Contributions to Passivity Theory and Dissipative Control Synthesis

This thesis contains contributions to some relevant problems in the field of control theory and controller design technology, namely to the areas of passivity analysis and dissipative control synthesis for linear and nonlinear dynamical systems. The first of our contributions consists in presenting a solution to a problem which had been unsolved for many years: the problem of the equivalence between the notions of strict positive realness and strict passivity of linear systems. Both properties imply the asymptotic stability of a linear system, although the former is a frequency-domain concept and the latter is a time-domain concept. Subsequently, we approach the equally classical topic of static output feedback stabilization of linear systems, a problem to which a definite solution remains to be given. We present a new necessary and sufficient LMI condition for stabilization based on the notion of strict dissipativity, and we propose a new noniterative strategy for controller design which consists in solving a single convex optimization problem. In addition, we also introduce a new dissipativity-based strategy for feedback stabilization of nonlinear systems using the notion of linear annihilators and the celebrated Finsler’s Lemma. This approach allows for analysing the dissipativity properties of rational nonlinear plants in terms of a polytopic LMI condition. A new stabilizability condition that would not be feasible in the case of a passive representation of the system is presented as well, making it possible to derive a closed-form expresion for the controller’s feedthrough term as a direct consequence of the local dissipativity analysis of the plant. This feature simplifies the remaing steps of the controller design procedure considerably, both in the case of a static or a dynamic output feedback

Identification of nonlinear interconnected systems

This thesis was submitted for the degree of Master of Philosophy and awarded by Brunel University.In this work we address the problem of identifying a discrete-time nonlinear system composed of a linear dynamical system connected to a static nonlinear component. We use linear fractional representation to provide a united framework for the identification of two classes of such systems. The first class consists of discrete-time systems consists of a linear time invariant system connected to a continuous nonlinear static component. The identification problem of estimating the unknown parameters of the linear system and simultaneously fitting a math order spline to the nonlinear data is addressed. A simple and tractable algorithm based on the separable least squares method is proposed for estimating the parameters of the linear and the nonlinear components. We also provide a sufficient condition on data for consistency of the identification algorithm. Numerical examples illustrate the performance of the algorithm. Further, we examine a second class of systems that may involve a nonlinear static element of a more complex structure. The nonlinearity may not be continuous and is approximated by piecewise a±ne maps defined on different convex polyhedra, which are defined by linear combinations of lagged inputs and outputs. An iterative identification procedure is proposed, which alternates the estimation of the linear and the nonlinear subsystems. Standard identification techniques are applied to the linear subsystem, whereas recently developed piecewise affine system identification techniques are employed for the estimation of the nonlinear component. Numerical examples show that the proposed procedure is able to successfully profit from the knowledge of the interconnection structure, in comparison with a direct black box identification of the piecewise a±ne system.Funding was obtained as a Marie Curie Early Stage Researcher Training fellowship, under the NET-ACE project (MEST-CT-2004-6724)