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Modern Control Systems 12th edition Richard C. Dorf, Robert H. Bishop - Solutions
Determine the root sensitivity of the dominant roots of Problem P7.1(a) when K is set so that the damping ratio of the unperturbed roots is 0.707. Evaluate and compare the sensitivity as a function of the poles and zeros of Gc(s)G(s).
Repeat Problem P7.22 for the loop transfer function Gc(s)G(s) of Problem P7.1(c). Problem P7.22 Determine the root sensitivity of the dominant roots of Problem P7.1(a) when K is set so that the damping ratio of the unperturbed roots is 0.707. Evaluate and compare the sensitivity as a function of
For systems of relatively high degree, the form of the root locus can often assume an unexpected pattern. The root loci of four different feedback systems of third order or higher are shown in Figure P7.24. The open-loop poles and zeros of KG(s) are shown, and the form of the root loci as K varies
Solid-state integrated electronic circuits are composed of distributed R and C elements. Therefore, feedback electronic circuits in integrated circuit form must be investigated by obtaining the transfer function of the distributed RC networks. It has been shown that the slope of the attenuation
A single-loop negative feedback system has a loop transfer function(a) Sketch the root locus for 0 ‰¤ K ‰¤ ˆž to indicate the significant features of the locus. (b) Determine the range of the gain K for which the system is stable. (c) For what value of K in the
A unity negative feedback system has a loop transfer functionSketch the root locus as a function of K. Carefully calculate where the segments of the locus enter and leave the real axis.
To meet current U.S. emissions standards for automobiles, hydrocarbon (HC) and carbon monoxide (CO) emissions are usually controlled by a catalytic converter in the automobile exhaust. Federal standards for nitrogen oxides (NOx) emissions are met mainly by exhaust-gas recirculation (EGR)
A unity feedback control system has a transfer functionWe desire the dominant roots to have a damping ratio equal to 0.707. Find the gain K when this condition is satisfied. Show that the complex roots are s = -3.56 ± j3.56 at this gain.
A unity feedback system has the loop transfer functionFind (a) The breakaway point on the real axis and the gain K for this point (b) The gain and the roots when two roots lie on the imaginary axis, and (c) The roots when K = 6. (d) Sketch the root locus.
An RLC network is shown in Figure P7.30. The nominal values (normalized) of the network elements are L - C = 1 and R = 2.5. Show that the root sensitivity of the two roots of the input impedance Z(s) to a change in R is different by a factor of 4.Figure P7.30RLC network.
The development of high-speed aircraft and missiles requires information about aerodynamic parameters prevailing at very high speeds. Wind tunnels are used to test these parameters. These wind tunnels are constructed by compressing air to very high pressures and releasing it through a valve to
A mobile robot suitable for nighttime guard duty is available. This guard never sleeps and can tirelessly patrol large warehouses and outdoor yards. The steering control system for the mobile robot has a unity feedback with the loop transfer function(a) Find K for all breakaway and entry points on
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 90° from a vertical position for takeoffs and landings as shown in Figure P7.33(a), and then to switch the engines to a horizontal position for cruising as an
The fuel control for an automobile uses a diesel pump that is subject to parameter variations. A unity negative feedback has a loop transfer function(a) Sketch the root locus as K varies from 0 to 2000. (b) Find the roots for K equal to 400, 500, and 600. (c) Predict how the percent overshoot to a
A powerful electro hydraulic forklift can be used to lift pallets weighing several tons on top of 35-foot scaffolds at a construction site. The negative unity feedback system has a loop transfer function(a) Sketch the root locus for K > 0. (b) Find the gain K when two complex roots have a
A micro robot with a high-performance manipulator has been designed for testing very small particles, such as simple living cells [6]. The single-loop unity negative feedback system has a loop transfer function(a) Sketch the root locus for K > 0. (b) Find the gain and roots when the characteristic
Identify the parameters K, a, and b of the system shown in Figure P7.37. The system is subject to a unit step input, and the output response has an overshoot but ultimately attains the final value of 1. When the closed-loop system is subjected to a ramp input, the output response follows the ramp
A unity feedback system has the loop transfer functionThis system is open-loop unstable. (a) Determine the range of K so that the closed-loop system is stable. (b) Sketch the root locus. (c) Determine the roots for K = 10. (d) For K = 10, predict the percent overshoot for a step input using Figure
High-speed trains for U.S. railroad tracks must traverse twists and turns. In conventional trains, the axles are fixed in steel frames called trucks. The trucks pivot as the train goes into a curve, but the fixed axles stay parallel to each other, even though the front axle tends to go in a
The analysis of a large antenna was presented in Problem P4.5. Sketch the root locus of the system as 0 < ka < ∞. Determine the maximum allowable gain of the amplifier for a stable system.
Automatic control of helicopters is necessary because, unlike fixed-wing aircraft which possess a fair degree of inherent stability, the helicopter is quite unstable. A helicopter control system that utilizes an automatic control loop plus a pilot stick control is shown in Figure P7.5. When the
An attitude control system for a satellite vehicle within the earth's atmosphere is shown in Figure P7.6. The transfer functions of the system areand(a) Draw the root locus of the system as K varies from 0 to ˆž.(b) Determine the gain K that results in a system with a settling time (with a 2%
The speed control system for an isolated power system is shown in Figure P7.7. The valve controls the steam flow input to the turbine in order to account for load changes ΔL(s) within the power distribution network. The equilibrium speed desired results in a generator frequency equal to
Consider again the power control system of Problem P7.7 when the steam turbine is replaced by a hydro turbine. For hydro turbines, the large inertia of the water used as a source of energy causes a considerably larger time constant. The transfer function of a hydro turbine may be approximated
The achievement of safe, efficient control of the spacing of automatically controlled guided vehicles is an important part of the future use of the vehicles in a manufacturing plant [14, 15]. It is important that the system eliminate the effects of disturbances (such as oil on the floor) as well as
The top view of a high-performance jet aircraft is shown in Figure AP7.1(a) [20]. Sketch the root locus and determine the gain K so that the ζ of the complex poles near the jω-axis is the maximum achievable. Evaluate the roots at this K and predict the response to a step
A feedback system is shown in Figure AP7.10. Sketch the root locus as K varies when K ‰¥ 0. Determine a value for K that will provide a step response with an overshoot less than 5% and a settling time (with a 2% criterion) less than 2.5 seconds.
A control system is shown in Figure AP7.11. Sketch the root locus, and select a gain K so that the step response of the system has an overshoot of less than 10% and the settling time (with a 2% criterion) is less than 4 seconds.Figure AP7.11A control system with parameter K.
A control system with PI control is shown in Figure AP7.12.(a) Let KI/KP = 0.2 and determine KP so that the complex roots have maximum damping ratio.(b) Predict the step response of the system with KP set to the value determined in part (a).Figure AP7.12A control system with a PI controller.
The feedback system shown in Figure AP7.13 has two unknown parameters K1 and K2. The process transfer function is unstable. Sketch the root locus for 0 ‰¤ K1, K2 Figure AP7.13An unstable plant with two parameters K1 and K2.
A unity feedback control system shown in Figure AP7.14 has the processDesign a PID controller using Ziegler-Nichols methods. Determine the unit step response and the unit disturbance response. What is the maximum percent overshoot and settling time for the unit step input? Figure AP7.14 Unity
A magnetically levitated high-speed train "flies" on an air gap above its rail system, as shown in Figure AP7.2(a) [24]. The air gap control system has a unity feedback system with a loop transfer functionThe feedback control system is illustrated in Figure AP7.2(b). The goal is to select K so that
A compact disc player for portable use requires a good rejection of disturbances and an accurate position of the optical reader sensor. The position control system uses unity feedback and a loop transfer functionThe parameter p can be chosen by selecting the appropriate DC motor. Sketch the root
A remote manipulator control system has unity feedback and a loop transfer functionWe want the steady-state position error for a step input to be less than or equal to 10% of the magnitude of the input. Sketch the root locus as a function of the parameter α. Determine the range of
A unity feedback system has a loop transfer function(a) Sketch the root locus and determine K for a stable system with complex roots with ζ equal to l/ˆš2. (b) Determine the root sensitivity of the complex roots of part (a). (c) Determine the percent change in K (increase or
A unity feedback system has a loop transfer functionSketch the root locus for K > 0, and select a value for K that will provide a closed step response with settling time less than 1 second.
A feedback system with positive feedback is shown in Figure AP7.7. The root locus for K > 0 must meet the conditionSketch the root locus for 0
A position control system for a DC motor is shown in Figure AP7.8. Obtain the root locus for the velocity feedback constant K, and select K so that all the roots of the characteristic equation are real (two are equal and real). Estimate the step response of the system for the K selected. Compare
A control system is shown in Figure AP7.9. Sketch the root loci for the following transfer functions Gc(s):(a) Gc(s) = K(b) Gc(s) = K(s + 3)(c)(d) Figure AP7.9 A unity feedback control system.
The drive motor and slide system uses the output of a tachometer mounted on the shaft of the motor as shown in Figure CDP4.1 (switch-closed option). The output voltage of the tachometer is vT = K1θ. Use the velocity feedback with the adjustable gain K1. Select the best values for the
A high-performance aircraft, shown in Figure DP7.1(a), uses the ailerons, rudder, and elevator to steer through a three-dimensional flight path [20]. The pitch rate control system for a fighter aircraft at 10.000 m and Mach 0.9 can be represented by the system in Figure DP7.1(b), where(a) Sketch
The four-wheel-steering automobile has several benefits. The system gives the driver a greater degree of control over the automobile. The driver gets a more forgiving vehicle over a wide variety of conditions. The system enables the driver to make sharp, smooth lane transitions. It also prevents
A pilot crane control is shown in Figure DP7.11(a). The trolley is moved by an input F(t) in order to control x(t) and Ï•(t) [13]. The model of the pilot crane control is shown in Figure DP7.11(b). Design a controller that will achieve control of the desired variables when Gc(s) =
A rover vehicle designed for use on other planets and moons is shown in Figure DP7.12(a) [21]. The block diagram of the steering control is shown in Figure DP7.12(b), where(a) When Gc(s) = K, sketch the root locus as K varies from 0 to 1000. Find the roots for K equal to 100, 300, and 600. (b)
The automatic control of an airplane is one example that requires multiple-variable feedback methods. In this system, the attitude of an aircraft is controlled by three sets of surfaces: elevators, a rudder, and ailerons, as shown in Figure DP7.I3(a). By manipulating these surfaces, a pilot can set
Consider the feedback system shown in Figure DP7.14. The process transfer function is marginally stable. The controller is the proportional-derivative (PD) controllerGc(s) = KP + KDs.(a) Determine the characteristic equation of the closed-loop system.(b) Let Ñ‚ = KP/KD. Write the
A large helicopter uses two tandem rotors rotating in opposite directions, as shown in Figure P7.33(a). The controller adjusts the tilt angle of the main rotor and thus the forward motion as shown in Figure DP7.2.The helicopter dynamics are represented byand the controller is selected as (a) Sketch
The vehicle Rover has been designed for maneuvering at 0.25 mph over Martian terrain. Because Mars is 189 million miles from Earth and it would take up to 40 minutes each way to communicate with Earth [22, 27], Rover must act independently and reliably. Resembling a cross between a small flatbed
A welding torch is remotely controlled to achieve high accuracy while operating in changing and hazardous environments [21]. A model of the welding arm position control is shown in Figure DP7.4, with the disturbance representing the environmental changes.(a) With Td(s) = 0, select K1 and K to
A high-performance jet aircraft with an autopilot control system has a unity feedback and control system, as shown in Figure DP7.5. Sketch the root locus and select a gain K that leads to dominant poles. With this gain K, predict the step response of the system. Determine the actual response of the
A system to aid and control the walk of a partially disabled person could use automatic control of the walking motion [25]. One model of a system that is open-loop unstable is shown in Figure DP7.6. Using the root locus, select K for the maximum achievable ζ of the complex roots.
A mobile robot using a vision system as the measurement device is shown in Figure DP7.7(a) [36]. The control system is shown in Figure DP7.7(b) whereG(s) = 1 / (s + 1)(0.5s + 1).and Gc(.f) is selected as a PI controller so that the steady-state error for a step input is equal to zero. We then
Most commercial op-amps are designed to be unity-gain stable [26]. That is, they are stable when used in a unity-gain configuration. To achieve higher bandwidth, some op-amps relax the requirement to be unity-gain stable. One such amplifier has a DC gain of 105 and a bandwidth of 10 kHz. The
A robotic arm actuated at the elbow joint is shown in Figure DP7.9(a), and the control system for the actuator is shown in Figure DP7.9(b). Plot the root locus for K ‰¥ 0. Select Gp(s) so that the steady-state error for a step input is equal to zero. Using the Gp(s) selected, plot y(t) for
Using the rlocus function, obtain the root locus for the following transfer functions of the system shown in Figure CP7.1 when 0 (a)(b) (c) (d) Figure CP7.1 A single-loop feedback system with parameter K.
Consider the system represented in state variable formx = Ax + Buy = Cx + Du,where(a) Determine the characteristic equation. (b) Using the Routh-Hurwitz criterion, determine the values of k for which the system is stable. (c) Develop an m-file to plot the root locus and compare the results to those
A unity negative feedback system has the loop transfer functionDevelop an m-file to plot the root locus and show with the rlocfind function that the maximum value of K for a stable system is K = 0.79.
Compute the partial fraction expansion ofand verify the result using the residue function.
A unity negative feedback system has the loop transfer functionDevelop an m-file to obtain the root locus as p varies; 0
Consider the feedback system shown in Figure CP7.1, whereFor what value of K is ζ = 0.707 for the dominant closed-loop poles?
Large antenna, as shown in Figure CP7.6(a), is used to receive satellite signals and must accurately track the satellite as it moves across the sky. The control system uses an armature-controlled motor and a controller to be selected, as shown in Figure CP7.6(b). The system specifications require a
Consider the feedback control system in Figure CP7.7. We have three potential controllers for our system:1. Gc(s) = K (proportional controller)2. Gc(s) = K/s (integral controller)3. Gc(s) = K(1 + 1/s) (proportional, integral (PI) controller)The design specifications are Ts ‰¤ 10 seconds
Consider the spacecraft single-axis attitude control system shown in Figure CP7.8. The controller is known as a proportional-derivative (PD) controller. Suppose that we require the ratio of KP/KD = 5. Then, develop an m-file using root locus methods find the values of KD/J and KP/J so that the
Consider the feedback control system in Figure CP7.9. Develop an m-file to plot the root locus for 0 Figure CP7.9Unity feedback system with parameter K.
Increased track densities for computer disk drives necessitate careful design of the head positioning control [1]. The loop transfer function isPlot the frequency response for this system when K = 4. Calculate the phase and magnitude at ω = 0.5, 1, 2, 4, and ˆž.
The dynamic analyzer shown in Figure E8.10(a) can be used to display the frequency response of a system. Also shown is the signal analyzer used to measure the mechanical vibration in the cockpit of an automobile. Figure E8.10(b) shows the actual frequency response of a system. Estimate the poles
Consider the feedback control system in Figure E8.11. Sketch the Bode plot of G(s) and determine the crossover frequency, that is the frequency when 20 1og10|G(jω)| = 0dB.Figure E8.11Unity feedback system.
Consider the system represented in state variable form(a) Determine the transfer function representation of the system. (b) Sketch the Bode plot.
Determine the bandwidth of the feedback control system in Figure E8.I3.Figure E8.13Third-order feedback system.
Consider the non unity feedback system in Figure E8.14, where the controller gain is K = 2. Sketch the Bode plot of the loop transfer function. Determine the phase of the loop transfer function when the magnitude 20 log|L(jω)| = 0 dB. Recall that the loop transfer function is L(s) =
Consider the single-input, single-output system described byx(t) = Ax(t) + Bu(t)y(t) = Cx(t)whereCompute the bandwidth of the system for K = 1, 2, and 10. As K increases, does the bandwidth increase or decrease?
A tendon-operated robotic hand can be implemented using a pneumatic actuator [8]. The actuator can be represented byPlot the frequency response of G(jω). Show that the magnitude of G(jω) is -17 dB at ω = 10 and -27.1 dB at ω = 200. Show also that the
A robotic arm has a joint-control loop transfer functionShow that the frequency equals 28.3 rad/s when the phase angle of L(jω) is -180°. Find the magnitude of L(jω) at that frequency.
The frequency response for a process of the formis shown in Figure E8.4. Determine K and a by examining the frequency response curves. Figure E8.4 Bode diagram.
The magnitude plot of a transfer functionis shown in Figure E8.5. Determine K, a, and b from the plot. Figure E8.5 Bode diagram.
Several studies have proposed an extravehicular robot that could move around in a NASA space station and perform physical tasks at various worksites [9]. The arm is controlled by a unity feedback control with loop transfer functionDraw the Bode diagram for K = 20, and determine the frequency when
Consider a system with a closed-loop transfer functionThis system will have no steady-state error for a step input. (a) Plot the frequency response, noting the two peaks in the magnitude response. (b) Predict the time response to a step input, noting that the system has four poles and cannot be
A feedback system has a loop transfer function(a) Determine the corner frequencies (break frequencies) for the Bode plot. (b) Determine the slope of the asymptotic plot at very low frequencies and at high frequencies. (c) Sketch the Bode magnitude plot.
The Bode diagram of a system is shown in Figure E8.9. Determine the transfer function G(s).Figure E8.9Bode diagram.
Sketch the polar plot of the frequency response for the following loop transfer functions:(a)(b) (c) (d)
A linear actuator is used in the system shown in Figure P8.10 to position a mass M. The actual position of the mass is measured by a slide wire resistor, and thus H{s) = 1.0. The amplifier gain is selected so that the steady-state error of the system is less than 1 % of the magnitude of the
Automatic steering of a ship would be a particularly useful application of feedback control theory [20]. In the case of heavily traveled seas, it is important to maintain the motion of the ship along an accurate track. An automatic system would be more likely to maintain a smaller error from the
The block diagram of a feedback control system is shown in Figure P8.12(a).The transfer functions of the blocks are represented by the frequency response curves shown in Figure P8.12(b).
A position control system may be constructed by using an AC motor and AC components, as shown in Figure P8.13.The syncro and control transformer may be considered to be a transformer with a rotating winding. The syncro position detector rotor turns with the load through an angle 80. The syncro
(a) Sketch the Bode diagram of G(jω).(b) Find the mid band gain (in dB).(c) Find the high and low frequency - 3 dB points.
To determine the transfer function of a process G(s), the frequency response may be measured using a sinusoidal input. One system yields the data in the following table:Determine the transfer function G(s).
The space shuttle has been used to repair satellites and the Hubble telescope. Figure P8.16 illustrates howa crew member, with his feet strapped to the platform on the end of the shuttle's robotic arm, used his arms to stop the satellite's spin. The control system of the robotic arm has a
The experimental Oblique Wing Aircraft (OWA) has a wing that pivots, as shown in Figure P8.17. The wing is in the normal un skewed position for low speeds and can move to a skewed position for improved supersonic flight [11]. The aircraft control system loop transfer function is(a) Sketch the Bode
Remote operation plays an important role in hostile environments, such as those in nuclear or high temperature environments and in deep space. In spite of the efforts of many researchers, a teleportation system that is comparable to the human's direct operation has not been developed. Research
A DC motor controller used extensively in automobiles is shown in Figure P8.19(a). The measured plot of Q(s)/I(s) is shown in Figure P8.19(b). Determine the transfer function of Q(s)/I(s).FIGURE P8.19 (a) Motor controller. (b) Measured plot.
Sketch the Bode diagram representation of the frequency response for the transfer functions given in Problem P8.1.
For the successful development of space projects, robotics and automation will be a key technology. Autonomous and dexterous space robots can reduce the workload of astronauts and increase operational efficiency in many missions. Figure P8.20 shows a concept called a free-flying robot [9,13]. A
Low-altitude wind shear is a major cause of air carrier accidents in the United States. Most of these accidents have been caused by either micro bursts (small-scale, low-altitude, intense thunderstorm downdrafts that impact the surface and cause strong divergent outflows of wind) or by the gust
The frequency response of a process G(j(o). Deduce the type number (number of integrations) for the system. Determine the transfer function of the system, G(s). Calculate the error to a unit step input.
The Bode diagram of a closed-loop film transport system. Assume that the system transfer function T(s) has two dominant complex conjugate poles, (a) Determine the best second- order model for the system, (b) Determine the system bandwidth, (c) Predict the percent overshoot and settling time (with
Determine the transfer function of the op-amp circuit shown in Figure P8.26. Assume an ideal op-amp. Plot the frequency response when R1 = 9k . R2 = 1 k, and C = μF
A unity feedback system has the loop transfer functionSketch the Bode plot of the loop transfer function and indicate how the magnitude 20 log |L(jω)| plot varies as K varies. Develop a table for K = 0.75,2, and 10, and for each K determine the crossover frequency (ωc. for
A rejection network that can be used instead of the twin-T network of Example 8.4 is the bridged-T network shown in Figure P8.3. The transfer function of this network iswhereCompute the bandwidth of the system for K - 1,2, and 10. As K increases, does the bandwidth increase or decrease?FIGURE P8.3
A control system for controlling the pressure in a closed chamber is shown in Figure P8.4. The transfer function for the measuring element isand the transfer function for the valve is The controller transfer function is Gc(s) = 2s + 1. Obtain the frequency response characteristics for the loop
The robot industry in the United States is growing at a rate of 30% a year [8]. A typical industrial robot has degrees of freedom. A unity feedback position control system for a force-sensing joint has a loop transfer functionwhere K = 10. Sketch the Bode diagram of this system.
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