On the ℓ2-stability of time-varying linear and nonlinear discrete-time MIMO systems

New conditions are derived for the ℓ2-stability of time-varying linear and nonlinear discrete-time multiple-input multiple-output (MIMO) systems, having a linear time time-invariant block with the transfer function Γ(z), in negative feedback with a matrix of periodic/aperiodic gains A(k), k = 0, 1, 2, … and a vector of certain classes of non-monotone/monotone nonlinearities φ̲(·̲), without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features: i) They involve the positive definiteness of the real part (as evaluated on |z| = 1) of the product of Γ(z) and a matrix multiplier function of z. ii) For periodic A(k), one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A(k), but for aperiodic A(k), which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of (A(k + 1), A(k)), k = 1, 2, …. iii) They are distinct from and less restrictive than recent results in the literature. © 2014, South China University of Technology, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg.