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.     

Non-fragile output feedback H∞ vehicle suspension control using genetic algorithm

This paper presents an approach to design static output feedback and non-fragile static output feedback H_∞ controllers for active vehicle suspensions by using linear matrix inequalities and genetic algorithms. A quarter-car model with active suspension system is considered in this paper. By suitably formulating the minimization problem of the sprung mass acceleration, suspension deflection and tyre deflection, a static output feedback H_∞ controller and a non-fragile static output feedback H_∞ controller are obtained. The controller gain is naturally constrained in the design process. The approach is validated by numerical simulation which shows that the designed static output feedback H_∞ controller can achieve good active suspension performance in spite of its simplicity, and the non-fragile static output feedback H_∞ controller has significantly improved the non-fragility characteristics over controller gain variations.     

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).     

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.     

Remarks on the time-varying H∞ Riccati equations

In this paper we establish some useful properties of three Riccati equations appearing in the standard H_∞-control problems for continuous and discrete-time time-varying systems. We then give necessary and sufficient conditions for the existence of a suboptimal controller by three conditions involving two independent Riccati equations with a coupling inequality.     

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.     

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.     

Direct computation of infimum in discrete-time H∞-optimization using measurement feedback

A direct and non-iterative method for the computation of the infimum for a class of discrete-time H_∞ optimal control problem is considered in this paper. The problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems. The method is applicable to systems where (i) the transfer function from the disturbance input to the measurement output is free of unit circle invariant zeros and left invertible, and (ii) the transfer function from the control input to the controlled output of the given system is free of the unit circle invariant zeros and right invertible.     

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.     

Reduced order H∞ control of an autonomous underwater vehicle

It is well known that the reduced-order (least-order) H_∞ control problem is a rank minimization problem subject to linear matrix inequality (LMI) constraints which is hard to solve due to the non-convexity of the objective and the trace heuristic is simple and efficient one among various heuristics that approximate the rank minimization problem. In this paper, a theoretical insight into the trade-off between the controller order and the performance of the closed-loop system is given. Autopilots (forward speed, heading and depth) for an autonomous underwater vehicle are synthesized based upon the trace heuristic. Simulation results as well as experimental results of heading control show that the reduced-order autopilots are desirable for practical implementation due to their simple structures and slight performance degradation.     

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.     

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.     

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.     

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.     

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.     

An inequality governing nonlinear H∞ control

This note gives necessary and sufficient conditions for solving a reasonable version of the nonlinear H^∞ control problem. The most objectionable hypothesis is elegant and holds in the linear case, but every possibly may not be forced for nonlinear systems. What we discover in distinction to Isidori and Astolfi (1992) and Ball et al. (1993) is that the key formula is not a (nonlinear) Riccati partial differential inequality, but a much more complicated inequality mixing partial derivatives and an approximation theoretic construction called the best approximation operator. This Chebeshev-Riccati inequality when specialized to the linear case gives the famous solution to the H^∞ control problem found in Doyle et al. (1989). While complicated the Chebeshev-Riccati inequality is (modulo a considerable number of hypotheses behind it) a solution to the nonlinear H^∞ control problem. It should serve as a rational basis for discovering new formulas and compromises. We follow the conventions of Ball et al. (1993) and this note adds directly to that paper.     

An L∞-convergent method to reduce frequency-weighted infinite-dimensional discrete-time systems

In this paper, we present a method based on balanced realizations truncations to reduce frequency-weighted infinite-dimensional discrete-time systems.The method is shown to be L_∞-convergent for a subclass of the nuclear systems' class. As its numerical computation is not easy, we propose a practical method to reduce frequency-weighted infinite-dimensional systems and prove that it is L_∞-convergent for the class of systems whose transfer function has its derivative in the Wiener algebra.     

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.     

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.