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

A mobile robot for toxic waste cleanup is shown in Figure DP9.1(a) [23]. The closed-loop speed control is represented by Figure 9.1 with H{s) = 1. The Nichols chart in Figure DP9.1(b) shows the plot of Gc{jÏ‰) G(jÏ‰)/K versus Ï‰. The value of the frequency at the
Consider the system is described in state variable form byy(t) = Cx(r) where Assume that the input is a linear combination of the states, that is, u(t) = -Kx(r) + r(f), where r{t) is the reference input and the gain matrix is K = [K1 K2]. Substituting u(t) into the state variable equation yields
The primary control loop of a nuclear power plant includes a time delay due to the need to transport the fluid from the reactor to the measurement point. The transfer function of the controller isThe transfer function of the reactor and time delay is where T = 0.4 s and T = 0.2 s. Using frequency
Flexible-joint robotic arms are constructed of lightweight materials and exhibit lightly damped open-loop dynamics [15]. A feedback control system for a flexible arm. Select K so that the system has maximum phase margin. Predict the overshoot for a step input based on the phase margin attained, and
An automatic drug delivery system is used in the regulation of critical care patients suffering from cardiac failure [24]. The goal is to maintain stable patient status within narrow bounds. Consider the use of a drug delivery system for the regulation of blood pressure by the infusion of a drug.
A robot tennis player is shown, and a simplified control system for θ2(t) is shown. The goal of the control system is to attain the best step response while attaining a high Kv for the system. Select Kv1 = 0.4 and Kv2 = 0.75, and determine the phase margin, gain margin, bandwidth. percent
An electro hydraulic actuator is used to actuate large loads for a robot manipulator. The system is subjected to a step input, and we desire the steady-state error to be minimized. However, we wish to keep the overshoot less than 10%. Let T = 0.8 s. (a) Select the gain K when Gc(s) - K, and
The physical representation of a steel strip-rolling mill is a damped-spring system [8]. The output thickness sensor is located a negligible distance from the output of the mill, and the objective is to keep the thickness as close to a reference value as possible. Any change of the input strip
Vehicles for lunar construction and exploration work will face conditions unlike anything found on Earth. Furthermore, they will be controlled via remote control. A block diagram of such a vehicle and the control are shown in Figure DP9.7. Select a suitable gain K when T = 0.5 s. The goal is to
The control of a high-speed steel-rolling mill is a challenging problem. The goal is to keep the strip thickness accurate and readily adjustable. The model of the control system is shown. Design a control system by selecting K so that the step response of the system is as fast as possible with an
A two-tank system containing a heated liquid has the model, where T() is the temperature of the fluid flowing into the first tank and T2 is the temperature of the liquid flowing out of the second tank. The block diagram model is shown in Figure DP9.9(b). The system of the two tanks has a heater in
Consider a unity negative feedback control system WithVerify that the gain margin is co and that the phase margin is 10°.
A closed-loop feedback system is shown in Figure CP9.10.(a) Obtain the Nyquist plot and determine the phase margin. Assume that the time delay T = 0 s.(b) Compute the phase margin when T = 0.05 s.(c) Determine the minimum time delay that destabilizes the closed-loop system.
Using the nyquist function, obtain the polar plot for the following transfer functions:(a)(b) (c)
Using the nichols function, obtain the Nichols chart with a grid for the following transfer functions:(a)(b) (c)
A negative feedback control system has the loop transfer function(a) When T = 0.2 s, find K such that the phase margin is 40° using the margin function, (b) Obtain a plot of phase margin versus T for K as in part (a), with 0 ‰¤ T ‰¤ 0.3 s.
Consider the paper machine control, Develop an m-file to plot the bandwidth of the closed-loop system as K varies in the interval 1 ≤ K ≤50.
A block diagram of the yaw acceleration control system for a bank-to-turn missile is shown. The input is yaw acceleration command (in g's), and the output is missile yaw acceleration (in g's). The controller is specified to be a proportional, integral (PI) controller. The nominal value of b0 is
An engineering laboratory has presented a plan to operate an Earth-orbiting satellite that is to be controlled from a ground station. A block diagram of the proposed system is shown. It takes T seconds for a signal to reach the spacecraft from the ground station and the identical delay for a return
Consider the system represented in state variable formy = [8 0]x + [0]Î¼ Using the nyquist function, obtain the polar plot.
For the system in CP9.8, use the nichols function to obtain the Nichols chart and determine the phase margin and gain margin.
A negative feedback control system has a transfer functionWe select a compensator in order to achieve zero steady-state error for a step input. Select a and K so that the overshoot to a step is approximately 5% and the settling time (with a 2% criterion) is approximately 1 second.
A control system with a controller is shown. We will select Kf = 2 in order to provide a reasonable steady-state error to a step [8], Find KP to obtain a phase margin of 60°. Find the peak time and percent overshoot of this system.
A unity feedback system hasA lead network is selected so that Determine the peak magnitude and the bandwidth of the closed-loop frequency response using (a) the Nichols chart, and (b) a plot of the closed-loop frequency response.
The control of an automobile ignition system has unity negative feedback and a loop transfer function Ge(s)G(s), whereA designer selects Kf/KP ~ 0.5 and asks you to determine KKP so that the complex roots have a £ of 1/V2.
The design of Example 10.3 determined a lead network in order to obtain desirable dominant root locations using a cascade compensator Gc(s) in the system configuration shown in Figure 10.1 (a). The same lead network would be obtained if we used the feedback compensation configuration. Determine the
A robot will be operated by NASA to build a permanent lunar station. The position control system for the gripper tool is shown in Figure 10.1(a), where //(5) = I, andDetermine a compensator lag network Gc(s) that will provide a phase margin of 45°.
A unity feedback control system has a plant transfer functionWe desire to attain a steady-state error to a ramp r(f) = At of less than 0.05/1 and a phase margin of 30°. We desire to have a crossover frequency coc of 10 rad/s. Use the methods of Section 10.9 to determine whether a lead or a lag
Consider again the system and specifications of Exercise E10.15 when the required crossover frequency is 2 rad/s.
Consider again the system of Exercise 10.9. Select KP and Kf so that the step response is deadbeat and the settling time (with a 2% criterion) is less than 2 seconds.
The non unity feedback control system shown in Figure E10.18 has the transfer functionsDesign a compensator Gt.(s) and prefilter Gp(s) so that the closed-loop system is stable and meets the following specifications: (i) a percent overshoot to a unit step input of less than 10%, (ii) a settling time
A unity feedback control system has the plant transfer functionDesign a PID controller of the form so that the closed-loop system has a settling time less than 1 second to a unit step input.
A control system with negative unity feedback has a process and we wish to use proportional plus integral compensation, where the steady-state error of this system for a ramp input is zero, (a) Set K/ = 1 and find a suitable value of KP so the step response will have an overshoot of approximately
Consider the system shown in Figure E 10.20. Design the proportional-derivative controller Gc(s) = KP + KDs such that the system has a phase margin of 40°
Consider the unity feedback system shown in Figure E10.21. Design the controller gain, K, such that the maximum value of the output y(t) in response to a unit step disturbance T(i(s) = l/s is less than 0.1.
A unity negative feedback control system in a manufacturing system has a process transfer functionand it is proposed that we use a compensator to achieve a 5% overshoot to a step input. The compensator is [4]
Consider a unity negative feedback system withwhere K is set equal to 100 in order to achieve a specified Kv = 2. We wish to add a lead-lag compensator Show that the gain margin of the compensated system is 28.6 dB and that the phase margin is 75.4°.
Consider a unity feedback system with the transfer functionWe desire to obtain the dominant roots with co" = 3 and £ = 0.5. We wish to obtain a Kv = 2.7. Show that we require a compensator Determine the value of K that should be selected.
Consider again the wind tunnel control system of Problem P7.31. When K = 326, find T(s) and estimate the expected overshoot and settling time (with a 2% criterion). Compare your estimates with the actual overshoot of 60% and a settling time of 4 seconds. Explain the discrepancy in your estimates.
NASA astronauts retrieved a satellite and brought it into the cargo bay of the space shuttle, as shown in Figure E10.7(a). A model of the feedback control system is shown in Figure. E10.7(b). Determine the value of K that will result in a phase margin of 40° when T = 0.6 s.
A negative unity feedback system has a plantwhere T = 2.8 ms. Select a compensator Gc(s) = KP + Kt/s, so that the dominant roots of the characteristic equation have I equal to 1/V2. Plot y(t) for a step input.
A control system with a controller is shown. Select KP and Kf so that the overshoot to a step input is equal to 5% and the velocity constant Kv is equal to 5. Verify the results of your design.
The design of a lunar excursion module (LEM) is an interesting control problem. The attitude control system for the lunar vehicle is shown in Figure PI 0.1. The vehicle damping is negligible, and the attitude is controlled by gas jets. The torque, as a first approximation, will be considered to be
A unity feedback system of the form shown in Figure 10.1(a) has a plant(a) Determine the step response when Gc(s) = l, and calculate the settling time and steady state for a ramp input r(t) = t, t > 0. (b) Design a lag network using the root locus method so that the velocity constant is increased
A unity feedback control system of the form shown in Figure 10.1(a) has a plantSelect a lead-lag compensator so that the percent overshoot for a step input is less than 5% and the settling time (with a 2% criterion) is less than 1 second. It also is desired that the acceleration constant Ka be
A unity feedback system has a plantSelect a compensator Gc(s) so that the phase margin is at least 75°. Use a two-stage lead compensator It is required that the error for a ramp input be 0.5% of the magnitude of the ramp input (Kv = 200).
Materials testing requires the design of control systems that can faithfully reproduce normal specimen operating environments over a range of specimen parameters [23]. From the control system design viewpoint, a materials-testing machine system can be considered a servomechanism in which we want to
For the system described in Problem P10.13, the goal is to achieve a phase margin of 50° with the additional requirement that the time to settle (to within 2% of the final value) be less than 4 seconds. Design a lead network to meet the specifications. As before, we require Kv = 2.
A robot with an extended arm has a heavy load, whose effect is a disturbance, as shown in Figure P10.15 [22]. Let R(s) = 0 and design Gc(s) so that the effect of the disturbance is less than 20% of the open loop system effect.
A driver and car may be represented by the simplified model shown. The goal is to have the speed adjust to a step input with less than 10% overshoot and a settling time (with a 2% criterion) of 1 second. Select a proportional plus integral (PI) controller to yield these specifications. For the
A unity feedback control system for a robot submarine has a plant with a third-order transfer function [20]:We want the overshoot to be approximately 7.5% for a step input and the settling time (with a 2% criterion) of the system be 400 ms. Find a suitable phase-lead compensator by using root locus
NASA is developing remote manipulators that can be used to extend the hand and the power of humankind through space by means of radio. A concept of a remote manipulator is shown. The closed-loop control is shown schematically. Assuming an average distance of 238,855 miles from Earth to the Moon,
Tliere have been significant developments in the application of robotics technology to nuclear power plant maintenance problems. Thus far, robotics technology in the nuclear industry has been used primarily on spent-fuel reprocessing and waste management. Today, the industry is beginning to apply
A magnetic tape recorder transport for modern computers requires a high-accuracy, rapid-response control system. The requirements for a specific transport are as follows: (1) The tape must stop or start in 10 ms, and (2) it must be possible to read 45,000 characters per second. This system was
An uncompensated control system with unity feedback has a plant transfer functionWe want to have a velocity error constant of Kv - 20. We also want to have a phase margin of approximately 45° and a closed-loop bandwidth greater than a) = 4 rad/s. Use two identical cascaded phase-lead networks to
For the system of Problem P10.20, design a phase lag network to yield the desired specifications, with the exception that a bandwidth equal to or greater than 2 rad/s will be acceptable.
For the system of Problem P10.20, we wish to achieve the same phase margin and Kv, but in addition, we wish to limit the bandwidth to less than 10 rad/s but greater than 2 rad/s. Use a lead-lag compensation network to compensate the system. The lead lag network could be of the formwhere a is to be
A system of the form of Figure 10.1 (a) with unity feedback has
The stability and performance of the rotation of a robot (similar to waist rotation) presents a challenging control problem. The system requires high gains in order to achieve high resolution; yet a large overshoot of the transient response cannot be tolerated. The block diagram of an electro
The possibility of overcoming wheel friction, wear, and vibration by contactless suspension for passenger-carrying mass-transit vehicles is being investigated throughout the world. One design uses a magnetic suspension with an attraction force between the vehicle and the guide way with an
A computer uses a printer as a fast output device. We desire to maintain accurate position control while moving the paper rapidly through the printer. Consider a system with unity feedback and a transfer function for the motor and amplifier ofDesign a lead network compensator so that the system
An engineering design team is attempting to control a process shown in Figure PI0.27. The system has a controller Gc(s), but the design team is unable to select Gc(s) appropriately. It is agreed that a system with a phase margin of 50° is acceptable, but Gc (s) is unknown. Determine
An adaptive suspension vehicle uses a legged locomotion principle. The control of the leg can be represented by a unity feedback system with [12]We desire to achieve a steady-state error for a ramp input of 10% and a damping ratio of the dominant roots of 0.707. Determine a suitable lag
A liquid-level control system (see Figure 9.32) has a loop transfer functionL(s) = Gc(s)G(s)H(s),where H(s) = 1, Gc(s) is a compensator, and the plant iswhere T - 50 ms. Design a compensator so that Mpu does not exceed 3.5 dB and
A simplified version of the attitude rate control for the F-94 or X-15 type aircraft is shown in Figure PI0.3. When the vehicle is flying at four times the speed of sound (Mach 4) at an altitude of 100,000 ft, the parameters are [26]Design a compensator Gc(s) so that the response to a step input
An automated guided vehicle (AGV) can be considered as an automated mobile conveyor designed to transport materials. Most AGVs require some type of guide path. The steering stability of the guidance control system has not been fully solved. The slight "snaking" of the AGV about the track generally
For the system of Problem PI0.30, use a phase-lag compensator and attempt to achieve a phase margin of approximately 50°. Determine the actual overshoot and peak time for the compensated system.
When a motor drives a flexible structure, the structure's natural frequencies, as compared to the bandwidth of the servo drive, determine the contribution of the structural flexibility to the errors of the resulting motion. In current industrial robots, the drives are often relatively slow, and the
Consider the extender robot presented in AP6.7. The block diagram of the system is shown in Figure P10.33 [14]. The goal is that the compensated system will have a velocity constant K " equal to 80, so that the settling time (with a 2% criterion) will be 1.6 seconds, and that the overshoot will be
A magnetically levitated train is operating in Berlin, Germany. The M-Bahn 1600-m line represents the current state of worldwide systems. Fully automated trains can run at short intervals and operate with excellent energy efficiency. The control system for the levitation of the car is shown. Select
A unity feedback system has the loop transfer functionwhere T is a time delay and K is the controller proportional gain. The block diagram is illustrated in Figure P10.35. The nominal value of K =2. Plot the phase margin of the system for 0
A system's open-loop transfer function is a pure time delay of 0.5 s, so that G(s) = e-s/2. Select a compensator Gc(s) so that the steady-state error for a step input is less than 2% of the magnitude of the step and the phase margin is greater than 30°. Determine the bandwidth of the compensated
A unity feedback system of the form hasDesign a compensator Gc(s) so that the overshoot for a step input R(s) is less than 5% and the steady-state error is less than 1 %. Determine the bandwidth of the system.
A unity feedback system has a plantWe desire to have a phase margin of 30° and a relatively large bandwidth. Select the crossover frequency ct>c = 10 rad/s, and design a lead compensator. Verify the results.
A unity feedback system has a plantWe desire that the phase margin be equal to 30°. For a ramp input r(t) ~ t, we want the steady-state error to be equal to 0.05. Design a lag compensator to satisfy the requirements. Verify the results.
Magnetic particle clutches are useful actuator devices for high power requirements because they can typically provide a 200-W mechanical power output. The particle clutches provide a high torque-to-inertia ratio and fast time-constant response. A particle clutch positioning system for nuclear
For the system and requirements of Problem P10.39, determine the required compensator when the steady-state error for the ramp input must be equal to 0.02.
Repeat Example 10.12 when we want the 100% rise time Tr = l second.
Consider again the design for Example 10.4. Using a system as shown and the compensator determined in Equation (10.46), select an appropriate prefilter. Compare the response of the system with and without the prefilter.
Consider the system shown in Figure P10.43 and let R(s) = 0 and T(i(s) = 0. Design the controller Gc (s) = Ksuch that, in the steady-state, the response of the system y(t) is less than -40 dB when the noise N(s) is a sinusoidal input at a frequency of ≥ 100 rad/s.
A unity feedback system has a loop transfer functionPlot the percent overshoot of the closed-loop system response to a unit step input for K in the range 0
A stabilized precision rate table uses a precision tachometer and a DC direct-drive torque motor, as shown in Figure P10.5. We want to maintain a high steady-state accuracy for the speed control. To obtain a zero steady-state error for a step command design, select a proportional plus integral
Repeat Problem P10.5 by using a lead network compensator, and compare the results.
A chemical reactor process whose production rate is a function of catalyst addition is shown in block diagram form in Figure P10.7 [10]. The time delay is T = 50 s, and the time constant T is approximately 40 s. The gain of the process is K = 1. Design a compensation by using Bode diagram methods
A numerical path-controlled machine turret lathe is an interesting problem in attaining sufficient accuracy [2, 23]. A block diagram of a turret lathe control system is shown. The gear ratio is n = 0.1. J - 10-3, and b - 10-2. It is necessary to attain an accuracy of 5 x 10-4 in., and therefore a
The Avemar ferry, shown in Figure PI0.9(a), is a large 670-ton ferry hydrofoil built for Mediterranean ferry service. It is capable of 45 knots (52 mph) [29]. The boat's appearance, like its performance, derives from the innovative design of the narrow "wave piercing" hulls which move through the
A three-axis pick-and-place application requires the precise movement of a robotic arm in three dimensional space, as shown APlO.l for joint2. The arm has specific linear paths it must follow to avoid other pieces of machinery. The overshoot for a step input should be less than 13%.(a) Let Gc (s) =
The system of Advanced Problem APlO.l is to have a percent overshoot less than 13%. In addition, we desire that the steady-state error for a unit ramp input will be less than 0.125 (Kv = 8) [24]. Design a lag network to meet the specifications. Check the resulting percent overshoot and settling
The system of Advanced Problem AP 10.1 is required to have a percent overshoot less than 13% with a steady state error for a unit ramp input less than 0.125(Kv = 8).Design a proportional plus integral (PI) controller to meet the specifications.
A DC motor control system with unity feedback has the form. Select K1 and K2 so that the system response has a settling time (with a 2% criterion) less than 1 second and an overshoot less than 5% for a step input.
A unity feedback system is shown in Figure AP10.5. We want the step response of the system to have an overshoot of about 10% and a settling time (with a 2% criterion) of about 4 seconds.(a) Design a lead compensator Gc (s) to achieve the dominant roots desired, (b) Determine the step response of
Consider again Example 10.12 when we wish to minimize the settling time of the system while requiring that K < 52. Determine the appropriate compensator that will minimize the settling time. Plot the system response.
A system has the form shown in Figure 10.22, withA lead compensator is used, with(a) Determine the overshoot and rise time for Gp (s) = 1 and for p = 3. (b) Select an appropriate value for p that will give an overshoot of 1 % and compare the results.
The Manutec robot has large inertia and arm length resulting in a challenging control problem, as shown in Figure AP10.8(a). The block diagram model of the system is shown in Figure AP10.8(b).The plant dynamics are represented byThe percentage overshoot for a step input should be less than 20% with
Tire plant dynamics of a chemical process are represented byWe desire that the system have a small steady-state error for a ramp input so that Kv = 100. For stability purposes, we desire a gain margin of 10 dB or greater and a phase margin of 40° or greater. Determine a lead-lag compensator
Two robots are shown cooperating with each other to manipulate a long shaft to insert it into the hole in the block resting on the table. Long part insertion is a good example of a task that can benefit from cooperative control. The unity feedback control system of one robot joint has the process
A unity feedback system has the process transfer functionDesign the controller Gc(s) such that the Bode magnitude plot of the loop transfer function L(s) = Gc(s)G(s) is greater than 20 dB for «
Modern micro analytical systems used for polymerase chain reaction (PCR) require fast, damped tracking response. The control of the temperature of the PCR reactor can be represented as shown. The controller is chosen to be PID controller, denoted by Gc(s), with a prefilter, denoted by Gp(s). The
The heading control of the traditional bi-wing aircraft, is represented by the block diagram.(a) Determine the minimum value of the gain K when Gc{s) = K, so that the steady-state effect of a unit step disturbance Td(s) = 1/sis less than or equal to 5% of the unit step (y(oo) = 0.05).(b) Determine
NASA has identified the need for large deployable space structures, which will be constructed of lightweight materials and will contain large numbers of joints or structural connections. This need is evident for programs such as the space station. These deployable space structures may have

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