Control Lab 1 Notes

Control Lab 1 Submission 1.1

$$V_i(t)=CR\dot{V_o}(t)+V_o(t)$$

1.2

$$G(s)=\frac{Y(s)}{U(s)}=\frac{V_o(s)}{V_i(s)}=e^{-{\tau }_{d}s}\frac{K}{s+\alpha }$$ V_{i}(s) = CRsV_{o}(s) + V_{o}(s) $$ $$ V_{i}(s) = V_{o}(s)*(CRs+1) $$\frac{V_o(s)}{V_i(s)}=\frac{1}{CRs+1}=\frac{\frac{1}{CR}}{s+\frac{1}{CR}}$$ $$\frac{V_o(s)}{V_i(s)}=\frac{\alpha }{s+\alpha }$$ $$\underset{s\to 0}{\lim }[sY(s)]=\frac{\alpha }{0+\alpha }=1$$

let $\alpha =\frac{1}{CR}$ ,

$$\frac{V_o(t)}{V_i(t)}=e^{-\frac{1}{RC}t}=e^{-\frac{t}{\tau }}$$ $$e^{-{\tau }_{d}s}\frac{K}{s+\alpha }=\frac{\frac{1}{CR}}{s+\frac{1}{CR}}$$ $$e^{-{\tau }_{d}s}K=\alpha$$ $$\alpha =\frac{1}{RC}$$ $$K=\frac{1}{RC}\ast \frac{1}{e^{-{\tau }_{d}s}}=\frac{1}{RC{e}^{-{\tau }_{d}s}}$$

Q3

$$y(t)=G_{o}V[1-e^{-at}]$$ $$\underset{t\to \infty }{\lim }y(t)=V[1-0]$$ V[1-\frac{1}{e}] = 0.63212055883V $$q4 SSgain = 1 0.632 reached at 0.5066t tD = 0.066t For 2 step soze 0.5066 SS gain = 1 for 0.5 0.5066 Q6

\omega_{n} = \frac{1}{\sqrt{LC}}

$$Q7$$

\frac{R}{L} = 2* \frac{1}{\sqrt{LC}} * \zeta

\zeta = \frac{R\sqrt{LC}}{2L} = \frac{R}{2} * \sqrt{\frac{C}{L}}

Q8 + Q9 lower -> higher oscillations Q10 overdamped b = 1.005 - 0.952 = 0.053 a = 1.246-1.005 = 0.241 $20\log_{10}(\frac{0.241}{0.053}) = 13.1546$ $\frac{\sigma}{\omega} = \frac{13.1546}{-27.287} = -0.48208$ $\omega = 698.132$

\sigma = -0.48208 * 698.132 = -336.555

$$FOR5T/2=0.0045,T=0.009$$

\omega = 2\pi * 100

FOR 1 a = 1;719 b = 0.509 20log10 .... 3.3129 $\sigma / \omega = -0.12141122$ T = 0.009 $\omega = 2\pi / 0.009 = 698.132$ $\sigma = -0.12141122 * 698.132 = -84.76102$ FOR 3 a = 1.395 b= 0.863 0.009 20log ... 9.19753 sigm/omeg = 0.337066 $\omega = 698.132$ \sigma = -235.316 ## EMAIL: 2692894d@student.