Question

# Consider the system described in "Stability Practice Problems" in the class Google Drive. Rather than using P control, consider instead Pl control. 5) Show that the overall transfer function for

the system is \frac{c}{R}=\frac{K_{C}\left(\tau_{\text {mtut }} s+1\right)\left(\tau_{M} s+1\right)}{\tau_{\text {Int }} s\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)\left(\tau_{M} s+1\right)+K_{C}\left(\tau_{I} s+1\right)} b. (15 points) Let t = 1, t2 = 2, Tm = 0.1, Kc = 5, and Tint = 3. Determine the roots of the characteristic equation. Is the system stable? c. (10 points) Confirm the results of Part b by plotting the output if the input were a unitstep change. Plot the output from 0 to 25 time units. Use the Routh test to confirm the results of Part b. hts) If K, = 5, what are the stability criteria for taumt? Use the Routh test. f.(15 points) See Chapter 18 of the textbook. Implement the Ziegler Nichols tuning method to determine the best Kc and tint. In summary, setTmt to zero. Starting with a low Kc, increase Kc until steady oscillations are obtained. (HINT: You can use the Routhtest to guide this process). Once you determine which K. results in steady oscillations,that is known as Ky. From this, you can determine the optimal values for Kc and tint: g. (10 points) Determine the output of the system using the results from Part e if the input were a unit step increase. Generate a plot from 0 to 25 time units. h. (10 points) In a few sentences, comment on the effect of TM with respect to stability.Does a large ty make the system more or less likely to be stable?

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