Robust H∞ output‐feedback control of retarded state‐multiplicative stochastic systems

Linear, state‐delayed, continuous‐time systems are considered with both stochastic and norm‐bounded deterministic uncertainties in the state–space model. The problem of robust dynamic H_∞ output‐feedback control is solved, for the stationary case, via the input–output approach where the system is replaced by a nonretarded system with additional deterministic norm‐bounded uncertainties. A delay‐dependent result is obtained which involves the solution of a simple linear matrix inequality. In this problem, a cost function is defined which is the expected value of the standard H_∞ performance cost with respect to the stochastic parameters. A practical example taken from the field of guidance control is given that demonstrates the applicability of the theory. Copyright © 2010 John Wiley & Sons, Ltd.     

L1 adaptive output‐feedback control of multivariable nonlinear systems subject to constraints using online optimization

In this paper, an L₁ adaptive output‐feedback controller is developed for multivariable nonlinear systems subject to constraints using online optimization. In the L₁ adaptive architecture, an adaptive law will update the adaptive parameters that represent the nonlinear uncertainties such that the estimation error between the predicted state and the real state is driven to zero at every integration time step. Of course, neglection of the unknowns for solving the error dynamic equations will introduce an estimation error in the adaptive parameters. The magnitude of this error can be lessened by choosing a proper sampling time step. A control law is designed to compensate the nonlinear uncertainties and deliver a good tracking performance with guaranteed robustness. Model predictive control is introduced to solve a receding horizon optimization problem with various constraints maintained. Numerical examples are given to illustrate the design procedures, and the simulation results demonstrate the availability and feasibility of the developed framework.     

Simultaneous robust normalization and delay‐dependent robust H∞ stabilization for singular time‐delay systems with uncertainties in the derivative matrices

This paper investigates the problem of simultaneous robust normalization and delay‐dependent H_∞ control for a class of singular time‐delay systems with uncertainties. Not only the state and input matrices but also the derivative matrices of the considered systems are assumed to have uncertainties. New sufficient conditions for the existence of a proportional plus derivative state feedback H_∞ controller are derived as LMIs such that the closed‐loop singular system is normal, stable, and guarantee a specific level of performance. Specially, a static state feedback H_∞ controller alone or a state‐derivative feedback H_∞ controller alone can unite to be dealt with by applying our proposed method. Two simulation examples are provided to demonstrate the effectiveness of the proposed approach in this paper. Copyright © 2014 John Wiley & Sons, Ltd.     

H∞ output feedback control for stochastic systems with randomly occurring convex‐bounded uncertainties and channel fadings

In this paper, the H_∞ output feedback control problem for a class of stochastic discrete‐time systems with randomly occurring convex‐bounded uncertainties and channel fadings is investigated. A sequence of mutually independent random variables with known probabilistic distributions are utilized to describe the randomness that convex‐bounded uncertainties appear in practical systems. The measurements with channel fadings are given by a stochastic Rice fading model which is regulated by a set of random variables with certain probability density functions. The purpose of this paper is to design an output feedback controller such that the closed‐loop control system is asymptotically stable with a prescribed H_∞ performance level. The less conservative results are obtained by employing the stochastic Lyapunov technique. Numerical examples are presented to illustrate effectiveness of the proposed approach.     

Singular L2‐gain control for switched nonlinear control systems under arbitrary switching

This article concerns with the synthesis of L₂ ‐gain state feedback controllers, without the standard regular assumption, for multi‐input switched nonlinear control‐affine systems under arbitrary switching. A common control storage function approach is developed for deriving sufficient conditions for the existence of uniform L₂ ‐gain controllers. Moreover, an explicit formula for constructing L₂ ‐gain controllers is presented. A numerical example is given for illustration.     

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.     

H∞ feedback‐control theory in biochemical systems

In this paper we study the possible optimality of biochemical pathways in the H_∞ sense. We start by presenting simple linearized models of single enzymatic reaction systems, where we apply classical and modern tools of feedback‐control theory. We then apply the results obtained by our analysis to a linearly unbranched enzyme pathway system, where we explore the effect of a negative feedback loop internally exerted on the system by a self‐product of the pathway. We then probe the sensitivity of the enzymatic system to variations in certain variables and we deal with the problem of assessing the optimality of the static‐output feedback control, in the H_∞ sense, inherent to the closed‐loop system. In this point we demonstrate the applicability of our results via a theoretical example that provides an open‐loop and closed‐loop analysis of a four‐block enzymatic system. We then apply the various tools we developed to the optimal analysis of the Threonine synthesis pathway which is regulated by three feedback loops. We demonstrate that this pathway is optimal in the H_∞ sense, in the face of considerable uncertainties in the various enzyme concentrations of the pathway. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Robust H∞ control for uncertain discrete-time systems with circular pole constraints

This paper investigates the problem of robust H_∞ control for uncertain discrete-time systems with circular pole constraints. The system under consideration is subject to norm-bounded time-invariant uncertainties in both the state and input matrices. The problem we address is to design state feedback controllers such that the closed poles are located within a prespecified circular region, and the H_∞ norm of the closed-loop transfer function is strictly less than a given positive scalar for all admissible uncertainties. By introducing the notion of quadratic d stabilizability with an H_∞ norm-bound, the problem is solved. Necessary and sufficient conditions for quadratic d stabilizability with an H_∞ norm-bound are derived. Our results can be regarded as extensions of existing results on robust H_∞ control and robust pole assignment of uncertain systems.     

Adaptive H∞ sliding mode control of uncertain neutral‐type stochastic systems based on state observer

This paper is focused on the problem of adaptive sliding mode control design for uncertain neutral‐type stochastic systems under a prescribed H_∞ performance. A simplified state observer is put forward to estimate the unknown state variables, which could be properly incorporated for establishing a new linear‐type switching surface and the associated adaptive variable structure controller. By virtue of the adaptive control design, unknown matched perturbation and potential uncertainties can be counteracted, and the system trajectories are guaranteed to reach the predefined switching surface within finite moment in almost surely sense, and performance analysis of the closed‐loop dynamics during the sliding surface is carried out with a specified H_∞ performance. At last, two illustrative examples through computer simulations are provided to verify the effectiveness and applicability of the proposed scheme.     

Decentralized state feedback robust H ∞ control using a differential evolution algorithm

This paper presents a new method to synthesize a decentralized state feedback robust H ^∞ controller for a class of large‐scale linear uncertain systems satisfying integral quadratic constraints. The decentralized controller is constructed by taking only block‐diagonal elements of a nondecentralized state feedback controller and treating neglected off‐diagonal blocks as uncertainties. A solution to this controller synthesis problem is given in terms of a stabilizing solution to a parametrized algebraic Riccati equation where the parameters are obtained using a differential evolution algorithm.Copyright © 2012 John Wiley & Sons, Ltd.     

Robust H ∞ control of stochastic linear switched systems with dwell time

The theory of H _∞ control of switched systems is extended to stochastic systems with state‐multiplicative noise. Sufficient conditions are obtained for the mean square stability of these systems where dwell time constraint is imposed on the switching. Both nominal and uncertain polytopic systems are considered. A Lyapunov function, in a quadratic form, is assigned to each subsystem that is nonincreasing at the switching instants. During the dwell time, this function varies piecewise linearly in time following the last switch, and it becomes time invariant afterwards. Asymptotic stochastic stability of the set of subsystems is thus ensured by requiring the expected value of the infinitesimal generator of this function to be negative between switchings, resulting in conditions for stability in the form of LMIs. These conditions are extended to the case where the subsystems encounter polytopic‐type parameter uncertainties. The method proposed is applied to the problem of finding an upper bound on the stochastic L₂‐gain of the system. A solution to the robust state‐feedback control problem is then derived, which is based on a modification of the L₂‐gain bound result. Two examples are given that demonstrate the applicability of the proposed theory. Copyright © 2013 John Wiley & Sons, Ltd.     

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 stability and H∞ control for uncertain discrete Markovian jump singular systems with mode‐dependent time‐delay

The robust stochastic stability, stabilization and H_∞ control for mode‐dependent time‐delay discrete Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent (r.s.e.) transformation and by introducing new state vectors, the singular system is transformed into a standard linear system, and delay‐dependent linear matrix inequalities (LMIs) conditions for the mode‐dependent time‐delay discrete Markovian jump singular systems to be regular, causal and stochastically stable, and stochastically stable with γ‐disturbance attenuation are obtained, respectively. With these conditions, robust stabilization problem and robust H_∞ control problem are solved, and the LMIs sufficient conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust H∞ control for uncertain singular systems with state delay

This paper addresses the problem of robust H_∞ control for uncertain continuous singular systems with state delay. The singular system under consideration involves state time delay and time‐invariant norm‐bounded uncertainty. Based on the linear matrix inequality (LMI) approach, we design a memoryless state feedback controller law, which guarantees that, for all admissible uncertainties, the resulting closed‐loop system is not only regular, impulse free and stable, but also meets an H_∞‐norm bound constraint on disturbance attenuation. A numerical example is provided to demonstrate the applicability of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Delay‐dependent robust H∞ control for uncertain stochastic time‐delay systems

This paper investigates the robust H_∞ control problem for stochastic systems with a delay in the state. Sufficient delay‐dependent conditions for the existence of state‐feedback controllers are proposed to guarantee mean‐square asymptotic stability as well as the prescribed H_∞ performance for the closed‐loop systems. Moreover, the results are further extended to the stochastic time‐delay systems with parameter uncertainties, which are assumed to be time‐varying norm‐bounded appearing in both the state and the input matrices. The appealing idea is to partition the delay, which differs greatly from the most existing results and reduces conservatism by thinning the delay partitioning. Numerical examples are provided to show the advantages of the proposed techniques. Copyright © 2009 John Wiley & Sons, Ltd.     

Composite hierarchical antidisturbance control for a class of discrete‐time stochastic systems

In this paper, antidisturbance control and estimation problem are discussed for a class of discrete‐time stochastic systems with nonlinearity and multiple disturbances, which include the disturbance with partially known information and a sequence of random vectors. A disturbance observer is constructed to estimate the disturbance with partially known information. A composite hierarchical antidisturbance control scheme is proposed by combining disturbance observer and H_∞ control. It is proved that the 2 different disturbances can be rejected and attenuated, and the corresponding desired performances can be guaranteed for discrete‐time stochastic systems with known and unknown nonlinear dynamics, respectively. Simulation examples are given to demonstrate the effectiveness of the proposed scheme.