115 年 國立臺灣聯合大學系統（清華、政治、陽明交通、中央四校聯招）研究所電機類《控制系統》

Consider the following RLC circuit (Figure 1):
🖼️【此處有附圖，請對照原卷】
(a) Please calculate the transfer function $\frac{V_o(s)}{V_i(s)}$.
(b) Define the state variables $v_C$ and $i_L$, input and output variables to be $v_i$ and $v_o$, respectively, please find the state space representation (in terms of $x_1 = v_C$, $x_2 = i_L$, $y = v_o$, and $u = v_i$).
(c) Let $R=L=C=1$. Is the system controllable? Observable?
(d) Let $R=L=C=1$, one utilizes the linear state feedback control $u = -k(x_1+x_2)$ to stabilize the system, please find the stability condition of the closed-loop system.
(e) Please calculate the natural frequency and damping ratio of the corresponding closed-loop system for $k=2$.

Consider a unit feedback system with open-loop transfer function as $G(s) = \frac{k}{(s+2)(s+3)(s^2+2s+5)}$.
Please sketch the corresponding root locus plot for $K>0$. Note: (2% for each term) The following terms should be identified for the root loci if there exists- poles, zeros, number of separate root loci, intersection of asymptotes, asymptotes, root loci on the real axis, angle of departure, intersection on the imaginary axis, breakaway points, and the corresponding values of $K$.
🖼️【此處有附圖，請對照原卷】

Consider the feedback system in Figure 2, where $K > 0$ is a constant gain. For $K = 1$, the closed-loop system is stable. The Nyquist plot (for $K = 1$) and its enlarged views are shown in Figure 3, where the Nyquist path is in the clockwise direction. Note that in the bottom-left corner of Figure 3, the small circle centered at $(-1,0)$ with radius 0.1288 is tangent to the Nyquist plot at $-0.8939 \pm 0.073j$.
🖼️【此處有附圖，請對照原卷】
(a) What are the gain margin and the phase margin of this system?
(b) Let $r$ be the unit impulse. Then, what is the value of $y(0)$?
(c) Let $r = \cos(\omega t)$. The steady-state error $e_{ss}$ is also a sinusoidal function with the same frequency $\omega$, i.e. $e_{ss} = A \cos(\omega t + \phi)$, where $A$ and $\phi$ are the magnitude and the phase of $e_{ss}$, respectively. Note that $A$ and $\phi$ depend on $\omega$. Find $\omega$ such that $A$ is maximum, and find this maximal $A$.
(d) Find the ranges of $K$ such that the closed-loop system is stable.
(e) Find the values of $K$ such that the closed-loop system has pure imaginary poles. What are these pure imaginary poles?

Consider the feedback system in Figure 4, where $G(s) = \frac{1}{s(s+1)}$ and $C(s) = K_p + \frac{K_i}{s}$, $K_p, K_i \ge 0$.
🖼️【此處有附圖，請對照原卷】
(a) Show that if $K_p = K_i > 0$, the closed-loop system always has a pair of pure imaginary poles.
(b) Let $K_i = 0$ and $K_p > 0$. Find the gain crossover frequency and the phase margin of the system in terms of $K_p$. What is the phase margin when $K_p = 1$?
(c) Find $K_p$ and $K_i$ such that the phase margin is at least $60^\circ$.