SUMMARYThe primary aim of this paper is to investigate the practical interest of the incremental norm approach for analysing (realistic) nonlinear dynamical systems. In this framework indeed, incremental stability, a stronger notion than L -gain stability, ensures suitable qualitative and quantitative properties. On the one hand, the qualitative properties essentially correspond to (steady-state) input/output properties, which are not necessarily obtained when ensuring only L -gain stability. On the other hand, it is possible to analyse quantitative robustness performance properties using the notion of (nonlinear) incremental performance, the latter being de"ned in the continuity of the (linear) H performance (i.e. through the use of a weighting function). As testing incremental properties is a di$cult problem, stronger, but computationally more attractive, notions are introduced, namely quadratic incremental stability and performance. Testing these properties reduces indeed to solving convex optimization problems over Linear Matrix Inequalities (LMIs). As an illustration, we consider a classical missile problem, which was already treated using several (linear and nonlinear) approaches. We focus here on the analysis of the nonlinear behavior of this PI controlled missile: using the notions of quadratic incremental stability and performance, the closed loop nonlinear missile is proved to meet desirable control speci"cations.

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SUMMARYWhen analysing the robustness properties of a flexible system, the classical solution, which consists of computing lower and upper bounds of the structured singular value (s.s.v.) at each point of a frequency gridding, appears unreliable. This paper describes two algorithms, based on the same technical result: the first one directly computes an upper bound of the maximal s.s.v. over a frequency interval, while the second one eliminates frequency intervals, inside which the s.s.v. is guaranteed to be below a given value. Various strategies are then proposed, which combine these two techniques, and also integrate methods for computing a lower bound of the s.s.v. The computational efficiency of the scheme is illustrated on a realworld application, namely a telescope mock-up which is significant of a high order flexible system.

The aim of this paper is to study the existence of limit-cycles in a closed loop, which simultaneously contains nonlinearities and parametric uncertainties. Three methods are presented: we ® rst consider the problem of detecting a limit-cycle using a necessary condition of oscillation. A graphical method and a ® rst ¹ based method are proposed. We then consider the problem of checking the absence of limit-cycles despite parametric uncertainties. A second ¹ based method is proposed, which uses a su cient condition of non-oscillation. An example is ® nally presented: the idea is to use the necessary condition of oscillation, so as to synthesize a controller which modi® es the characteristics (magnitude and frequency) of the limit-cycle.

A practical method is proposed for the convex design of robust feedforward controllers, which ensure H∞/L2 performance in the face of LTI and arbitrarily time-varying model uncertainties. A technique which computes the global minimum of this difficult infinite dimensional optimization problem is proposed, as well as a suboptimal but computationally less involving algorithm. A missile example illustrates the efficiency of the scheme: a robust feedforward controller is designed, either on the continuum of linearised time invariant models (corresponding to trim points), or on a quasi-LPV model representing the non-linear one.