Pramod P. Khargonekar

State-space solutions to standard H/sub 2/ and H/sub infinity / control problems

Simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: For a given number gamma >0, find all controllers such that the H/sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma . It is known that a controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than gamma /sup 2/. Under these conditions, a parameterization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable, free parameter. The state dimension of the coefficient matrix for the LFT, constructed using the two Riccati solutions, equals that of the plant and has a separation structure reminiscent of classical LQG (i.e. H/sub 2/) theory. This paper is intended to be of tutorial value, so a standard H/sub 2/ solution is developed in parallel. >

Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity / control theory

The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H/sup infinity / optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H/sup infinity / control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H/sup infinity / control theory-based methods. >

Mixed H/sub 2//H/sub infinity / control: a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H/sub 2//H/sub infinity / performance measure subject to an inequality constraint on the H/sub infinity / norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H/sub 2//H/sub infinity / performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n*n and n*q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H/sub infinity / estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant. >

Robust control of linear time-invariant plants using periodic compensation

This paper considers the use and design of linear periodic time-varying controllers for the feedback control of linear time-invariant discrete-time plants. We will show that for a large class of robustness problems, periodic compensators are superior to time-invariant ones. We will give explicit design techniques which can be easily implemented. In the context of periodic controllers, we also consider the strong and simultaneous stabilization problems. Finally, we show that for the problem of weighted sensitivity minimization for linear time-invariant plants, time-varying controllers offer no advantage over the time-invariant ones.

An algebraic Riccati equation approach to H ∞ optimization

Abstract This paper considers the general (so-called four block) H ∞ optimal control problem with the assumption that system states are available for feedback. It is shown that infimization of the H ∞ norm of the closed loop transfer function over all linear constant, i.e., nondynamic, stabilizing state feedback laws can be completely characterized via an algebraic Riccati equation. It is further shown that the optimal norm is not improved by allowing feedback to be dynamic. Thus, the general state-feedback H ∞ optimal control problem can be solved by iteratively solving one ARE and the controller can be chosen to be static gain.

A time-domain approach to model validation

In this paper we offer a novel approach to control-oriented model validation problems. The problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input-output data. Our approach directly uses time-domain input-output data to validate uncertainty models. The algorithms we develop are computationally tractable and reduce to (generally nondifferentiable) convex feasibility problems or to linear programming problems. In special cases, we give analytical solutions to these problems. >

Robust stabilization of linear systems with norm-bounded time-varying uncertainty

Abstract In this paper, robust stabilization of a class of linear systems with norm-bounded time-varying uncertainties is considered. It is shown that for this class of uncertain systems quadratic stabilizability via linear control is equivalent to the existence of a positive definite symmetric matrix solution to a (parameter-dependent) Riccati equation. Also, a construction for the stabilizing feedback law is given in terms of the solution to the Riccati equation.

Stability robustness bounds for linear state-space models with structured uncertainty

In this note, we consider the robust stability analysis problem in linear state-space models. We consider systems with structured uncertainty. Some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived. These bounds are shown to be less conservative than the existing ones.

Solution to the positive real control problem for linear time-invariant systems

In this paper we study the problem of synthesizing an internally stabilizing linear time-invariant controller for a linear time-invariant plant such that a given closed-loop transfer function is extended strictly positive real. Necessary and sufficient conditions for the existence of a controller are obtained. State-space formulas for the controller design are given in terms of solutions to algebraic Riccati equations (or inequalities). The order of the constructed controller does not exceed that of the plant. >

H/sub infinity / control and filtering for sampled-data systems

H/sub infinity / control and filtering problems for sampled-data systems are studied. Necessary and sufficient conditions are obtained for the existence of controllers and filters that satisfy a specified H/sub infinity / performance bound. When these conditions hold, explicit formulas for a controller and a filter satisfying the H/sub infinity / performance bound are also given. >