110 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 two-mass system in Figure 1, where $k_1$ and $k_2$ are spring constants; $x_1$ and $x_2$ are displacements of $m_1$ and $m_2$, assuming that both springs have their natural lengths when $x_1 = x_2 = 0$. The surface is frictionless and $f$ is an external force applied to $m_1$.

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Figure 1: Two-mass system

(a) (6%) Choose $x_1$ as the output of the system. Write down the state space representation using $x = [x_1, x_2, \dot{x}_1, \dot{x}_2]^T$ as the state vector, where $\dot{x}_i$ denotes the time derivative of $x_i$ for $i = 1,2$.

(b) (8%) Find the transfer function from $f$ to $x_1$.

(c) (6%) Suppose that $\frac{k_2}{m_1} = r$ and $\frac{k_2}{m_2} = r$. Find the resonant frequencies of the system in terms of $r$.

(d) (8%) Let $f(t) = \sin(\omega_0 t)$ and $x_1(0) = 0$. Find $\omega_0, \ddot{x}_2(0)$, and $\ddot{x}_2(0)$ such that $x_1(t) = 0$ for all $t \ge 0$.

Consider the feedback control system in Figure 2, where
$G(s) = \frac{10}{(s + 1)(s + 10)}$, $C(s) = \frac{k(as + 1)}{s}$, $k, a > 0$.

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Figure 2: Feedback Control System

(a) (6%) Find the conditions on $k$ and $a$ such that the closed-loop system is stable.

(b) (8%) Find $k$ and $a$ to satisfy the following requirements simultaneously:
i. The closed-loop transfer function from $r$ to $y$ is a stable, 2nd order rational function.
ii. The steady-state error with respect to the unit-ramp input $r(t) = t$, $t \ge 0$, is less than 0.2.
iii. The percent maximum overshoot with respect to the unit-step input $r(t) = 1$, $t \ge 0$, is less than 10%.

(c) (8%) Let $a = \frac{1}{20}$. Draw the root locus when $k$ increases from 0 to $\infty$. If $k = 1$, what is the gain margin of the system?

Consider the following feedback system with plant $G_p(s)$ and controller $G_c(s)$.

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Let $G_p(s) = \frac{s^2}{s+2}$, where $G_p(j\omega) = \frac{-2j\omega^3}{4+\omega^2}$. The controller $G_c(s) = K$.

(a) (6%) Plot the root locus for $K>0$ and $K<0$, respectively.

(b) (5%) Sketch the Nyquist plot for $K>0$.

(c) (4%) Analyze the stability from Nyquist plot for all $K$.

(d) (6%) To stabilize the system, a control engineer chooses $G_c(s) = \frac{K}{s(s+p)}$, where $p>0$. Show that the root locus contains a circle for $K>0$.

(e) (4%) As in part (d), design the closed-loop poles at $-1 \pm j$, design the controller $G_c(s)$.

(f) (3%) From the root locus in part (d), what are the Gain margin?

(g) (4%) In part (e) and input $r(t)=1+\cos(\sqrt{2}t-30^\circ)$, what is the steady state output?