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

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.