Christian Commault

Survey Generic properties and control of linear structured systems: a survey

In this survey paper, we consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A,B,C and D, is either a fixed zero or a free parameter. The location of the fixed zeros in these matrices constitutes the structure of the system. Indeed a lot of man-made physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly, a lot of classical properties of linear systems can be studied in terms of genericity. Moreover, these generic properties can, in general, be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system-specific properties like controllability, the finite and infinite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input-output decoupling, and so on. In this paper, we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool. We emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only. Moreover, the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well-known polynomially bounded combinatorial techniques for larger systems.

Observer-based fault detection and isolation for structured systems

Fault detection and isolation (FDI) problems are here consid- ered for linear systems with faults and disturbances. We design a set of ob- server-based residuals in such a way that the transfer from the disturbances to the residuals is zero, and the transfer from the faults to the residuals al- lows fault isolation. We are interested in obtaining a transfer function from faults to residuals with either a diagonal structure (i.e., a dedicated struc- tured residuals set) or a triangular one. We deal with this problem when the system under consideration is structured. That is, the entries of the system matrices are either fixed zeros or free parameters. To a structured system, one can associate in a natural way a directed graph. We can then provide necessary and sufficient conditions under which the FDI problems have a solution for almost any value of the free parameters. These conditions are simply expressed in terms of input-output paths in the associated graph. Index Terms—Fault detection and isolation, graph theory, model-based approach, structured residuals, structured systems.

New decoupling invariants: the essential orders

The aim of this paper is to introduce a new list of integers which is invariant under the action of biproper compensation and which plays a key role in the solution of the row-by-row decoupling problem. Several equivalent characterizations are provided within both the geometric and the transfer matrix approaches.

Feedback decoupling of structured systems

The feedback decoupling of structured linear systems is considered. Using results gathered on decoupling and the graph characterization of the structure at infinity, a necessary and sufficient decoupling condition for structured systems is developed. This condition has a graphical interpretation in terms of shortest input-output paths. >

Modeling and analysis of closed-loop production lines with unreliable machines and finite buffers

The purpose of this paper is to propose an analytical method for the performance evaluation of closed-loop production lines with unreliable machines and finite buffers. Such a system consists of a set of K machines separated by buffers of finite capacity. There is a fixed number of pallets circulating in the closed loop. We assume that machines have deterministic processing times but are subject to failures. Failures and repair times are exponentially distributed. We approximate the behavior of this system by a continuous flow model. The continuous flow model is then analyzed with a decomposition technique, which is similar to that used for (open) production lines. Numerical experiments are reported that show that the results provided by this method are in general fairly accurate.

Sensor Location for Diagnosis in Linear Systems: A Structural Analysis

We consider here the fault detection and isolation (FDI) problem for linear systems. We are interested in designing a set of observer-based residuals, in such a way that the transfer from the faults to the residuals is diagonal and the transfer from the disturbances to the residuals is zero. We deal with this problem when the system under consideration is structured, that is, the entries of the system matrices are either fixed zeros or free parameters. This problem can be solved in terms of the graph that can be associated in a natural way with a structured system. When the FDI solvability conditions are not satisfied, we assume that internal variables can be measured at a cost and look into the question of wether the problem is solvable with these new measurements. We give solvability conditions for a solution with a minimal number of additional sensors and among such solutions provide a minimal cost solution for the sensor location problem under consideration. We pay particular attention to the internal analysis of the system, and we propose a structural decomposition of the system associated graph based on some particular separators. This analysis leads to the definition of a reduced system. We prove that some potential additional sensors are inefficient for solving our FDI problem and that the FDI problem can be solved using only measurements on the reduced system

Input addition and leader selection for the controllability of graph-based systems

In this paper, we consider dynamical graph-based models, which are well fitted for the structural analysis of complex systems. A significant amount of work has been devoted to the controllability of such graph based models, e.g. recently for multi-agent systems or complex networks. We study here the controllability through input addition in this framework. We present several variants of this problem depending on the freedom which is left to the designer on the additional inputs. We use a unified framework, which allows us to encompass the different applications and representations (large scale systems, complex communications networks, multi-agent systems, ...) and provide convenient graph tools for their analysis. Our contribution is to characterize the structural modifications of the system resulting from an input addition (or a leader selection) and of the mechanisms which lead to controllability. We provide information on the possible location of additional inputs and on the minimal number of inputs to be added for controllability.

Observability Preservation Under Sensor Failure

This paper is concerned with the study of observability in a structural framework. It turns out that the system is structurally observable if and only if the system is output connected and contains no contraction. We focus our attention on the observability preservation under sensor failure. We consider linear observable systems and we wonder if a given system remains observable in case of sensor failure. More precisely, we will characterize among the sensors those which are critical, i.e., which failure leads to observability loss, those which are useless for observability purpose and the set of those which are useful without being critical. Using a graph approach we classify the sensors with respect to their importance for output connection preservation, contraction avoidance and then observability preservation under sensor failure.

Phase-type distributions and representations: Some results and open problems for system theory

In this paper we consider phase-type distributions. These distributions correspond to the random hitting time of an absorbing Markov chain. They are used for modelling various random times, in particular, those which appear in manufacturing systems as processing times, times to failure, repair times, etc. The Markovian nature of these distributions allows the use of very efficient matrix based computer methods for performance evaluation. In this paper we give a system theory oriented introduction to phase-type distributions. We concentrate mainly on the representation problem which consists of finding a Markov chain associated with some phase-type distribution. This is a realization problem in the sense of system theory with a lot of links with the classical linear system theory but also with a number of constraints which make the problem harder but more interesting. Indeed this problem has strong connections with the positive realization problem in control theory. The paper recalls known results, gives some new results, and points out the main remaining problems.

Sparse representations of phase-type distributions

A phase-type distribution is the distribution of hitting time in a finite-state Markov chain. A phase-type distribution is triangular if one can find an upper triangular markovian representation for this distribution. We introduce in this paper an extension of the triangular phase-type distributions, which we call monocyclic distributions. We will show that any phase-type distribution can be represented as a mixture of these simple sparse distributions