EXPLICIT CONSTRUCTION OF QUADRATIC LYAPUNOV FUNCTIONS FOR THE SMALL GAIN, POSITIVITY, CIRCLE, AND POPOV THEOREMS AND THEIR APPLICATION TO ROBUST STABILITY .2. DISCRETE-TIME THEORY

In a companion paper ('Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. Part I: Continuous-time theory'), Lyapunov functions were constructed in a unified framework to prove sufficiency in the small gain, positivity, circle, and Popov theorems. In this Part II, analogous results are developed for the discrete-time case. As in the continuous-time case, each result is based upon a suitable Riccati-like matrix equation that is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Multivariable versions of the discrete-time circle and Popov criteria are obtained as extensions of known results. Each result is specialized to the linear uncertainty case and connections with robust stability for state-space systems is explored.