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Modern Control Systems

Dorf

Chapter 5

The Performance of Feedback Control Systems - all with Video Answers

Educators

Chapter Questions

Problem 1

An important problem for television systems is the jumping or wobbling of the picture due to the movement of the camera. This effect occurs when the camera is mounted in a moving truck or airplane. The Dynalens system has been designed to reduce the effect of rapid scanning motion; see Figure P5.1. A maximum scanning motion of $25 \%$ is expected. Let $K_g=K_t=1$ and assume that $\tau_g$ is negligible. (a) Determine the error of the system $E(s)$. (b) Determine the necessary loop gain $K_a K_m K_t$ when a $1 \%$ steady-state error is allowable. (c) The motor time constant is $0.40 \mathrm{~s}$. Determine the necessary loop gain so that the settling time (to within $2 \%$ of the final value of $v_b$ ) is less than or equal to $0.03 \mathrm{~s}$.

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Problem 2

A specific closed-loop control system is to be designed for an underdamped response to a step input. The specifications for the system are as follows:
$10 \%<$ percent overshoot $<20 \%$, Settling time $<0.6 \mathrm{~s}$.
(a) Identify the desired area for the dominant roots of the system. (b) Determine the smallest value of a third root $r_3$ if the complex conjugate roots are to represent the dominant response. (c) The closedloop system transfer function $T(s)$ is third-order, and the feedback has a unity gain. Determine the forward transfer function $G(s)=Y(s) / E(s)$ when the settling time to within $2 \%$ of the final value is $0.6 \mathrm{~s}$ and the percent overshoot is $20 \%$.

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03:20

Problem 3

A laser beam can be used to weld, drill, etch, cut, and mark metals, as shown in Figure P5.3(a) [16]. Assume we have a work requirement for an accurate laser to mark a parabolic path with a closed-loop control system, as shown in Figure P5.3(b). Calculate the necessary gain to result in a steady-state error of $5 \mathrm{~mm}$ for $r(t)=t^2 \mathrm{~cm}$.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:12

Problem 4

The final value of step response of a II order system is unity and percentage overshoot is $9 \%$. If the step response settled in $2 s$, find the transfer function of the system. Also find the poles of the system.

Arpit Gupta
Arpit Gupta
Numerade Educator
23:23

Problem 5

A space telescope is to be launched to carry out astronomical experiments [9]. The pointing control system is desired to achieve 0.01 minute of arc and track solar objects with apparent motion up to 0.21 arc minute per second. The system is illustrated in Figure P5.5(a). The control system is shown in Figure P5.5(b). Assume that $\tau_1=1$ second and $\tau_2=0$ (an approximation). (a) Determine the gain $K=K_1 K_2$ required so that the response to a step command is as rapid as reasonable with an overshoot of less than $5 \%$. (b) Determine the steadystate error of the system for a step and a ramp input. (c) Determine the value of $K_1 K_2$ for an ITAE optimal system for (1) a step input and (2) a ramp input.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 6

A robot is programmed to have a tool or welding torch follow a prescribed path $[8,13]$. Consider a robot tool that is to follow a sawtooth path, as shown in Figure P5.6(a). The transfer function of the plant is
$$
G(s)=\frac{75(s+1)}{s(s+5)(s+20)}
$$
for the closed-loop system shown in Figure 5.6(b). Calculate the steady-state error.

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07:32

Problem 7

Astronaut Bruce McCandless II took the first untethered walk in space on February 7,1984, using the gas-jet propulsion device illustrated in Figure P5.7(a). The controller can be represented by a gain $K_2$, as shown in Figure P5.7(b). The moment of inertia of the equipment and man is $25 \mathrm{~kg} \mathrm{~m}^2$. (a) Determine the necessary gain $K_3$ to maintain a steady-state error equal to $1 \mathrm{~cm}$ when the input is a ramp $r(t)=t$ (meters). (b) With this gain $K_3$, determine the necessary gain $K_1 K_2$ in order to restrict the percent overshoot to $10 \%$. (c) Determine analytically the gain $K_1 K_2$ in order to minimize the ISE performance index for a step input.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
03:10

Problem 8

Photovoltaic arrays (solar cells) generate a DC voltage that can be used to drive DC motors or that can be converted to $\mathrm{AC}$ power and added to the distribution network. It is desirable to maintain the power out of the array at its maximum available as the solar incidence changes during the day. One such closed-loop system is shown in Figure P5.8. The transfer function for the process is
$$
G(s)=\frac{K}{s+10},
$$
where $K=20$. Find (a) the time constant of the closed-loop system and (b) the settling time to within $2 \%$ of the final value of the system when disturbances such as clouds occur.

Chai Santi
Chai Santi
Numerade Educator
02:17

Problem 9

The antenna that receives and transmits signals to the Telstar communication satellite is the largest horn antemna ever built. The microwave antenna is $177 \mathrm{ft}$ long, weighs 340 tons, and rolls on a circular track. A photo of the antenna is shown in Figure P5.9. The Telstar satellite is 34 inches in diameter and moves about $16,000 \mathrm{mph}$ at an altitude of 2500 miles. The antenna must be positioned accurately to $1 / 10$ of a degree, because the microwave beam is $0.2^{\circ}$ wide and highly attenuated by the large distance. If the antenna is following the moving satellite, determine the $K_r$ necessary for the system.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:47

Problem 10

A speed control system of an armature-controlled DC motor uses the back emf voltage of the motor as a feedback signal. (a) Draw the block diagram of this system (see Equation (2.69)). (b) Calculate the steady-state error of this system to a step input command setting the speed to a new level. Assume that $R_a=L_a=J=$ $b=1$, the motor constant is $K_m=1$, and $K_b=1$. (c) Select a feedback gain for the back emf signal to yield a step response with an overshoot of $15 \%$.

Vishal Gupta
Vishal Gupta
Numerade Educator

Problem 11

A simple unity feedback control system has a process transfer function
$$
\frac{Y(s)}{E(s)}=G(s)=\frac{K}{s} .
$$

The system input is a step function with an amplitude $A$. The initial condition of the system at time $t_0$ is $y\left(t_0\right)=Q$, where $y(t)$ is the output of the system. The performance index is defined as
$$
I=\int_0^{\infty} e^2(t) d t .
$$
(a) Show that $I=(A-Q)^2 /(2 K)$. (b) Determine the gain $K$ that will minimize the performance index $I$. Is this gain a practical value? (c) Select a practical value of gain and determine the resulting value of the performance index.

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10:15

Problem 12

Train travel between cities will increase as trains are developed that travel at high speeds, making the travel time from city center to city center equivalent to airline travel time. The Japanese National Railway has a train called the Bullet Express that travels between Tokyo and Osaka on the Tokaido line. This train travels the 320 miles in 3 hours and 10 minutes, an average speed of 101 $\mathrm{mph}[20]$. This speed will be increased as new systems are used, such as magnetically levitated systems to float vehicles above an aluminum guideway. To maintain a desired speed, a speed control system is proposed that yields a zero steady-state error to a ramp input. A thirdorder system is sufficient. Determine the optimum system transfer function $T(s)$ for an ITAE performance criterion. Estimate the settling time (with a $2 \%$ criterion) and overshoot for a step input when $\omega_n=10$.

Averell Hause
Averell Hause
Carnegie Mellon University

Problem 13

We want to approximate a fourth-order system by a lower-order model. The transfer function of the original system is
$$
\begin{aligned}
H(s) & =\frac{s^3+7 s^2+24 s+24}{s^4+10 s^3+35 s^2+50 s+24} \\
& =\frac{s^3+7 s^2+24 s+24}{(s+1)(s+2)(s+3)(s+4)} .
\end{aligned}
$$

Show that if we obtain a second-order model by the method of Section 5.8 , and we do not specify the poles and the zero of $L(s)$, we have
$$
\begin{aligned}
L(s) & =\frac{0.2917 s+1}{0.399 s^2+1.375 s+1} \\
& =\frac{0.731(s+3.428)}{(s+1.043)(s+2.4)} .
\end{aligned}
$$

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Problem 14

For the original system of Problem P5.13, we want to find the lower-order model when the poles of the second-order model are specified as -1 and -2 and the model has one unspecified zero. Show that this low-order model is
$$
L(s)=\frac{0.986 s+2}{s^2+3 s+2}=\frac{0.986(s+2.028)}{(s+1)(s+2)} .
$$

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04:00

Problem 15

A magnetic amplifier with a low-output impedance is shown in Figure P5.15 in cascade with a low-pass filter and a preamplifier. The amplifier has a high-input impedance and a gain of 1 and is used for adding the signals as shown. Select a value for the capacitance $C$ so that the transfer function $V_0(s) / V_{\text {in }}(s)$ has a damping ratio of $1 / \sqrt{2}$. The time constant of the magnetic amplifier is equal to 1 second, and the gain is $K=10$. Calculate the settling time (with a $2 \%$ criterion) of the resulting system.

Narayan Hari
Narayan Hari
Numerade Educator
04:24

Problem 16

Electronic pacemakers for human hearts regulate the speed of the heart pump. A proposed closed-loop system that includes a pacemaker and the measurement of the heart rate is shown in Figure P5.16 [2,3]. The transfer function of the heart pump and the pacemaker is found to be
$$
G(s)=\frac{K}{s(s / 12+1)} .
$$

Design the amplifier gain to yield a system with a settling time to a step disturbance of less than 1 second. The overshoot to a step in desired heart rate should be less than $10 \%$. (a) Find a suitable range of $K$. (b) If the nominal value of $K$ is $K=10$, find the sensitivity of the system to small changes in $K$. (c) Evaluate the sensitivity of part (b) at $D C$ (set $s=0$ ). (d) Evaluate the magnitude of the sensitivity at the normal heart rate of 60 beats/minute.

Amany Waheeb
Amany Waheeb
Numerade Educator

Problem 17

Consider the original third-order system given in Example 5.9. Determine a first-order model with one pole unspecified and no zeros that will represent the third-order system.

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Problem 18

A closed-loop control system with negative unity feedback has a plant with a transfer function
$$
G(s)=\frac{8}{s\left(s^2+6 s+12\right)} .
$$
(a) Determine the closed-loop transfer function $T(s)$. (b) Determine a second-order approximation for $T(s)$ using the method of Section 5.10. (c) Plot the response of $T(s)$ and the second-order approximation to a unit step input and compare the results.

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Problem 19

A system is shown in Figure P5.19.
(a) Determine the steady-state error for a unit step input in terms of $K$ and $K_1$, where $E(s)=$ $R(s)-Y(s)$.
(b) Select $K_1$ so that the steady-state error is zero.

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Problem 20

Consider the closed-loop system in Figure P5.20. Determine values of the parameters $k$ and $a$ so that the following specifications are satisfied:
(a) The steady-state error to a unit step input is zero.
(b) The closed-loop system has a percent overshoot of less than $5 \%$.

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Problem 21

Consider the closed-loop system in Figure P5.21, where
$$
G_c(s) G(s)=\frac{2}{s+0.2 K} \text { and } H(s)=\frac{2}{2 s+\tau} .
$$
(a) If $\tau=2.43$, determine the value of $K$ such that the steady-state error of the closed-loop system response to a unit step input, $R(s)=1 / s$, is zero.
(b) Determine the percent overshoot $P . O$. and the time to peak $T_p$ of the unit step response when $K$ is as in part (a).

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