Paper:

# A Study on Controlling Algorithm to Realize High-Speed & High-Accuracy Control Systems – Proposal of modified Delta Operator –

## Tatsu Aoki^{*}, Yuji Furukawa^{**} and Nobuyuki Moronuki ^{*}

^{*}Department of Electrical Engineering, Tokyo Metropolitan College of Technology, 1-10-40 Higashi-Ohi, Shinagawa-ku, Tokyo 140, Japan

^{**}Department of precision engineering, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji-shi, Tokyo 192-03, Japan

*z*-transform, Mechatronics, Real-time control

In high-speed sampling, in terms of the shift operator *z*, digital control algorithms easily become numerically illconditioned. In such situations, control algorithms based on the delta operator *δ* are useful. However, when the delta operator *δ* is applied to a control algorithm based on fixed-point arithmetic, coefficient quantization errors and product quantization errors appear. As a result, the control algorithm again becomes numerically ill-conditioned. To solve this problem, a modified delta operator *δ’* is proposed. It is defined by subtracting unity from the shift operator *z* and then dividing by an arbitrary constant *T _{δ}*. This definition allows an easy transformation from the shift form to the delta form. Then, its applicability to a control algorithm is demonstrated and its physical ramifications are considered. Also, it is shown that the sensitivity function approaches unity as the sampling period

*T*approaches zero. Furthermore, the numbers of calculations are the same as those in the usual delta model. Simulations show that the modified delta form can decrease the numerical errors in fixedpoint arithmetic, in particular, with short word length. Thus, the modified delta operator

*δ’*can be applied to a control algorithm in exactly the same manner as the usual delta operator.

*J. Robot. Mechatron.*, Vol.9, No.6, pp. 446-454, 1997.