Some qualitative properties of multirate digital control systems

We consider multirate digital control systems which consist of an interconnection of a continuous-time nonlinear plant (described by ordinary differential equations), a digital controller (described by ordinary difference equations) which has quantizers (but is otherwise linear), along with the required interface elements (A/D and D/A converters). The input to the digital controller consists of the multirate sampled output of the plant. In the present note we show that when quantizer nonlinearities are neglected, then under reasonable conditions (which exclude the critical cases), the stability properties (in the Lyapunov sense) of the trivial solution of the nonlinear multirate digital control systems can be deduced from the stability properties of the trivial solution of its linearization. For such systems we also present a result concerning the existence and construction of stabilizing multirate-output digital controllers. In the present note we also show that the solutions of multirate digital feedback control systems with nonlinear plant and quantizers are uniformly ultimately bounded if the trivial solution of the corresponding linear systems consisting of the linearization of the plant and with the quantization removed from the digital controller, is asymptotically stable. We also provide a result which compares the response of multirate digital control systems with nonlinear plant and quantizers in the controller with the response of the corresponding nonlinear multirate digital control systems without quantizers in the digital controller.