Robust performance analysis with LMI-based methods for realparametric uncertainty via parameter-dependent Lyapunov functions

Robust performance analysis for linear time-invariant systems with linear fractional transformation real parametric uncertainty is considered. New conditions of robust stability/performance based on parameter-dependent Lyapunov functions are proposed. The robust stability/performance measures are: robust pole location, robust H_∞ performance and robust H₂ performance. Linear matrix inequality (LMI)-based sufficient conditions for the existence of parameter-dependent Lyapunov functions are derived by using the quadratic separation concept. The performances of the proposed conditions are compared with existing tests     

Ellipsoidal approximation of the stability domain of a polynomial

The stability domain in the space of coefficients of a polynomial is a nonconvex set in general. In this note, we propose a new convex ellipsoidal inner approximation of this set derived via optimization over linear matrix inequalities. As a by-product, we obtain new simple sufficient conditions for stability that may prove useful in robust control design.     

Global asymptotic stabilization of systems satisfying two different sector conditions

Global asymptotic stabilization for a class of nonlinear systems is addressed. The dynamics of these systems are composed of a linear part to which is added some nonlinearities which satisfy two different sector bound conditions depending on whether the state is near or far from the origin. The proposed approach is based on uniting control Lyapunov functions. In this framework, the stabilization problem may be recast as an LMI optimization problem for which powerful semidefinite programming softwares exist. This is illustrated by means of three numerical examples.     

Periodically time-varying memory state-feedback controller synthesis for discrete-time linear systems

In this paper, we deal with discrete-time linear periodic/time-invariant systems with polytopic-type uncertainties and propose a new linear matrix inequality (LMI)-based method for robust state-feedback controller synthesis. In stark contrast with existing approaches that are confined to memoryless static controller synthesis, we explore dynamical controller synthesis and reveal a particular periodically time-varying memory state-feedback controller (PTVMSFC) structure that allows LMI-based synthesis. In the context of robust controller synthesis, we prove rigorously that the proposed design method encompasses the well-known extended-LMI-based static controller synthesis methods as particular cases. Through numerical experiments, we demonstrate that the suggested design method is indeed effective in achieving less conservative results, under both periodic and time-invariant settings. We finally derive a viable test to verify that the designed robust PTVMSFC is “exact” in the sense that it attains the best achievable robust performance. This exactness verification test works fine in practice, and we will show via a numerical example that exact robust control is indeed attained by designing PTVMSFCs, even for such a problem where the standard memoryless static state-feedback fails.     

Pole assignment of linear uncertain systems in a sector via aLyapunov-type approach

The problem of designing robust control laws, in performance and in stability, for uncertain linear systems is considered. Performances are taken into account using root clustering of the closed-loop dynamic matrix in a sector of the complex plane. A synthesis procedure, based on a sufficient condition for quadratic stabilization and root clustering, such as stabilizability, is given, using an auxiliary convex problem. The results are illustrated by a significant example from the literature     

Quadratic guaranteed cost control for uncertain dissipative models: A Riccati equation approach

The problem of H₂ guaranteed cost control and dynamic output-feedback for linear uncertain systems with dissipative uncertainty is addressed. The problem of robust H₂ synthesis has been open for the last two decades. In this paper, a problem of H₂ quadratic guaranteed cost control is defined for uncertain systems affected by LTI quadratic dissipative model uncertainty. A necessary and sufficient condition of quadratic stabilizability via output-feedback is derived in terms of two coupled parameter-dependent Riccati equations. Then, a method is given to design controllers which minimize an upper bound for the worst-case H₂ norm of the uncertain system. It therefore assesses a guaranteed level of robust performance where in literature, only nominal performance is ensured in most cases. A reliable numerical iterative procedure based on Riccati solvers and one-dimensional convex parameter search is provided. With this uncertainty modelling and the developed numerical procedure, we hope to reduce the usual conservatism of quadratic designs.     

Robust control of a bimorph mirror for adaptive optics systems

We apply robust control techniques to an adaptive optics system including a dynamic model of the deformable mirror. The dynamic model of the mirror is a modification of the usual plate equation. We propose also a state-space approach to model the turbulent phase. A continuous time control of our model is suggested, taking into account the frequential behavior of the turbulent phase. An H(infinity) controller is designed in an infinite-dimensional setting. Because of the multivariable nature of the control problem involved in adaptive optics systems, a significant improvement is obtained with respect to traditional single input-single output methods.     

Ellipsoidal sets for resilient and robust static output-feedback

The implementation of a controller, if not exact, may lead to the so-called fragility problem, i.e., the loss of expected closed-loop properties. In the present note, this difficult problem is dealt with considering robust static-output feedback (SOF) control for uncertain linear time-invariant systems. By analogy with robust analysis theory based on quadratic separation, a new formulation for the SOF design is shown to be a valuable way to tackle fragility issues. Indeed, the use of a quadratic separator for design purpose allows to define a whole resilient (nonfragile) set of SOF control laws. Results are formulated as matrix inequalities one of which is nonlinear. A numerical algorithm based on nonconvex optimization is provided ant its running is illustrated on classical examples from literature.     

Robust performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback control law is tackled here, along with performance optimization. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on a numerical example from the literature.     

A sliding mode control approach for systems subjected to a norm‐bounded uncertainty

This paper proposes a design approach of continuous sliding mode control of uncertain systems, the uncertainty being norm bounded. The two steps of the design methodology are investigated. The existence step, in which we choose the sliding surface that gives good behaviour during the sliding mode, is formulated as a pole assignment of linear uncertain system in a sector through convex optimization. The solution to this problem is therefore numerically tractable via linear matrix inequalities (LMI) optimization. In the reaching step, we propose a continuous nonlinear control strategy ensuring a bounded motion about the ideal sliding mode, thus approximating the ideal dynamic behaviour in the presence of uncertainty. Finally, the validity and the applicability of this approach are illustrated by a flight stabilization benchmark example. Copyright © 2006 John Wiley & Sons, Ltd.