Question 13.2: A plant has a nominal model given by Go(s) = 1/(s - 1)² (13....
A plant has a nominal model given by
G_{o}(s) =\frac{1}{(s-1)^{2}} (13.5.7)
13.2.1 Synthesize a continuous time PID controller such that the dominant closed loop poles are the roots of the polynomial s^{2}+ 3s + 4.
13.2.2 Using the above result, obtain a discrete time PID controller. Assume that the sampling frequency can be as large as required and that a zero order hold is present at the plant input.
13.2.3 Using SIMULINK compare the response to a unit step reference for the continuous and the discrete time loop.
13.2.1 The closed loop characteristic polynomial A_{cl}(s) is chosen as
A_{cl}(s) = (s^{2}+ 3s + 4)(s^{2}+ 10s + 25). (13.5.8)
where the factor s^{2}+ 10s + 25has been added to ensure that the degree of A_{cl}(s) is 4, which is the minimum degree required for an arbitrarily chosen A_{cl}(s) . On solving the pole assignment equation we obtain P(s) = 88s^{2} + 100s + 100 and \bar{L}(s) = s + 15.. This leads to the following PID controller
C(s) = \frac{88s^{2} + 100s +100}{s(s+15)} . (13.5.9)
13.2.2 Since we can use high sampling rates , a straightforward procedure to obtain a discrete time PID controller is to substitute s by γ in (13.5.9), i.e.
C_{\delta }(\gamma ) = \frac{88 \gamma ^{2} + 100\gamma +100}{\gamma (\gamma +15)} . (13.5.10)
or, in Z transform form
C_{q }(z) = \frac{88 z^{2} – 166z + 79}{(z-1) (z+0.5)} . (13.5.11)
where we have assumed a sampling period Δ = 0.1.
13.2.3 The continuous and the discrete time loops are simulated with SIMULINK for a unit step reference at t = 1 and a unit step input disturbance at t = 10. The difference of the plant outputs is shown in Figure 13.3 on the next page. The reader is invited to verify that if a smaller sampling period is used, say Δ = 0.01,t hen the difference would be barely noticeable. Both loops are included in the SIMULINK scheme in file lcodi.mdl.
However, the fact that none of the ad-hoc transformations listed above is entirely satisfactory with more modest sampling rates is illustrated in the following example.
