This is the discrete-time version of a previous paper by the same authors [Math. Probl. Eng. 6, 461-493 (2000; Zbl 0968.93075)]. A linear system with additive nonlinear time-varying state dynamics is considered, and provided the nonlinearity is not too much structured (namely, provided it is bounded by some given positive quadratic function of the state), it is shown that stability can be checked (analysis) or ensured (state-feedback design) via convex optimization over linear matrix inequalities (LMIs).
Stability is ensured with the help of a quadratic Lyapunov function. The main result, Theorem 1, is obtained by applying a standard algebraic trick on convexity of quadratic functions, called the S-procedure or also Finsler’s lemma.
Applications to the stability of interconnected systems composed of linear subsystems with uncertain nonlinear couplings are also described.