Optimum quantization in dynamic systems

In this paper the problem of optimally designing a quantizer imbedded in a closed-loop dynamic system is considered. The criterion for the design is that the overall system performance as expressed by a variational criterion is optimized. The function of quantization is thus related to the functions of control and estimation that are performed in the system. First, a procedure is described for optimally designing a quantizer in a static open-loop system, where the design criterion is the expected value of a function of the instantaneous error between the input and output of the quantizer. This procedure reduces the search over all quantizer parameters to an iterative search over a single parameter. Next, the existing methods for finding the optimal design of a quantizer imbedded in a dynamic system are reviewed. The most general method found in the literature involves a combination of dynamic programming with an exhaustive search for all quantizer parameters. The computational requirements of this procedure are quite large even for low-order systems with few quantizer parameters. Finally, a new result is presented that leads to greatly reduced computational requirements for the dynamic system case. It is shown that under certain conditions an overall optimum system design is obtained by first optimizing the system with all quantizers removed and then applying the procedure for the static open-loop case mentioned above. This result is analogous to the separation of the functions of estimation and control that occurs under similar conditions. The computational savings over the existing procedures are very extensive, and the new procedure is computationally feasible for a large class of practical systems.