Second Moment Polytopic Systems: Generalization of Uncertain Stochastic Linear Dynamics

TL;DR

This paper introduces second moment polytopic (SMP) systems, a new framework for stabilizing uncertain stochastic linear systems by analyzing second moments and employing convex optimization techniques.

Contribution

It generalizes uncertain stochastic linear systems using second moments and develops a stability analysis and controller design method via linear matrix inequalities and convex optimization.

Findings

SMP systems unify uncertain and stochastic linear dynamics.

Stability conditions are expressed as linear matrix inequalities.

Iterative convex optimization yields stabilizing feedback gains.

Abstract

This paper presents a new paradigm to stabilize uncertain stochastic linear systems. Herein, second moment polytopic (SMP) systems are proposed that generalize systems with both uncertainty and randomness. The SMP systems are characterized by second moments of the stochastic system matrices and the uncertain parameters. Further, a fundamental theory for guaranteeing stability of the SMP systems is established. It is challenging to analyze the SMP systems owing to both the uncertainty and randomness. An idea to overcome this difficulty is to expand the SMP systems and exclude the randomness. Because the expanded systems contain only the uncertainty, their stability can be analyzed via robust stability theory. The stability of the expanded systems is equivalent to statistical stability of the SMP systems. These facts provide sufficient conditions for the stability of the SMP systems as linear matrix inequalities (MIs). In controller design for the SMP systems, the linear MIs reduce to cubic MIs whose solutions correspond to feedback gains. The cubic MIs are transformed into simpler quadratic MIs that can be solved using optimization techniques. Moreover, solving such non-convex MIs is relaxed into the iteration of a convex optimization. Solutions to the iterative optimization provide feedback gains that stabilize the SMP systems. As demonstrated here, the SMP systems represent linear dynamics with uncertain mean and covariance and other existing systems such as independently identically distributed dynamics and random polytopes. Finally, a numerical simulation shows the effectiveness of the proposed method.