New frequency-domain ℓ2-stability criteria for nonlinear MIMO discrete-time systems with constant and varying time delays

New frequency-domain $ \ell _2 $ ℓ2-stability criteria are derived for nonlinear discrete-time MIMO systems, having a linear time-invariant block with the transfer function $ \Gamma (z) $ Γ(z), in negative feedback with an aperiodic matrix gain $ A (k),\ k = 0, 1, 2, \ldots $ A(k), k=0,1,2,…, and a linear combination of a vector of certain classes of (generalised) first-and-third-quadrant non-monotone nonlinearities $ \underline {\varphi } (\underline {\cdot }) $ ϕ_(⋅_), having arguments with constant and time-varying delays, but without restrictions on their slopes. The framework does not employ Lyapunov–Krasovskii functionals involving linear matrix inequalities (LMIs) or their equivalent. The new stability criteria seem to be the most general for nonlinear and time variying time-delay systems, and have the following structure: (1) positive definiteness of the real part (as evaluated on $ |z| =1 $ |z|=1) of the product of $ \Gamma (z) $ Γ(z) and an algebraic sum of general causal and anticausal matrix multiplier functions of z. (2) An upper bound on the $ L_1 $ L1-norm of the inverse Fourier transform of the multiplier function, the $ L_1 $ L1-norm being weighted by certain novel, quantitative characteristic parameters (CPs) of the nonlinearities $ \underline {\varphi } (\underline {\cdot }) $ ϕ_(⋅_) without the asumptions of monotonicity, slope restrictions and the like. And (3) constraints on certain global averages of the generalised eigenvalues of $ (A (k+1), A (k)),\ k=1, 2, \ldots $ (A(k+1),A(k)), k=1,2,…, that are expressed in terms of (i) CPs of the nonlinearities, (ii) their coefficients, and, in general, (iii) time-delays in their arguments, a trade-off among all the three being possible. These global averages imply a restriction on the rate of variation of $ A (k) $ A(k) in a new sense. The literature results turn out to be special cases of the results of the present paper. Examples illustrate the stability theorems.