Consider the position control servomechanism described in Subsec. 3.13.2. For simplicity, let La≃0, Kp=1. and K = A = 1 and B = 2. Then, the transfer function of the motor–gear–load system becomes Gp(s)=1/s(s+2).To this servomechanism, a continuous-time controller Gc(s) is introduced, as shown in

Question

Consider the position control servomechanism described in Subsec. 3.13.2. For simplicity,
let [latex]L_a\simeq 0, K_p=1[/latex]. and K = A = 1 and B = 2. Then, the transfer function of the motor–gear–load system becomes [latex]G_p(s)=1/s(s+2)[/latex]. To this servomechanism, a continuous-time controller [latex]G_c(s)[/latex] is introduced, as shown in Figure 12.20, which satisfies certain design requirements. Of course, if a discrete-time controller [latex]G_c(z)[/latex] is introduced instead of the continuous-time controller [latex]G_c(s)[/latex], then the closed-loop system would be as shown in Figure 12.21.
Let the continuous-time controller [latex]G_c(s)[/latex] satisfying the design requirements have the following form:

[latex]G(s)=\frac{K}{sT_f+1}, where K=\left[\frac{R_LR_f}{R_g+R_L}\right]K_aK_gω and T_f=L_f/R_f[/latex] (3.13.2)

[latex]G_c(s)=K_s\left[\frac{s+a}{s+b}\right]=101\left[\frac{s+2}{s+6.7}\right][/latex] (12.7-8)

Then the problem at hand is to determine [latex]G_c(z)[/latex] of the closed-loop system shown in Figure 12.21 satisfying the same design requirements, where the sampling time T = 0.2 sec.

Accepted Answer

The transfer function H(s) of the closed-loop system of Figure 12.20 is

[latex]H(s)=\frac{Θ_y(s)}{Θ_r(s)}=\frac{G_c(s)G_p(s)}{1+G_c(s)G_p(s)}=\frac{101(s+2)}{s(s+2)(s+6.7)+101(s+2)}[/latex]

[latex]=\frac{101s+202}{s³+8.7s²+114.4s+202}[/latex] (12.7-9)

To find [latex]G_c(z)[/latex] of the closed-loop system of Figure 12.21, it is sufficient to
discretize [latex]G_c(s)[/latex] given in Eq. (12.7-8). To this end, we will use the method of pole-zeromatching presented in Subsec. 12.3.3 [relations (12.3-20)–(12.3-23)]. According to this method, [latex]G_c(z)[/latex] has the form

[latex]U^{*}(s)=\sum\limits_{k=0}^{∞}{u(kT)e^{-skT}}[/latex] (12.3-3)

[latex]G(s)=K_s\frac{(s+μ_1)(s+μ_2)…(s+μ_m)}{(s+π_1)(s+π_2)…(s+π_n)}, m≤n[/latex] (12.3-20)

[latex]G(z)|_{z=1}=K_z2^{n-m}\frac{(1+z_1)(1+z_2)…(1+z_m)}{(1+p_1)(1+p_2)…(1+p_n)}=G(s)|_{s=0}=K_s\frac{μ_1μ_2…μ_m}{π_1π_2…π_n}[/latex] (12.3-23)

[latex]G_c(z)=K_z\left[\frac{z+z_1}{z+p_1}\right][/latex] (12.7-10)

where the pole [latex]s=-6.7[/latex] of [latex]G_c(s)[/latex] is mapped into the pole [latex]z=-p_1[/latex] of [latex]G_c(z)[/latex] and the zero [latex]s=-2[/latex] of [latex]G_c(s)[/latex] is mapped into the zero [latex]z=-z_1[/latex] of [latex]G_c(z)[/latex]. That is, we have that

[latex]p_1=e^{-bT}=-e^{-6.7(0.2)}=-e^{-1.34}=-0.264[/latex]

[latex]z_1=e^{-aT}=-e^{-2(0.2)}=-e^{-0.4}=-0.67[/latex]

The constant [latex]K_z[/latex] of Eq. (12.7-10) is calculated so that the zero frequency amplification constants of [latex]G_c(z)[/latex] and [latex]G_c(s)G_h(s)[/latex] are the same, i.e., so that the following holds (see relation (12.3-23)):

[latex]G_c(z=1)=K_z\left[\frac{1-0.67}{1-0.264}\right]=G_c(s=0)G_h(s=0)=101\left[\frac{0+2}{0+6.7}\right]\left[\frac{2}{0+10}\right][/latex]

Thus [latex]K_z = 13.6[/latex]. It is noted that in the relation above the value of [latex]G_h(s)[/latex] for s = 0 was taken into consideration to obtain the total zero frequency amplification for the continuous-time controller. Hence

[latex]G_c(z)=13.6\left[\frac{z-0.67}{z-0.264}\right][/latex] (12.7-11)

In what follows, the responses of the closed-loop systems of Figures 12.20 and 12.21 are compared. To this end, the discrete-time transfer function of [latex]\hat{G}(s)= G_h(s)G_p(s)[/latex] is determined for T = 0.2. We have

[latex]\hat{G}(z)=Z\left\{G_h(s)G_p(s)\right\}=Z\left\{\left[\frac{1-e^{-0.2s}}{s}\right]\left[\frac{1}{s(s+2)}\right]\right\} [/latex]

[latex]=(1-z^{-1})Z\left\{\frac{1}{s^{2}(s+2)}\right\}=(1-z^{-1})Z\left\{\frac{0.5}{s^{2}}-\frac{0.25}{s}+\frac{0.25}{s+2}\right\}[/latex]

[latex]=\frac{z-1}{z}\left[\frac{0.1}{(z-1)²}-\frac{0.25z}{z-1}+\frac{0.25z}{z-e^{-0.4}}\right][/latex]

[latex]=\frac{0.0176(z+0.876)}{(z-1)(z-0.67)}[/latex] (12.7-12)

The transfer function of the closed-loop system of Figure 12.21 would be

[latex]H(z)=\frac{G_c(z)\hat{G}(z)}{1+G_c(z)\hat{G}(z)}=\frac{0.239z^{-1}(1+0.876z^{-1})}{(1-0.264z^{-1})(1-z^{-1})+0.239z^{-1}(1+0.876z^{-1})}[/latex]

[latex]=\frac{0.239z^{-1}+0.209z^{-2}}{1-1.025z^{-1}+0.473z^{-2}}[/latex] (12.7-13)

Figures 12.22 and 12.23 show the response of the closed-loop systems of Figures 12.20 and 12.21, respectively, where it can be seen that the two step responses are almost the same.

Question 12.7.1: Consider the position control servomechanism described in Su...

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