## Bonnie S. Heck, Edward W. Kamen

## Chapter 9

## Application to Control - all with Video Answers

## Educators

Chapter Questions

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.

Figure Can't Copy

(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.

Check back soon!

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.

Figure Can't Copy

Check back soon!

A rocket is drawn in Figure P9.3a, where $\theta(t)$ represents the angle between the rocket's orientation and its velocity, $\phi(t)$ represents the angle of the thrust engines, and $w(t)$ represents wind gusts, which act as a disturbance to the rocket. The goal of the control design is to have the angle $\theta(t)$ track a reference angle $\theta_r(t)$. The angle of the thrust engines can be directly

Figure Can't Copy controlled by motors that position the engines; therefore, the plant output is $\theta(t)$ and the controlled input is $\phi(t)$. The system can be modeled by the following equation:

$$

\Theta(s)=\frac{1}{s(s-1)} \Phi(s)+\frac{0.5}{s(s-1)} W(s)

$$

(a) Consider an open-loop control $\Theta(s)=G_c(s) X(s)$, where $G_c(s)$, the controller transfer function, and $x(t)$, the command signal, can be chosen as desired. Is such a controller practical for having $\theta(t)$ track $\theta_r(t)$ ? Justify your answer.

(b) Now consider a feedback controller as shown in Figure P9.3b, where

$$

G_c(s)=K(s+2)

$$

Find an expression for the output $\Theta(s)$ of the closed-loop system in terms of $W(s)$ and $\Theta_r(s)$. Consider the part of the response due to $W(s)$; the lower this value is, the better the disturbance rejection will be. How does the magnitude of this response depend on the magnitude of $K$ ?

(c) Suppose that $\Theta_r(s)=0$ and $w(t)$ is a random signal uniformly distributed between 0 and 1. Define a vector $w$ in MATLAB as $w=$ rand $(201,1)$, and define the time vector as $t=0: 0.05: 10$. Use $w$ as the input to the closed-loop system, and simulate the response for the time interval $0 \leq t \leq 10$. Perform the simulation for $K=5,10$, and 20, and plot the responses. Explain how the magnitude of the response is affected by the magnitude of $K$. Does this result match your prediction in part (b)?

Check back soon!

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 $e(t)=r(t)-y(t)$.

Figure Can't Copy

(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 $t \rightarrow \infty$.

Check back soon!

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.

Check back soon!

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.

Check back soon!

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) $G_p(s)=\frac{s+4}{(s+6)^2+64} ; \quad G_c(s)=K$

Check back soon!

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.

Check back soon!

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.

Check back soon!

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) $G_p(s)=\frac{1}{(s+1)(s+3)} ; \quad G_c(s)=\frac{K(s+6)}{s+10}$

Check back soon!

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

Check back soon!

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?

Figure Can't Copy

Check back soon!

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 $K_P$.

Check back soon!

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).

Check back soon!

A third-order system has the transfer function

$$

G_p(s)=\frac{1}{(s+1)(s+3)(s+5)}

$$

The performance specifications are that the dominant second-order poles have a damping ratio of $0.4 \leq \zeta \leq 0.707$ and $\zeta \omega_n>1$.

(a) Plot the root locus for $G_c(s)=K_P$.

(b) From the root locus, find the value(s) of $K_P$ that satisfy the criteria.

Check back soon!

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.

Check back soon!

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).

Check back soon!

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).

Figure Can't Copy

Check back soon!

A system is given by the transfer function

$$

G_p=\frac{10}{s(s+1)}

$$

Suppose that the desired closed-loop poles are at $-3 \pm j 3$.

(a) Design a PD controller to obtain the desired poles. Use the angle criterion (9.60) evaluated at the desired closed-loop pole (i.e., $p=-3+j 3$ ) to determine the zero position.

(b) Simulate the step response of the closed-loop system.

Check back soon!

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).

Check back soon!

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.

Check back soon!

An inverted pendulum shown in Figure P9.22 has the transfer function

$$

G_p(s)=\frac{\Theta(s)}{T(s)}=\frac{2}{s^2-2}

$$

where $\Theta(s)$ represents the angle of the rod and $T(s)$ represents the torque applied by a motor at the base.

Figure Can't Copy

(a) Sketch the root locus for a proportional controller, $G_c(s)=K_p$. What type of closedloop response would you expect for different values of $K_P$ ?

(b) Design a controller of the form

$$

G_c(s)=K_L \frac{s-z_c}{s-p_c}

$$

Choose $z_c=-3$, and solve for $p_c$ from the angle criterion so that the dominant closedloop poles are at $-3 \pm 3 j$. (Note: The resulting controller is called a lead controller.) Draw the resulting root locus for this system, and calculate the gain $K_L$ that results in the desired closed-loop poles.

(c) Simulate the impulse response of the closed-loop system with the controller designed in part (b). (The impulse is equivalent to someone bumping the pendulum.)

Check back soon!

A system has the transfer function

$$

G_p(s)=\frac{1}{s(s+2)}

$$

(a) Sketch the root locus for a proportional controller, $G_c(s)=K_P$ -

(b) Design a controller of the form

$$

G_c(s)=K_L \frac{s-z_c}{s-p_c}

$$

Select the zero of the controller to cancel the pole at -2 . Solve for $p_c$ from the angle criterion so that the dominant closed loop poles are at $-2 \pm 3 j$. Draw the resulting root locus for this system, and calculate the gain $K_L$ that results in the desired closed-loop poles.

(c) Design another controller by using the method described in part (b); except choose the zero to be at $z_c=-3$. Draw the resulting root locus for this system, and calculate the gain $K_L$ that results in the desired closed-loop poles.

(d) Compare the two controllers designed in parts (b) and (c) by simulating the step response of the two resulting closed-loop systems. Since both systems have the same dominant poles at $-2 \pm 3 j$, speculate on the reason for the difference in the actual response.

Check back soon!

Design a feedback controller that sets the position of a table tennis ball suspended in a plastic tube, as illustrated in Figure P9.24. Here, $M$ is the mass of the ball, $g$ the gravity constant, $y(t)$ the position of the ball at time $t$, and $x(t)$ the wind force on the ball due to the fan. The position $y(t)$ of the ball is continuously measured in real time by an ultrasonic sensor. The system is modeled by the differential equation

$$

M \ddot{y}(t)=x(t)-M g

$$

The objective is to design the feedback controller so that $y(t) \rightarrow y_0$ as $t \rightarrow \infty$, where $y_0$ is the desired position (set point).

(a) Can the control objective be met by the use of a proportional controller given by $G_c(s)=K_P$ ? Justify your answer.

(b) Can the control objective be met by the use of a PI controller given by $G_c(s)$ $=K_P+K_I / s$ ? Justify your answer.

(c) Design a PID controller that achieves the desired objective when $M=1$ and $g=9.8$.

Figure Can't Copy

Check back soon!

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.

Figure Can't Copy

Check back soon!