Olivier Bachelier

Static output feedback design for uncertain linear discrete time systems

In this paper we deal with the class of uncertain discrete-time linear systems. The polytopic uncertainty type is considered in this work. For the certain and uncertain systems a new alternative to design static or dynamic output feedback controllers is developed. The design of the corresponding controller is formulated as an LMI problem that includes some slack variables. For the feasibility of the LMI problem, the controller is obtained by simple matrix calculation. The proposed design is performed into two steps. The first step is devoted to a classical state feedback controller design whereas the second one is the solution of the LMI problem.

D-stability of polynomial matrices

Necessary and sufficient conditions are formulated for the zeros of an arbitrary polynomial matrix to belong to a given region D of the complex plane. The conditions stem from a general optimization methodology mixing quadratic and semidefinite programming, LFRs and rank-one LMIs. They are expressed as an LMI feasibility problem that can be tackled with widespread powerful interior-point methods. Most importantly, the D-stability conditions can be combined with other LMI conditions arising in robust stability analysis.

On Pole Placement via Eigenstructure Assignment Approach

This note comes back to the hard problem of pole placement by static output feedback: let a triplet of matrices {A;B;C} be given with n state variables, m inputs and p ouputs, find a matrix K such that the spectrum of A+BKC equals a specified set. When mp>m+p, a simple noniterative technique based upon the notion of eigenstructure that, in most cases, assigns m+p roots is proposed. It, therefore, enables the designer to assign the whole of the desired spectrum when m+p=n<mp

On the Kalman---Yakubovich---Popov lemma and the multidimensional models

This paper focuses on Kalman–Yakubovich–Popov lemma for multidimensional systems described by Roesser model that possibly includes both continuous and discrete dynamics. It is shown that, similarly to the standard 1-D case, this lemma can be studied through the lens of S-procedure. Furthermore, by virtue of this lemma, we will examine robust stability, bounded and positive realness of multidimensional systems.

Robust Root-Clustering of a Matrix in Intersections or Unions of Regions

This paper considers robust stability analysis for a matrix affected by LFT-based complex uncertainty (LFT for linear fractional transformation). A method is proposed to compute a bound on the amount of uncertainty ensuring robust root-clustering in a combination (intersection and/or union) of several possibly nonsymmetric half planes, discs, and exteriors of discs. In some cases to be detailed, this bound is not conservative. The conditions are expressed in terms of (linear matrix inequalities) LMIs and derived through Lyapunov’s second method. As a distinctive feature of the approach, the Lyapunov matrices proving robust root-clustering (one per subregion) are not necessarily positive definite, but have prescribed inertias depending on the number of roots in the corresponding subregions. As a special case, when root-clustering in a single half plane, disc or exterior of a disc is concerned, the whole clustering region reduces to only one convex subregion and the corresponding unique Lyapunov matrix has to be positive definite as usual. The extension to polytopic LFT-based uncertainty is also addressed.

Pole placement in a union of regions with prespecified subregion allocation

This paper proposes a method to compute a static state feedback control law which achieves a non-strict pole assignment. The specification on the closed-loop poles is given in terms of a clustering region. The originality of this work is the choice of the region which can result from the union of disjoint and non-symmetric subregions. Such a choice is made possible by a technique that enables a partial pole placement via aggregation. The distribution of poles in various subregions can be chosen. The proposed conditions are formulated through an LMI approach and some robustness aspects are also considered from an analytical point of view.

Strong practical stability and stabilization of differential linear repetitive processes

Abstract Differential linear repetitive processes evolve over a subset of the upper-right quadrant of the 2D plane where the unique feature is a series of sweeps or passes through a set of dynamics governed by the solution of a linear matrix differential equation over a finite duration t ∈ [ 0 , α ] where α is termed the pass length or duration. On each pass an output, termed the pass profile, is produced, which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. The result can be oscillations in the pass-to-pass direction that cannot be controlled by direct application of standard, or 1D linear systems theory. The existing stability theory demands a bounded-input bounded-output property uniformly, which in the case of the along-the-pass dynamics means for t ∈ [ 0 , ∞ ] and for ( k , t ) ∈ [ 0 , ∞ ] × [ 0 , ∞ ] ⊃ [ 0 , ∞ ] × [ 0 , α ] where the integer k ≥ 0 denotes the pass number or index. The pass length is always finite, however, and hence this stability theory could well be too strong in many cases and, in particular, impose very strong conditions in terms of control law design. This paper develops an alternative in such cases by relaxing the requirement for the bounded-input bounded-output property to hold when k → ∞ and t → ∞ simultaneously, provides an explanation of the implications of this in the frequency domain, and then develops control law design algorithms.

Robust Matrix Root-Clustering Analysis through Extended KYP Lemma

This paper is dedicated to robust matrix eigenvalue clustering in a subregion

LMI Stability Conditions for 2D Roesser Models

{\mathcal D}

Control of discrete linear repetitive processes using strong practical stability and H? disturbance attenuation

of the complex plane. A norm-bounded uncertainty is considered.