Covariance Stabilization for a class of Stochastic Discrete-time Linear Systems using the S-Variable Approach

TL;DR

This paper introduces a less conservative LMI-based method using the S-variable approach for covariance stabilization in stochastic discrete-time linear systems within the SMPC framework, accommodating unbounded uncertainties.

Contribution

It presents a new LMI-based design condition for covariance stabilization that reduces conservatism and computational complexity compared to existing methods.

Findings

The proposed method is more numerically tractable with smaller LMIs.

It effectively stabilizes covariance in systems with unbounded stochastic uncertainties.

Numerical examples demonstrate the method's robustness and efficiency.

Abstract

This paper deals with the problem of covariance stabilization for a class of linear stochastic discrete-time systems in the Stochastic Model Predictive Control (SMPC) framework. The considered systems are affected by independent and identically distributed (i.i.d.) additive and parametric stochastic uncertainties (potentially unbounded), in addition to polytopic deterministic uncertainties bounding the mean of the state and input parameters. The design conditions presented in this paper are formulated as Linear Matrix Inequalities (LMIs), using the S-variable approach in order to reduce the potential conservatism. These conditions are derived using a deterministic exact characterization of the covariance dynamics, the latter involves bilinear terms in the control gain. A technique to linearize such dynamics is presented, it results in a descriptor representation allowing to derive sufficient conditions for the design of a covariance-stabilizing controller. The derived condition is first compared with a known necessary and sufficient stability condition for systems without deterministic uncertainties and additive stochastic noise. Although more conservative, the proposed condition is more numerically tractable, with an LMI size scaling as O(n^2) instead of O(n^3). Then, the same condition is used to design controllers that are robust to both deterministic and stochastic uncertainties. Several numerical examples are presented for comparison and illustration.