Question 13.6: Consider the servo system of Examples 3.4 on page 47 and 13....
Consider the servo system of Examples 3.4 on page 47 and 13.5 on page 364, i.e. a system with transfer function
G_{o}(s) =\frac{1}{s(s+1)} (13.6.33)
Synthesize a minimum time dead-beat control with sampling period Δ = 0.1[s].
From Example 13.5 on page 364, we have that the system pulse transfer function (with Δ =0.1[s]) is given by (13.6.21),i.e.
G_{oq}(z) =0.0048 \frac{z + 0.967}{(z-1)(z-0.905)} (13.6.21)
G_{oq}(z) =0.0048 \frac{z + 0.967}{(z-1)(z-0.905)}\Longrightarrow \alpha = 105.49 (13.6.34)
Hence, applying (13.6.32) we have that
C_{q}(z) =\frac{\alpha A_{oq}(z)}{z^{n} -\alpha B_{oq}(z)} (13.6.32)
C_{q}(z) =\frac{\alpha A_{oq}(z)}{z^{n} -\alpha B_{oq}(z)} = \frac{105.49z – 95.47}{z+ 0.4910} (13.6.35)
The performance of this controller is shown in Figure 13.7 for a unit step reference applied at time t = 0. Observe that the continuous time plant output, y(t), reaches its steady state value after two sampling periods , as expected. Furthermore, the intersample response is now quite acceptable. To evaluate the control effort we apply (13.6.31) to obtain
U_{q}(z) =\frac{\alpha A_{oq}(z)}{z^{n}} R_{q}(z) (13.6.31)
U_{q}(z) =\frac{\alpha A_{oq}(z)}{z^{n}} R_{q}(z) = 105.49 \frac{(z-1)(z-0.905)}{z^{2}} R_{q}(z) (13.6.36)
Solving, we obtain the control sequence: u[0] = 105.49, u[1] = −95.47 and u[k] = 0 ∀k ≥ 2. Note that the zero value is due to the fact that the plant exhibits integration. Also observe the magnitudes of the first two sample values of the control signal; they run the danger of saturating the actuator in practical applications. In this respect, de ad-beat control is not different to minimal prototype and to continuous time control, since fast control (with respect to the plant bandwidth) will always demand large control magnitudes (as happens in this example) since this is a tradeoff which does not depend on the control architecture or control philosophy. (See Chapter 8.)
Remark 13.1. The controller presented above has been derived for stable plants or plants with at most one pole at the origin, since cancellation of A_{oq}(z) is allowed. However, the dead-beat philosophy can also be applied to unstable plants, provided that dead-beat is attained in more than n sampling periods. To do this we simply use pole assignment and place all of the closed loop poles at the origin.
