Michel Malabre

Technical Communique: Fixed poles of simultaneous disturbance rejection and decoupling: a geometric approach

When the simultaneous disturbance rejection and regular block decoupling problem by static state feedback is solvable, and under some unrestrictive minimality assumption, all the poles of the closed-loop system can be placed, except the so-called fixed poles (which are present for any solution). We present here a geometric characterization for this set of fixed poles as well as a constructive procedure (based on geometric tools) for designing a feedback solution of the problem which assigns the remaining poles to arbitrary prespecified locations.

Brief Fixed poles of disturbance rejection by dynamic measurement feedback: a geometric approach

This paper deals with the problem of disturbance rejection by dynamic measurement feedback. It aims at proposing new geometric characterizations for the fixed poles of this problem. We also propose solutions to the problem for which all the poles, except the fixed poles of the problem, are freely placed. These fixed poles are also characterized as some particular invariant zeros. As a corollary, we propose a necessary and sufficient condition for the existence of internally stable solutions, which generalizes the sufficient condition proposed by Basile and Marro.

Brief Simultaneous model matching and disturbance rejection with stability by state feedback

This work deals with the Model Matching Problem and the Disturbance Rejection Problem with stability by state feedback. Each one of these control problems has, separately, received a lot of contributions, but, to our best knowledge, we propose here for the first time a complete structural solution for square (same number of control inputs and outputs) strictly proper systems. Our structural conditions nicely enhance the role played by the zeros, both finite and infinite.

Some issues of structure and control for linear time-invariant systems

Abstract The objective of this paper is to emphasise the role played by particular structures in the solution of some control problems. The so-called “structural approach” relies on various indicators of dynamical systems such as, for instance, finite and infinite zeros, kernel indices, …. The fundamental invariance properties of these structures under the action of some transformations groups (e.g. feedback) are at the origin of their key role. Structural solutions to “classical” control problems, such as disturbance rejection, model matching and non-interaction are now rather well known: zeros at infinity play a role in the existence of “proper” solutions, while finite (invariant) zeros allow for the characterisation of “fixed poles”, whose location in the complex plane gives answer to pole placement limitations (including stability). Among the recent contributions to this structural approach, a particular attention is here devoted to: - “Partial” versions of some of these control problems: The control objective only concerns a finite number of (and not necessarily all) the first Markov parameters of the transfer function matrix of the controlled system (e.g. to be zero for disturbance rejection or model matching, to be diagonal for non-interaction). Some interesting new issues in the dual context of failure detection are also sketched. - Generalised solutions: Based on proportional and derivative feedback laws, with new issues in the context of systems with variable internal structures, and also for systems with delays. Geometric concepts, such as invariant and almost invariant subspaces, and algebraic counterparts, such as factorisations on some special rings, are intermediary tools which support the characterisations of those particular structures and which allow for a structural treatment of the considered control and/or observation problems. The results are here presented without proof: references are given to previous published results (in most cases in books and journals which are easily available), and some simple examples are used to illustrate non-standard notions (among which systems with variable internal structure, and time domain left invertibility). Most of the results here presented rely on long and intensive collaborations between the author and various colleagues.

Simultaneous model matching and disturbance rejection by state feedback

The model matching problem (MMP) and the disturbance rejection problem (DRP) have received much attention respectively, but to the best of our knowledge we propose here for the first time a complete structural solution to the simultaneous MMP and DRP by state feedback. This problem is of high practical and theoretical importance, as for instance the simultaneous I/O decoupling and DRP. Structural conditions nicely enhance the role played by the zeros, both finite and infinite.

New geometric insight into the almost disturbance decoupling problem by state feedback

Abstract The objective of this paper is to enhance new geometric properties which hold when the Almost Disturbance Decoupling Problem is solvable by state feedback (ADDP). The most important ones are summarized in a lattice description which, as usually, expresses algebraic relations based on sums and intersections, but also points out fundamental properties like exact or almost invariance, as well as exact or almost controllability.

FIXED POLES FOR NON MINIMAL SYSTEMS: A GEOMETRIC APPROACH

Abstract When considering the set of all possible solutions, e.g. by state or measurement feedback, of a given control problem ( e.g. disturbance decoupling), complete pole placement is usually not possible, due to the so-called fixed poles: these are present as dynamics of the compensated system, whatever be the way the solution has been designed (within the chosen class). These fixed poles have two different origins: some are present, whatever be the considered control problem, because of possible non minimality of the state description (in the Kalman sense, i.e. controllability and observability). The other are due to the very particular control problem for which the feedback law is a solution. We show here how non minimality impacts the corresponding geometric solvability conditions and how the global set of the fixed poles of such control problems can be characterised in the general case (i.e. without any controllability or observability assumption).

ON THE FAUILURE DETECTION PROBLEM BY ADMISSIBLE POST-COMPENSATION

Abstract The block decoupling and failure detection problems are analyzed together. The aim is to introduce new solvability conditions for the failure detection problem deduced by dualization of recently obtained block decoupling conditions by admissible precompensation.

Structural Necessary and Sufficient Conditions for the Design of Observers with Prescribed Error Dynamics

Abstract Necessary and sufficient conditions for the design of observers whose error dynamics has prescribed invariant factors are given for full and minimal state observers. The result is obtained through standard algebraic tools along with Rosenbrocks theorem.

On decoupling of linear time delay systems by output feedback

Abstract We consider the row-by-row decoupling problelm for linear delay systems by output feedback. The characterization of the solvability of this problem is given in tenns of some easily checkable structural conditions. The main contribution is, in particular, to use generalized output feedback laws which may incorporate derivatives of the delayed new reference.