Abstract
This paper presents a design procedure for suboptimal quantizers of states in linear discrete feedback control systems and discusses the tradeoffs of the number of samples per unit time and the word length of digitalized states. In computer control systems of industrial processes, it is more efficient from the standpoint of computer control if the state signals or the error signals of the processes are taken in sparse samples and quantized coarsely, because the information load to be processed in the computer will thereby be reduced. On quantizing the sampled data, the usual uniform quantization is not necessarily suitable for transferring, processing and storing the data also for determining the control efficiently. The word length of the data can be shortened by the use of non-uniform quantization.
In this paper the process is described by linear differential equations and the cost functional is given by an ordinary quadratic performance index. This system is discretized by zero-order holding the control. The optimal control law is given by a linear combination of states, and the minimum cost is given by a quadratic form of states. When the sampled states are quantized, the control law is also given by a linear combination of quantized states and the minimum cost is approximated by a quadratic form of states. Then, the suboptimal quantizer is designed by minimizing the trace of the coefficient matrix of the cost with respect to input and output levels of the quantizer.
The suboptimal quantizer obtained here has a rather general nature depending only on the order of the processes. It quantizes the state finely near the zero and coarsely far away from the zero. In some 1st and 2nd order systems the information to be processed becomes about 20% less for the 1st order system and about 15% less for the 2nd order system with the suboptimal quantizers as compared with using the ordinary uniform quantizers.
