Question 13.9: A continuous time plant is being digitally controlled. The p...
A continuous time plant is being digitally controlled. The plant output should track a sinusoidal reference of frequency 0.2[rad/s],in the presence of step disturbances.
Determine the polynomial to be included in the transfer function denominator of the digital controller, necessary to achieve zero steady state errors. Assume that the sampling period is 1[s].
We first note that the sampled reference is r[k] = K_{1}\sin (0.2k + K_{2} ), where K_{1} and K_{2} are unknown constants. Thus, the reference generating polynomials, in shift and delta form, correspond to the denominators of the Z-transform and the Delta transform of r[k], respectively. We then obtain
\Gamma _{rq}(z) = z^{2} – 1.96z + 1; \Gamma _{r\delta }(\gamma ) = \gamma ^{2} + 0.04\gamma + 0.04 (13.7.1)
Similarly,t he disturbance generating polynomials are given by
\Gamma _{dq}(z) = z- 1; \Gamma _{d\delta }(\gamma ) = \gamma (13.7.2)
To obtain zero steady state errors the IMP must be satisfied. This requires that the denominator of the controller must include either the factor \Gamma _{rq}(z) \Gamma _{dq}(z) = (z^{2} – 1.96z +1)(z-1); (for the shift form) or the factor\Gamma _{r\delta }(\gamma ) \Gamma _{d\delta }(\gamma ) = (\gamma ^{2} + 0.04\gamma + 0.04)\gamma (for the delta form).