A Study on Controlling Algorithm to Realize High-Speed & High-Accuracy Control Systems (1st Report)

Proposal of Modified Delta Operator

Abstract

In high-speed sampling, a digital control algorithm in terms of the shift operator Z easily becomes numerically illconditioned. In such situations, a control algorithm in terms of the delta operator δ is useful. However, when the delta operator δ is applied to a control algorithm based on fixed-point arithmetic, there appear coefficient-quantization errors and product-quantization errors. As a result, a control algorithm becomes numerically illconditioned again. To solve this problem, the modified delta operator δ' is proposed, which is defined as subtracting one from the shift operator Z. This definition can make an easy transformation from the shift form to the delta form. Then, its applicability to a control algorithm is proved and its physical meaning is considered. Also, it is shown that sensitivity function approaches one, as the sampling period T approaches zero. And the number of multiplications becomes about two-thirds of that of a usual delta operation. Simulations show that the modified delta form can decrease the numerical errors in fixed-point arithmetic, in particular, with short word length. Thus, the modified delta operator δ' can be applied to a control algorithm just like the usual delta operator.