A Finite Settling Time Response of the 2nd-order Control System with Backlash

This paper is concerned with a consideration of a finite settling time response of the 2nd-order sampled data control system with backlash by the aid of a state-transition method. The transfer-function of the linear part of the control process is G(s)=K/s(s+1), and sampling period is T(sec). The slope and the total displacement associated with the backlash are assumed to be p and 2h respectively. The control system is designed on the basis of transient behavior in response to a step-function input whose magnitude is R. The desired digital controller such that the system will respond with deadbeat performance, and the backlash-element are treated as a variable gain element, which has different values during different sampling periods. To simplify, the authors assumed zero initial conditions. The pulse-transfer function of the desired digital controller is obtained as D(s)=(1+az^<-1>+bz^<-2>)/(1+cz^<-1>+dz^<-2>) where a, b, c and d are as follows : a=f_1(R, p, h, T) b=f_2(p, T) c=f_3(T) d=f_4(T) where f_i, i=1,2,3,4,denotes a function. The system compensated by the digital controller could be made to settle in two sampling periods to a step function input.