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Modern Control Systems

Dorf

Chapter 9

Stability in the Frequency Domain - all with Video Answers

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Chapter Questions

Problem 1

For the polar plots of Problem P8.1, use the Nyquist criterion to ascertain the stability of the various systems. In each case, specify the values of $N, P$, and $Z$.

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Problem 2

Sketch the polar plots of the following loop transfer functions $G_c(s) G(s)$, and determine whether the system is stable by applying the Nyquist criterion:
$$
G_c(s) G(s)=\frac{K}{s\left(s^2+s+4\right)} .
$$
(b)
$$
G_c(s) G(s)=\frac{K(s+2)}{s^2(s+4)}
$$

If the system is stable, find the maximum value for $K$ by determining the point where the polar plot crosses the $u$-axis.

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01:20

Problem 3

(a) Find a suitable contour $\Gamma_s$ in the $s$-plane that can be used to determine whether all roots of the characteristic equation have damping ratios greater than $\zeta_1$. (b) Find a suitable contour $\Gamma_s$ in the $s$-plane that can be used to determine whether all the roots of the characteristic equation have real parts less than $s=-\sigma_1$. (c) Using the contour of part (b) and Cauchy's theorem, determine whether the following characteristic equation has roots with real parts less than $s=-1$ :
$$
q(s)=s^3+11 s^2+56 s+96 .
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator

Problem 4

The polar plot of a conditionally stable system is shown in Figure P9.4 for a specific gain K. (a) Determine whether the system is stable, and find the number of roots (if any) in the right-hand $s$-plane. The system has no poles of $G_c(s) G(s)$ in the right half-plane. (b) Determine whether the system is stable if the -1 point lies at the dot on the axis.

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17:41

Problem 5

A speed control for a gasoline engine is shown in Figure P9.5. Because of the restriction at the carburetor intake and the capacitance of the reduction manifold, the lag $\tau_t$ occurs and is equal to 1 second. The engine time constant $\tau_e$ is equal to $J / b=3 \mathrm{~s}$. The speed measurement time constant is $\tau_m=0.4 \mathrm{~s}$. (a) Determine the necessary gain $K$ if the steady-state speed error is required to be less than $10 \%$ of the speed reference setting. (b) With the gain determined from part (a), apply the Nyquist criterion to investigate the stability of the system. (c) Determine the phase and gain margins of the system.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
01:15

Problem 6

A direct-drive arm is an innovative mechanical arm in which no reducers are used between motors and their loads. Because the motor rotors are directly coupled to the loads, the drive systems have no backlash, small friction, and high mechanical stiffness, which are all important features for fast and accurate positioning and dexterous handling using sophisticated torque control.
The goal of the MIT direct-drive arm project is to achieve arm speeds of $10 \mathrm{~m} / \mathrm{s}$ [15]. The arm has torques of up to $660 \mathrm{~N} \mathrm{~m}$ (475 ft lb). Feedback and a set of position and velocity sensors are used with each motor. The frequency response of one joint of the arm is shown in Figure P9.6(a). The two poles appear at 3.7 $\mathrm{Hz}$ and $68 \mathrm{~Hz}$. Figure P9.6(b) shows the step response with position and velocity feedback used. The time constant of the closed-loop system is $82 \mathrm{~ms}$. Develop the block diagram of the drive system and prove that $82 \mathrm{~ms}$ is a reasonable result.

Manish Jain
Manish Jain
Numerade Educator
07:09

Problem 7

A vertical takeoff (VTOL) aircraft is an inherently unstable vehicle and requires an automatic stabilization system. An attitude stabilization system for the K-16B U.S. Army VTOL aircraft has been designed and is shown in block diagram form in Figure P9.7 [16]. At 40 knots, the dynamics of the vehicle are approximately represented by the transfer function
$$
G(s)=\frac{10}{s^2+0.36} .
$$
The actuator and filter are represented by the transfer function
$$
G_c(s)=\frac{K_1(s+7)}{s+3} .
$$
(a) Obtain the Bode diagram of the loop transfer function $L(s)=G_c(s) G(s) H(s)$ when the gain is $K_1=2$. (b) Determine the gain and phase margins of this system. (c) Determine the steady-state error for a wind disturbance of $T_d(s)=1 / s$. (d) Determine the maximum amplitude of the resonant peak of the closed-loop frequency response and the frequency of the resonance. (e) Estimate the damping ratio of the system from $M_{p w}$ and the phase margin.

Vipender Yadav
Vipender Yadav
Numerade Educator

Problem 8

Electrohydraulic servomechanisms are used in control systems requiring a rapid response for a large mass. An electrohydraulic servomechanism can provide an output of $100 \mathrm{~kW}$ or greater [17]. A photo of a servovalve and actuator is shown in Figure P9.8(a). The output sensor yields a measurement of actuator position, which is compared with $V_{\text {in }}$. The error is amplified and controls the hydraulic valve position, thus controlling the hydraulic fluid flow to the actuator. The block diagram of a closed-loop electrohydraulic servomechanism using pressure feedback to obtain damping is shown in Figure P9.8(b) $[17,18]$. Typical values for this system are $\tau=0.02 \mathrm{~s}$; for the hydraulic system they are $\omega_2=7(2 \pi)$ and $\zeta_2=0.05$. The structural resonance $\omega_1$ is equal to $10(2 \pi)$, and the damping is $\zeta_1=0.05$. The loop gain is $K_A K_1 K_2=1.0$. (a) Sketch the Bode diagram and determine the phase margin of the system. (b) The damping of the system can be increased by drilling a small hole in the piston so that $\zeta_2=0.25$. Sketch the Bode diagram and determine the phase margin of this system.

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03:03

Problem 9

The space shuttle, shown in Figure P9.9(a), carries large payloads into space and returns them to earth for reuse [19]. The shuttle uses elevons at the trailing edge of the wing and a brake on the tail to control the flight during entry. The block diagram of a pitch rate control system is shown in Figure P9.9(b). The sensor is represented by a gain, $H(s)=0.5$, and the vehicle by the transfer function
$$
G(s)=\frac{0.30(s+0.05)\left(s^2+1600\right)}{\left(s^2+0.05 s+16\right)(s+70)} .
$$

The controller $G_c(s)$ can be a gain or any suitable transfer function. (a) Sketch the Bode diagram of the system when $G_c(s)=2$ and determine the stability margin. (b) Sketch the Bode diagram of the system when
$$
G_c(s)=K_P+K_l / s \text { and } K_l / K_P=0.5 .
$$

The gain $K_P$ should be selected so that the gain margin is $10 \mathrm{~dB}$.

James Kiss
James Kiss
Numerade Educator
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Problem 10

Machine tools are often automatically controlled as shown in Figure P9.10. These automatic systems are often called numerical machine controls [9]. On each axis, the desired position of the machine tool is compared with the actual position and is used to actuate a solenoid coil and the shaft of a hydraulic actuator. The transfer function of the actuator (see Table 2.7) is
$$
G_a(s)=\frac{X(s)}{Y(s)}=\frac{K_a}{s\left(\tau_a s+1\right)} .
$$
where $K_a=1$ and $\tau_a=0.4 \mathrm{~s}$. The output voltage of the difference amplifier is
$$
E_0(s)=K_1\left(X(s)-X_d(s)\right),
$$
where $x_d(t)$ is the desired position input. The force on the shaft is proportional to the current $i$, so that $F=K_2 i(t)$, where $K_2=3.0$. The spring constant $K_5$ is equal to $1.5, R=0.1$, and $L=0.2$.
(a) Determine the gain $K_1$ that results in a system with a phase margin of $30^{\circ}$. (b) For the gain $K_1$ of part (a), determine $M_{p \omega}, \omega_r$, and the closed-loop system bandwidth. (c) Estimate the percent overshoot of the transient response to a step input $X_d(s)=1 / s$, and the settling time (to within $2 \%$ of the final value).

Lainey Roebuck
Lainey Roebuck
Numerade Educator

Problem 11

A control system for a chemical concentration control system is shown in Figure P9.11. The system receives a granular feed of varying composition, and we want to maintain a constant composition of the output mixture by adjusting the feed-flow valve. The transfer function of the tank and output valve is
$$
G(s)=\frac{5}{5 s+1},
$$
and that of the controller is
$$
G_c(s)=K_1+\frac{K_2}{s} .
$$
The transport of the feed along the conveyor requires a transport (or delay) time, $T=1.5 \mathrm{~s}$. (a) Sketch the Bode diagram when $K_1=K_2=1$, and investigate the stability of the system. (b) Sketch the Bode diagram when $K_1=0.1$ and $K_2=0.04$, and investigate the stability of the system. (c) When $K_1=0$, use the Nyquist criterion to calculate the maximum allowable gain $K_2$ for the system to remain stable.

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Problem 12

A simplified model of the control system for regulating the pupillary aperture in the human eye is shown in Figure P9.12 [20]. The gain $K$ represents the pupillary gain, and $\tau$ is the pupil time constant, which is $0.6 \mathrm{~s}$. The time delay $T$ is equal to $0.15 \mathrm{~s}$. The pupillary gain is equal to 3.8 .
(a) Assuming the time delay is negligible, sketch the Bode diagram for the system. Determine the phase margin of the system. (b) Include the effect of the time delay by adding the phase shift due to the delay. Determine the phase margin of the system with the time delay included.

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Problem 13

A controller is used to regulate the temperature of a mold for plastic part fabrication, as shown in Figure P9.13. The value of the delay time is estimated as $1.2 \mathrm{~s}$. (a) Using the Nyquist criterion, determine the stability of the system for $K_a=K=1$. (b) Determine a suitable value for $K_a$ for a stable system that will yield a phase margin greater than $50^{\circ}$ when $K=1$.

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Problem 14

Electronics and computers are being used to control automobiles. Figure P9.14 is an example of an automobile control system, the steering control for a research automobile. The control stick is used for steering. A typical driver has a reaction time of $T=0.2 \mathrm{~s}$.
(a) Using the Nichols chart, determine the magnitude of the gain $K$ that will result in a system with a peak magnitude of the closed-loop frequency response $M_{p o}$ less than or equal to $2 \mathrm{~dB}$.
(b) Estimate the damping ratio of the system based on (1) $M_{p \omega}$ and (2) the phase margin. Compare the results and explain the difference, if any.
(c) Determine the closed-loop 3-dB bandwidth of the system.

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Problem 15

Consider the automatic ship-steering system discussed in Problem P8.11. The frequency response of the open-loop portion of the ship steering control system is shown in Figure P8.11. The deviation of the tanker from the straight track is measured by radar and is used to generate the error signal, as shown in Figure P9.15. This error signal is used to control the rudder angle $\delta(s)$.
(a) Is this system stable? Discuss what an unstable ship-steering system indicates in terms of the transient response of the system. Recall that the system under consideration is a ship attempting to follow a straight track.
(b) Is it possible to stabilize this system by lowering the gain of the transfer function $G(s)$ ?
(c) Is it possible to stabilize this system? Suggest a suitable feedback compensator?
(d) Repeat parts (a), (b), and (c) when switch $S$ is closed.

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Problem 16

An electric carrier that automatically follows a tape track laid out on a factory floor is shown in Figure P9.16(a) [15]. Closed-loop feedback systems are used to control the guidance and speed of the vehicle. The cart senses the tape path by means of an array of 16 phototransistors. The block diagram of the steering system is shown in Figure P9.16(b). Select a gain $K$ so that the phase margin is approximately $30^{\circ}$.

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Problem 17

The primary objective of many control systems is to maintain the output variable at the desired or reference condition when the system is subjected to a disturbance [23]. A typical chemical reactor control scheme is shown in Figure P9.17. The disturbance is represented by $U(s)$, and the chemical process by $G_3$ and $G_4$. The controller is represented by $G_1$ and the valve by $G_2$. The feedback sensor is $H(s)$ and will be assumed to be equal to 1 . We will assume that $G_2, G_3$, and $G_4$ are all of the form
$$
G_i(s)=\frac{K_i}{1+\tau_i s},
$$
where $\tau_3=\tau_4=4 \mathrm{~s}$ and $K_3=K_4=0.1$. The valve constants are $K_2=20$ and $\tau_2=0.5 \mathrm{~s}$. We want to maintain a steady-state error less than $5 \%$ of the desired reference position.
(a) When $G_1(s)=K_1$, find the necessary gain to satisfy the error-constant requirement. For this condition, determine the expected overshoot to a step change in the reference signal $r(t)$.
(b) If the controller has a proportional term plus an integral term so that $G_1(s)=K_1(1+1 / s)$, determine a suitable gain to yield a system with an overshoot less than $30 \%$, but greater than $5 \%$. For parts (a) and (b), use the approximation of the damping ratio as a function of phase margin that yields $\zeta=0.01 \phi_{\mathrm{pm}}$. For these calculations, assume that $U(s)=0$.
(c) Estimate the settling time (with a $2 \%$ criterion) of the step response of the system for the controller of parts (a) and (b).
(d) The system is expected to be subjected to a step disturbance $U(s)=A / s$. For simplicity, assume that the desired reference is $r(t)=0$ when the system has settled. Determine the response of the system of part (b) to the disturbance.

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09:19

Problem 18

A model of an automobile driver attempting to steer a course is shown in Figure P9.18, where $K=5.3$. (a) Find the frequency response and the gain and phase margins when the reaction time $T$ is zero. (b) Find the phase margin when the reaction time is $0.1 \mathrm{~s}$. (c) Find the reaction time that will cause the system to be borderline stable (phase margin $=0^{\circ}$ ).

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 19

In the United States, billions of dollars are spent annually for solid waste collection and disposal. One system, which uses a remote control pick-up arm for collecting waste bags, is shown in Figure P9.19. The loop transfer function of the remote pick-up arm is
$$
G_c(s) G(s)=\frac{0.25}{s(4 s+1)(s+3)} .
$$
(a) Plot the Nichols chart and show that the gain margin is approximately $32 \mathrm{~dB}$. (b) Determine the phase margin and the $M_{p \omega}$ for the closed loop. Also, determine the closed-loop bandwidth.

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04:19

Problem 20

The Bell-Boeing V-22 Osprey Tiltrotor is both an airplane and a helicopter. Its advantage is the ability to rotate its engines to a vertical position, as shown in Figure P7.33(a), for takeoffs and landings and then switch the engines to a horizontal position for cruising as an airplane. The altitude control system in the helicopter mode is shown in Figure P9.20. (a) Obtain the frequency response of the system for $K=100$. (b) Find the gain margin and the phase margin for this system. (c) Select a suitable gain $K$ so that the phase margin is $40^{\circ}$. (Decrease the gain above $K=100$.) (d) Find the response $y(t)$ of the system for the gain selected in part (c).

Anand Jangid
Anand Jangid
Numerade Educator

Problem 21

The open-loop transfer function of a unity feedback system is,
$$
\frac{K}{(1+0.1 s)(s+2)}
$$

Find the value of $K$ so that the phase margin is $40^{\circ}$.

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Problem 22

The Nichols diagram for $G_c(j \omega) G(j \omega)$ of a closedloop system is shown in Figure P9.22. The frequency for each point on the graph is given in the following table:
$$
\begin{array}{lccccccccc}
\text { Point } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline \omega & 1 & 2.0 & 2.6 & 3.4 & 4.2 & 5.2 & 6.0 & 7.0 & 8.0
\end{array}
$$

Determine (a) the resonant frequency, (b) the bandwidth, (c) the phase margin, and (d) the gain margin. (e) Estimate the overshoot and settling time (with a $2 \%$ criterion) of the response to a step input.

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Problem 23

Consider the system given in figure P9.23.
(a) If $K_I=1$, find gain Margin using Bode Plot. Also find steady state error to a ram input.
(b) Find $K_I$ so that gain margin is 5 .
(c) Corresponding to the $K_I$ obtained in (b) find the steady-state error to a ramp input.

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Problem 24

A closed-loop system with unity feedback has a loop transfer function
$$
G_c(s) G(s)=\frac{K(s+20)}{s^2} .
$$
(a) Determine the gain $K$ so that the phase margin is $45^{\circ}$. (b) For the gain $K$ selected in part (a), determine the gain margin. (c) Predict the bandwidth of the closed-loop system.

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Problem 25

A closed-loop system has the loop transfer function
$$
G_c(s) G(s)=\frac{K e^{-T s}}{s} .
$$
(a) Determine the gain $K$ so that the phase margin is $60^{\circ}$ when $T=0.2$. (b) Plot the phase margin versus the time delay $T$ for $K$ as in part (a).

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Problem 26

A specialty machine shop is improving the efficiency of its surface-grinding process [22]. The existing machine is mechanically sound, but manually operated. Automating the machine will free the operator for other tasks and thus increase overall throughput of the machine shop. The grinding machine is shown in Figure P9.26(a) with all three axes automated with motors and feedback systems. The control system for the $y$-axis is shown in Figure P9.26(b). To achieve a low steady-state error to a ramp command, we choose $K=10$. Sketch the Bode diagram of the open-loop system and obtain the Nichols chart plot. Determine the gain and phase margin of the system and the bandwidth of the closed-loop system. Estimate the $\zeta$ of the system and the predicted overshoot and settling time (with a $2 \%$ criterion)

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Problem 27

Consider the system shown in Figure P9.27. Determine the maximum value of $K=K_{\max }$ for which the closed-loop system is stable. Plot the phase margin as a function of the gain $1 \leq K \leq K_{\max }$. Explain what happens to the phase margin as $K$ approaches $K_{\max }$.

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Problem 28

Consider the feedback system shown in Figure P9. 28 with the process transfer function given as
$$
G(s)=\frac{1}{s(s+1)} .
$$

The controller is the proportional controller
$$
G_c(s)=K_P .
$$
(a) Determine a value of $K_P$ such that the phase margin is approximately P.M. $\approx 45^{\circ}$.
(b) Using the P.M. obtained, predict the percent overshoot of the closed-loop system to a unit step input.
(c) Plot the step response and compare the actual percent overshoot with the predicted percent overshoot.

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