Stability Conditions in Multiple-Input Multiple-Output Systems

This paper investigates the stabilization of unstable third-order Multiple-Input Multiple-Output (MIMO) systems whose interaction structure is described by a doubly stochastic combined matrix, also known as the Relative Gain Array (RGA). Starting from systems with negative Niederlinski index, we derive necessary and sufficient conditions under which stability can be recovered through diagonal perturbations while preserving the doubly stochastic structure of the combined matrix. By exploiting the canonical representation of matrices associated with a prescribed combined matrix and the invariance properties under diagonal equivalence, the problem is reduced to a structured parametric form that allows a complete algebraic characterization. Special attention is given to perturbations involving the (1, 1) entry and one additional diagonal entry, leading to explicit bounds on the perturbation parameters that guarantee stabilization. The results extend previous papers on diagonal perturbations of combined matrices and provide a constructive method for stabilizing MIMO systems without altering their interaction pattern. Numerical examples illustrate the applicability of the proposed approach.