On the ?2 stability of time varying linear and nonlinear discrete time MIMO systems

On the ${\ell}_2$-stability of time-varying linear and nonlinear discrete-time MIMO systems

Key words:  Circle criterion  Discrete-time MIMO system  ₂-stability  Feedback system stability  Linear matrix inequalities(LMI)  Lur’e problem  Multiplier functions  Nyquist’s criterion  Periodic coefficient systems  Popov’s criterion  Time-varyingsystems

摘要:

Abstract:

New conditions are derived for the ?₂-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput (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 φ_{_}(·_{_}), 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.