113 年 國立清華大學動力機械工程學系碩士班乙組《控制系統》

Q1. The system block diagram is shown on the right (Fig. 1). The Bode plot of the loop gain $G_c(s)G_p(s)$ is given below (Fig. 2) for $G_c(s) = kG_{cc}(s)$ at $k=1$.

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Fig. 1

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Fig. 2.

(a) Write the transfer function of the loop gain $G_c(s)G_p(s)$. (5 pts)
(b) (b1)Draw the Nyquist plot. (10 pts) (b2)"Calculate" the real axis crossing if there is. (5 pts)
(c) Use Nyquist Criterion to figure out the stability of the closed-loop system. (need to give the values of Z, N, P to get points). (5 pts)
(d) If tuning $k$ as a variable from 0 up to $\infty$. What is the range for $k$ to result in a stable closed-loop system, if there is? (5 pts)

Q2. A system has the open-loop plant transfer function $G(s) = \frac{10(s-0.1)}{(s+1)(s-1)}$.

(a) Sketch a root locus for the system (a simple root locus sketch is sufficient here). (5 pts)
(b) Based on looking at the root locus, can we use high gain proportional control to obtain a stable closed-loop system? (5 pts)
(c) Suggest a dynamic compensator, $D(s)$, that will result in a stable closed-loop system. Sketch the compensated root locus of $D(s)G(s)$ (again, a simple root locus sketch is sufficient). (5 pts)
Comment on any practical difficulty of implementing your solution. (5 pts)

Q3. Fig. 3 shows a hydraulic system where liquid is stored in an open tank. The cross-sectional area of the tank, $A(h) = A_0 + a_0h$ where $h$ is the height of the liquid level above the bottom of the tank and $A_0$ and $a_0$ are constants. For a liquid of density $\rho$, the absolute pressure $p$ at the bottom of the tank is given by $p = \rho gh + p_a$, where $p_a$ is the atmospheric pressure (assumed constant) and $g$ is the acceleration due to gravity. The tank receives liquid at a flow rate $w_i$ and loses liquid through a valve at a flow rate $w_o$ in which $w_o = k\sqrt{\Delta p}$. In the current case, $\Delta p = p - p_a$ and $k$ is a positive constant. Take $u = w_i$ to be the control input and $y = h$ to be the output.

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Fig. 3

(a) Compute the volume of the liquid as a function of $h$. (3%)
(b) Using $h$ as the state variable, determine the state model. (Hint: Rate of volume change $w_i - w_o$) (7%)
(c) Find $u_0$ (steady-state input $u$) that is needed to maintain the output at a constant $h_0$. (3%)
(d) Assume $u = u_0 + \Delta u$ and $h = h_0 + \delta h$ in which $\delta$ represents a small perturbation. Linearize the state model in (b) around $u = u_0$ and $h = h_0$. What is the open-loop pole of the system? Is the system open-loop stable? (12%)

Q4. To devise the balancing controller for the autonomous bicycle, Fig. 4 models the lateral dynamics of the bicycle as an inverted-pendulum system. In the model, $\theta$ denotes the tilt angle of the bicycle, and $\delta$ is the steering angle of the handle. The task is to design a controller to make the tilt angle $e(t)$ follow a reference command $r(t)$.

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Fig. 4

(a) The differential equation describing the inverted-pendulum dynamics is given by $\ddot{\theta} = 2\dot{\theta} + \delta + \dot{\delta}$. Define $\dot{\theta}$ as the control input, $x = [\theta \ \dot{\theta}]^T$ as the state vector, and the measurement output $y = \theta$. Derive the state equations for the system. Examine its controllability and observability. (3 pts)
(b) Assume that all of the states $[\theta \ \dot{\theta}]^T$ can be measured. Design a control law $\delta = N \cdot r - K_1\theta - K_2\dot{\theta} - K_3\delta$, (1) where $N, K_1 \sim K_3$ are control gains, so that the closed-loop transfer $G_{re}(s)$ is given by $G_{re}(s) = \frac{1}{s^2+2s+1}$. (7 pts)
(c) Consider the dynamic equation $\ddot{\theta} = 2\dot{\theta} + \delta + \dot{\delta}$ using $u = \delta + \dot{\delta}$ as the control input, and $y = \theta + \dot{\theta}$ as the measurement in which $\theta$ is an unknown, constant offset. Construct an observer to estimate $\theta, \dot{\theta}, \delta$ using $u$ and $y$. The observer you design should have poles located at $-10, -10, -10$. (10 pts) (Hint: Consider the state equation for $[\theta \ \dot{\theta} \ \delta]^T$.)
(d) Shows that the control law $u = -K_1\theta - K_2\dot{\theta} - K_3\delta$, where $\hat{\theta}$ and $\hat{\dot{\theta}}$ are the estimates of $\theta$ and $\dot{\theta}$ by the observer designed in (c), can stabilize the system (i.e., $\theta(\infty) = \dot{\theta}(\infty) = 0$, for any $\theta(0)$ and $\dot{\theta}(0)$) if $K_1 > 2, K_2 > 0$. (5 pts)