Question

1.b) for system given in figure below, find;
i. calculate the maximum value for k for the system that will result in a stable system, (6 pt)
ii. for half of the maximum value of k found in part (i), find the roots of the characteristic equation, (2 pt)
iii. estimate the overshoot of the system of part (ii) when it is subjected to a step input. (4 pt)

Explanation:

Step1: Find the open - loop transfer function

The open - loop transfer function $G(s)H(s)$ is the product of the transfer functions of the controller, the wire - melting process, and the vision system.

$$G(s)H(s)=\frac{K}{(s + 2)}\cdot\frac{1}{(0.5s + 1)(s + 1)}\cdot\frac{1}{(0.005s+1)}$$

$$G(s)H(s)=\frac{K}{(s + 2)(0.5s + 1)(s + 1)(0.005s + 1)}$$

Expand the denominator:
\((s + 2)(0.5s + 1)(s + 1)(0.005s + 1)\)
\(=(0.5s^{2}+s + s+2)(s + 1)(0.005s + 1)\)
\(=(0.5s^{2}+2s + 2)(s + 1)(0.005s + 1)\)
\(=(0.5s^{3}+0.5s^{2}+2s^{2}+2s+2s + 2)(0.005s + 1)\)
\(=(0.5s^{3}+2.5s^{2}+4s + 2)(0.005s + 1)\)
\(=0.0025s^{4}+0.5s^{3}+0.0125s^{3}+2.5s^{2}+0.02s^{2}+4s+0.01s+2\)
\(=0.0025s^{4}+0.5125s^{3}+2.52s^{2}+4.01s + 2\)
So, \(G(s)H(s)=\frac{K}{0.0025s^{4}+0.5125s^{3}+2.52s^{2}+4.01s + 2}\)
The closed - loop transfer function \(T(s)=\frac{G(s)}{1 + G(s)H(s)}=\frac{K}{0.0025s^{4}+0.5125s^{3}+2.52s^{2}+4.01s+(2 + K)}\)
The characteristic equation is \(0.0025s^{4}+0.5125s^{3}+2.52s^{2}+4.01s+(2 + K)=0\)
For a fourth - order system, we use the Routh - Hurwitz criterion.
The Routh array is:

$$
LATEXBLOCK0
$$

First, simplify \(\frac{0.5125\times2.52-0.0025\times4.01}{0.5125}=\frac{1.2915 - 0.010025}{0.5125}=\frac{1.281475}{0.5125}=2.5\)
For stability, all elements in the first column of the Routh array must be positive.
From the \(s^{1}\) row, we set the first - element of the \(s^{1}\) row greater than 0.
\(\frac{2.5\times4.01-0.5125\times(2 + K)}{2.5}>0\)
\(2.5\times4.01-0.5125\times(2 + K)>0\)
\(10.025-1.025 - 0.5125K>0\)
\(9 - 0.5125K>0\)
\(0.5125K<9\)
\(K < \frac{9}{0.5125}\approx17.56\)

Step2: Find the roots of the characteristic equation for \(K=\frac{K_{max}}{2}\)

When \(K=\frac{17.56}{2}=8.78\), the characteristic equation is \(0.0025s^{4}+0.5125s^{3}+2.52s^{2}+4.01s+(2 + 8.78)=0\)
\(0.0025s^{4}+0.5125s^{3}+2.52s^{2}+4.01s + 10.78=0\)
We can use numerical methods (e.g., MATLAB's roots function) to find the roots. Let \(a = [0.0025,0.5125,2.52,4.01,10.78]\), and the roots are the solutions of \(a(1)s^{4}+a(2)s^{3}+a(3)s^{2}+a(4)s + a(5)=0\)

Step3: Estimate the overshoot for step - input

For a second - order approximation (if possible), we first need to rewrite the system in a form close to a second - order system. However, for a higher - order system, we can also use the relationship between the damping ratio \(\zeta\) and overshoot \(M_p\) which is \(M_p=e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^{2}}}}\times100\%\)
We can use the dominant poles (the poles closest to the imaginary axis) to estimate the damping ratio. After finding the roots of the characteristic equation in step 2, we identify the dominant poles and calculate the damping ratio \(\zeta\) corresponding to the dominant - pole pair. Then we use the above formula to calculate the overshoot.

Answer:

i. \(K_{max}\approx17.56\)
ii. Roots can be found using numerical methods (e.g., in MATLAB with the roots function) for the equation \(0.0025s^{4}+0.5125s^{3}+2.52s^{2}+4.01s + 10.78 = 0\)
iii. First, find the dominant poles from the roots in part (ii), calculate the damping ratio \(\zeta\) corresponding to the dominant - pole pair, and then use \(M_p=e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^{2}}}}\times100\%\) to estimate the overshoot.