Chapter 9, Application to Control Video Solutions, Fundamentals of Signals and Systems Using the Web and MATLAB | Numerade (2024)

Consider the following system transfer function:
$$
G_p(s)=\frac{1}{s+0.1}
$$
(a) An open-loop control is shown in Figure P9.1a. Design the control, $G_c(s)$, so that the combined plant and controller $G_c(s) G_p(s)$ has a pole at $p=-2$, and the output $y(t)$ tracks a constant reference signal $r(t)=r_0 u(t)$ with zero steady-state error, where $e_{\mathrm{ss}}=r_0-y_{\mathrm{ss}}$.
(b) Now, suppose that the plant pole at $p=-0.1$ was modeled incorrectly and that the actual pole is $p=-0.2$. Apply the control designed in part (a) and the input $r(t)=r_0 \mu(t)$ to the actual plant, and compute the resulting steady-state error.
(c) A feedback controller $G_c(s)=2(s+0.1) / s$ is used in place of open-loop control, as shown in Figure P9.1b. Verify that the closed-loop pole of the nominal system is at $p=-2$. (The nominal system has the plant pole at $p=-0.1$.) Let the input to the closedloop system be $r(t)=r_0 u(t)$. Verify that the steady-state error $e_{s s}=r_0-y_{\mathrm{ss}}$ is zero.
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(d) Compute the steady-state error of the actual closed-loop system (with plant pole at $p=-0.2)$ when $r(t)=r_0 u(t)$. Compare this error with that of the actual open-loop system computed in part (b).
(e) Simulate the responses of the systems in parts (a) to (d) when $r_0=1$. Explain the differences (and similarities) in responses.

Examine the effect of a disturbance on the performance of open- and closed-loop control systems by performing the following analysis:
(a) Consider an open-loop control system with a disturbance $D(s)$, as shown in Figure P9.2a. Define an error $E(s)=R(s)-Y(s)$, where $R(s)$ is a reference signal. Derive an expression for $E(s)$ in terms of $D(s), X(s)$, and $R(s)$. Suppose that $D(s)$ is known. Can its effect be removed from $E(s)$ by proper choice of $X(s)$ and/or $G_c(s)$ ? Now, suppose that $D(s)$ represents an unknown disturbance. Can its effect be removed (or reduced) from $E(s)$ by proper choice of $X(s)$ and/or $G_c(s)$ ? Justify your answers
(b) Now consider the feedback system shown in Figure P9.2b. Derive an expression for $E(s)$ in terms of $D(s)$ and $R(s)$. Suppose that $D(s)$ represents an unknown disturbance and that $G_c(s)=K$. Can the effect of $D(s)$ be removed (or reduced) from $E(s)$ by proper choice of $K$ ? Justify your answer.
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Consider the feedback control system shown in Figure P9.4. Assume that the initial conditions are zero.
(a) Derive an expression for $E(s)$ in terms of $D(s)$ and $R(s)$, where $E(s)$ is the Laplace transform of the error signal $e(t)=r(t)-y(t)$.
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(b) Suppose that $r(t)=u(t)$ and $d(t)=0$ for all $t$. Determine all (real) values of $K$ so that $e(t) \rightarrow 0$ as $t \rightarrow \infty$.
(c) Suppose that $r(t)=u(t)$ and $d(t)=u(t)$. Determine all (real) values of $K$ so that $e(t) \rightarrow 0$ as $t \rightarrow \infty$.
(d) Suppose that $r(t)=u(t)$ and $d(t)=(\sin t) u(t)$. Determine all (real) values of $K$ so that $e(t) \rightarrow 0$ as $t \rightarrow \infty$.
(e) Again suppose that $r(t)=u(t)$ and $d(t)=(\sin t) u(t)$. With the controller transfer function given by
$$
G_c(s)=\frac{7 s^3+K_1 s+K_2}{s\left(s^2+1\right)}
$$
determine all (real) values of $K_1$ and $K_2$ so that $e(t) \rightarrow 0$ as $t \rightarrow \infty$.

Consider a feedback connection as shown in Figure P9.1(b). The impulse response of the system with transfer function $G_p(s)$ is $h(t)=(\sin t) u(t)$.
(a) Determine the transfer function $G_c(s)$ so that the impulse response of the feedback connection is equal to $(\sin t) e^{-t} u(t)$.
(b) For $G_c(s)$ equal to your answer in part (a), compute the step response of the feedback connection.

Each of the following systems is to be controlled by feedback:
(i) $G_p(s)=\frac{s+5}{s+1}$
(ii) $G_p(s)=\frac{1}{s(s+4)}$
For each system, do the following:
(a) Use the angle condition to determine which part of the real axis is on the root locus when $G_c(s)=K$. For the system in (ii), verify by using the angle condition that $s=-2+j \omega$ is on the root locus for all real $\omega$.
(b) Calculate the closed-loop poles for specific values of $K>0$, then use this information to plot the root locus.
(c) Verify the answers in parts (a) and (b) by using MATLAB to plot the root locus.

Use MATLAB to plot the root locus for each of the following systems:
(a) $G_p(s)=\frac{1}{(s+1)(s+10)} ; \quad G_c(s)=K$
(b) $G_p(s)=\frac{1}{(s+1)(s+4)(s+10)} ; \quad G_c(s)=K$
(c) $G_p(s)=\frac{(s+4)^2+4}{\left[(s+2)^2+16\right](s+8)} ; \quad G_c(s)=K$
(d) $G_p(s)=\frac{s+4}{(s+6)^2+64} ; \quad G_c(s)=K$

For each of the systems given in Problem 9.7, determine the following:
(a) The range of $K$ that gives a stable response.
(b) The value of $K$ (if any) that gives a critically damped response.
(c) The value(s) of $K$ that gives the smallest time constant.

For each of the closed-loop systems defined in Problems 9.7, do the following:
(a) Compute the steady-state error $e_{s s}$ to a unit step input when $K=100$.
(b) Verify your answer in part (a) by simulating the responses of the closed-loop systems to a step input.

Use MATLAB to plot the root locus for each of the following systems:
(a) $G_p(s)=\frac{1}{(s+1)(s+10)} ; \quad G_c(s)=\frac{K(s+1.5)}{s}$
(b) $G(s)=\frac{1}{(s+1)(s+10)} ; \quad G_c(s)=K(s+15)$
(c) $G_p(s)=\frac{1}{s(s-2)} ; \quad G_c(s)=K(s+4)$
(d) $G_p(s)=\frac{1}{(s+2)^2+9} ; \quad G_c(s)=\frac{K(s+4)}{s+10}$
(e) $G_p(s)=\frac{1}{(s+1)(s+3)} ; \quad G_c(s)=\frac{K(s+6)}{s+10}$

Repeat Problem 9.9 for the closed-loop systems defined in Problem 9.10.

The transfer function of a dc motor is
$$
G_p(s)=\frac{\Theta(s)}{V_a(s)}=\frac{60}{s(s+50)}
$$
where $\theta(t)$ is the angle of the motor shaft and $v_a(t)$ is the input voltage to the armature. A closed-loop system is used to try to make the angle of the motor shaft track a desired motor angle $\theta_r(t)$. Unity feedback is used, as shown in Figure $\mathrm{P} 9.12$, where $G_c(s)=K_P$ is the gain of an amplifier. Let $K=K_P(60)$.
(a) Plot the root locus for the system.
(b) Calculate the closed-loop transfer function for the following values of $K: K=500$, 625,5000 , and 10,000 . For each value of $K$, identify the corresponding closed-loop poles on the root locus plotted in part (a).
(c) Plot the step response for each value of $K$ in part (b). For which value of $K$ does the closed-loop response have the smallest time constant? The smallest overshoot?
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A system has the transfer function
$$
G_p(s)=\frac{1}{(s+1)(s+7)}
$$
(a) Sketch the root locus for a closed-loop system with a proportional controller, $G_c(s)=K_P$.
(b) Compute the closed-loop poles for $K_P=5,9,73,409$, and mark these pole positions on the root locus. Describe what type of closed-loop behavior you will expect for each of these selections of $K_P$. Calculate the steady-state error to a unit step function for each of these values of $K_P$.
(c) Verify the results of part (b) by using MATLAB to compute and plot the closed-loop step response for each value of $K_P$.

A system has the transfer function
$$
G_p(s)=\frac{s+4}{s(s+2)(s+8)}
$$
(a) Sketch the root locus, using MATLAB for a proportional controller, $G_c(s)=K_P$.
(b) Find a value of $K_P$ that yields a closed-loop damping ratio of $\zeta=0.707$ for the dominant poles. Give the corresponding closed-loop pole.
(c) Use MATLAB to compute and plot the closed-loop step response for the value of $K_P$ found in part (b).

A system has the transfer function
$$
G_p(s)=\frac{1}{s^2}
$$
(a) Sketch the root locus for a closed-loop system with a proportional controller, $G_c(s)=K_P$. Describe what type of closed-loop response you will expect.
(b) Sketch the root locus for a PD controller of the form $G_c(s)=K_D s+K_P=$ $K_D(s+2)$. Describe what type of closed-loop response you will expect as $K_D$ is varied.
(c) Give the steady-state error of the closed-loop system with a PD controller for a step input when $K_D=10$.
(d) Verify your result in part (c) by simulating the system.

A system has the transfer function
$$
G_p(s)=\frac{s+4}{(s+1)(s+2)}
$$
(a) Sketch the root locus for a closed-loop system with a proportional controller, $G_c(s)=K_P$. Determine the value of $K_P$ that will give closed-loop poles with a time constant of $\tau=0.5$ seconds.
(b) Compute the steady-state error of the step response for the value of $K_P$ chosen in part (a).
(c) Design a PI controller so that the closed-loop system has a time constant of approximately $0.5 \mathrm{sec}$. For simplicity of design, select the zero of the controller to cancel the pole of the system. What is the expected steady-state error to a step input?
(d) Simulate the closed-loop system with two different controllers designed in parts (a) and (c) to verify the results of parts (b) and (c).

The system shown in Figure P9.18 is a temperature control system where the output temperature $T(t)$ should track a desired set-point temperature $r(t)$. The open-loop system has the transfer function
$$
G_p(s)=\frac{0.05}{s+0.05}
$$
(a) Sketch the root locus for a closed-loop system with a proportional controller, $G_c(s)=K_P$. Suppose that the desired temperature is $70^{\circ} \mathrm{F}$. Let $r(t)=70 u(t)$, and compute the gain required to yield a steady-state error of $2^{\circ}$. What is the resulting time constant of the closed-loop system?
(b) Design a PI controller so that the closed-loop system has the same time constant as that computed in part (a). For simplicity of design, select the zero of the controller to cancel the pole of the system.
(c) To verify the results, simulate the response of the closed-loop system to $r(t)=70 u(t)$ for the two different controllers designed in parts (a) and (b).
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A dc motor has the transfer function
$$
\frac{\Omega(s)}{V_i(s)}=G_p(s)=\frac{2}{(s+2)(s+10)}
$$
where $\Omega(s)$ represents the motor speed and $V_i(s)$ represents the input voltage.
(a) Design a proportional controller to have a closed-loop damping ratio of $\zeta=0.707$. For this value of $K_P$, determine the steady-state error for a unit step input.
(b) Design a PID controller so that the dominant closed-loop poles are at $-10 \pm 10 j$. For simplicity, select one of the zeros of the controller to cancel the pole at -2 . Then, use the angle criterion (9.60) with $p=-10+j 10$ to determine the other zero position. What is the expected steady-state error to a step input?
(c) To verify your results, simulate the step response of the closed-loop system with the two different controllers designed in (a) and (b).

Consider the rocket described in Problem 9.3. A feedback loop measures the angle $\theta(t)$ and determines the corrections to the thrust engines.
(a) Design a PD controller to have the closed loop poles at $-0.5 \pm 0.5 j$. (Hint: See the comment regarding the use of the angle criterion in Problem 9.19.)
(b) Simulate the response of the resulting closed-loop system to a unit impulse input.

A proportional controller can be implemented by the use of a simple amplifier. However, PD, PI, and PID controllers require a compensating network. Often, this is achieved in analog with the use of operational amplifier (op amp) circuits. Consider the ideal op amp in Figure P9.25a. This op amp is an infinite impedance circuit element, so that $v_a=0$ and $i_a=0$. These relationships also hold when the op amp is embedded in a circuit, as shown in Figure P9.25b.
(a) Suppose that $R_1=1000 \Omega, R_2=2000 \Omega, C_1=C_2=0$ in Figure P9.25b. Compute the transfer function between the input $v_1$ and the output $v_2$. (This circuit is known as an inverting circuit.)
(b) Suppose that $R_1=10 \mathrm{k} \Omega, R_2=20 \mathrm{k} \Omega, C_1=10 \mu \mathrm{F}$, and $C_2=0$ in Figure 9.25b. The resulting circuit is a PD controller. Compute the transfer function of the circuit.
(c) Suppose that $R_1=10 \mathrm{k} \Omega, R_2=\infty$ (removed from circuit), $C_1=200 \mu \mathrm{F}$, and $C_2=10 \mu \mathrm{F}$ in Figure 9.25b. The resulting circuit is a PI controller. Compute the transfer function of the circuit.
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