Exact covariance method simplifies state-feedback design for stochastic systems

A new paper from academic researchers tackles one of control theory's hardest problems: designing state-feedback controllers for linear discrete-time systems with i.i.d. multiplicative uncertainties and additive noise. Kaouther Moussa, Dimitri Peaucelle (LAAS-CNRS), Yohei Hosoe (Kyoto University), and Mirko Fiacchini (CNRS/GIPSA) establish a definitive link between covariance recursions, Kronecker-based matrix spectra, and mean-square stability. Their key contribution is an exact deterministic covariance recursion built on a Kronecker-product matrix augmentation, which simplifies analysis of stochastic systems within the tube-based Stochastic Model Predictive Control (SMPC) framework.

For systems with multiplicative uncertainty and no additive noise, the authors prove that the full-space augmented matrix from the covariance recursion has the same spectral radius as its symmetric-space counterpart. This yields an important equivalence: Schur stability of the full-space matrix is both necessary and sufficient for mean-square stability. They also propose new sufficient Linear Matrix Inequality (LMI) conditions for state-feedback design that are significantly smaller than the standard necessary-and-sufficient formulations, cutting computational burden while remaining conservative in practice. Numerical tests confirm the covariance characterization can recursively estimate covariance without relying on Monte Carlo sampling, and quantify the trade-off between computational cost and conservatism.