the Schauder’s fixed point theorem are employed to develop our main results. Finally, we illustrate the analytical outcome by providing a simulated example.

Interval stability/stabilization and [formula omitted] feedback control for linear impulsive stochastic systems

This paper considers the interval stability/stabilization as well as the H∞-performance constrained by convergence rate of the linear impulsive stochastic systems (LISSs). Firstly, based on the relationship between the system pole configuration and the system performance, a sufficient condition for the distribution of the system eigenvalues in a specific region is given, i.e., a new criterion for the interval stability/stabilization of LISSs. Meanwhile, a new algorithm is designed to precisely control the convergence speed of the target system by combining the interval stabilization method. In addition, an H∞ control criterion constrained by the convergence rate of the system is also proposed. Finally, the effectiveness and applicability of the method are verified by numerical examples.

Characterization and computation of control invariant sets for linear impulsive control systems

Impulsive control systems are suitable to describe and control a venue of real-life problems, going from disease treatment to aerospace guidance. The main characteristic of such systems is that they evolve freely in-between impulsive actions, which makes it difficult to guarantee its permanence in a given state-space region. In this work, we develop a method for characterizing and computing approximations to the maximal control invariant sets for linear impulsive control systems, which can be explicitly used to formulate a set-based model predictive controller. We approach this task using a tractable and non-conservative characterization of the admissible state sets, namely the states whose free response remains within given constraints, emerging from a spectrahedron representation of such sets for systems with rational eigenvalues. The so-obtained impulsive control invariant set is then explicitly used as a terminal set of a predictive controller, which guarantees the feasibly asymptotic convergence to a target set containing the invariant set. Necessary conditions under which an arbitrary target set contains an impulsive control invariant set (and moreover, an impulsive control equilibrium set) are also provided, while the controller performance are tested by means of two simulation examples.

Sliding mode control with integral sliding surface for linear uncertain impulsive systems with time delays

This paper devotes to the problem of sliding mode control (SMC) for linear uncertain impulsive systems with time delays and matched disturbances. An SMC scheme based on a novel integral-type sliding function with an impulse regulation term is proposed. The sliding function is designed so that the finite-time reachability of the sliding surface can be guaranteed for any given impulses. A linear delayed state feedback control law with switching gains is introduced to stabilize the resulting sliding mode dynamics, which is able to improve robust performance against parameter uncertainty. The existence conditions of the delayed state feedback control laws are derived by employing a piecewise discontinuous Lyapunov functional. It is shown that the desired switching gains can be obtained by solving a set of LMIs through a convex optimization procedure. Finally, the correctness and validity of the theoretical results are fully verified in a practical application with several different types of impulse inputs.