111 年 國立清華大學動力機械工程學系碩士班乙組《控制系統》

For the system in Figure 1,

🖼️【此處有附圖，請對照原卷】

(a) find the transfer function $\frac{Y(s)}{R(s)}$. (5 pts)

(b) Sketch the root locus with respect to $K$, $K>0$. (5 pts)

Ziegler-Nichols Tuning for the Regulator
$D_c(s) = K_p(1 + \frac{1}{T_I s} + T_{D}s)$, Based on the Ultimate Sensitivity Method

Table 1 Ziegler-Nichols Tuning method

(c) Let's replace $K$ with a PI controller $D_c(s)$. Based on the Ziegler-Nichols Tuning method shown as Table 1, design $D_c(s)$. Give the values of your $(K_p, T_I)$. (10 pts)

(d) With above PI controller $D_c(s)$, determine the relevant error constant of closed-loop system with respect to reference input $R(s)$. (5 pts)

For the system shown in Figure 2,

🖼️【此處有附圖，請對照原卷】

The plant $G_p(s)$ in Figure 2 results in $y(t) = 0$ for $0 \leq t < 1$ and $y(t) = u(t-1)$ for $t \geq 1$.

(a) What is the phase margin PM for the system shown in Figure 2? (5 pts)

(b) Draw the Nyquist plot. Specify the crossover points on the real axis to get points. (10 pts)

(c) Use Nyquist criterion to decide the closed-loop system stability. Give the values of $N, P, Z$ to get points. (5 pts)
(Notes: (c) will be counted only when (b) is correct!)

For the system shown in Figure 3,

🖼️【此處有附圖，請對照原卷】

the plant $G_p(s)$ is the same described above and $k > 0$. What is the range of $k$ to result in a stable closed-loop system? (5 pts)

For the model depicted in Figure 4,

🖼️【此處有附圖，請對照原卷】

(a) derive the (two) differential equations describing the dynamics of the system (8 pts), and

(b) derive the transfer function $G(s) = \frac{X_2(s)}{F_1(s)}$. Assuming $0 < \delta k << k$ and $0 < c << 2\sqrt{km}$, roughly sketch the frequency response of $G(s)$. (4 pts)

Consider the plant shown in Figure 5.

🖼️【此處有附圖，請對照原卷】

(a) Using $u(t)$ and $b(t)$ as the inputs, $x(t)$ as the state, and $y(t)$ as the output, establish a state-space model of the system. (3 pts)

(b) In Figure 5, $u(t)$ is the control input and $b(t)$ can be treated as the sensor bias. Assume that $b(t)$ is constant but unknown (i.e. $\dot{b} = 0$). Using $\begin{bmatrix} x \\ b \end{bmatrix}$ as the new state vector, derive the state equation and output equation of the augmented system. (5 pts)

(c) Examine the observability of the augmented system in (b). For such an augmented system, construct an observer that generates estimates for $x$ and $b$ using $u$ and $y$. The observer's characteristic polynomial should be chosen as $E(s) = s^2 + 8s + 16$. Write down the state equation of the observer as the answer. (7 pts)

(d) Use the results in (c) to implement state-estimate feedback control in the form $u(t) = -k\hat{x}(t)$, where $\hat{x}(t)$ is the observer's estimate of $x$. What is the controller's transfer function (i.e., $\frac{U(s)}{Y(s)}$)? (7 pts) What is the range of $k$ for the closed-loop system to be stable and what are the closed-loop poles? (8 pts) (Hint: Separation principle.)