On improving control-loop robustness of model-matching controllers

This note first characterizes the class of all stabilizing controllers for a two-degree-of-Freedom control system which achieve a prescribed achievable transfer function. The characterization is in terms of an arbitrary proper stable transfer function. With this characterization, robust model matching is formulated as a standard H^∞-optimization problem. This means that standard controller designs for a nominal plant, such as linear-quadratic Gaussian ones, can be enhanced to give improved robustness properties using H^∞-design techniques.     

Stabilization and H control of symmetric systems: an explicit solution

In this paper we address the H^∞ control analysis, the output feedback stabilization, and the output feedback H^∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H^∞ norm of the system. For the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the H^∞ control synthesis problem we derive an explicit expression for the optimally achievable closed-loop H^∞ norm and the optimal control gains. Extension to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H^∞ control synthesis problems using simple matrix algebraic tools.     

An operator-theoretic approach to the mixed-sensitivity minimization problem

We consider the mixed-sensitivity minimization problem (scalar case). It gives rise to the so-called two-block problem on the algebra H^∞; we analyze this problem from an operator point of view, using Krein space theory. We obtain a necessary and sufficient condition for the uniqueness of the solution and a parameterization of all solutions in the non-uniqueness case. Moreover, an interpolation interpretation is given for the finite-dimensional case.     

On pure H reduced-order dynamic controllers: an erratum

There are at least two approaches advocated to obtain a pure H^∞ reduced-order dynamic controller for a given augmented plant. One approach is to eliminate completely the H² aspect from a standard H²/H^∞ setting. A second approach is to equate the H² aspect with the H^∞ aspect in that same setting. This paper invalidates the first approach but affirms the second approach and produces the correct equations resulting therefrom.     

State-space formulae for all stabilizing controllers that satisfy an H∞-norm bound and relations to relations to risk sensitivity

Given a linear system, all stabilizing controllers such that a specified closed-loop transfer function has H_∞ norm less than a given scalar, are parametrized. This characterization involves the solution to two algebraic Riccati equations, each with the same order as the system, and further gives gives feasible controllers also with this order. The relationship to the risk-sensitive LQG stochastic control problem is established, giving an equivalence between robust and stochastic control.     

On the weighted sensitivity minimization problem for delay systems

We consider the H^∞-optimal sensitivity problem for delay systems. In particular, we consider computation of μ:= inf {|W-φq|_∞ : q ε H^∞(j )} where W(s) is any function in RH^∞(j ), and φ in H^∞(j ) is any inner function. We derive a new explicit solution in the pure delay case where φ = e^−sh, h > 0.     

A Hamiltonian-based solution to the two-block H problem for general plants in H and rational weights

A complete skew-Toeplitz-type solution to the two-block H^∞ problem for infinite-dimensional stable plants with rational weights is derived with a basis-free proof. The solution consists of one Riccati equation with a rank criterion for a transcendental function of a certain Hamiltonian. This gives a natural extension of the well-known formula for the one-block case. An example is given to illustrate the result.     

Bounded real lemma and robust control of 2-D singular Roesser models

This paper discusses the problem of robust H_∞ control for linear discrete time two-dimensional (2-D) singular Roesser models (2-D SRM) with time-invariant norm-bounded parameter uncertainties. The purpose is the design of static output feedback controllers such that the resulting closed-loop system is acceptable, jump modes free, stable and satisfies a prescribed H_∞ performance level for all admissible uncertainties. A version of bounded realness of 2-D SRM is established in terms of linear matrix inequalities. Based on this, a sufficient condition for the solvability of the robust H_∞ control problem is solved, and a desired output feedback controller can be constructed by solving a set of matrix inequalities. A numerical example is provided to demonstrate the applicability of the proposed approach.     

On stability, -gain and control for switched systems

This paper addresses the issues of stability, L₂-gain analysis and H_∞ control for switched systems via multiple Lyapunov function methods. A concept of general Lyapunov-like functions is presented. A necessary and sufficient condition for stability of switched systems is given in terms of multiple generalized Lyapunov-like functions, which enables derivation of improved stability tests, an L₂-gain characterization and a design method for stabilizing switching laws. A solution to the H_∞ control problem for switched systems is also provided.     

Factorization and feedback stabilization for nonlinear systems

We establish the equivalence of internal input-out stability for two feedback configurations of a nonlinear, time-varying plant P for which a related plant G is assumed to have a factorization G = R with both R and R⁻¹ incrementally stable; this extends a factorization principle for stabilizability previously given only for the linear, time-invariant case. As an application of a special case we recover a version of the Youla parametrization of stabilizing compensators for the nonlinear case previously presented in the literature. We use degree theory to parametrize a collection of solutions of the H_∞-control problem for the case of a 1-gain stable or lossless plant. In the case of a plant G having a J-inner-outer factorization, this last result combined with the above-mentioned factorization principle leads to results on the H_∞-control problem for P.     

Reduced-order filtering for linear systems with Markovian jump parameters

This paper addresses the reduced-order H_∞ filtering problem for continuous-time Makovian jump linear systems, where the jump parameters are modelled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order H_∞ filter are proposed in terms of linear matrix inequalities (LMIs) and a coupling non-convex matrix rank constraint. In particular, the sufficient conditions for the existence of the zero-order H_∞ filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.     

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.     

Achievable H performance in sampled-data smoothing: beyond the -barrier

The lifting technique is a powerful tool for handling the periodically time-varying nature of sampled-data systems. Yet all known solutions of sampled-data H^∞ problems are limited to the case when the feedthrough part of the lifted system, , satisfies , where γ is the required H^∞ performance level. While this condition is always necessary in feedback control, it might be restrictive in signal processing applications, where some amount of delay or latency between measurement and estimation can be tolerated. In this paper, the sampled-data H^∞ fixed-lag smoothing problem with a smoothing lag of one sampling period is studied. The problem corresponds to the a-posteriori filtering problem in the lifted domain and is probably the simplest problem for which a smaller than performance level is achievable. The necessary and sufficient solvability conditions derived in the paper are compatible with those for the sampled-data filtering problem. This result extends the scope of applicability of the lifting technique and paves the way to the application of sampled-data methods in digital signal processing.     

H robust stabilizability

In this paper, we improve on the results of robust stabilizability obtained by Kimura. We do this in a constructive way by using an H^∞-approach, and exhibit an upper bound for the order of robust controllers.     

output feedback control for uncertain stochastic systems with time-varying delays

This paper deals with the problem of H_∞ output feedback control for uncertain stochastic systems with time-varying delays. The parameter uncertainties are assumed to be time-varying norm-bounded. The aim is the design of a full-order dynamic output feedback controller ensuring robust exponential mean-square stability and a prescribed H_∞ performance level for the resulting closed-loop system, irrespective of the uncertainties. A sufficient condition for the existence of such an output feedback controller is obtained and the expression of desired controllers is given.     

H optimization problem with sign-indefinite quadratic form

This paper presents the solution to min-max control problem arising when the matrix C₁^TC₁ of the cost function in the standard H^∞ control problem (Doyle et al., 1989) is replaced by an arbitrary matrix Q 0. This difference is proved to be sufficient for results obtained in (Doyle et al., 1989) not to cover such the case. Their derivations essentially base on the cost function being H^∞ norm and can not be adjusted to deal with sign-indefinite quadratic form. With some sort of strict frequency condition assumed, state space technique is fruitful to obtain the necessary and sufficient conditions of the solvability of the problem. The solution is given by two Riccati equations and has some difference when compared to that of (Doyle et al., 1989).     

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.     

LMI characterization of reduced-order H∞ controllers via invariant zero structure

In this paper, we clarify a new relationship between invariant zeros of a generalized plant and the order reduction of H_∞ controllers by using linear matrix inequalities in both continuous-time and discrete-time cases. In contrast with our recent paper, where a relationship between an unstable transmission-zero structure and the H_∞ controller order reduction is initiated in a fundamental manner, results obtained in this paper are more flexible in two senses: assumptions that are made for the generalized plant are relaxed, and stable as well as unstable invariant zeros are characterized to obtain a reduced-order H_∞ controller.     

Robust output-feedback control of linear discrete-time systems

The problem of designing H_∞ dynamic output-feedback controllers for linear discrete-time systems with polytopic type parameter uncertainties is considered. Given a transfer function matrix of a system with uncertain real parameters that reside in some known ranges, an appropriate, not necessarily minimal, state-space model of the system is described which permits reconstruction of all its states via the delayed inputs and outputs of the plant. The resulting model incorporates the uncertain parameters of the transfer function matrix in the state-space matrices. A recently developed linear parameter-dependent LMI approach to state-feedback H_∞ control of uncertain polytopic systems is then used to design a robust output-feedback controllers that are of order comparable to the one of the plant. These controllers ensure the stability and guarantee a prescribed performance level within the uncertainty polytope.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.