Discrete-Time Pairwise Connected Switched Systems and Lur’e Systems. Tsypkin’s Criterion for Systems with Two Nonlinearities

We study the stability of discrete-time switched systems for any laws of switching between linear subsystems. Pairwise connected systems are distinguished among such systems. A sufficient frequency-domain stability condition has been obtained for them. Two sufficient conditions and two criteria for the existence of a Lyapunov quadratic function are obtained for switched systems whose stability is equivalent to the absolute stability of Lur’e systems with two nonlinearities. These conditions amount to checking the solvability of special matrix inequalities whose dimensions are considerably lower than the dimension of the original system of matrix inequalities that defines the necessary and sufficient conditions. The resulting conditions are compared with the conditions of the Tsypkin criterion and with the necessary and sufficient conditions using the examples of systems of the third and sixth orders.