108 年 國立臺灣大學工程科學及海洋工程學系碩士班丙組《控制系統（B）》

1. A PI-controller is used to improve the steady state error of a unity feedback control system as shown in Fig.1, where $D(s) = K_p + \frac{K_i}{s}$ is the controller and $G(s) = \frac{48500}{s(s+2.89)}$ is the plant.

(a) Determine the value of $K_i$ to achieve $e_{ss}=2\%$ for a parabolic input. (7%)
(b) Plot the root locus of the system as a function of $K_p$ by using the value of $K_i$ in (a). (10%)
(c) Please find the value of $K_p$ to achieve a real pole at -1, and solve the other two poles. (8%)

🖼️【此處有附圖，請對照原卷】
R(s)
+
E(s)
C(s)
D(s)
G(s)
Fig.1

2. A DC motor drive system is shown in Fig. 2, where $J$ is moment of inertia; $N$ is gear number; $D$ is damping ratio; $R_a$ is armature resistance; $e_a$ is input voltage; $\theta_l$ is output rotational angle. The torque $T_m$ versus rotational speed $\omega_m$ relation of the DC motor is also given.

(a) Derive the differential equations of mathematical model for the overall system. (10%)
(b) Find the transfer function $G(s) = \theta_l(s) / e_a(s)$. (15%)

🖼️【此處有附圖，請對照原卷】
R(s)
+
E(s)
C(s)
D(s)
G(s)
Fig.1

🖼️【此處有附圖，請對照原卷】
$e_a(t)$
+
$T_m$
$J_1 = 2$ kg-m²
$D_1 = 2$ N-m-s/rad
$N_1 = 50$
$\omega_m$
$\theta_l(t)$
$N_2 = 150$
$J_2 = 18$ kg-m²
$D_2 = 36$ N-m-s/rad
$T_m$ (N-m)
100
50 V
150
$\omega_m$ (rad/s)
Fig. 2

3. A linear time-invariant causal control system can be described by the 2nd order differential equation as
   $m\ddot{y}(t) + d\dot{y}(t) + ky(t) = u(t) + \dot{u}(t)$

(a) Derive the state equation with the state variables $x_1 = y(t)$, $x_2 = \dot{y}(t)$.
$\dot{x}(t) = Ax(t) + Bu(t)$
$y(t) = Cx(t) + Du(t)$
(10%)
(b) Derive the initial state $x(0)$ in terms of $\dot{y}(0)$, $y(0)$ and $u(0)$. (15%)