Exponential stabilization of a class of nonlinear systems: A generalized Gronwall–Bellman lemma approach

In this paper, stabilizing control design for a class of nonlinear affine systems is presented by using a new generalized Gronwall–Bellman lemma approach. The nonlinear systems under consideration can be non Lipschitz. Two cases are treated for the exponential stabilization: the static state feedback and the static output feedback. The robustness of the proposed control laws with regards to parameter uncertainties is also studied. A numerical example is given to show the effectiveness of the proposed method.     

Input-output linearization with simultaneous decoupling by restricted state feedback

The problem of input-output linearization with simultaneousdecoupling via restricted state feedback for nonsquare nonlinearsystems is investigated in this paper. The problem is treatedon the basis of a new algebraic approach whose main featureis that it reduces the determination of the admissible state-feedbackcontrol laws to the solution of an algebraic and a first-orderpartial differential system of equations. Verifiable necessaryand sufficient conditions of an algebraic nature are establishedfor the solvability of the aforementioned problem, which relyon the solvability of these systems of equations. Moreover,an explicit expression for a special admissible restricted state-feedbackcontroller is analytically derived.     

Robust guaranteed cost control for uncertain linear differential systems of neutral type

In this paper, the robust guaranteed cost control problem for a class of uncertain linear differential systems of neutral type with a given quadratic cost functions is investigated. The uncertainty is assumed to be norm-bounded and time-varying nonlinear. The problem is to design a state feedback control laws such that the closed-loop system is robustly stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainty and time delay. A criterion for the existence of such controllers is derived based on the matrix inequality approach combined with the Lyapunov method. A parameterized characterization of the robust guaranteed cost controllers is given in terms of the feasible solutions to the certain matrix inequalities. A numerical example is given to illustrate the proposed method.     

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.     

Decentralized Adaptive Robust State Feedback for Uncertain Large-Scale Interconnected Systems with Time Delays

The problem of the decentralized robust control is considered for a class of large-scale time-varying systems withdelayed state perturbations and external disturbances in the interconnections. Here, the upper bounds of the delayed stateperturbations and external disturbances in the interconnections are assumed to be unknown. Adaptation laws areproposed to estimate such unknown bounds; by making use of the updated values of the unknown bounds, decentralized linear and nonlinear memoryless robust state feedback controllers are constructed. Based on Lyapunov stability theoryand Lyapunov–Krasovskii functionals, as well as employing the proposed decentralized nonlinear robust state feedback controllers, it is shown that the solutions of the resulting adaptive closed-loop large-scale time-delay system can be guaranteed to be uniformly bounded and that the states converge uniformly and asymptotically to zero. It is also shown that the proposed decentralized linear robust state feedback controllers can guarantee the uniform ultimate boundedness of the resulting adaptive closed-loop large-scale time-delay system. Finally, a numerical example is given to demonstrate the validity of the results.     

Stable fuzzy logic design of point-to-point control for mechanical systems

An approach for the development of fuzzy point-to-point control laws for second-order mechanical systems is presented. Asymptotic stability of the resulting closed-loop system is proved using Lyapunov stability theory. Closed-loop performance and robustness are quantified in terms of the parameters of membership functions. As opposed to most existing fuzzy control laws, the closed-loop stability of the proposed controller does not depend on the knowledge of the entire dynamics. Moreover, the approach does not require the plant to be open-loop stable. The proposed approach is demonstrated on design and simulation study of a fuzzy controller for a two-link robotic arm.     

New Approach to Linear Exact Model Matching for a Class of Nonlinear Systems

In this paper, a new approach to the linear exact model matching problem for a class of nonlinear systems, using static state feedback, is presented. This approach reduces the problem of determining the state feedback control law to that of solving a system of first-order partial differential equations. Based on these equations, two major issues are resolved: the necessary and sufficient conditions for the problem to have a solution and the general analytical expression for the feedback control law. Furthermore, the proposed approach is extended to solve the same problem via static output feedback.     

Global asymptotical synchronization of chaotic neural networks by output feedback impulsive control: An LMI approach

In this paper, impulsive control for master–slave synchronization schemes consisting of identical chaotic neural networks is studied. Impulsive control laws are derived based on linear static output feedback. A sufficient condition for global asymptotic synchronization of master–slave chaotic neural networks via output feedback impulsive control is established, in which synchronization is proven in terms of the synchronization errors between the full state vectors. An LMI-based approach for designing linear static output feedback impulsive control laws to globally asymptotically synchronize chaotic neural networks is discussed. With the help of LMI solvers, linear output feedback impulsive controllers can be easily obtained along with the bounds of the impulsive intervals for global asymptotic synchronization. The method is finally illustrated by numerical simulations.     

Robust Stability and Stabilization of a Class of Nonlinear Itô-Type Stochastic Systems via Linear Matrix Inequalities

This article presents a new approach to robust quadratic stabilization of nonlinear stochastic systems. The linear rate vector of a stochastic system is perturbed by a nonlinear function, and this nonlinear function satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of stochastic systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The new formulation provides a suitable setting for robust stabilization of nonlinear stochastic systems where the underlying deterministic systems satisfy the generalized matching conditions. Our sufficient conditions are written in matrix forms, which are determined by solving linear matrix inequalities (LMIs), which have significant computational advantage over any other existing techniques. Examples are given to demonstrate the results.     

线性系统的输出反馈特征结构配置——具有不可控不可观模态的情形

本文考虑具有不可控不可观模态的线性系统的输出反馈特征结构配置问题。根据文中给出的矩阵方程AV＋BW＝VF的一种显式参数解，得到了线性系统的输出反馈特征结构配置的一种参数化方法，本文没有对系统附加能控能观条件，在[AB]能控的条件下，本文得到的矩阵方程的解是文[5]中定理2的结果，在[AB]能控且[AC]能观的条件下，本文得到的输出反馈特征结构配置方法是文[6]在系统为定常时算法Ⅱ的结果。     

时变时滞切换系统动态回馈控制器设计分析

研究了一类时变时滞切换控制系统的动态回馈控制器的设计问题.其中,控制输入具有导数有界的时变时滞.利用多Lyapunov泛函方法,结合不等式的放缩技巧,用线性矩阵不等式(LMI)的形式给出了确保系统渐近稳定的控制器存在的充分条件,同时设计出了相应的有效切换律.最后给出实例说明了本文控制器设计的有效性.     

Constructing Robust Feedback Laws by Set Oriented Numerical Methods

In [8, 6] a numerical method for the construction of optimally stabilizing feedback laws was proposed. The method is based on a set oriented discretization of phase space in combination with graph theoretic algorithms for the computation of shortest paths in directed weighted graphs. The resulting approximate optimal value function is piecewise constant, yielding an approximate optimal feedback which might not be robust with respect to perturbations of the system. In this contribution we extend the approach to the case of perturbed control systems. Based on the concept of a multivalued game we show how to derive a directed weighted hypergraph from the original system and generalize the corresponding shortest path algorithm. The resulting optimal value function yields a robustly stabilizing approximate optimal feedback law. This note is an abbreviated version of [5]. For the proofs of the statements here we refer to the full paper. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discrete valued feedback laws and the Zeno phenomenon

We consider closed loop control systems in which feedback laws take values in a discrete set U$U$ and we study the Zeno phenomenon, i.e. the accumulation of discontinuities with respect to time. The results generalize those obtained in [F. Ceragioli, Finite valued feedback laws and piecewise classical solutions, Nonlinear Analysis 65 (2006) 984–998] for the case where U$U$ is finite to the case where U$U$ may be infinite. An application to a quantized control problem is also shown.     

Global feedback stabilization of multi-input bilinear systems

This paper mainly investigates the problem of stabilization of homogeneous bilinear systems with multiple inputs. Explicit state feedback laws are given to stabilize the bilinear systems. Meanwhile, an estimate of the convergence speed is obtained under the given feedback laws. Besides, sufficient conditions, which are easy to be verified, are presented for the stabilization of the bilinear systems.     

A novel feedback controller design with robust fault isolation ability

This paper proposes a novel feedback controller design scheme which can achieve fault isolation based on the control signal or its embedded signal, i.e., with the self-fault-isolation ability. First of all, according to the well-known Youla parameterization, a controller structure consisting of state and residual joint feedback is developed. Then, the residual feedback gain and observer gain are designed to make the cooperative structured residual feedback signal have fault isolation ability. Some free design parameters in the two gains are further utilized for robust self-fault-isolation. Moreover, the state feedback gain is designed, in the framework of switched system, to realize the self-fault-diagnosis and isolation, based on the control signal directly. The proposed control structure also has the advantage of cooperative fault tolerance. Finally, the simulation of HVAC (Heating, Ventilation and Air Conditioning) system, composed of four rooms in one story building scenario, is carried out to demonstrate the effectiveness and superiority of the proposed feedback controller design approach.     

State feedback and output feedback control of a class of nonlinear systems with delayed measurements

This paper is concerned with the problem of state feedback and output feedback control of a class of nonlinear systems with delayed measurements. This class of nonlinear systems is made up of continuous-time linear systems with nonlinear perturbations. The nonlinearity is assumed to satisfy a global Lipschitz condition and the time delay is assumed to be time-varying and have no restriction on its derivative. On the basis of the Lyapunov–Krasovskii approach, sufficient conditions for the existence of the state feedback controller and the output feedback controller are derived in terms of linear matrix inequalities. Methods of calculating the controller gain matrices are also presented. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

A HJB-POD feedback synthesis approach for the wave equation

We propose a computational approach for the solution of an optimal control problem governed by the wave equation. We aim at obtaining approximate feedback laws by means of the application of the dynamic programming principle. Since this methodology is only applicable for low-dimensional dynamical systems, we first introduce a reduced-order model for the wave equation by means of Proper Orthogonal Decomposition. The coupling between the reduced-order model and the related dynamic programming equation allows to obtain the desired approximation of the feedback law. We discuss numerical aspects of the feedback synthesis and providenumerical tests illustrating this approach.     

Optimal persistent disturbance attenuation control for linear hybrid systems

In this paper, the disturbance attenuation properties for a class of linear hybrid systems are investigated, and a hybrid optimal persistent disturbance attenuation control problem is studied. First, a procedure is developed to determine the minimal l^∞$l^{\infty }$ induced gain of linear hybrid systems. However, for general hybrid systems, the termination of the procedure is not guaranteed. Then, the decidability issues are discussed. The termination of the procedure in a finite number of steps is shown for a subclass of hybrid systems with simplified discrete event dynamics, called switched linear systems. Finally, the optimal persistent disturbance attenuation controller synthesis problem is studied. It is shown that the optimal performance level can be achieved by a piecewise linear state feedback control law, and a systematic approach is proposed to design such feedback control.     

Closed-form recursive formula for an optimal tracker with terminal constraints

Feedback control laws are derived for a class of optimal finite time tracking problems with terminal constraints. Analytical solutions are obtained for the feedback gain and the closed-loop response trajectory. Such formulations are expressed in recursive forms so that a real-time computer implementation becomes feasible. An example involving the feedback slewing of a flexible spacecraft is given to illustrate the validity and usefulness of the formulations.     

基于输出反馈的一类大型互联非线性系统的鲁棒全局指数稳定

研究基于输出反馈的一类新的大型互联非线性不确定系统的鲁棒全局指数稳定问题,通过构造每个子系统收敛的状态观测器,并对观测器的状态作线性变换,得到鲁棒分散输出反馈控制器.当该反馈控制律作用于该系统时,闭环系统是全局指数稳定的.