The position control system of an object holder robot is shown in Figure 9.36. The block diagram is given in Figure 9.37. Use the stability limit method of Ziegler- Nichols to determine the parameters of the PID controller. Use the Routh stability criterion to calculate the values of Kp and T.

Question

The position control system of an object holder robot is shown in Figure 9.36. The block diagram is given in Figure 9.37. Use the stability limit method of Ziegler- Nichols to determine the parameters of the PID controller. Use the Routh stability criterion to calculate the values of [latex] ilde{K}_p[/latex] and [latex] ilde{T}[/latex].

Accepted Answer

Using only a proportional controller, as in Figure 9.35a, yields the following closedloop transfer function

[latex]\frac{Y(s)}{R(s)}=\frac{K_p}{s(s+1)(s+4)+K_p}[/latex]

The value of [latex]K_p[/latex] that makes the system marginally stable, in which case sustained oscillations occur, can be obtained using Routh’s stability criterion. The characteristic equation of the closed-loop system is given by

[latex]s³+5s²+4s+K_p=0[/latex]

The Routh array is as follows:

[latex]\begin{matrix}s^{3}\s^{2}\s^{1}\s^{0} \end{matrix} \left|\begin{matrix}\ \ 1&4\ 5&K_p\ \frac{20-K_p}{5}\\ \ K_p\end{matrix} ight.[/latex]

Examining the coefficients of the first column of the Routh table, we find that sustained oscillation will occur when [latex]K_p=20[/latex]. Thus, the critical gain [latex] ilde{K}_p[/latex] is 20. With
the gain [latex]K_p[/latex] set equal to [latex] ilde{K}_p[/latex], the characteristic equation becomes

s²+5s²+4s+20=0

Substituting s = jω into the characteristic equation, we can find the frequency of the sustained oscillations. We have

(jω)³+5(jω)²+4jω+20=0

or

5(4-ω²)+jω(4-ω²)=0 and hence ω²=4 or ω=2

The period [latex] ilde{T}[/latex] of the sustained oscillations is

[latex] ilde{T}=\frac{2π}{ω}=\frac{2π}{2}=π=3.14[/latex]

Referring to Table 9.2, we determine [latex]K_p[/latex], [latex]T_i[/latex], and [latex]T_d[/latex] as follows:

[latex]K_p=0.6 ilde{K}_p=(0.6)(20)=12[/latex]

[latex]T_i=0.5 ilde{T}=(0.5)(3.14)=1.57[/latex]

[latex]T_d=0.125 ilde{T}=(0.125)(3.14)=0.3925[/latex]

Hence, the transfer function of the PID controller is the following:

[latex]G_c(s)=K_p\left[1+\frac{1}{T_is}+T_ds ight]=12\left[1+\frac{1}{1.57s}+0.3925s ight][/latex]

[latex]=\frac{4.71(s+1.27389)²}{s}[/latex]

The PID controller has a pole at the origin and double zero at s = -1.27389. The
transfer function H(s) of the closed-loop system is given by

[latex]H(s)=\frac{G_c(s)G_r(s)}{1+G_c(s)G_r(s)}=\frac{\left[\frac{4.71(s+1.27389)²}{s} ight]\left[\frac{1}{s(s+1)(s+4)} ight]}{1+\left[\frac{4.71(s+1.27389)²}{s} ight]\left[\frac{1}{s(s+1)(s+4} ight]}[/latex]

[latex]=\frac{4.71s²+12s+7.643}{s^{4}+5s³+8.71s²+12s+7.643}[/latex]

The unit step response Y(s) of the closed-loop system is given by

[latex]Y(s)=H(s)R(s)=H(s)\left[\frac{1}{s} ight]=\left[\frac{4.71s²+12s+7.643}{s^{4}+5s³+8.71s²+12s+7.643} ight]\left[\frac{1}{s} ight][/latex]

The unit step response of the closed-loop system is shown in Figure 9.38. If the maximum overshoot is excessive, it can be reduced by fine tuning the controller parameters.

Question 9.6.6: The position control system of an object holder robot is sho...

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