Consider the closed-loop system shown in Figure 12.25. Construct the root locus of the system for K 0 and for several values of the sampling time T. Note that for a = 2, the system under control is the position control system presented in Example 12.7.1.

Question

Consider the closed-loop system shown in Figure 12.25. Construct the root locus of the system for K > 0 and for several values of the sampling time T. Note that for a = 2, the system under control is the position control system presented in Example 12.7.1.

Accepted Answer

Let [latex]\hat{G}(s)=G_h(s)G_p(s)[/latex]. Then

[latex]\hat{G}(s)=\left[\frac{1-e^{-sT}}{s} ight]\left[\frac{1}{s(s+a)} ight]=\frac{1}{s²(s+a)}[1-e^{-sT}][/latex]

[latex]=\frac{1}{a²}\left[\frac{a}{s²}-\frac{1}{s}+\frac{1}{s+a} ight](1-e^{-sT})[/latex]

Hence

[latex]\hat{g}(t)=L^{-1}[/latex]{[latex]\hat{G}(s)[/latex]}= p(t) - p(t-T)β(t-T)

where

[latex]p(t)=L^{-1}\left\{\frac{1}{a²}\left[\frac{a}{s²}-\frac{1}{s}-\frac{1}{s+a} ight] ight\}=\frac{1}{a²} [at-1+e^{-at}][/latex]

Therefore

[latex]\hat{g}(kT)=p(kT)-p(kT-T)β(kT-T)[/latex]

Applying relation (B.3-12b) of Appendix B, we have

[latex]\begin{aligned} Z[f(kT-σT)β(kT-σT)]&=\sum\limits_{k=0}^{∞}{f[(k-σ)T]β[(k-σ)T]z^{-k}} \ &=z^{-σ}\left[\sum\limits_{k=0}^{∞}{f[(k-σ)T]β[(k-σ)T]z^{-(k-σ)}} ight] \ &=z^{-σ}\sum\limits_{m=-σ}^{∞}{f(mT)β(mT)z^{-m}} \ &=z^{-σ}\sum\limits_{m=0}^{∞}{f(mT)z^{-m}} \ &=z^{-σ}F(z) \end{aligned}[/latex] (B.3-12b)

[latex]\hat{G}(z)=Z[/latex]{[latex]\hat{g}(kT)[/latex]}[latex]=P(z)-z^{-1}P(z)=(1-z^{-1})P(z)[/latex]

where

[latex]P(z)=Z[/latex]{p(kT)}[latex]=\frac{1}{a²}Z[/latex]{[latex]akT-β(kT)+e^{-akT}[/latex]}

[latex]=\frac{1}{a²}\left[\frac{aTz}{(z-1)²}-\frac{z}{z-1}+\frac{z}{z-e^{-aT}} ight][/latex]

Therefore

[latex]\hat{G}(z)=Z[/latex]{[latex]\hat{G}(s)[/latex]} = Z{[latex]G_h(s)G_p(s)[/latex]}=Z{[latex]\hat g(kT)[/latex]}[latex]=(1-z^{-1})P(z)[/latex]

[latex]=\frac{1}{a²}(1-z^{-1})\left[\frac{aTz}{(z-1)²}-\frac{z}{z-1}+\frac{z}{z-e^{-aT}} ight][/latex]

[latex]=\frac{(aT+e^{-aT}-1)}{a²}\left[\frac{z+\frac{[1-aTe^{-aT}-e^{-aT}]}{[-1+aT+e^{-aT}]}}{(z-1)(z-e^{-aT})} ight][/latex]

or

[latex]\hat{G}(z)=K_0\left[\frac{z+z_1)}{(z-1)(z-p_1)} ight][/latex]

where

[latex]K_0=\frac{1}{a²}(aT+e^{-aT}-1), p_1=e^{-aT}, and z_1=\frac{1-Tae^{-aT}-e^{-aT}}{-1+aT+e^{-aT}}[/latex]

Hence, the open-loop transfer function G(z)F(z) for the present example is

[latex]G(z)F(z)=K\hat{G}(z)=K\left[\frac{K_0(z+z_1)}{(z-1)(z-p_1)} ight][/latex]

Clearly, the constant [latex]K_0[/latex], the pole [latex]p_1[/latex], and the root [latex]-z_1[/latex] are changing with the sampling period T. For this reason, as T changes, so will the root locus of G(z)F(z). Figure 12.26 presents the root loci for three different values [latex]T_1, T_2,[/latex] and [latex]T_3[/latex] of T, where [latex]T_1 < T_2 < T_3[/latex].

Next, the special case where a = 1 and T = 1, 2, and 4 sec will be studied. For a = 1 and T = 1, the open-loop transfer function G(z)F)z) becomes

[latex]G(z)F(z)=K\left[\frac{0.368(z+0.718)}{(z-1)(z-0.368)} ight][/latex]

For a = 1 and T + 2, the open-loop transfer function G(z)F(z) becomes

[latex]G(z)F(z)=K\left[\frac{1.135(z+0.523)}{(z-1)(z-0.135)} ight][/latex]

Finally, for a = 1 and T = 4, the open-loop transfer function G(z)F(z) becomes

[latex]G(z)F(z)=K\left[\frac{3.018(z+0.3)}{(z-1)(z-0.018)} ight][/latex]

Figure 12.27 presents the root loci for a = 1 and for the three cases of T = 1, 2, and 4 sec. The influence of the parameter T can be observed here in greater detail than in Figure 12.26. Figure 12.27 shows that, for a fixed value of K, an increase in the sampling time T would result in a less stable closed-loop system. On the contrary, a decrease in T results in a more stable system. As a matter of fact, the more the sampling period T goes to zero [latex](T → 0)[/latex], the more the behavior of the closed-loop system approaches that of the continuous-time system (here, the continuous-time closed-loop system is stable for all positive values of K). It is also noted that as the value of T increases, the critical value of [latex]K_c[/latex] decreases and vice versa, where by critical value of K we mean that particular value of K where the system becomes unstable.

Question 12.8.1: Consider the closed-loop system shown in Figure 12.25. Const...

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