Exact Covariance Characterization for Controlled Linear Systems subject to Stochastic Parametric and Additive Uncertainties
Kaouther Moussa, Mirko Fiacchini
TL;DR
This work derives an exact, vectorized characterization of the state-covariance dynamics for discrete-time linear systems subject to both additive and stochastic parametric uncertainties, enabling precise analysis within stochastic MPC. By decomposing the state into a nominal and error component and leveraging Kronecker products, the authors obtain a linear covariance map $M(K)$ influenced by the second-order moment matrix $C_p=\mathbb{E}[\bar{A}(p_k)\otimes\bar{A}(p_k)]$, along with a closed-form expression for the evolution of $\textrm{cov}(e_k)$. They establish a conservative yet tractable LMI condition: there exists $K$ and $\alpha\in(0,1)$ with $(\alpha-\sigma_{max}(C_p)) I \succ (A_0+BK)^T(A_0+BK)$ ensuring $\rho(M(K))<\alpha$, providing a practical route to covariance stabilization in higher dimensions. A numerical example confirms agreement between theoretical and empirical covariances, demonstrates the lower-dimensionality and computational benefits of the proposed LMI, and compares with existing MSS-based conditions. Overall, the results enable tighter SMPC formulations and scalable covariance control for systems affected by both additive and multiplicative uncertainties.
Abstract
This work addresses the exact characterization of the covariance dynamics related to linear discrete-time systems subject to both additive and parametric stochastic uncertainties that are potentially unbounded. Using this characterization, the problem of control design for state covariance dynamics is addressed, providing conditions that are conservative yet more tractable compared to standard necessary and sufficient ones for the same class of systems. Numerical results assess this new characterization by comparing it to the empirical covariance and illustrate the control design problem.