Commuting matrices, equilibrium points for control systems with single saturated input

In this paper, commutative matrices of multiple input multiple output (MIMO) linear systems are considered. The existence of the feedback matrices of a commutative state matrix set in the MIMO closed-loops is reduced to the existence of an invariant subspace of a matrix A. The existence of feedback matrices in systems in open-loop is equivalent to the existence of the solution of matrix equations denoted by Kronecker products. By defining new equilibrium points, the relationship between equilibrium points are discussed for a linear system with a single saturated input. Four criteria for equilibrium points are outlined for such linear systems. Finally, four interesting examples, including their corresponding simulink plots, are shown to illustrate the above results.