A multivariable extension of the Tsypkin criterion using a Lyapunov function approach

For analyzing the stability of discrete-time systems containing a feedback nonlinearity, the Tsypkin criterion is the closest analog to the Popov criterion which is used for analyzing such systems in continuous-time. Traditionally, the proof of this criterion is based upon input-output properties and function analytic methods. We extend the Tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector-bounded memoryless time-invariant nonlinearities, along with providing a Lyapunov function proof for this classical result