Deals with several notions of incremental stability. In other words, the focus is on stability of trajectories with respect to one another, rather than with respect to some attractor. The aim is to present a framework for understanding such questions fully compatible with the well-known input-to-state stability approach. Applications of the newly introduced stability notions are also discussed.

A Lyapunov approach to incremental stability properties

Convergent dynamics, a tribute to Boris Pavlovich Demidovich

The authors show how the extended Kalman filter, used as an observer for nonlinear discrete-time systems or extended Kalman observer (EKO), becomes a useful state estimator when the arbitrary matrices, namely R/sub k/ and Q/sub k/, are adequately chosen. As a first step, we use the linearization technique given by Boutayed et al. (1997), which consists of introducing unknown diagonal matrices to take the approximation errors into account. It is shown that the decreasing Lyapunov function condition leads to a linear matrix inequality (LMI) problem, which points out the connection between a good convergence behavior of the EKO and the instrumental matrices R/sub k/ and Q/sub k/. In order to satisfy the obtained LMI, a particular design of Q/sub k/ is given. High performances of the proposed technique are shown through numerical examples under the worst conditions.

A strong tracking extended Kalman observer for nonlinear discrete-time systems

All approaches to optimal experiment design for control have so far focused on deriving an input signal (or input signal spectrum) that minimizes some control-oriented measure of plant/model mismatch between the nominal closed-loop system and the actual closed-loop system, typically under a constraint on the total input power. In practical terms, this amounts to finding the (constrained) input signal that minimizes a measure of a control-oriented model uncertainty set. Here we address the experiment design problem from a "dual" point of view and in a closed-loop setting: given a maximum allowable control-oriented model uncertainty measure compatible with our robust control specifications, what is the cheapest identification experiment that will give us an uncertainty set that is within the required bounds? The identification cost can be measured by either the experiment time, the performance degradation during experimentation due to the added excitation signal, or a combination of both. Our results are presented for the situation where the control objective is disturbance rejection only.

Least costly identification experiement for control

https://hal.archives-ouvertes.fr/hal-00413370/document

Least costly identification experiment for control

The 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 ℒ2-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 ℒ2-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 defined in the continuity of the (linear) H∞ performance (i.e. through the use of a weighting function). As testing incremental properties is a difficult 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 specifications. Copyright © 1999 John Wiley & Sons, Ltd.

Nonlinear performance of a PI controlled missile: an explanation

Considers the practical interest of an approach to nonlinear system analysis. Based on the incremental norm, this approach proposes performance and robustness analysis criteria. Using computationally efficient tests involving linear matrix inequality optimization, these criteria enable us to guarantee the performance and the robustness of a realistic PI controlled nonlinear missile.

The aim of this freeware is to provide computational (skew) mu methods for analysing the robust stability and performance properties of an uncertain closed loop. It could also be considered as a software complement to the book. The toolbox contains low-level routines as well as fully automated procedures which allow a non specialist to obtain guaranteed robustness margins. Different realistic engineering applications are included (missile, rigid and flexible aircraft, telescope mock-up), which illustrate the efficiency and reliability of the proposed tools

Gilles Ferreres

Papers

Nonlinear performance of a PI controlled missile: an explanation

Nonlinear performance of a PI controlled missile: an explanation

A skew mu toolbox (SMT) for robustness analysis

Reliable computation of the robustness margin for a flexible aircraft

A robust self-scheduled missile autopilot: design by multi-model eigenstructure assignment