A Computational Approach to Nonlinear System Analysis

Jorge Tierno, Electrical Engineering, Caltech

Most practical control systems have significant nonlinear components. However, in practice these systems are analyzed either through robustness analysis of their linearizations, or through extensive simulation of their nonlinear models. Other forms of analysis of nonlinear systems have not as yet led to computationally tractable solutions. The aim of this thesis is to extend the analysis methodology for linear systems given by the structured singular value framework to nonlinear systems.

We study the question: Given an uncertain nonlinear system, driven by a nominal command signal over a finite time horizon, and subject to bounded noise, norm bounded feedback components and uncertain parameters, how far from the nominal trajectory will the actual one be? In order to inherit the properties of the structured singular value we will use the 2-norm as measure for noise signals and undermodeled feedback components. As is the case for robustness analysis of linear systems, we can only find efficient computation algorithms for upper and lower bounds to the answer to this question.

To compute the lower bound we develop a power algorithm similar in nature to the one developed for the structured singular value, and with similar behavior. Since, as was the case for linear systems, the algorithm is not guaranteed to converge in general, its analysis has to be done empirically. We test this algorithm by applying it to simulations of real systems, and show that it performs better than other available optimization methods. To develop an upper bound, we study a class of rational nonlinear systems. We show that for problems in this class, an uncertain, constrained linear system can be constructed that achieves the same performance level. Upper bounds on the performance of these systems can be computed by solving linear matrix inequalities. Finally, we study extensions that can be obtained to these analysis methods when the system is linear but time varying.