Pascal Gahinet

Multiobjective output-feedback control via LMI optimization

This paper presents an overview of a linear matrix inequality (LMI) approach to the multiobjective synthesis of linear output-feedback controllers. The design objectives can be a mix of H/sub /spl infin// performance, H/sub 2/ performance, passivity, asymptotic disturbance rejection, time-domain constraints, and constraints on the closed-loop pole location. In addition, these objectives can be specified on different channels of the closed-loop system. When all objectives are formulated in terms of a common Lyapunov function, controller design amounts to solving a system of linear matrix inequalities. The validity of this approach is illustrated by a realistic design example.

Self-scheduled H ∞ control of linear parameter-varying systems: a design example

This paper is concerned with the design of gain-scheduled controllers with guaranteed H∞ performance for a class of linear parameter-varying (LPV) plants. Here the plant state-space matrices are assumed to depend affinely on a vector θ of time-varying real parameters. Assuming real-time measurement of these parameters, they can be fed to the controller to optimize the performance and robustness of the closed-loop system. The resulting controller is time-varying and automatically ‘gain-scheduled’ along parameter trajectories. Based on the notion of quadratic H∞ performance, solvability conditions are obtained for continuous- and discrete-time systems. In both cases the synthesis problem reduces to solving a system of linear matrix inequalities (LMIs). The main benefit of this approach is to bypass most difficulties associated with more classical schemes such as gain-interpolation or gain-scheduling techniques. The methodology presented in this paper is applied to the gain scheduling of a missile autopilot. The missile has a large operating range and high angles of attack. The difficulty of the problem is reinforced by tight performance requirements as well as the presence of flexible modes that limit the control bandwidth.

A convex characterization of gain-scheduled H/sub /spl infin// controllers

An important class of linear time-varying systems consists of plants where the state-space matrices are fixed functions of some time-varying physical parameters /spl theta/. Small gain techniques can be applied to such systems to derive robust time-invariant controllers. Yet, this approach is often overly conservative when the parameters /spl theta/ undergo large variations during system operation. In general, higher performance can be achieved by control laws that incorporate available measurements of /spl theta/ and therefore adjust to the current plant dynamics. This paper discusses extensions of H/sub /spl infin// synthesis techniques to allow for controller dependence on time-varying but measured parameters. When this dependence is linear fractional, the existence of such gain-scheduled H/sub /spl infin// controllers is fully characterized in terms of linear matrix inequalities. The underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available. The formalism and derivation techniques developed here apply to both the continuous- and discrete-time problems. Existence conditions for robust time-invariant controllers are recovered as a special case, and extensions to gain-scheduling in the face of parametric uncertainty are discussed. In particular, simple heuristics are proposed to compute such controllers. >

The LMI control toolbox

This paper describes a new MATLAB-based toolbox for control design via linear matrix inequality (LMI) techniques. After a brief review of LMIs and of some of their applications to control, the toolbox contents and capabilities are presented.<<ETX>>

Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions

In this paper, the problem of robust stability of systems subject to parametric uncertainties is considered. Sufficient conditions for the existence of parameter-dependent Lyapunov functions are given in terms of a criterion which is reminiscent of, but less conservative than, Popovs stability criterion. An equivalent frequency-domain criterion is demonstrated. The relative sharpness of the proposed test and existing stability criteria is then discussed. The use of parameter-dependent Lyapunov functions for robust controller synthesis is then considered. It is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant. A possible synthesis procedure and a numerical example are then discussed.

Affine parameter-dependent Lyapunov functions for real parametric uncertainty

A new test of robust stability/performance is proposed for linear systems with uncertain real parameters. This test is an extension of the notion of quadratic stability where the fixed quadratic Lyapunov function is replaced by a Lyapunov function with affine dependence on the uncertain parameters. Admittedly with some conservatism, the construction of such parameter-dependent Lyapunov functions can be reduced to an linear matrix inequality (LMI) problem, hence is numerically tractable. This LMI-based test can be used for both fixed or time-varying uncertain parameters and is always less conservative than the quadratic stability test whenever the parameters cannot vary arbitrarily fast. Its also completely bypasses the frequency sweep required in real /spl mu/-analysis.<<ETX>>

Robust pole placement in LMI regions

This paper discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality (LMI) regions, a class of convex regions of the complex plane that embraces most practically useful stability regions. The focus is on linear systems with static uncertainty on the state matrix. For this class of uncertain systems, the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary LMI regions. The resulting tests for robust pole clustering are all numerically tractable since they involve solving LMIs, and cover both unstructured and parameter uncertainty. These analysis results are then applied to the synthesis of dynamic output-feedback controllers that robustly assign the closed-loop poles in a prescribed LMI region. With some conservatism, this problem is again tractable via LMI optimization. In addition, robust pole placement can be combined with other control objectives such as H/sub 2/ or H/sub /spl infin// performance to capture realistic sets of design specifications. Physically-motivated examples demonstrate the effectiveness of the approaches for robust analysis and synthesis.

General-Purpose LMI solvers with benchmarks

This paper presents the software package LMI-LAB for the manipulation and resolution of linear matrix inequalities (LMI). Fairly general systems of LMIs can be handled as well as two important optimization problems under LMI constraints. The polynomial-time projectile method of Nesterov and Nemirovsky is used to solve the underlying convex optimization programs. Several benchmark examples demonstrate that the complexity and running time of these algorithms are by no means prohibitive. This confirms that LMI formulations constitute a computationally viable and reasonable approach to control system design.<<ETX>>

The projective method for solving linear matrix inequalities

Numerous problems in control and systems theory can be formulated in terms of linear matrix inequalities (LMI). Since solving an LMI amounts to a convex optimization problem, such formulations are known to be numerically tractable. However, the interest in LMI-based design techniques has really surged with the introduction of efficient interior-point methods for solving LMIs with a polynomial-time complexity. This paper describes one particular method called the Projective Method. Simple geometrical arguments are used to clarify the strategy and convergence mechanism of the Projective algorithm. A complexity analysis is provided, and applications to two generic LMI problems (feasibility and linear objective minimization) are discussed.

On the game Riccati equations arising in H/sub infinity / control problems

The properties of the stabilizing solutions of H/sub infinity /-type game Riccati equations (GRE) are summarized. Special attention is given to the dependence on the parameter gamma , and convexity properties are established in the most general regular H/sub infinity / framework (D/sub 11/ not=0).<<ETX>>