State feedback performance recovery via an observer in H∞ robust control

The paper is concerned with H_∞ robust suboptimal control problem for linear time-invariant systems with structural uncertainties. The main result shows that if there exists a state feedback gain matrix which ensures H_∞robust suboptimally of the closed-loop system, then the robust performance achieved by the state feedback can be asymptotically recovered by a controller with high-gain observer.     

Computational complexity of a problem arising in fixed order output feedback design

This paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design. This problem involves finding two symmetric and positive definitive matrices X and Y such that each satisfies a linear matrix inequality and that XY=I. It is well-known that many control problems such as fixed order output feedback stabilization, H_∞ control, guaranteed H₂ control, and mixed H₂/H_∞ control can all be converted into the matrix inequality problem above, including static output feedback problems as a special case. We show, however, that this matrix inequality problem is NP-hard.     

H∞ control via output feedback for linear systems with time-varying delays in state and control input

In this paper, an observer-based H_∞ controller for systems with time-varying delays in state and control input is proposed. Using the solution of proposed modified algebraic Riccati equation, we can construct a controller which depends on the maximum value of the time derivative of time-varying delay and guarantees a prescribed H_∞ norm bound of the closed-loop transfer matrix from the disturbance to the controlled output. A given example illustrates the availability of the proposed design method.     

Robust H∞ control design for systems with structured parameter uncertainty

In a recent paper a unification of the H₂ (LQG) and H_∞ control-design problems was obtained in terms of modified algebraic Riccati equations. In the present paper these results are extended to guarantee robust H₂ and H_∞ performance in the presence of structured real-valued parameter variiations (ΔA, ΔB, ΔC) in the state space model. For design flexibility the paper considers two distinct types of uncertainty bounds for both full- and reduced-order dynamic compensation. An important special case of these results generates H₂/H_∞ controller designs with guaranteed gain margins.     

Simultaneous H2/H∞ optimal control The state feedback case

A simultaneous H₂/H_∞ control problem is considered. This problem seeks to minimize the H₂ norm of a closed-loop transfer matrix while simultaneously satisfying a prescribed H_∞ norm bound on some other closed-loop transfer matrix by utilizing dynamic state feedback controllers. Such a problem was formulated earlier by Rotea and Khargonekar (Automatica, 27, 307–316, 1991) who considered only so called regular problems. Here, for a class of singular problems, necessary and sufficient conditions are established so that the posed simultaneous H₂/H_∞ problem is solvable by using state feedback controllers. The class of singular problems considered have a left invertible transfer function matrix from the control input to the controlled output which is used for the H₂ norm performance measure. This class of problems subsumes the class of regular problems.     

Optimal H∞ model reduction via linear matrix inequalities: continuous- and discrete-time cases

Necessary and sufficient conditions are derived for the existence of a solution to the continuous-time and discrete-time H_∞ model reduction problems. These conditions are expressed in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. In addition, an explicit parametrization of all reduced-order models that correspond to a feasible solution is provided in terms of a contractive matrix. These results follow from the recent solution of the H_∞ control design problem using LMIs. Particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth-order H_∞ approximation problem, and the convexity of this problem is demonstrated. Computational issues are discussed and an iterative procedure is proposed to solve the H_∞ model reduction problem using alternating projections, although global convergence of the algorithm is not guaranteed.     

H∞-control design for systems with multiple delays

In this paper, H_∞-control design is developed for a class of linear dynamical systems with multiple state and input delays. Both the asymptotic stability conditions and the H_∞-norm bound of the closed-loop transfer function are established. Feedback control synthesis is developed using state feedback. In addition, when the class of linear systems with multiple state and input delays undergoes norm-bound uncertainties, H_∞-control synthesis is established. The developed results are conveniently cast into linear matrix inequality (LMI) framework. Simulation examples are given to illustrate the theoretical developments.     

Robust memoryless H∞ controller design for linear time-delay systems with norm-bounded time-varying uncertainty

We consider the robust H_∞ control problem for linear time-delay systems with norm-bounded time-varying uncertainty. Based on the Riccati-equation approach, we present a method for designing a robust memoryless linear time-invariant state feedback control law, which stabilizes the system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties.     

Asymptotic expansions for game-theoretic Riccati equations and stabilization with disturbance attenuation for singularly perturbed systems

If the systems to be controlled are singularly perturbed, then the Riccati equations which appear in H_∞ problems are difficult to solve due to the presence of the small parameter.

The present paper describes and validates asymptotic expansions, showing how they may be used in H_∞ control. Furthermore, we construct a composite controller based on the solution of slow and fast problems, which guarantees a desired achievable performance level for the full-order plant as approaches zero.     

H∞-control by state feedback: An iterative algorithm and characterization of high-gain occurence

We present an iterative algorithm to compute the achievable H_∞-norm by state feedbak for a standard regular problem and give a characterization when high-gain feedback is necessary to approach the optimal value.

If there is no need for high-gain components to approximate the optimal value, we can reduce the problem to the computation of the H_∞-norm of a certain stable transfer matrix.     

A comment on “L∞ optimal control of SISO continuous-time systems”

The paper “L_∞ optimal control of SISO continuous-time systems” by Wang and Sznaier (Wang & Sznaier (1997). Automatica, 33 (1), 85–90) studies the problem of designing a controller that optimally minimizes the peak absolute value of system output, due to a fixed input signal. With a newly defined function space A_∞, it was claimed that the set of all L_∞-bounded outputs could be parameterized and that the problem could be transformed to a minimal distance problem on L_∞ space. We believe, however, their formulation has essential flaws.     

Nonlinear H∞-control and estimation of optimal H∞-gain

The structure of nonlinear H_∞-controller and the estimation of optimal H_∞-gain are investigated in this paper. The essential problem boils down to the existence of semipositive solution to the Hamilton-Jacobi-Isaacs inequality. Further, the solvability of this first-order fully nonlinear differential inequality is discussed. We look for one kind of special semipositive radial solution to the Hamilton-Jacobi-Isaacs inequality. An explicit estimation of optimal H_∞-gain and explicit formulas of semipositive radial solutions to the Hamilton-Jacobi-Isaacs inequality are obtained. The results are quite simple and intuitive. They even shed a new insight on the linear H_∞-theory.     

Simultaneous H control of a finite collection of linear plants with a single nonlinear digital controller

This paper considers the problem of simultaneous H^∞ control of a finite collection of linear time-invariant systems via a nonlinear digital output feedback controller. The main result is given in terms of the existence of suitable solutions to Riccati algebraic equations and a dynamic programming equation. Our main result shows that if the simultaneous H^∞ control problem for k linear time-invariant plants of orders n₁,n₂,…,n_k can be solved, then this problem can be solved via a nonlinear time-invariant controller of order nn₁+n₂++n_k.     

The singular minimum entropy H∞ control problem

In this paper we search for controllers which minimize an entropy function of the closed loop transfer matrix under the constraint of internal stability and under the constraint that the closed loop transfer matrix has H_∞ norm less than some a priori given bound γ. We find an explicit expression for the infimum. Moreover, we give a characterization when the infimum is attained (contrary to the regular case, for the singular minimum entropy H_∞ control problem the infimum is not always attained).     

H∞ control of an electromechanical drive with nonlinearities using a multi-block criterion

H_∞ optimisation was used to design a controller for an electromechanical drive, subjected to imperfectly known memoryless nonlinearities and uncertain inertia and elasticity of the load. After classical two- and four-block criteria had been tested, more satisfying results were obtained by using a multi-block criterion, which was built by taking advantage of the physical knowledge of the plant.     

On the characterization of sampled-data controllers in the lifted domain

In this paper, we address sampled-data control problems on which the sampling and hold devices are not fixed before-hand, but are chosen as part of the design process. The idea is to apply the lifting transformation to the plant and then characterize the set of generalized sampled-data controllers, i.e., those that can be implemented as the cascade of a generalized sampler, a discrete-time controller and a generalized hold. We argue that our approach offers significant advantages over previous works, and will illustrate this by considering three applications: the input/output stabilization by generalized sampled-data controllers, the solution of the H_∞ optimal control problem and the combination of analog and digital controllers.     

Robust analysis and synthesis of linear time-delay systems with norm-bounded time-varying uncertainty

The problems of robust analysis and synthesis of systems with state-delay and norm-bounded time-varying parameter uncertainty are studied. It is shown that these problems are equivalent to the H_∞ analysis and synthesis problems of an auxiliary system, which is independent of the time-delay and the uncertainty in the system. The necessary and sufficient conditions for the equivalence are developed. Thus any standard H_∞ analysis and synthesis method can be used to solve this problem.     

Robust dissipative control for linear systems with dissipative uncertainty and nonlinear perturbation

This paper focuses on the problem of dissipative control for linear systems which are subjected to dissipative uncertainty and matched nonlinear perturbation. Specifically, quadratic dissipative uncertainty is considered, which contains norm-bounded uncertainty, positive real uncertainty and uncertainty satisfying integral quadratic constraints (IQCs) as special cases. We develop a linear matrix inequality (LMI) approach for designing a robust nonlinear state feedback controller such that the closed-loop system is quadratic dissipative for all admissible uncertainties. Furthermore, under some condition on the dissipative uncertainty, we show that the controller also guarantees the asymptotic stability of the closed-loop system. As special cases, robust H_∞ control and robust passive control problems for systems with nonlinear perturbation and norm-bounded uncertainty (respectively, generalized positive real uncertainty) are solved using the LMI approach.     

A measure of worst-case H∞ performance and of largest acceptable uncertainty

The structured singular value (SSV or μ) is known to be an effective tool for assessing robust performance of linear time-invariant models subject to structured uncertainty. Yet all single μ analysis provides is a bound β on the uncertainty under which stability as well as H_∞ performance level of κ/β are guaranteed, where κ is preselectable. In this paper, we introduce a related quantity ν which provides answers for the following questions: (i) given β, determine the smallest with the property that, for any uncertainty bounded by β, an H∞ performance level of is guaranteed; (ii) conversely, given , determined the largest β with the property that, again, for any uncertainty bounded by β, an H_∞ performance level of is guaranteed. Properties of this quantity are established and approaches to its computation are investigated. Both unstructured uncertainty and structured uncertainty are considered.