Finite word-length effects in multirate direct digital control systems

When a digital computer acts as the compensating element of a sampled-data feedback system, the system performance is degenerated by the computational errors inherent in the finite word-length machine. The statistical analysis presented in this paper is an extension of the authors´ previous work to the case of a closed-loop, linear, multirate direct digital control system. A slide-rule calculation provides an upper bound for the loss of steady-state control-system performance for all computer word lengths and inputs. Furthermore, experimental results regarding the second-order probability-density functions of typical steady-state quantisation and roundoff-error processes are given, which allow the actual increase in steady-state mean-square system error to be evaluated by a simple iterative computer programme for single and multirate systems. An application of the analysis is to specify the precision required in the digital compensator and peripheral equipment for consistency with the overall accuracy demanded of the closed-loop system. It also enables the selection of the programming technique which requires the shortest computer word length for this accuracy. These calculations can result in better utilisation of computer facilities, thereby implying an economic saving. As a means of verifying the analysis, the performance of two different triple-rate digital compensators for a practical system are each compared with a counterpart having negligible quantisation and roundoff errors.