Filtering and Control by means of LMI optimisation

Chapter 1 Filtering and control via LMI optimisation 1.1 Introduction Several well-known problems of system and control theory can be reformulated as convex optimisation problems. This possibility appears advantageous when an analytical solution is not available. Indeed, the algorithms as the Interior Point Methods, which have asymptotic polynomial complexity, can be applied. Despite the fact that these techniques have been assuming a great relevance in the field of filtering and control of dynamic systems, many problems are still open: for this reason this particular research subject is very active in our days. These techniques, for which a fundamental reference is [Boy94], are known with the acronym LMI (Linear Matrix Inequality). LMI theory originates in Lyapunov s work of the fourties; it has been rediscovered in the sixties by a number of outstanding contributors, such as Kalman, Yakubovich, Popov and J. W. Willems. The importance of LMI can be appreciated recalling the possibility of tackling in a very simple and computationally advantageous way multi-objective problems by means of techniques based on a single quadratic Lyapunov function (see e.g. [Sch97a]). Nevertheless, it is well known that the issue of reducing the conservatism of these LMI techniques remains open to this date. There are many applications of LMI theory in the field of dynamic systems such as: uncertain systems ([Boy94]), robust least squares ([Elg97]), model predictive control ([Kot96]), failure detectability ([Fai98]), fuzzy control ([Kim97]). Nowadays, LMI is considered a fundamental tool for filtering and control of both Linear Time Invariant (LTI) systems and polytopic systems. A polytopic system is an uncertain system whose uncertainty, possibly time-varying, belongs to a polytope: the problems of filtering and control of these systems can be faced imposing for each vertex defining the uncertainty polytope an LMI constraint. In this case, the classical LMI techniques based on a single quadratic Lyapunov function turn out to be conservative: all the time, some new results towards the reduction of the conservatism for this particular class of uncertain systems have been already obtained (see e.g. [Shk99]). A further field in which LMI seems to be an effective tool is represented by periodic systems. Periodic systems have received an increasing attention in the last decades in a wide set of fields, see e.g. [Bit99b]. Also in the periodic context, LMI is recognised to be a valid and flexible tool to tackle multi-objective problems since it allows to alleviate required the computational effort. In fact the techniques today available in order to face control and filtering problems of periodic systems are mostly based on the necessity of finding the periodic solution of a Periodic Differential Riccati Equality (PDRE). This task is computationally heavy, whereas finding the periodic solution of a periodic LMI (see e.g. [Bit99a]) is decidedly computationally lighter. Furthermore, another significant feature of periodic LMI theory is represented by the fact that it represents a very effective tool in order to face periodic mixed filtering and control problems. On the contrary, mixed objectives can be hardly faced by means of classical filtering and control periodic techniques.