Generalized Input-to-State ℓ2-Gains of Discrete-Time Switched Linear Control Systems

A generalized notion of input-to-state $\ell_2$-gain is proposed for discrete-time switched linear control systems (SLCSs). Dependent on a discount factor of subsystem matrices, this generalized $\ell_2$-gain provides new insight into input-to-state behaviors of the SLCSs under parameter variations. After establishing analytical properties of the generalized $\ell_2$-gain, this paper focuses on the generating function approach to the study of the generalized $\ell_2$-gain. Important properties of generating functions are derived, and it is shown that their radii of convergence characterize the generalized $\ell_2$-gain. Furthermore, iterative algorithms are developed for computing the generating functions with proven uniform or exponential convergence. Numerical results show that these algorithms yield efficient estimates of both the generalized and classical $\ell_2$-gains.