Approximation of The Constrained Joint Spectral Radius via Algebraic Lifting
For researchers studying stability of constrained switching systems, this provides a method to approximate a previously intractable quantity, though it is an incremental theoretical reduction.
The paper tackles the problem of computing the constrained joint spectral radius for constrained switching systems, which is known to be difficult. By using semi-tensor products, they show equivalence to the joint spectral radius of an arbitrary switching system, enabling approximation via existing algorithms.
This paper studies the constrained switching (linear) system which is a discrete-time switched linear system whose switching sequences are constrained by a deterministic finite automaton. The stability of a constrained switching system is characterized by its constrained joint spectral radius that is known to be difficult to compute or approximate. Using the semi-tensor product of matrices, the matrix-form expression of a constrained switching system is shown to be equivalent to that of a lifted arbitrary switching system. Then the constrained joint/generalized spectral radius of a constrained switching system is proved to be equal to the joint/generalized spectral radius of its lifted arbitrary switching system which can be approximated by off-the-shelf algorithms.