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Discrete-Time Implementation of Arbitrary Delay-Free Feedback Networks

The delay-free feedback loop can be directly implemented in discrete time by separately discretizing the forward and backward transfer functions and simultaneously solving the resulting linear system for the nodes connecting the filters within the loop. The ability to form the solutions rests upon the fact that, at sample n, the output of a discrete-time linear system is a linear function of the input to the system at sample n. This technique allows for relatively simple calculation of coefficients for certain time-varying feedback systems, and allows for inclusion of memoryless nonlinearities inside feedback loops. We show that the technique can be generalized to discretize an arbitrary network of LTI systems arranged in multiple-loop feedback networks. Two examples are presented: one time-varying system and one nonlinear system.

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