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Modern Control Systems 12th edition Richard C. Dorf, Robert H. Bishop - Solutions

The asymptotic log-magnitude curves for two transfer functions are given in Figure P8.6. Sketch the corresponding asymptotic phase shift curves for each system. Determine the transfer function for each system. Assume that the systems have minimum phase transfer functions.
Driverless vehicles can be used in warehouses, airports, and many other applications. These vehicles follow a wire embedded in the floor and adjust the steerable front wheels in order to maintain proper direction, as shown in Figure P8.7(a) [10]. The sensing coils, mounted on the front wheel
A feedback control system is shown in Figure P8.8.The specification for the closed-loop system requires that the overshoot to a step input be less than 15%. (a) Determine the corresponding specification MpÏ‰ in the frequency domain for the closed-loop transfer function(b) Determine the
Sketch the logarithmic-magnitude versus phase angle curves for the transfer functions (a) and (b) of Problem P8.1.
A spring-mass-damper system is shown in Figure AP8.1(a).The Bode diagram obtained by experimental means using a sinusoidal forcing function is shownFIGURE AP8.1 A spring-mass damper system.
A system is shown in Figure AP8.2. The nominal value of the parameter b is 4.0. Determine the sensitivityFIGURE AP8.2 System with parameter b.
As an automobile moves along the road, the vertical displacements at the tires act as the motion excitation to the automobile suspension system [16]. FigureFIGURE AP8.3 Auto suspension system model. AP8.3 is a schematic diagram of a simplified automobile suspension system, for which we assume the
A helicopter with a load on the end of a cable is shown in Figure AP8.4 (a).The position control system is shown in Figure AP8.4 (b), where the visual feedback is represented by H(s). Sketch the Bode diagram of the loop transfer function L(jω) = G(j(ω)H(j(ω).
A closed-loop system with unity feedback has a transfer function(a) Determine the loop transfer function Gc(s) G(s). (b) Plot the log-magnitude-phase (similar to Figure 8.27), and identify the frequency points for co equal to 1,10,50,110, and 500. (c) Is the open-loop system stable? Is the
Î¼(t) = sin(Ï‰t) is less than 1 for all co. For the values you selected for k and b, what is the frequency at which the peak response occurs?
An op-amp circuit is shown in Figure AP8.7. The circuit represents a lead compensator discussed in more detail in Chapter 10.(a) Determine the transfer function of this circuit.(b) Sketch the frequency response of the circuit
In this chapter, we wish to use a PD controller such that Gc(s) = K(s + 2). The tachometer is not used (see Figure CDP4.1). Plot the Bode diagram for the system when K = 40. Determine the step response of this system and estimate the overshoot and settling time (with a 2% criterion).
Understanding the behavior of a human steering an automobile remains an interesting subject [14,15, 16, 21]. The design and development of systems for four-wheel steering, active suspensions, active, independent braking, and "drive-by-wire" steering provide the engineer with considerably more
The unmanned exploration of planets such as Mars requires a high level of autonomy because of the communication delays between robots in space and their Earth-based stations. This affects all the components of the system: planning, sensing, and mechanism. In particular, such a level of autonomy can
A table is used to position vials under a dispenser head, as shown in Figure DP8.3(a). The objective is speed, accuracy, and smooth motion in order to eliminate spilling. The position control system is shown in Figure DP8.3(b). Since we want small overshoot for a step input and yet desire a short
Anesthesia can be administered automatically by a control system. For certain operations, such as brain and eye surgery, involuntary muscle movements can be disastrous. To ensure adequate operating conditions for the surgeon, muscle relaxant drugs, which block involuntary muscle movements, are
Consider the control system depicted where the plant is a "black box" for which little is known in the way of mathematical models. The only information available on the plant is the frequency response. Design a controller Gc(s) to meet the following specifications: (i) The crossover frequency is
A single-input, single-output system is described b y (t) = [0 l]x(/) (a) Determine p and K such that the unit step response exhibits a zero steady-state error and the percent overshoot meets the requirement P.O. ‰¤ 5%. (b) For the values of p and K determined in part (a),
Consider the system of Figure DPS.7. Consider the controller to be a proportional plus integral plus derivative (PID) given byDesign the PID controller gains to achieve (a) an acceleration constant Ka = 2, (b) a phase margin of P.M.‰¥ 45°, and (c) a bandwidth Ï‰b > 3.0. Plot
Consider the closed-loop transfer functionDevelop an m-file to, obtain the Bode plot and verify that the resonant frequency is 5 rad/s and that the peak magnitude M pÏ‰ is 14 dB.
For the following transfer functions, sketch the Bode plots, then verify with the bode function:
For each of the following transfer functions, sketch the Bode plot and determine the crossover frequency (that is, the frequency at which 20 log10 |G(jÏ‰) | = OdB):
A unity negative feedback system has the loop transfer functionDetermine the closed-loop system bandwidth. Using the bode function obtain the Bode plot and label the plot with the bandwidth.
A block diagram of a second-order system is shown in Figure CP8.5.FIGURE CP8.5 A second-order feedback controlsystem.
Consider the feedback system in Figure CP8.6. Obtain the Bode plots of the loop and closed-loop transfer functions using an m-file.FIGURE CP8.6 Closed-loop feedback system.
A unity feedback system has the loop transfer functionGenerate a plot of the bandwidth versus the parameter p as 0
Consider the problem of controlling an inverted pendulum on a moving base, as shown in Figure CP8.8 (a).The transfer function of the system isThe design objective is to balance the pendulum (i.e., 0(f) ‰ˆ 0) in the presence of disturbance inputs. A block diagram representation of the
Design a filter, G(s), with the following frequency response: 1. For ω < 1 rad/s, the magnitude 20 log1() |G(jω)|< OdB 2. For 1 < ω < 1000 rad/s, the magnitude 201og10 | G(jω) | ≥ OdB 3. For co > 1000 rad/s, the magnitude 20 logI0 |G(jω) | < OdB Try to maximize the peak magnitude as close
A system has the loop transfer functionPlot the Bode diagram. Show that the phase margin is approximately 17.5° and that the gain margin is approximately 26.2 dB.
Consider the wind tunnel control system of Problem P7.31 for K = 326. Obtain the Bode diagram and show that the P.M. = 25° and that the G.M. = 10 dB. Also, show that the bandwidth of the closed-loop system is 6 rad/s.
Consider a unity feedback system with the loop transfer function(a) Plot the Bode diagram, (b) Find the gain margin and the phase margin.
A unity feedback system with the loop transfer functionwhere T1 = 0.02 and T2 = 0.2 s. (a) Select a gain K so that the steady-state error for a ramp input is 10% of the magnitude of the ramp function A, where r(l) = At, ‰¥ 0. (b) Plot the Bode plot of Gc(s)G(s), and determine the phase and
A unity feedback system has a loop transfer function(a) Find the maximum magnitude of the closed-loop frequency response using the Nichols chart, (b) Find the bandwidth and the resonant frequency of this system. (c) Use these frequency measures to estimate the overshoot of the system to a step
A Nichols chart is given in Figure E9.14 for a system with Gc(jÏ‰)G(jÏ‰)). Using the following table, find (a) the peak resonance MpÏ‰) in dB; (b) the resonant frequency cor; (c) the 3-dB bandwidth; and (d) the phase margin of the system.
Consider a unity feedback system with the loop transfer functionFind the bandwidth of the closed-loop system.
The pure time delay e-sT may be approximated bya transfer function asfor 0
A unity feedback system has a loop transfer function(a) Plot the Bode diagram and (b) determine the gain K required to obtain a phase margin of 30°. What is the steady-state error for a ramp input for the gain of part(b)?'
An actuator for a disk drive uses a shock mount to absorb vibrational energy at approximately 60 Hz [14]. The Bode diagram of G(.(s)G(s) of the control system is shown in Figure E9.18. (a) Find the expected percent overshoot for a step input for the closed-loop(a) Plot the Bode diagram and (b)
A unity feedback system with Gc(s) = K hasSelect a gain K so that the phase margin of the system is 50°. Determine the gain margin for the selected gain,K.
A system has the loop transfer functionwhere K = 10.5. Show that the system crossover (O dB) frequency is 5 rad/s and that the phase margin is 40°.
Consider a simple model of an automobile driver following another car on the highway at high speed. The model shown in Figure E9.20 incorporates the driver's reaction time, T. One driver has T = 1 s, and another has T = 1.5 s. Determine the time response y(t) of the system for both drivers for a
A unity feedback control system has a loop transferFunctionDetermine the phase margin, the crossover frequency, and the gain margin when K - 1300.
A unity feedback system has a loop transfer function(a) Using a Bode diagram for K = 10, determine the system phase margin, (b) Select a gain K so that the phase margin is at least 60°.
Consider again the system of E9.21 when K = 438. Determine the closed-loop system bandwidth, resonant frequency, and Mpω using the Nichols chart.
A unity feedback system has a loop transfer functionwhere K = 1 / 2and T = 1. The polar plot for Gc(jÏ‰)G(jaÏ‰) is shown in Figure E9.24. Determine whether the system is stable by using the Nyquist criterion. FIGURE E9.24 Polar plot for Gc(s)G(s) = K/(-1 + TS).
A unity feedback system has a loop transfer functionDetermine the phase margin and the crossover frequency.
For the system of E9.25, determine Mpω, and ωB for the closed-loop frequency response by using the Nichols chart.
A unity feedback system has a loop transfer function
A unity feedback system has the loop transfer function(a) Determine the phase margin of the system when K = 0.16. (b) Use the phase margin to estimate £ and predict the overshoot, (c) Calculate the actual response for this second-order system, and compare the result with the part (b) estimate.
A loop transfer function isUsing the contour in the .y-plane shown in Figure E9.29, determine the corresponding contour in the F(s)-plane (B = - 1 + /). FIGURE E9.29 Contour in the s-plane.
An integrated circuit is available to serve as a feedback system to regulate the output voltage of a power supply. The Bode diagram of the required loop transfer function Gc(J(ω)G(J(ω). Estimate the gain and phase margins of the regulator.
Consider the system represented in state variable formv = Cx + Du, where C = [1000 0]. and D = [0] Sketch the Bode plot.
A closed-loop feedback system is shown in Figure E9.31. Sketch the Bode plot and determine the phase margin.FIGURE E9.31 Non unity feedback system.
Consider the system described in state variable form by y(t) = Cx(t) where Compute the phase margin.
Consider the system that compute the loop transfer function L(s), and sketch the Bode plot. Determine the phase margin and gain margin when the controller gain K = 5.
Consider a system with a loop transfer functionWe wish to obtain a resonant peak MpÏ‰) = 3.0 dB for the closed-loop system. The peak occurs between 6 and 9 rad/s and is only 1.25 dB. Plot the Nichols chart for the range of frequency from 6 to 15 rad/s. Show that the system gain needs to
An integrated CMOS digital circuit can be represented by the Bode diagram shown in Figure E9.5.(a) Find the gain and phase margins of the circuit.(b) Estimate how much we would need to reduce the system gain (dB) to obtain a phase margin of 60°.
A system has a loop transfer function(a) For K = 4, show that the gain margin is 3.5 dB. (b) If we wish to achieve a gain margin equal to 16 dB, determine the value of the gain K.
A unity feedback system has a loop transfer functionDetermine the range of K for which the system is stable using the Nyquist plot.
Consider a unity feedback system with the loop transfer function(a) For K = 4, show that the gain margin is 3.5 dB. (b) If we wish to achieve a gain margin equal to 16 dB, determine the value of the gain K.
For the system of E9.8, find the phase margin of the system for K - 5.
For the Nyquist 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.
Machine tools are often automatically controlled. 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
A control system for a chemical concentration control system. 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 The transport of the
A simplified model of the control system for regulating the pupillary aperture in the human eye. The gain K represents the pupillary gain, and r is the pupil time constant, which is 0.5 s. The time delay 7" is equal to 0.5 s. The papillary gain is equal to 2. (a) Assuming the time delay is
A controller is used to regulate the temperature of a mold for plastic part fabrication. The value of the delay time is estimated at 1.2 s. (a) Using the Nyquist criterion, determine the stability of the system for Ka = K = 1. (b) Determine a suitable value for Ka for a stable system that will
Electronics and computers arc being used to control automobiles. 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 s. (a) Using the Nichols chart, determine the
Consider the automatic ship-steering system discussed in Problem P8.ll. The frequency response of the open-loop portion of the ship steering control system is shown in Figure P8.ll. The deviation of the tanker from the straight track is measured by radar and is used to generate the error signal, as
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
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 [22]. A typical chemical reactor control scheme. The disturbance is represented by U(s), and the chemical process by G3 and G4. The
A model of an automobile driver attempting to steer a course, 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 s. (c) Find the reaction time that will cause the system to be
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(a) Plot the Nichols chart and show
Sketch the Nyquist plots of the following loop transfer functions L(s) - Gc(s)G(s), and determine whether the system is stable by applying the Nyquist criterion:If the system is stable, find the maximum value for K by determining the point where the Nyquist plot crosses the w-axis.
The Bell-Boeing V-22 Osprey Tilt rotor 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
Consider a unity feedback system with the loop transfer function(a) Sketch the Bode diagram for K = 4. Determine (b) the gain margin, (c) the value of K required to provide a gain margin equal to 12 dB, and (d) the value of K to yield a steady-state error of 25% of the magnitude A for the ramp
The Nichols diagram for Gc(jÏ‰)G(jÏ‰) of a closed loop system is shown in Figure P9.22.The frequency for each point on the graph is given in the following table:Determine (a) the resonant frequency, (b) the bandwidth, (c) The phase margin, and (d) the gain margin. (e) Estimate
A closed-loop system has a loop transfer function(a) Determine the gain K so that the phase margin is 60°. (b) For the gain K selected in part (a), determine the gain margin of the system.
A closed-loop system with unity feedback has a loop transfer function(a) Determine the gain K so that the phase margin is 45°. (b) For the gain K selected in part (a), determine the gain margin, (c) Predict the bandwidth of the closed-loop system.
A closed-loop system has the loop transfer function(a) Determine the gain K so that the phase margin is 60° when T - 0.2. (b) Plot the phase margin versus the time delay T for K as in part (a).
A specialty machine shop is improving the efficiency of its surface-grinding process [21]. 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
Consider the system that determine the maximum value of K = Kmax for which the closed-loop system is stable. Plot the phase margin as a function of the gain 1 ≤ K ≤ Kmax. Explain what happens to the phase margin as K approaches K max.
Consider the feedback system shown in Figure P9.28 with the process transfer function given asThe controller is the proportional controller Gc(s) = KP (a) Determine a value of KP such that the phase margin is approximately P.M ‰ˆ 45°. (b) Using the P.M. obtained, predict the percent
(b) Find a suitable contour Ts in the .v-plane that can be used to determine whether all the roots of the characteristic equation have real parts less than s = -Ïƒ1.(c) Using the contour of part (b) and Cauchy's theorem, determine whether the following characteristic equation has roots
The Nyquist plot of a conditionally stable system 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 Gc (s)G(s) in the right half-plane, (b) Determine whether the system is stable if the - 1
A speed control for a gasoline engine because of the restriction at the carburetor intake and the capacitance of the reduction manifold, the lag T1, occurs and is equal to 1 second. The engine time constant re is equal to J/b = 3 s. The speed measurement time constant is Te = 0.4 s. (a) Determine
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
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. At 40 knots, the dynamics of the vehicle are approximately
Electro hydraulic servomechanisms are used in control systems requiring a rapid response for a large mass. An electro hydraulic servomechanism can provide an output of 100 kW or greater [17]. A photo of a servo valve and actuator.
The space shuttle 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
Operational spacecraft undergo substantial mass property and configuration changes during their lifetime [25]. For example, the inertias change considerably during operations. Consider the orientation control system shown in Figure AP9.1.(a) Plot the Bode diagram, and determine the gain and phase
A multi loop block diagram. (a) Compute the transfer function T(s) = Y(s)/R(s). (b) Determine K such that the steady-state tracking error to a unit step input R(s) = 1/s is zero. Plot the unit step response. (c) Using K from part (b). compute the system bandwidth ωb.
Patients with a cardio logical illness and less than normal heart muscle strength can benefit from an assistance device. An electric ventricular assist device (EVAD) converts electric power into blood flow by moving a pusher plate against a flexible blood sac. The pusher plate reciprocates to eject
Anesthesia is used in surgery to induce unconsciousness. One problem with drug-induced unconsciousness is large differences in patient responsiveness. Furthermore, the patient response changes during an operation. A model of drug-induced anesthesia control is shown in Figure AP9.2. The proxy for
Welding processes have been automated over the past decades. Weld quality features, such as final metallurgy and joint mechanics, typically are not measurable online for control. Therefore, some indirect way of controlling the weld quality is necessary. A comprehensive approach to in-process
The control of a paper-making machine is quite complex [27]. The goal is to deposit the proper amount of fiber suspension (pulp) at the right speed and in a uniform way. Dewatering, fiber deposition, rolling, and drying then take place in sequence. Control of the paper weight per unit area is very
NASA is planning many Mars missions with rover vehicles. A typical rover is a solar-powered vehicle which will see where it is going with TV cameras and will measure distance to objects with laser range finders. It will be able to climb a 30° slope in dry sand and will carry a spectrometer that
Building elevators are limited to about 800 meters. Above that height, elevator cables and too heavy for practical use. One solution is to eliminate the cable. The key to the cordless elevator is the linear motor technology now being applied to the development of magnetically levitated rail
A control system is shown. The gain K is greater than 500 and less than 3000. Select a gain that will cause the system step response to have an overshoot of less than 20%. Plot the Nichols diagram, and calculate the phase margin.
Consider again the system which uses a PI controller. Letand determine the gain KP that provides the maximum phase margin.
The system of Figure CDP4.1 uses a controller Gc (s) = K0. Determine the value of Ku so that the phase margin is 70°. Plot the response of this system to a step input.

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