Data-Driven Probabilistic Finite $\mathcal{L}_2$-Gain Stabilization of Stochastic Linear Systems

TL;DR

This paper introduces a data-driven probabilistic approach to stabilize stochastic linear systems by controlling their finite $ ext{L}_2$-gain, overcoming traditional challenges with stochastic uncertainties.

Contribution

It develops a novel probabilistic $ ext{L}_2$-gain concept and a convex data-driven control design method using noisy measurements and disturbance forecasts.

Findings

The approach successfully stabilizes stochastic systems with noisy data.

The method provides a convex LMI-based offline controller synthesis.

Numerical example demonstrates the effectiveness of the proposed approach.

Abstract

In process operations, it is desirable to manage the sensitivity of the system output against external disturbance in the form of finite $\mathcal{L}_2$-gain stabilization. This matter is, however, nonsensical for stochastic systems because the stochastic uncertainties in the control input almost always lead to an unbounded $\mathcal{L}_2$ gain from the disturbance to the output. To address this issue, this article develops a novel concept that characterizes the $\mathcal{L}_2$ gain of stochastic systems in a probabilistic way. Combined with a large data set, we formulate a data-driven probabilistic finite $\mathcal{L}_2$-gain stabilization design using noisy trajectory measurements and the disturbance forecast that does not necessarily agree with the actual future disturbance. The design approach consists of a data-driven trajectory estimation algorithm, whose resulting estimation error covariance is nicely integrated into the feasibility conditions for controller synthesis, leading to a convex offline design in the form of linear matrix inequalities. The effectiveness of the proposed design, along with the additional insights provided by the approach, is illustrated via a numerical example.