Question 13.9: For step, ramp, and parabolic inputs, find the steady-state ...

First find G(s), the product of the z.o.h. and the plant.

G (s) = \frac{10  (1  −  e^{−Ts} )}{s²  (s  +  1)} = 10  (1  −  e^{−Ts} ) \left[\frac{1}{s²}   –   \frac{1}{s}  +  \frac{1}{s  +  1} \right]                     (13.77)

The z-transform is then

G (z) = 10 (1  −  z^{−1} ) \left[\frac{Tz}{(z  −  1)²}  –  \frac{z}{z  –  1}  +  \frac{z}{z  –  e^{−T} } \right]                   (13.78)

= 10 \left[\frac{T}{z  −  1}  –  1  +  \frac{z  –  1}{z  –  e^{−T} } \right]

For a step input,

K_{p} = \underset{z\rightarrow 1}{lim}  G (z)  = \infty ;        e^{*} (\infty ) = \frac{1}{1  +  K_{p}} = 0                       (13.79)

For a ramp input,

K_{v} = \frac{1}{T}  \underset{z\rightarrow 1}{lim}  (z  −  1)  G (z) = 10 ;     e^{*} (\infty ) = \frac{1}{ K_{v}} = 0.1                   (13.80)

For a parabolic input,

K_{a} = \frac{1}{T²}  \underset{z\rightarrow 1}{lim}  (z  −  1)²  G (z) = 0 ;     e^{*} (\infty ) = \frac{1}{ K_{a}} = \infty             (13.81)

You will notice that the answers obtained are the same as the results obtained for the analog system. However, since stability depends upon the sampling interval, be sure to check the stability of the system after a sampling interval is established before making steady-state error calculations.

Students who are using MATLAB should now run ch13apB6 in Appendix B. You will learn how to use MATLAB to determine K_{p} , K_{v} , and K_{a} in a digital system as well as check the stability. This exercise solves Example 13.9 using MATLAB.