Discrete time approximation of continuous time nonlinear state-space models

In this paper we discuss the problem of approximating a continuous time nonlinear state space model with an explicit discrete time state space model. Two steps are made to do so. The first one is to replace the classical zero-order-hold (ZOH) excitation signal assumption by a more general concept of low pass (LP) signals. It will be shown that opposed to the ZOH-property, the LP-property is maintained for a wide class of (non)linear systems, including cascaded and closed loop systems. The second step is based on a result from the signal processing theory that states that a BL-signal can be exactly predicted from its past samples by a fixed causal linear filter. This result is generalized to the discrete time integration of LP-signals by introducing an error bound that can be made arbitrarily small by increasing the sample frequency. Both ideas are then combined to bound the approximation errors of an explicit discrete time nonlinear state space approximation for a continuous time nonlinear state space model. The order of the decay of the approximation error as a function of the sample frequency is given. It is shown that eventually the aliasing error is the dominating error, at a cost of a (slightly) increased model complexity. These results are directly applicable to nonlinear system identification, and will be experimentally verified on the identification of a closed loop nonlinear system (the silverbox) using a discrete time nonlinear state space model.