Model‐Free H∞ Control Design for Unknown Continuous‐Time Linear System Using Adaptive Dynamic Programming

In this paper, a new online model‐free adaptive dynamic programming algorithm is developed to solve the H_∞ control problem of the continuous‐time linear system with completely unknown system dynamics. Solving the game algebraic Riccati equation, commonly used in H_∞ state feedback control design, is often referred to as a two‐player differential game where one player tries to minimize the predefined performance index while the other tries to maximize it. Using data generated in real time along the system trajectories, this new method can solve online the game algebraic Riccati equation without requiring the full knowledge of system dynamics. A rigorous proof of convergence of the proposed algorithm is given. Finally, simulation studies on two examples demonstrate the effectiveness of the proposed method.     

RECEDING HORIZON H∞ CONTROL FOR SYSTEMS WITH A STATE‐DELAY

This paper proposes the receding horizon H_∞ control (RHHC) for linear systems with a state‐delay. We first proposes a new cost function for a finite horizon dynamic game problem. The proposed cost function includes two terminal weighting terms, each of which is parameterized by a positive definite matrix, called a terminal weighting matrix. Secondly, we derive the RHHC from the solution to the finite dynamic game problem. Thirdly, we propose an LMI condition under which the saddle point value satisfies the nonincreasing monotonicity. Finally, we show the asymptotic stability and H_∞‐norm boundedness of the closed‐loop system controlled by the proposed RHHC. Through a numerical example, we show that the proposed RHHC is stabilizing and satisfies the infinite horizon H_∞‐norm bound.     

Numerical computation of cross‐coupled algebraic Riccati equations related to H2/H∞ control problem for singularly perturbed systems

In this paper, we present a numerical algorithm to the cross‐coupled algebraic Riccati equations(CARE) related to H₂/H_∞ control problems for singularly perturbed systems (SPS) by means of Newton's method. The resulting algorithm can be widely used to solve Nash game problems and robust control problems because the CARE is solvable even if the quadratic term has an indefinite sign. We prove that the resulting iterative algorithm has the property of the quadratic convergence. Using the solution of the CARE, we construct the high‐order approximate H₂/H_∞ controller. Copyright © 2004 John Wiley & Sons, Ltd.     

Delay‐dependent adaptive reliable H∞ control of linear time‐varying delay systems

This paper considers the problem of delay‐dependent adaptive reliable H_∞ controller design against actuator faults for linear time‐varying delay systems. Based on the online estimation of eventual faults, the parameters of adaptive reliable H_∞ controller are updating automatically to compensate the fault effects on the system. A new delay‐dependent reliable H_∞ controller is established using a linear matrix inequality technique and an adaptive method, which guarantees the stability and adaptive H_∞ performance of closed‐loop systems in normal and faulty cases. A numerical example and its simulation results illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

State feedback H∞ control for quantized discrete‐time systems

This paper is concerned with the quantized state feedback H_∞ control problem for discrete‐time linear time‐invariant systems. The quantizer considered here is dynamic and composed of an adjustable “zoom” parameter and a static quantizer. Static quantizer ranges are with practical significance and fully considered here. A quantized H_∞ controller design strategy is proposed with taking quantizer errors into account, where an iterative linear matrix inequality (LMI) based optimization algorithm is developed to minimize static quantizer ranges with meeting H_∞ performance requirement for quantized closed‐loop systems. An example is presented to illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new finite sum inequality approach to delay‐dependent H∞ control of discrete‐time systems with time‐varying delay

This paper deals with delay‐dependent H_∞ control for discrete‐time systems with time‐varying delay. A new finite sum inequality is first established to derive a delay‐dependent condition, under which the resulting closed‐loop system via a state feedback is asymptotically stable with a prescribed H_∞ noise attenuation level. Then, an iterative algorithm involving convex optimization is proposed to obtain a suboptimal H_∞ controller. Finally, two numerical examples are given to show the effectiveness of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.     

H∞ Model Reduction for Positive Fractional Order Systems

This paper focuses on the H_∞ model reduction problem of positive fractional order systems. For a stable positive fractional order system, we aim to construct a positive reduced‐order fractional system such that the associated error system is stable with a prescribed H_∞ performance. Then, based on the bounded real lemma for fractional order systems, a sufficient condition is given to characterize the model reduction problem with a prescribed H_∞‐norm error bound in terms of a linear matrix inequality (LMI). Furthermore, by introducing a new flexible real matrix variable, the desired reduced‐order system matrices are decoupled with the complex matrix variable and further parameterized by the new matrix variable. A corresponding iterative LMI algorithm is also proposed. Finally, several illustrative examples are given to show the effectiveness of the proposed algorithms.     

Robust passivity‐based H∞ control for uncertain switched nonlinear systems

In this paper, we investigate the H_∞ control problem for uncertain switched nonlinear systems with passive and non‐passive subsystems. For any given average dwell time, any given passivity rate and any given disturbance attenuation level, we design feedback controllers of subsystems, which may depend on the pre‐given constants, to solve the H_∞ control problem for the uncertain switched nonlinear systems for all admissible uncertainties. For linear systems, the exponential small‐time norm‐observability is shown to be preserved under disturbance. Two examples are provided to demonstrate the effectiveness of the proposed design method. Copyright © 2016 John Wiley & Sons, Ltd.     

A NEW INTEGRAL INEQUALITY APPROACH TO DELAY‐DEPENDENT ROBUST H∞ CONTROL

A design method for the robust H_∞ control of an uncertain linear system with a time‐varying state delay is proposed. First, an integral inequality that we recently obtained is employed to establish a new delay‐dependent bounded real lemma for a system with a time‐varying delay. The lemma uses neither a model transformation nor a bounding technique for cross terms. Then, the lemma is used in combination with a matrix decomposition method to derive delay‐dependent conditions for the existence of robust H_∞ control based on linear matrix inequalities. Finally, some numerical examples are given to demonstrate the validity of the method.     

Network‐based robust H∞ control of continuous‐time systems with uncertainty

This paper studies the problem of robust H_∞ control for continuous‐time networked control systems (NCSs). A new type of Lyapunov functionals is exploited to derive sufficient conditions for guaranteeing the robust exponential stability and H_∞ performance of the considered system. It is shown that the new result is less conservative than the existing corresponding ones. Meanwhile, a method of eliminating redundant variables to reduce computational complexity is given, which is also applied to design state feedback H_∞ controllers, and the design condition is given in terms of solutions to a set of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed methods. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H∞ control for linear systems with state saturation nonlinearities

The problem of H_∞ control for a class of linear systems with state saturation nonlinearities is considered in this paper. By introducing a row diagonally dominant matrix with negative diagonal elements and a diagonal matrix with positive elements, the H_∞ control problem is reduced to a matrix inequality feasibility problem that can be solved by the proposed iterative linear matrix inequality algorithm. The effectiveness of the presented method is demonstrated by a numerical example. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Distributed reliable H∞ consensus control for a class of multi‐agent systems under switching networks: A topology‐based average dwell time approach

This paper investigates the problem of distributed reliable H_∞ consensus control for high‐order networked agent systems with actuator faults and switching undirected topologies. The Lipschitz nonlinearities, several types of actuator faults, and exogenous disturbances are considered in subsystems. Suppose the communication network of the multi‐agent systems may switch among finite connected graphs. By utilizing the relative state information of neighbors, a new distributed adaptive reliable consensus protocol is presented for actuator failure compensations in individual nodes. Note that the Lyapunov function for error systems may not decrease as the communication network is time‐varying; as a result, the existing distributed adaptive control technique cannot be applied directly. To overcome this difficulty, the topology‐based average dwell time approach is introduced to deal with switching jumps. By applying topology‐based average dwell time approach and Lyapunov theory, the distributed controller design condition is given in terms of LMIs. It is shown that the proposed scheme can guarantee that the reliable H_∞ consensus problem is solvable in the presence actuator faults and external disturbance. Finally, two numerical examples are given the effectiveness of the proposed theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

An Alternative approach to H∞ control for fuzzy systems

In this paper the problem of H_∞ dynamic feedback control for fuzzy dynamic systems has been studied. First the problem of H_∞ dynamic feedback controller designs for complex nonlinear systems, which can be represented by Takagi‐Sugeno (T‐S) fuzzy systems, is presented. Second, based on a Lyapunov function, four new dynamic feedback H_∞ fuzzy controllers are developed by adequately considering the interactions among all fuzzy sub‐systems and these dynamic feedback H_∞ controllers can be obtained by solving a set of suitable linear matrix inequalities. Finally, two examples are given to demonstrate the effectiveness of the proposed design methods. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H∞ tracking control for linear discrete‐time systems via reinforcement learning

In this paper, the H_∞ tracking control of linear discrete‐time systems is studied via reinforcement learning. By defining an improved value function, the tracking game algebraic Riccati equation with a discount factor is obtained, which is solved by iteration learning algorithms. In particular, Q‐learning based on value iteration is presented for H_∞ tracking control, which does not require the system model information and the initial allowable control policy. In addition, to improve the practicability of algorithm, the convergence analysis of proposed algorithm with a discount factor is given. Finally, the feasibility of proposed algorithms is verified by simulation examples.     

A further result of the nonlinear mixed H2/H∞ tracking control problem for robotic systems

The design objective of a mixed H₂/H_∞ control is to find the H₂ optimal tracking control law under a prescribed disturbance attenuation level. With the help of the technique of completing the squares, a further result of the mixed H₂/H_∞ optimal tracking control problem is presented, by combining it with standard LQ optimal control technique. In this paper, only a nonlinear time‐varying Riccati equation is required to solve the problem in the design procedure—instead of two coupled nonlinear time‐varying Riccati equations, or two coupled linear algebraic Riccati‐Iike equations—with some assumptions made regarding the weighting matrices in the existing results. A closed‐form controller for the mixed H₂/H_∞ robotic tracking problem is simply constructed with a matrix inequality check. Moreover, it shows that the existing results are the special cases of these results. Finally, detailed comparison is performed by numerical simulation of a two‐link robotic manipulator. © 2002 John Wiley & Sons, Inc.     

H∞ Static Output Feedback Control of 2‐D Discrete Systems in FM Second Model

The problem of H_∞ static output feedback (SOF) control of two‐dimensional (2‐D) discrete systems described by the Fornasini‐Marchesini (FM) second model is investigated in this paper. First, by applying the 2‐D Bounded Real Lemma, the 2‐D H_∞ SOF control problem is formulated in terms of a bilinear matrix inequality (BMI). Then, by combining the slack variable technique with two kinds of existing LMI methods, respectively, less conservative sufficient LMI conditions are proposed for the BMI formulation. The relation of these two kinds of LMI conditions are revealed by analyzing the choices of coordinate transformation matrices involved in the first kind of LMI conditions. Finally, a numerical example is provided to demonstrate the effectiveness and merits of the proposed methods.     

New results on non‐fragile H∞ controller design for uncertain linear systems

This paper deals with the non‐fragile H_∞ control problem for uncertain linear systems. The uncertainties are of a linear fractional form and appear in both the state and control input matrices. The purpose is to design a state feedback controller, which is subject to linear fractional parametric uncertainties, such that the resulting closed‐loop system is quadratically stable with an H_∞ norm bound. A sufficient condition for the solvability of the problem is obtained in terms of linear matrix inequalities. A numerical example is provided to demonstrate the effectiveness of the proposed design method. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new static output feedback approach to the suboptimal mixed H2/H∞ problem

In this paper we propose a new approach to solve the static output feedback suboptimal mixed H₂/H_∞ control problem using a state fixed‐structure feedback design. We formulate the static output feedback problem as a constrained static state feedback problem and obtain three coupled design equations: one Riccati equation, one Lyapunov equation, and a gain equation. We will prove the equivalence of the proposed solution to the existing solution. A very simple iterative algorithm is then presented to solve the design equations for the stabilizing output feedback gain that minimizes an upper bound of H₂ norm while satisfying the H_∞ disturbance attenuation requirement. A unique feature of the new approach is that it admits the Kalman gain as an initial stabilizing gain to start the above iterative solution procedure, which is computationally attractive and advantageous compared to the direct approach, as the latter has to deal with the difficult algorithm initialization problem. Some illustrative numerical examples are given to demonstrate the effectiveness of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Tracking control for switched linear systems with time‐delay: a state‐dependent switching method

Tracking control for switched linear systems with time‐delay is investigated in this paper. Based on the state‐dependent switching method, sufficient conditions for the solvability of the tracking control problem are given. We use single Lyapunov function technique and a typical hysteresis switching law to design a tracking control law such that the H_∞ model reference tracking performance is satisfied. The controller design problem can be solved efficiently by using linear matrices inequalities. Since convex combination techniques are used to derive the delay independent criteria, some subsystems are allowed to be unstable. It is highly desirable that a non‐switched time‐delay system can not earn such property. Simulation example shows the feasibility and validity of the switching control law. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H∞ control for discrete‐time nonlinear switching systems

This paper formulates and solves the robust H^∞ control problem for discrete‐time nonlinear switching systems. The H^∞ control problem is interpreted as the l₂ finite gain control problem and is studied using a dissipative systems theory for switched systems. Both state and measurement feedback control problems are formulated as dynamic games and solved using dynamic programming. The partially observed dynamic game corresponding to the measurement feedback control problem is solved by transforming into a completely observed, full state infinite‐dimensional game problem using information states. Our results are illustrated with an example. Copyright © 2007 John Wiley & Sons, Ltd.