In Example 6.1 the response of a sampled-data system between sample instants was expressed as a sum

Question:

In Example 6.1 the response of a sampled-data system between sample instants was expressed as a sum of delayed step responses.

(a) Use this procedure to find the system output y(t) of Fig. P6.2-1 at t = 1 s .

(b) Repeat part (a) at t = 3 s .

(c) The equation for c(t) in part (a) and that for c(t) in part (b) should give the same value at t = 2 s . Why? Hint: Consider the step response of the analog plant.

(d) Show that the statement in part (c) is true.

Example 6.1

The unit-step response will be found for the first-order system in Fig. 6-1(a). Since the plant of a temperature control system is often modeled as a first-order system, this system might then be the model of a temperature control system. Using the techniques developed in Chapter 5, we can express the system output as where G(2) is defined as C(z) = G(z) 1 + G(z) -R(z) -Ts G(z) 0 = [1-8,4] - - 1 = S s +2 ss 4 +2) z - 1 2(1 - )z 0.3625 z (z- 1)(z - e-27) z - 0.8187' from the transform table in Appendix VI. Thus the R(s) + c(t) 0.6 0.4 0.2 T(z) = E(s) T= 0.1 s 0.667 G(z) 1 + G(z) 0.2 = (a) 1-g-Ts S 0.3625 Z 0.4562 Analog e (1) 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 (c) 0.6 0.8 t(s)