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.     

Nonlinear robust H∞ control via dynamic output feedback

The problem on robust H_∞ control for a class of nonlinear systems with parameter uncertainty is studied. Sufficient conditions for the existence of the dynamic output feedback controller are obtained. Under these conditions, the closed-loop systems have robust H_∞-performance. A numerical example is given to illustrate the design of a robust controller using the proposed approach.     

Static output feedback stabilisation with H∞ performance for a class of plants

In this paper, the problem of static output feedback control of a linear system is considered. The existence of a static output feedback control law is given in terms of the solvability of two coupled Lyapunov inequalities which result in a non-linear optimisation problem. However, using state-coordinate and congruence transformations and by imposing a block-diagonal structure on the Lyapunov matrix, we will see that the determination of a static output feedback gain reduces, for a specific class of plants, to finding the solution of a system of linear matrix inequalities. The class of plants considered is those which are minimum phase with a full row rank Markov parameter. The method is extended to incorporate H_∞ performance objectives. This results in a sub-optimal static H_∞ control law found by non-iterative means. The simplicity of the method is demonstrated by a numerical example.     

H∞ model reduction for discrete-time singular systems

This paper investigates the problem of H_∞ model reduction for linear discrete-time singular systems. Without decomposing the original system matrices, necessary and sufficient conditions for the solvability of this problem are obtained in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. When these conditions are feasible, an explicit parametrization of the desired reduced-order models is given. Particularly, a simple LMI condition without rank constraint is derived for the zeroth-order H_∞ approximation problem. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.     

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.     

Robust stabilization of nonlinear systems by H∞ state feedback

This paper is concerned with robust stabilization of nonlinear systems with unstructured uncertainty via state feedback. First, a robust stability condition is given for a closed loop system which is composed of a nonlinear nominal system and an unstructured uncertainty. Second, based on the obtained robust stability condition, a sufficient condition for robust stabilization by state feedback is given in terms of the solvability of some H_∞ state feedback control.     

Robust H∞ control for linear discrete-time systems with norm-bounded time-varying uncertainty

The problem of robust H_∞ analysis and synthesis for linear discrete-time systems with norm-bounded time-varying uncertainty is studied in this paper. It will be shown that this problem is equivalent to the problem of H_∞ analysis and synthesis of an auxiliary system. The necessary and sufficient conditions for the equivalency are proved. Thus the original problem can be solved by existing H_∞ control methods.     

Quadratic stabilisation with H∞-norm bound of non-linear discrete-time uncertain systems with bounded control

The paper addresses the problem of quadratic stabilisability with H_∞-norm bound of uncertain discrete-time control-affine systems by norm-bounded controls. Both structured parameter uncertainties and unstructured exogenous disturbances are taken into account. The given definition of quadratic stabilisability is a generalisation of that used for linear systems so far. A necessary condition of the stabilisability is formulated. A state feedback control satisfying an a priori constraint is proposed for the solution of the formulated H_∞ problem. The proposed method may be applicable even in such cases when the linearisation technique cannot be used.     

Robust H∞ control for uncertain discrete singular systems with pole placement in a disk

This paper addresses the design of robust H_∞ controllers for uncertain discrete singular systems with time-invariant uncertainty in both the state and measurement matrices. The singular system to be controlled is not assumed to be regular. A regular dynamic output feedback controller is designed such that a prescribed H_∞ performance condition is satisfied and the closed-loop poles are placed in a specified disk while the regularity, causality and stability of the closed-loop system can be guaranteed for all admissible uncertainties. The desired controller can be obtained by solving a set of matrix inequalities. A numerical example is given to demonstrate the application of the proposed method.     

Robust H∞ observer design of linear time-delay systems with parametric uncertainty

This paper deals with the problem of H_∞ observer design for a class of uncertain linear systems with delayed state and parameter uncertainties. This problem aims at designing the linear state observers such that, for all admissible parameter uncertainties, the observation process remains robustly stable and the transfer function from exogenous disturbances to error state outputs meets the prespecified H_∞ norm upper bound constraint, independently of the time delay. The time delay is assumed to be unknown, and the parameter uncertainties are allowed to be norm-bounded and appear in all the matrices of the state-space model. An effective matrix inequality methodology is developed to solve the proposed problem. We derive the conditions for the existence of the desired robust H_∞ observers, and then characterize the analytical expression of these observers in terms of some free parameters. A numerical example demonstrates the validity and applicability of the present approach.     

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.     

An algebraic approach to H∞ output feedback control problems

We derive an output feedback controller which stabilizes a system and satisfies a prescribed H_∞ norm bound of the closed loop transfer function. The proposed design method is available for any system, i.e. there are no restrictions on D₁₂ and D₂₁. This approach utilizes only algebraic operations, thus proofs are simple and clear.     

H∞ control for a class of structured time-delay systems

In this paper, we design an H_∞ controller for a class of lower-triangular time-delay systems. Backstepping is applied to construct an explicit feedback controller, and the closed-loop system maintains internal stability and an L₂-gain from the disturbance input to the output. The design is delay-dependent. Simulations on an example system demonstrate the good performance of the proposed design.     

H∞ control of two-time-scale systems

H_∞ control of linear time-invariant singularly perturbed systems is considered. A sequential procedure is described to decompose the problem into slow and fast subproblems. The fast problem is solved first. Then the slow problem is solved under a constraint on the value of the compensator at infinity. A composite compensator is formed as the parallel connection of the fast compensator with the strictly proper part of the slow compensator. The asymptotic validity of the composite compensator is established.     

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.     

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.     

H∞ controller synthesis for pendulum-like systems

This paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities, called pendulum-like systems. A notion of Lagrange stabilizability is introduced, which extends the concept of Lagrange stability to the case of controller synthesis. Based on this concept, we address the problem of designing a linear dynamic output controller which stabilizes (in the Lagrange sense) a pendulum-like system within the framework of the H_∞ control theory. Lagrange stabilizability conditions for uncertainty-free systems and systems with norm-bounded uncertainty in the linear part are derived, respectively. When these conditions are satisfied, the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities (LMIs).     

H∞-norm and invariant manifolds of systems with state delays

The problem of finding bounds on the H_∞-norm of systems with a finite number of point delays and distributed delay is considered. Sufficient conditions for the system to possess an H_∞-norm which is less or equal to a prescribed bound are obtained in terms of Riccati partial differential equations (RPDE’s). We show that the existence of a solution to the RPDE’s is equivalent to the existence of a stable manifold of the associated Hamiltonian system. For small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary differential equations that govern the flow on the slow manifold of the Hamiltonian system. This leads to an algebraic, finite-dimensional, criterion for systems with small delays.     

H∞ filtering for a class of uncertain nonlinear systems

This paper investigates the problem of H_∞ filtering for a class of uncertain continuous-time nonlinear systems with real time-varying parameter uncertainty and unknown initial state. We develop an infinite horizon H_∞ filtering methodology which provides both robust stability and a guaranteed H_∞ performance for the filtering error irrespective of the parameter uncertainty.     

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.