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with the matlab code

Plot the root locus of: \[ G(s)=\frac{1}{4(x+4)\left(x^{2}+4 t+8\right)} \] Assume that \( \sigma_{,}(a)=N \) in the unity facoback system shown in Figure 1 . Figure 1: Lnity feedosck system. After reading the PID and Ziegler-Nichols tuning rules background cetalled below, you are required (wSn the ate or MATLAB) to design a Proportionate + Derivative cortroller so that the below specifications are satic'ied on the hacit of the following decign proverturn: (a) What is the value of \( K \) such that persatent ascilation occurs in the stop response? Plot the step respanse and determine the freeuency \( w \) of the sersistent oscitation. (b) Choose \( K \) so that the gain margin and phase margin of the feedback syetem are \( G M=6 d B \) and \( P M=3 e^{\prime} \), nessectively. Plot the Bode diagram of the new shaced olant KG(u). (c) Choose the proportionate gain \( k \), according to the second method of Ziegier and Nichols tining niles, and bulld the Proportionate cortrolier. Find the compensated ciosed-Aooo sosten CLTF \( =H_{F}=\frac{K_{v} G}{1+k_{p} C^{2}} \) Piot the assoc ated step resporse, and then calculate the rise time \( t_{\gamma, \text {, bottling time }} \ell_{n} \) and peak cvershoot percentage OS\%. (d) Choose the proportionate gain 1 , and twe derivative time constaat \( T_{i} \) according to the soccnd method of Ziegler and Nichols turing rules, and bulld tre Proportionate \( + \) Derivatve controller \( \left(k_{i},(a)\right) \). It should be noted that the dervative gain k\( k_{i}=k_{p} T_{d} \). Plot the Bode diagram of the new shaped plant \( G_{0}(x) G(s) \), and compare it with the stability arargins obtained in part (b) Also find the newly compensated closed licop system CLTF = \( H_{v}= \) \( \frac{G E_{r_{H}}}{1+G G_{i+1}} \), plot the assoclated step response, ard compare it with the response obtained in part ic). (e) Starting trom the values calcuated nor is, and 4 in aart (d), nee tune the PD controller gains sach that: - the settlinj time is \( t, 54 \) seconds, and rise time is \( t, 51 \) second; - the peak overshoot peroertage is \& 3t: - the closec-loop gain margin ard phase margin are at seast \( 12 d B \) ane 65 : , reepoctively. Finally, realize the tuned PD controller by using coerational ampit ers:

From the given block diagram, The open loop transfer function is, Poles, Zeros, Define the intersect of the asymptotes, The intersect of the asymptote

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