H∞ Control of Discrete-Time Systems With Multiple Input Delays

This paper is concerned with the finite horizon H_infin full-information control for discrete-time systems with multiple control and exogenous input delays. We first establish a duality between the H_infin full-information control and the H₂ smoothing of a stochastic backward system in Krein space. Like the duality between the linear quadratic regulation (LQR) of linear systems without delays and the Kalman filtering, the established duality allows us to address complicated multiple input delay problems in a simple way. Indeed, by applying innovation analysis and standard projection in Krein space, in this paper we derive conditions under which the H_infin full-information control is solvable. An explicit controller is constructed in terms of two standard Riccati difference equations of the same order as the original plant (ignoring the delays). As special cases, solutions to the H_infin state feedback control problem for systems with delays only in control inputs and the H_infin control with preview are obtained. An example is given to demonstrate the effectiveness of the proposed H_infin control design     

On Necessary and Sufficient Conditions for H∞ Output Feedback Control of Markov Jump Linear Systems

This note addresses the output feedback H_infin control problem for continuous-time Markov jump linear systems. It is shown that the feasibility of a certain set of linear matrix inequalities is both sufficient and necessary for the existence of a solution. Under standard assumptions, we also give a Riccati-type sufficient and necessary condition for an H_infin-suboptimal controller to exist.     

Control Synthesis of Singularly Perturbed Fuzzy Systems

This paper considers the problem of designing stabilizing and H_infin controllers for nonlinear singularly perturbed systems described by Takagi-Sugeno fuzzy models with the consideration of the bound of singular perturbation parameter e. For the synthesis problem of simultaneously designing the bound of e and stabilizing or H_infin controllers, linear matrix inequalities (LMI)- based methods are presented. For evaluating the upper bound of e subject to stability or a prescribed H_infin performance bound constraint for the resulting closed-loop system, sufficient conditions are developed, respectively. For the stabilizing and H_infin control synthesis without the consideration of improving the bound of e, new design methods are also given in terms of solutions to a set of LMIs. Examples are given to illustrate the efficiency of the proposed methods.     

Adaptive Critic Designs for Discrete-Time Zero-Sum Games With Application to H∞ Control

In this correspondence, adaptive critic approximate dynamic programming designs are derived to solve the discrete-time zero-sum game in which the state and action spaces are continuous. This results in a forward-in-time reinforcement learning algorithm that converges to the Nash equilibrium of the corresponding zero-sum game. The results in this correspondence can be thought of as a way to solve the Riccati equation of the well-known discrete-time H_infin optimal control problem forward in time. Two schemes are presented, namely: 1) a heuristic dynamic programming and 2) a dual-heuristic dynamic programming, to solve for the value function and the costate of the game, respectively. An H_infin autopilot design for an F-16 aircraft is presented to illustrate the results     

H∞ Estimation for Uncertain Systems With Limited Communication Capacity

This paper investigates the problem of robust H_infin estimation for uncertain systems subject to limited communication capacity. The parameter uncertainty belongs to a given convex polytope and the communication limitations include measurement quantization, signal transmission delay, and data packet dropout, which appear typically in a network environment. The problem of H_infin filter design is first solved for a nominal system subject to the aforementioned information limitations, which is then extended to the uncertain case based on the notion of quadratic stability. To further reduce the overdesign in the quadratic framework, this paper also proposes a parameter-dependent filter design procedure, which is much less conservative than the quadratic approach. The quadratic and parameter-dependent approaches provide alternatives for designing robust H_infin filters with different degrees of conservativeness and computational complexity. Two examples, including a mass-spring system, are utilized to illustrate the design procedures proposed in this paper.     

Dynamic Output Feedback H∞ Control Problem for Linear Neutral Systems

This note presents the solvability conditions of the dynamic output feedback H_infin control problem for linear neutral systems with time- varying state delays in the delay-dependent case, where the delay is assumed to be time-varying continuous or differentiable uniformly bounded function, but no restriction on the derivative of the delay is needed, which means that the delay may be the fast time-varying function. An improved delay-dependent bounded real lemma (BRL) for a closed-loop system is established. Based on the obtained BRL, the dynamic output feedback H_infin controller is designed in terms of the linear matrix inequalities with inverse constraints. An iterative algorithm involving convex optimization methods is used to satisfy these constraints. The proposed results are illustrated in the examples.     

Existence of Similarity Transformation Converting BMIs to LMIs

Under a mild condition, we investigate the existence of a similarity transformation regarding robust H_infin stability analysis and observer-controller synthesis problems. The observer-based controller is capable of disturbance-rejection in the presence of unknown but bounded disturbance. We present results, which illustrate plainly the role of the similarity transformation in converting a BMI problem into an LMI problem in a unified fashion applicable to both continuous- and discrete-time problems with or without uncertainty. Finally the validity and applicability of the approach are illuminated by examples.     

Fixed-Order Robust H∞ Controller Design With Regional Pole Assignment

In this technical note, the problem of designing fixed-order robust H_infin controllers is considered for linear systems affected by polytopic uncertainty. A polynomial method is employed to design a fixed-order controller that guarantees that all the closed-loop poles reside within a given region of the complex plane. In order to utilize the freedom of the controller design, an H_infin performance specification is also enforced by using the equivalence between robust stability and H_infin norm constraint. The design problem is formulated as a linear matrix inequality (LMI) constraint whose decision variables are controller parameters. An illustrative example demonstrates the feasibility of the proposed design methods.     

Nonfragile H∞ Control for Uncertain Neutral Systems With Time-Varying Delays via the LMI Optimization Approach

The nonfragile H_infin control problem for a neutral system with time-varying delays is considered. Delay-dependent criteria are proposed to guarantee the stabilization and disturbance attenuation of systems. A linear matrix inequality optimization approach is used to solve the nonfragile H_infin control problem. Nonfragile H _infin control for an unperturbed neutral system is investigated first. Then, nonfragile H_infin control for an uncertain neutral system is derived directly from the unperturbed case. Finally, two examples are illustrated to show the improvement of this correspondence     

H∞-control of discrete-time nonlinear systems

This paper presents an explicit solution to the problem of disturbance attenuation with internal stability via full information feedback, state feedback, and dynamic output feedback, respectively, for discrete-time nonlinear systems. The H_∞-control theory is first developed for affine systems and then extended to general nonlinear systems based on the concepts of dissipation inequality, differential game, and LaSalle's invariance principle in discrete time. A substantial difficulty that V(A(x)+B(x)u+E(x)w) [respectively, V(f(x,u,w))] is no longer quadratic in [_w^u] arising in the case of discrete-time nonlinear systems has been surmounted in the paper. In the case of a linear system, we show how the results reduce to the well-known ones recently proposed in the literature     

Linear and nonlinear algorithms for identification in H∞ with error bounds

A linear algorithm and a nonlinear algorithm for the problem of system identification in H_∞ posed by Helmicki et al. (1990) for discrete-time systems are presented. The authors derive some error bounds for the linear algorithm which indicate that it is not robustly convergent. However, the worst-case identification error is shown to grow as log(n), where n is the model order. A robustly convergent nonlinear algorithm is derived, and bounds on the worst-case identification error (in the H_∞ norm) are obtained     

A necessary condition for local asymptotic stability of discrete-time nonlinear systems with exogenous disturbance

Local asymptotic stability of nonlinear systems with real-parametric uncertainty or disturbance is one of the important problems in the control systems literature. In this paper, we investigate the problem of asymptotic stability for discrete-time nonlinear systems with time-varying disturbance. We assume that the disturbance vector is generated by an exosystem, which is neutrally stable. Thus, the disturbances that we consider include both constant and periodic signals. For this class of nonlinear systems with time-varying disturbance, we derive a necessary condition for local asymptotic stability of equilibria. As corollaries of our general result, we deduce the necessary condition obtained by Sundarapandian [1] for discrete-time nonlinear systems with constant real parametric uncertainty, and the necessary condition obtained by Lin and Byrnes [2] for discrete-time nonlinear autonomous systems. We illustrate our result with several examples.     

$H_{infty}$ Controller Synthesis via Switched PDC Scheme for Discrete-Time T--S Fuzzy Systems

This paper is concerned with the problem of designing switched state feedback $H_{infty}$ controllers for discrete-time Takagi--Sugeno (T--S) fuzzy systems. New types of state feedback controllers, namely, switched parallel distributed compensation (PDC) controllers, are proposed, which are switched based on the values of membership functions. Switched quadratic Lyapunov functions are exploited to derive a new method for designing switched PDC controllers to guarantee the stability and $H_{infty}$ performances of closed-loop nonlinear systems. The design conditions are given in terms of solvability of a set of linear matrix inequalities. It is shown that the new method provides better or at least the same results of the existing design methods via the pure PDC scheme with a quadratic Lyapunov function or switched constant controller gain scheme. Numerical examples are given to illustrate the effectiveness of the proposed method.      

Robust H/sub /spl infin// control for uncertain discrete-time-delay fuzzy systems via output feedback controllers

This work investigates the problem of robust output feedback H/sub /spl infin// control for a class of uncertain discrete-time fuzzy systems with time delays. The state-space Takagi-Sugeno fuzzy model with time delays and norm-bounded parameter uncertainties is adopted. The purpose is the design of a full-order fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed-loop system and guarantees an H/sub /spl infin// norm bound constraint on disturbance attenuation for all admissible uncertainties. In terms of linear matrix inequalities (LMIs), a sufficient condition for the solvability of this problem is presented. Explicit expressions of a desired output feedback controller are proposed when the given LMIs are feasible. The effectiveness and the applicability of the proposed design approach are demonstrated by applying this to the problem of robust H/sub /spl infin// control for a class of uncertain nonlinear discrete delay systems.     

Robust predictor feedback for discrete-time systems with input delays

This work studies the design problem of feedback stabilisers for discrete-time systems with input delays. A backstepping procedure is proposed for disturbance-free discrete-time systems. The feedback law designed by using backstepping coincides with the predictor-based feedback law used in continuous-time systems with input delays. However, simple examples demonstrate that the sensitivity of the closed-loop system with respect to modelling errors increases as the value of the delay increases. The paper proposes a Lyapunov redesign procedure that can minimise the effect of the uncertainty. Specific results are provided for linear single-input discrete-time systems with multiplicative uncertainty. The feedback law that guarantees robust global exponential stability is a nonlinear, homogeneous of degree 1 feedback law.     

Quadratic stabilizability of switched linear systems with polytopic uncertainties

In this paper, we consider quadratic stabilizability via state feedback for both continuous-time and discrete-time switched linear systems that are composed of polytopic uncertain subsystems. By state feedback, we mean that the switchings among subsystems are dependent on system states. For continuous-time switched linear systems, we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is quadratically stabilizable via state feedback. For discrete-time switched linear systems, we derive a quadratic stabilizability condition expressed as matrix inequalities with respect to a family of non-negative scalars and a common positive definite matrix. For both continuous-time and discrete-time switched systems, we propose the switching rules by using the obtained common positive definite matrix.     

广义时变脉冲系统的输入输出时域稳定

研究了广义时变脉冲系统的输入输出时域稳定问题.基于矩阵微分不等式(Differential matrix inequalities,DMI),给出了两个上述系统输入输出时域稳定的充分条件分别对应 L2干扰输入和 L∞干扰输入.这样的条件要求矩阵微分不等式解的存在性.接下来根据给出的充分条件设计了控制器,使得闭环系统输入输出时域稳定.本文的结果对于一般情况下的广义时变系统同样适用.最后,给出了两个算例来验证结果的有效性.