Riccati-Based Discretization for Nonlinear Continuous-Time Systems

A discretization method is proposed for a rather general class of nonlinear continuous-time systems, which can have a piecewise-constant input, such as one under digital control via a zero-order-hold device. The resulting discrete-time model is expressed as a product of the integration-gain and the system function that governs the dynamics of the original continuous-time system. This is made possible with the use of the delta or Euler operator and makes comparisons of discrete and continuous time systems quite simple, since the difference between the two forms is concentrated into the integration-gain. This gain is determined in the paper by using the Riccati approximation of a certain gain condition that is imposed on the discretized system to be an exact model. The method is shown to produce a smaller error norm than one uses the linear approximation. Simulations are carried out for a Lotka–Volterra and an averaged van der Pol nonlinear systems to show the superior performance of the proposed model to ones known to be online computable, such as the forward-difference, Kahan's, and Mickens' methods. Insights obtained should be useful for developing digital control laws for nonlinear continuous-time systems, which is currently limited to the simplest forward-difference model.