Redesign of the DC Servo Compensator Using the SRL Design a compensator for the DC servo system of Example 7.30 using pole placement based on the SRL. For the control law, let the cost output z be the same as the plant output; for the estimator design, assume that the process noise enters at the

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Redesign of the DC Servo Compensator Using the SRL
Design a compensator for the DC servo system of Example 7.30 using pole placement based on the [latex] ext{SRL}.[/latex] For the control law, let the cost output [latex] z [/latex] be the same as the plant output; for the estimator design, assume that the process noise enters at the same place as the system control signal. Select roots for a control bandwidth of about [latex] 2.5 ext{ rad/sec}, [/latex] and choose the estimator roots for a bandwidth of about [latex] 2.5 [/latex] times faster than the control bandwidth ([latex] 6.3 ext{ rad/sec} [/latex]). Derive an equivalent discrete controller with a sampling period of [latex] T_s = 0.1 ext{ sec (}10 [/latex] times the fastest pole), and compare the continuous and digital control outputs and control efforts.

Accepted Answer

Because the problem has specified that [latex] \pmb G_1 = \pmb G [/latex] and [latex] \pmb H_1 = \pmb H,[/latex] then the [latex] ext{SRL} [/latex] is the same for the control as for the estimator, so we need to generate only one locus based on the plant transfer function. The [latex] ext{SRL} [/latex] for the system is shown in Fig. 7.44. From the locus, we select [latex] −2 ± 1.56j [/latex] and [latex] −8.04 [/latex] as the desired control poles [latex] (pc=[−2+1.56*j;−2−1.56*j;−8.04]) [/latex] and [latex] −4±4.9j [/latex] and [latex] −9.169 (pe=[−4+4.9*j;−4−4.9*j;−9.169]) [/latex] as the desired estimator poles. The state feedback gain is K=(F,G,pc), or

[latex] \pmb K = \begin{bmatrix} −0.285 & 0.219 & 0.204 \end{bmatrix}, [/latex]

and the estimator gain is Lt=place(F’,H’,pe), L=Lt’, or

[latex] \pmb L = \begin{bmatrix} 7.17 \ 97.4 \ 367 \end{bmatrix} .[/latex]

Notice that the feedback gains are much smaller than before. The resulting compensator transfer function is computed from Eq. (7.177) to be

[latex] D_c(s) = \frac{U(s)}{Y(s)} = −\pmb K(s \pmb I − \pmb F + \pmb {GK + LH})^{−1} \pmb L. [/latex] (7.177)

[latex] D_c(s) = − \frac{94.5(s + 7.98)(s + 2.52)}{(s + 4.28 ± 6.42j)(s + 10.6)}.[/latex]

We now take this compensator, put it in series with the plant, and use the compensator gain as the parameter. The resulting ordinary root locus of the closed-loop system is shown in Fig. 7.45. When the root-locus gain equals the nominal gain of [latex] 94.5, [/latex] the roots are at the closed-loop locations selected from the [latex] ext{SRL},[/latex] as they should be.

Note that the compensator is now stable and minimum phase. This improved design comes about in large part because the plant pole at [latex] s = −8 [/latex] is virtually unchanged by either controller or estimator. It does not need to be changed for good performance; in fact, the only feature in need of repair in the original [latex] G(s) [/latex] is the pole at [latex] s = 0. [/latex] Using the [latex] ext{SRL} [/latex] technique, we essentially discovered that the best use of control effort is to shift the two low-frequency poles at [latex] s = 0 [/latex] and [latex] −2 [/latex] and to leave the pole at [latex] s = −8 [/latex] virtually unchanged. As a result, the control gains are much lower and the compensator design is less radical. This example illustrates why [latex] ext{LQR} [/latex] design is typically preferable over pole placement.

The discrete equivalent for the controller is obtained from MATLAB with the [latex] ext{c2d } [/latex] command, as in the following code:

Question 7.EX.32: Redesign of the DC Servo Compensator Using the SRL Design a ...

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