H∞ control for discrete‐time Markovian jump linear systems with partly unknown transition probabilities

In this paper, the problem of H_∞ control for a class of discrete‐time Markovian jump linear system with partly unknown transition probabilities is investigated. The class of systems under consideration is more general, which covers the systems with completely known and completely unknown transition probabilities as two special cases. Moreover, in contrast to the uncertain transition probabilities studied recently, the concept of partly unknown transition probabilities proposed in this paper does not require any knowledge of the unknown elements. The H_∞ controllers to be designed include state feedback and dynamic output feedback, since the latter covers the static one. The sufficient conditions for the existence of the desired controllers are derived within the matrix inequalities framework, and a cone complementary linearization algorithm is exploited to solve the latent equation constraints in the output‐feedback control case. Two numerical examples are provided to show the validness and potential of the developed theoretical results. Copyright © 2008 John Wiley & Sons, Ltd.     

Asynchronous H∞ filtering of continuous‐time Markov jump systems

The paper investigates the asynchronous H_∞ filtering design problem for continuous‐time linear systems with Markov jump. The hidden Markov jump principle is applied to represent the asynchronous situation between the target system and the designed filter. Via a Lyapunov technique, two sufficient conditions are developed to guarantee that the filtering error system is stochastically stable with a prescribed H_∞ noise attenuation level. Furthermore, three filtering design approaches are developed in the form of linear matrix inequalities. Finally, one example is provided to show the effectiveness and feasibility of the developed methods.     

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 discrete‐time Markovian jump systems with partial information on transition probabilities

This paper focuses on mode‐dependent H_∞ state‐feedback control for a class of discrete‐time Markovian jump systems (MJSs) with partial information on transition probabilities (TPs). The augmented free‐connection weighting matrices are introduced by considering the influence of partial information of TPs on discrete‐time MJSs and the disturbance input on the state vector. As a result, the less conservative stability criterion and bounded real lemma (BRL) of MJSs with partly unknown TPs are obtained. Then the sufficient conditions for designing the mode‐dependent H_∞ controllers are derived in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness and the merits of the proposed method.     

Robust H∞ control of Markovian jump systems with uncertain switching probabilities

This paper is concerned with the robust H_∞ control problem for a class of Markovian jump systems with uncertain switching probabilities, whose uncertainties are assumed to be elementwise bounded. First, new criterion of H_∞ performance for such uncertain systems is given. Then, new sufficient condition for H_∞ controller is established as strict linear matrix inequalities. Finally, a numerical example is used to demonstrate the effectiveness of the proposed methods. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new approach to network‐based H∞ control for stochastic systems

This paper deals with the problem of network‐based H_∞ control for a class of uncertain stochastic systems with both network‐induced delays and packet dropouts. The networked control system under consideration is represented by a stochastic model, which consists of two successive delay components in the state. The uncertainties are assumed to be time varying and norm bounded. Sufficient conditions for the existence of H_∞ controller are proposed to ensure exponentially stable in mean square of the closed‐loop system that also satisfies a prescribed performance. The conditions are expressed in the frame of linear matrix inequalities (LMIs), which can be verified easily by means of standard software. Two practical examples are provided to show the effectiveness of the proposed techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

New iterative linear matrix inequality based procedure for H2 and H∞ state feedback control of continuous‐time polytopic systems

This article provides new linear matrix inequality (LMI) sufficient conditions for a generalized robust state feedback control synthesis problem for linear continuous‐time polytopic systems. This generalized problem includes the robust stability, H₂ ‐norm, and H_∞ ‐norm problems as special cases. Using a novel general separation result, which separates the state feedback gain from the Lyapunov matrix but with the state feedback gain synthesized from the slack variable, then allows the formulation of LMI sufficient conditions for the generalized problem. Compared to existing parameterized LMI based conditions, where auxiliary scalar parameters are introduced in order to include the quadratic stability conditions (ie, assuming a constant Lyapunov matrix) as a special case, the proposed new conditions are true LMIs and contain as a particular case the optimal quadratic stability solution. Utilizing any initial solution derived by the quadratic or some existing methods as a starting solution, we propose an algorithm based on an iterative procedure, which is recursively feasible in each update, to compute a sequence of nonincreasing upper bounds for the H₂ ‐norm and H_∞ ‐norm. In addition, if no feasible initial solution can be found for some uncertain systems using any existing methods, another algorithm is presented that offers the possibility of obtaining a robust stabilizing gain. Numerical examples from the literature demonstrate that our algorithms can provide less conservative results than existing methods, and they can also find feasible solutions where all other methods fail.     

Singular H∞ control for nonlinear systems

This paper presents a solution to the singular H_∞ control problem via state feedback for a class of nonlinear systems. It is shown that the problem of almost disturbance decoupling with stability plays a fundamental role in the solution of the considered problem. We also point out when the singular problem can be reduced to a regular one or solved via standard H_∞ technique. We must stress that the solution of the singular problem is obtained without making any approximation of it by means of regular problems. © 1997 John Wiley & Sons, Ltd.     

Network‐based H∞ control for stochastic systems

This paper investigates the problem of network‐based control for stochastic plants. A new model of stochastic time‐delay systems is presented where both network‐induced delays and packet dropouts are taken into consideration for a sampled‐data network‐based control system. This model consists of two successive delay components in the state, and we solve the network‐based H_∞ control problem based on this model by a new stochastic delay system approach. The controller design for the sampled‐data systems is carried out in terms of linear matrix inequalities. Finally, we illustrate the methodology by applying these results to an air vehicle control problem. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

H∞ estimation for uncertain systems

This paper deals with the problem of H_∞ estimation for linear systems with a certain type of time-varying norm-bounded parameter uncertainty in both the state and output matrices. We address the problem of designing an asymptotically stable estimator that guarantees a prescribed level of H_∞ noise attenuation for all admissible parameter uncertainties. Both an interpolation theory approach and a Riccati equation approach are proposed to solve the estimation problem, with each method having its own advantages. The first approach seems more numerically attractive whilst the second one provides a simple structure for the estimator with its solution given in terms of two algebraic Riccati equations and a parameterization of a class of suitable H_∞ estimators. The Riccati equation approach also pinpoints the ‘worst-case’ uncertainty.     

H∞ control of networked control systems based on Markov jump unified model

This paper focuses on a new H_∞ controller design issue for networked control systems with external disturbance as well as random time delays and packet dropouts in forward and feedback channels, which are modeled by multiple Markov chains in a unified style. The output feedback controller is designed to stabilize the networked control system and also achieves the prescribed H_∞ disturbance attenuation level. The addressed controller design problem is transformed into a nonlinear minimization problem with LMI constraints. An illustrative example is provided to show the effectiveness of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Delay‐dependent stability and H∞ control of stochastic time‐delay systems with Markovian jumping parameters

In this paper, the mean‐square exponential stability and H_∞ control problems are investigated for a general class of stochastic time‐delay systems with Markovian jumping parameters. First, a delay‐dependent result in terms of linear matrix inequalities (LMIs) for mean‐square exponential stability and H_∞ performance analysis is presented by constructing a modified Lyapunov‐Krasovskii functional. The decay rate can be chosen in a range to be a finite positive constant without equation constraint. Then, based on the proposed stability result, we derive sufficient condition to solve the H_∞ controller design problem. Finally, numerical examples are provided to illustrate the effectiveness of the theoretical results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Differential graphical games for H∞ control of linear heterogeneous multiagent systems

Differential graphical games have been introduced in the literature to solve state synchronization problem for linear homogeneous agents. When the agents are heterogeneous, the previous notion of graphical games cannot be used anymore and a new definition is required. In this paper, we define a novel concept of differential graphical games for linear heterogeneous agents subject to external unmodeled disturbances, which contain the previously introduced graphical game for homogeneous agents as a special case. Using our new formulation, we can solve both the output regulation and H_∞ output regulation problems. Our graphical game framework yields coupled Hamilton‐Jacobi‐Bellman equations, which are, in general, impossible to solve analytically. Therefore, we propose a new actor‐critic algorithm to solve these coupled equations numerically in real time. Moreover, we find an explicit upper bound for the overall ‐gain of the output synchronization error with respect to disturbance. We demonstrate our developments by a simulation example.     

Quadratic stabilizability and H∞ control of linear discrete‐time stochastic uncertain systems

This paper considers quadratic stabilizability and H_∞ feedback control for stochastic discrete‐time uncertain systems with state‐ and control‐dependent noise. Specifically, the uncertain parameters considered are norm‐bounded and external disturbance is an l²‐square summable stochastic process. Firstly, both quadratic stability and quadratic stabilization criteria are presented in the form of linear matrix inequalities (LMIs). Then we design the robust H_∞ state and output feedback H_∞ controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H_∞ controllable. Sufficient conditions for the existence of the desired robust H_∞ controllers are obtained via LMIs. Finally, some examples are supplied to illustrate the effectiveness of our results.     

Robust H∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching

This paper is concerned with the problems of robust stochastic stabilization and robust H_∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching. The parameter uncertainties are time‐varying norm‐bounded. For the robust stochastic stabilization problem, the purpose is the design of a state feedback controller which ensures the robust stochastic stability of the closed‐loop system irrespective of all admissible parameter uncertainties; while for the robust H_∞ control problem, in addition to the robust stochastic stability requirement, a prescribed level of disturbance attenuation is required to be achieved. Sufficient conditions for the solvability of these problems are obtained in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, explicit expressions of the desired state feedback controllers are also given. An illustrative example is provided to show the effectiveness of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Delay‐range‐dependent H∞ control for Markovian jump systems with mode‐dependent time delays

This paper considers mean‐square exponential stability and H_∞ control problems for Markovian jump systems (MJSs) with time delays which are time‐varying in an interval and depend on system mode. By exploiting a novel Lyapunov‐Krasovskii functional which takes into account the range of delay, and by making use of some techniques, new delay‐range‐dependent stability result and bounded real lemma for MJSs are obtained, where the introduction of the lower bound of delay is shown to be advantageous for reducing conservatism. Moreover, a sufficient condition for the solvability of the H_∞ control problem is derived in terms of linear matrix inequalities. Finally, illustrative examples are presented to show the advantage and effectiveness of the proposed approaches. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust H∞ estimation of uncertain linear quantum systems

We consider classical estimators for a class of physically realizable linear quantum systems. Optimal estimation using a complex Kalman filter for this problem has been previously explored. Here, we study robust H_∞ estimation for uncertain linear quantum systems. The estimation problem is solved by converting it to a suitably scaled H_∞ control problem. The solution is obtained in the form of two algebraic Riccati equations. Relevant examples involving dynamic squeezers are presented to illustrate the efficacy of our method. Copyright © 2016 John Wiley & Sons, Ltd.     

H∞ estimation for discrete-time linear uncertain systems

This paper is concerned with the problem of H_∞ estimation for linear discrete-time systems with time-varying norm-bounded parameter uncertainty in both the state and output matrices. We design an estimator such that the estimation error dynamics is quadratically stable and the induced operator norm of the mapping from noise to estimation error is kept within a prescribed bound for all admissible uncertainties. A Riccati equation approach is proposed to solve the estimation problem and it is shown that the solution is related to two algebraic Riccati equations.