Discrete-time counterparts of the RL and RC multipliers

The use of multipliers is an important technique for absolute stability analysis. In the continuous-time domain, the RL and RC multipliers preserve the positivity of memoryless and monotone nonlinearities. We classify their discrete-time counterparts and analyse their phase properties. The classification of the discrete-time counterparts of the multipliers is richer than that of their continuous-time counterparts. Some classes of multipliers that preserve the positivity of memoryless and monotone nonlinearities are parametrised in term of their zero-pole description. Similarly we discuss classes that preserve the positivity of memoryless, monotone and symmetric nonlinearities. Multipliers that lie outside such classes are illustrated with numerical counterexamples. We give and prove the phase properties of the discrete-time counterparts of the multipliers. The phase properties are less flexible compared to those of their
continuous-time counterparts.