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Modern Control Systems 12th edition Richard C. Dorf, Robert H. Bishop - Solutions
A capstan drive for a table slide is described in CDP2.1. The position of the slide x is measured with a capacitance gauge, as shown in Figure CDP4.1, which is very linear and accurate. Sketch the model of the feedback system and determine the response of the system when the controller is an
A closed-loop speed control system is subjected to a disturbance due to a load, as shown in Figure DP4.1. The desired speed is ωd(t) = 100 rad/s, and the load disturbance is a unit step input Td(s) = l/s. Assume that the speed has attained the no-load speed of 100 rad/s and is in steady
The control of the roll angle of an airplane is achieved by using the torque developed by the ailerons. A linear model of the roll control system for a small experimental aircraft is shown in Figure DP4.2, whereThe goal is to maintain a small roll angle θ due to disturbances. Select an
The speed control system of Figure DP4.1 is altered so that G(s) = l/(s + 5) and the feedback is K1 as shown in Figure DP4.3.(a) Determine the range of K1 allowable so that the steady state is ess ‰¤ 1%.(b) Determine a suitable value for K1 and K so that the magnitude of the
Lasers have been used in eye surgery for more than 25 years. They can cut tissue or aid in coagulation [17]. The laser allows the ophthalmologist to apply heat to a location in the eye in a controlled manner. Many procedures use the retina as a laser target. The retina is the thin sensory tissue
An op-amp circuit can be used to generate a short pulse. The circuit shown in Figure DP4.5 can generate the pulse v0(t) = 5e-100t, t > 0, when the input v(t) is a unit step [6]. Select appropriate values for the resistors and capacitors. Assume an ideal op-amp.
A hydrobot is under consideration for remote exploration under the ice of Europa, a moon of the giant planet Jupiter. Figure DP4.6(a) shows one artistic version of the mission. The hydrobot is a self-propelled underwater vehicle that would analyze the chemical composition of the water in a search
Interest in unmanned underwater vehicles (UUVs) has been increasing recently, with a large number of possible applications being considered. These include intelligence-gathering, mine detection, and surveillance applications. Regardless of the intended mission, a strong need exists for reliable and
A new suspended, mobile, remote-controlled video camera system to bring three-dimensional mobility to professional football is shown in Figure DP4.8(a) [29]. The camera can be moved over the field, as well as up and down. The motor control on each pulley is represented by the system in Figure
Consider a unity feedback system withObtain the step response and determine the percent overshoot. What is the steady-state error?
Consider the closed-loop system is depicted in Figure CP4.10. The controller gain K can be modified to meet the design specifications.(a) Determine the closed-loop transfer function T(s) = Y(s)/R(s).(b) Plot the response of the closed-loop system for K = 5, 10, and 50.(c) When the controller gain
Consider the non-unity feedback system is depicted in Figure CP4.11.(a) Determine the closed-loop transfer function T(s) = Y(s)/R(s).(b) For K = 10, 12, and 15, plot the unit step responses. Determine the steady-state errors and the settling times from the plots.For parts (a) and (b), develop an
Consider the transfer function (without feedback)When the input is a unit step, the desired steady-state value of the output is one. Using the step function, show that the steady-state error to a unit step input is 0.8.
Consider the closed-loop transfer functionObtain the family of step responses for K = 10, 200, and 500. Co-plot the responses and develop a table of results that includes the percent overshoot, settling time, and steady-state error.
Consider the feedback system in Figure CP4.4. Suppose that the controller isGc(s) = K = 10.Figure CP4.4Unity feedback system with controller gain K.(a) Develop an m-file to compute the closed-loop transfer function T(s) = Y(s)/R(s) and plot the unit step response. (b) In the same m-file, compute
Consider the closed-loop control system shown in Figure CP4.5. Develop an m-file script to assist in the search for a value of k so that the percent overshoot to a unit step input is greater than 1%, but less than 10%. The script should compute the closed-loop transfer function T(s) = Y(s)/R(s) and
Consider the closed-loop control system shown in Figure CP4.6. The controller gain is K = 2. The nominal value of the plant parameter is a = 1. The nominal value is used for design purposes only, since in reality the value is not precisely known. The objective of our analysis is to investigate the
Consider the torsional mechanical system in Figure CP4.7(a). The torque due to the twisting of the shaft is -kθ; the damping torque due to the braking device is -bθ; the disturbance torque is td(t); the input torque is r(t); and the moment of inertia of the mechanical
A negative feedback control system is depicted in Figure CP4.8. Suppose that our design objective is to find a controller Gc(s) of minimal complexity such that our closed-loop system can track a unit step input with a steady-state error of zero.(a) As a first try, consider a simple proportional
Consider the closed-loop system in Figure CP4.9, whose transfer function is(a) Obtain the closed-loop transfer function T(s) = Y(s)/R(s) and the unit step response; that is, let R(s) = l/s and assume that N(s) = 0. (b) Obtain the disturbance response when N(s) = 100 / s2 + 100 is a sinusoidal input
A second-order control system has the closed-loop transfer function T(s) = Y(s)/R(s). The system specifications for a step input follow: (1) Percent overshoot P.O. ≤ 5%. (2) Settling time Ts < 4s. (3) Peak time Tp < 1s. Show the permissible area for the poles of T(s) in order to achieve the
A system with unity feedback is shown in Figure E5.11. Determine the steady-state error for a step and a ramp input whenFigure E5.11 Unity feedback system.
We are all familiar with the Ferris wheel featured at state fairs and carnivals. George Ferris was born in Galesburg, Illinois, in 1859; he later moved to Nevada and then graduated from Rensselaer Polytechnic Institute in 1881. By 1891, Ferris had considerable experience with iron, steel, and
For the system with unity feedback shown in Figure E5.l l, determine the steady-state error for a step and a ramp input whenFigure E5.11 Unity feedback system.
A feedback system is shown in Figure E5.14.(a) Determine the steady-state error for a unit step when K = 0.4 and Gp(s) = 1.(b) Select an appropriate value for Gp(s) so that the steady-state error is equal to zero for the unit step input.
A closed-loop control system has a transfer function T(s) as follows:Plot y(t) for a step input R(s) when (a) the actual T(s) is used, and (b) using the relatively dominant complex poles. Compare the results.
A second-order system isConsider the case where 1
A closed-loop control system transfer function T(s) has two dominant complex conjugate poles. Sketch the region in the left-hand 5-plane where the complex poles should be located to meet the given specifications. (a) 0.6 ≤ ζ ≤ 0.8, ωn ≤ 10 (b) 0.5 ≤ ζ ≤ 0.707, ωn ≥ 10 (c) ζ ≥
A system is shown in Figure E5.18(a). The response to a unit step, when K = 1, is shown in Figure E5.18(b). Determine the value of K so that the steady-state error is equal to zero.
A second-order system has the closed-loop transfer function(a) Determine the percent overshoot P.O., the time to peak Tp, and the settling time Ts of the unit step response, R(s) = 1/s. To compute the settling time, use a 2% criterion. (b) Obtain the system response to a unit step and verify the
The engine, body, and tires of a racing vehicle affect the acceleration and speed attainable [9]. The speed control of the car is represented by the model shown in Figure E5.2.(a) Calculate the steady-state error of the car to a step command in speed.(b) Calculate overshoot of the speed to a step
Consider the closed-loop system in Figure E5.19, where(a) Determine the closed-loop transfer function T(s) = Y(s)/R(s). (b) Determine the steady-state error of the closed-loop system response to a unit ramp input, R(s) = 1/s2. (c) Select a value for Ka so that the steady-state error of the system
New passenger rail systems that could profitably compete with air travel are under development. Two of these systems, the French TGV and the Japanese Shinkansen, reach speeds of 160 mph [17]. The Transrapid, a magnetic levitation train, is shown in Figure E5.3(a).The use of magnetic levitation and
A feedback system with negative unity feedback has a loop transfer function(a) Determine the closed-loop transfer function T(s) = Y(s)/R(s). (b) Find the time response, y(t), for a step input r(t) = A for t > 0. (c) Using Figure 5.13(a), determine the overshoot of the response. (d) Using the
Consider the feedback system in Figure E5.5. Find K such that the closed-loop system minimizes the ITAE performance criterion for a step input.Figure E5.5Feedback system with proportional controller Gc(s) = K.
Consider the block diagram shown in Figure E5.6 [16].(a) Calculate the steady-state error for a ramp input.(b) Select a value of K that will result in zero overshoot to a step input. Provide the most rapid response that is attainable.Plot the poles and zeros of this system and discuss the dominance
Effective control of insulin injections can result in better lives for diabetic persons. Automatically controlled insulin injection by means of a pump and a sensor that measures blood sugar can be very effective. A pump and injection system has a feedback control as shown in Figure E5.7. Calculate
A control system for positioning the head of a floppy disk drive has the closed-loop transfer functionPlot the poles and zeros of this system and discuss the dominance of the complex poles. What overshoot for a step input do you expect?
A unity negative feedback control system has the loop transfer function(a) Determine the percent overshoot and settling time (using a 2% settling criterion) due to a unit step input. (b) For what range of K is the settling time less than 1 second?
An important problem for television systems is the jumping or wobbling of the picture due to the movement of the camera. This effect occurs when the camera is mounted in a moving truck or airplane. The Dynalens system has been designed to reduce the effect of rapid scanning motion; see Figure P5.1.
A speed control system of an armature-controlled DC motor uses the back emf voltage of the motor as a feedback signal. (a) Draw the block diagram of this system. (b) Calculate the steady-state error of this system to a step input command setting the speed to a new level. Assume that Ra = La = J = b
A simple unity feedback control system has a process transfer functionThe system input is a step function with an amplitude A. The initial condition of the system at time t0 is y(t0) = Q, where y(t) is the output of the system. The performance index is defined as (a) Show that I = (A -
Train travel between cities will increase as trains are developed that travel at high speeds, making the travel time from city center to city center equivalent to airline travel time. The Japanese National Railway has a train called the Bullet Express that travels between Tokyo and Osaka on the
We want to approximate a fourth-order system by a lower-order model. The transfer function of the original system isShow that if we obtain a second-order model by the method of Section 5.8, and we do not specify the poles and the zero of GL(s), we have
For the original system of Problem P5.13, we want to find the lower-order model when the poles of the second-order model are specified as -1 and -2 and the model has one unspecified zero. Show that this low-order model is
Consider a unity feedback system with loop transfer functionDetermine the value of the gain K such that the percent overshoot to a unit step is minimized.
A magnetic amplifier with a low-output impedance is shown in Figure P5.16 in cascade with a low-pass filter and a preamplifier. The amplifier has a high-input impedance and a gain of 1 and is used for adding the signals as shown. Select a value for the capacitance C so that the transfer function
Electronic pacemakers for human hearts regulate the speed of the heart pump. A proposed closed-loop system that includes a pacemaker and the measurement of the heart rate is shown in Figure P5.17 [2, 3]. The transfer function of the heart pump and the pacemaker is found to beDesign the amplifier
Consider the original third-order system given in Example 5.9. Determine a first-order model with one pole unspecified and no zeros that will represent the third-order system.
A closed-loop control system with negative unity feedback has a loop transfer function(a) Determine the closed-loop transfer function T(s). (b) Determine a second-order approximation for T(s). (c) Plot the response of T(s) and the second-order approximation to a unit step input and compare the
A specific closed-loop control system is to be designed for an under damped response to a step input. The specifications for the system are as follows: 10% < percent overshoot < 20%, Settling time < 0.6 s. (a) Identify the desired area for the dominant roots of the system. (b) Determine the
A system is shown in Figure P5.20.(a) Determine the steady-state error for a unit step input in terms of K and K1, where E(s) = R(s) - Y(s).(b) Select K1 so that the steady-state error is zero.
Consider the closed-loop system in Figure P5.21. Determine values of the parameters k and a so that the following specifications are satisfied:(a) The steady-state error to a unit step input is zero,(b) The closed-loop system has a percent overshoot of less than 5%.
Consider the closed-loop system in Figure P5.22, where(a) If Ñ‚ = 2.43, determine the value of K such that the steady-state error of the closed-loop system response to a unit step input, R(s) = l/s, is zero. (b) Determine the percent overshoot P.O. and the time to peak Tp of the unit
A laser beam can be used to weld, drill, etch, cut. And mark metals, as shown In Figure P5.3(a) [14]. Assume we have a work requirement for an accurate laser to mark a parabolic path with a closed-loop control system, as shown in Figure P5.3(b). Calculate the necessary gain to result in a
The loop transfer function of a unity negative feedback system (see Figure E5.11) isA system response to a step input is specified as follows: peak time Tp = 1.1 s, percent overshoot P.O. = 5%. (a) Determine whether both specifications can be met simultaneously. (b) If the specifications cannot be
A space telescope is to be launched to carry out astronomical experiments [8]. The pointing control system is desired to achieve 0.01 minute of arc and track solar objects with apparent motion up to 0.21 are minute per second. The system is illustrated in Figure P5.5(a). The control system is shown
A robot is programmed to have a tool or welding torch follow a prescribed path [7, 11]. Consider a robot tool that is to follow a sawtooth path, as shown in Figure P5.6(a). The transfer function of the plant isfor the closed-loop system shown in Figure 5.6(b). Calculate the steady-state error.
Astronaut Bruce McCandless II took the first un-tethered walk in space on February 7, 1984, using the gas-jet propulsion device illustrated in Figure P5.7(a). The controller can be represented by a gain K2, as shown in Figure P5.7(b). The moment of inertia of the equipment and man is 25 kg m2.(a)
Photovoltaic arrays (solar cells) generate a DC voltage that can be used to drive DC motors or that can be converted to AC power and added to the distribution network. It is desirable to maintain the power out of the array at its maximum available as the solar incidence changes during the day. One
The antenna that receives and transmits signals to the Telstar communication satellite is the largest horn antenna ever built. The microwave antenna is 177 ft long, weighs 340 tons, and rolls on a circular track. A photo of the antenna is shown in Figure P5.9. The Telstar satellite is 34 inches in
A closed-loop transfer function is(a) Determine the steady-state error for a unit step input R(s) = 1/s. (b) Assume that the complex poles dominate, and determine the overshoot and settling time to within 2% of the final value. (c) Plot the actual system response, and compare it with the estimates
A closed-loop system is shown in Figure AP5.2. Plot the response to a unit step input for the system for Ñ‚z = 0, 0.05, 0.1, and 0.5. Record the percent overshoot, rise time, and settling time (with a 2% criterion) as Ñ‚z varies. Describe the effect of varying
A closed-loop system is shown in Figure AP5.3. Plot the response to a unit step input for the system with Ñ‚p = 0, 0.5, 2, and 5. Record the percent overshoot, rise time, and settling time (with a 2% criterion) as Ñ‚p varies. Describe the effect of varying Ñ‚p.
The speed control of a high-speed train is represented by the system shown in Figure AP5.4 [17]. Determine the equation for steady-state error for K for a unit step input r(t). Consider the three values for K equal to 1, 10, and 100.(a) Determine the steady-state error.(b) Determine and plot the
A system with a controller is shown in Figure AP5.5. The zero of the controller may be varied. Let α = 0, 10, 100.(a) Determine the steady-state error for a step input r(t) for α = 0 and α ‰ 0.(b) Plot the response of the system to a step input
The block diagram model of an armature-current-controlled DC motor is shown in Figure AP5.6.(a) Determine the steady-state tracking error to a ramp input r(t) = t, t ‰¥ 0, in terms of K, Kb, and Km.(b) Let Km = 10 and Kb = 0.05, and select K so that steady-state tracking error is equal to
Consider the closed-loop system in Figure AP5.7 with transfer functionsGc(s) = 100/s + 100 and G(s) = K/s(s + 50).where1000 ‰¤ K ‰¤ 5000.(a) Assume that the complex poles dominate and estimate the settling time and percent overshoot to a unit step input for K = 1000, 2000,
A unity negative feedback system (as shown in Figure E5.11) has the loop transfer functionDetermine the gain K that minimizes the damping ratio ζ of the closed-loop system poles. What is the minimum damping ratio?Figure E5.11Unity feedback system.
The unity negative feedback system in Figure AP5.9 has the process given byG(s) = 1 / s(s + 15)(s + 25).The controller is a proportional plus integral controller with gains Kp and KI. The objective is to design the controller gains such that the dominant roots have a damping ratio ζ
The capstan drive system of the previous problems has a disturbance due to changes in the part that is being machined as material is removed. The controller is an amplifier Gc(s) - Ka. Evaluate the effect of a unit step disturbance, and determine the best value of the amplifier gain so that the
The roll control autopilot of a jet fighter is shown in Figure DP5.1. The goal is to select a suitable K so that the response to a unit step command Ï•d(t) = A, t ‰¥ 0, will provide a response Ï•(t) that is a fast response and has an overshoot of less than 20%.(a)
The design of the control for a welding arm with a long reach requires the careful selection of the parameters [13]. The system is shown in Figure DP5.2, where ζ = 0.6, and the gain K and the natural frequency ωn can be selected.(a) Determine K and ωn so that the
Active suspension systems for modern automobiles provide a comfortable firm ride. The design of an active suspension system adjusts the valves of the shock absorber so that the ride fits the conditions. A small electric motor, as shown in Figure DP5.3, changes the valve settings [13]. Select a
The space satellite shown in Figure DP5.4(a) uses a control system to readjust its orientation, as shown in Figure DP5.4(b).(a) Determine a second-order model for the closed-loop system.(b) Using the second-order model, select a gain K so that the percent overshoot is less than 15% and the
A deburring robot can be used to smooth off machined parts by following a preplanned path (input command signal). In practice, errors occur due to robot inaccuracy, machining errors, large tolerances, and tool wear. These errors can be eliminated using force feedback to modify the path online [8,
The model for a position control system using a DC motor is shown in Figure DP5.6. The goal is to select K1 and K2 so that the peak time is Tp ‰¤ 0.5 second and the overshoot P.O. for a step input is P.O. ‰¤ 2%.
A three-dimensional cam for generating a function of two variables is shown in Figure DP5.7(a). Both x and y may be controlled using a position control system [31].The control of x may be achieved with a DC motor and position feedback of the form shown in Figure DP5.7(b), with the DC motor and load
Computer control of a robot to spray-paint an automobile is accomplished by the system shown in Figure DP5.8(a) [7]. We wish to investigate the system when K = 1, 10, and 20. The feedback control block diagram is shown in Figure DP5.8(b).(a) For the three values of K, determine the percent
Consider the closed-loop transfer function T(s) = 15 / s2 + 8s + 15. Obtain the impulse response analytically and compare the result to one obtained using the impulse function.
Develop an m-file to simulate the response of the system in Figure CP5.10 to a ramp input R(s) = l/s2. What is the steady-state error? Display the output on an x-y graph.
Consider the closed-loop system in Figure CP5.11. Develop an m-file to accomplish the following tasks:(a) Determine the closed-loop transfer function T(s) = Y(s)/R(s),(b) Plot the closed-loop system response to an impulse input R(s) = 1, a unit step input R(s) = l/s, and a unit ramp input R(s) =
A closed-loop transfer function is given by(a) Obtain the response of the closed-loop transfer function T(s) = Y(s)/R(s) to a unit step input. What is the settling time Ts (use a 2% criterion) and percent overshoot P.O.? (b) Neglecting the real pole at s = -7, determine the settling time Ts and
A unity negative feedback system has the loop transfer functionUsing Isim, obtain the response of the closed-loop system to a unit ramp input, R(s) = 1/s2. Consider the time interval 0 ‰¤ t ‰¤ 50. What is the steady-state error?
A working knowledge of the relationship between the pole locations of the second-order system shown in Figure CP5.3 and the transient response is important in control design. With that in mind, consider the following four cases:1. ωn = 2, ζ = 0,2. ωn = 2,
Consider the control system shown in Figure CP5.4.(a) Show analytically that the expected percent overshoot of the closed-loop system response to a unit step input is about 50%.(b) Develop an m-file to plot the unit step response of the closed-loop system and estimate the percent overshoot from the
Consider the feedback system in Figure CP5.5. Develop an m-file to design a controller and prefiltersuch that the ITAE performance criterion is minimized. For ωn = 0.45 and ζ = 0.59, plot the unit step response and determine the percent overshoot and settling time. Figure
The loop transfer function of a unity negative feedback system is L(s) = Gc(s)G(s) = 25/s(s + 5). Develop an m-file to plot the unit step response and determine the values of peak overshoot Mp, time to peak Tp, and settling time Ts (with a 2% criterion).
An autopilot designed to hold an aircraft in straight and level flight is shown in Figure CP5.7.(a) Suppose the controller is a constant gain controller given by Gc(s) = 2. Using the Isim function, compute and plot the ramp response for θd(t) = at, where a = 0.5°/s. Determine the attitude error
The block diagram of a rate loop for a missile autopilot is shown in Figure CP5.8. Using the analytic formulas for second-order systems, predict Mpt, Tp, and Ts for the closed-loop system due to a unit step input. Compare the predicted results with the actual unit step response obtained with the
Develop an m-file that can be used to analyze the closed-loop system in Figure CP5.9. Drive the system with a step input and display the output on a graph. What is the settling time and the percent overshoot?
A system has a characteristic equation s3 + Ks2 + (1 + K)s + 6 = 0. Determine the range of K for a stable system.
A system has the second-order characteristic equation s2 + as + b = 0, where a and b are constant parameters. Determine the necessary and sufficient conditions for the system to be stable. Is it possible to determine stability of a second-order system just by inspecting the coefficients of the
Consider the feedback system in Figure E6.13. Determine the range of Kp and KD for stability of the closed-loop system.Figure E6.13Closed-loop system with a proportional plus derivative controller Gc(s) = KP + KDs.
By using magnetic bearings, a rotor is supported contactless. The technique of contactless support for rotors becomes more important in light and heavy industrial applications [14]. The matrix differential equation for a magnetic bearing system iswhere xT = [y, dy/dt, i], y = bearing gap, and i is
A system has a characteristic equation q(s) = s6 + 9s5 + 31.25s4 + 61.25s3 + 67.75s2 + 14.75s + 15 = 0. (a) Determine whether the system is stable, using the Routh-Hurwitz criterion. (b) Determine the roots of the characteristic equation.
A system has a characteristic equation q(s) = s4 + 9s3 + 45s2 + 87s + 50 = 0. (a) Determine whether the system is stable, using the Routh-Hurwitz criterion. (b) Determine the roots of the characteristic equation.
The matrix differential equation of a state variable model of a system has(a) Determine the characteristic equation. (b) Determine whether the system is stable. (c) Determine the roots of the characteristic equation.
A system has a characteristic equation q(s) = s3 + 20s2 + 5s + 100 = 0. (a) Determine whether the system is stable, using the Routh-Hurwitz criterion. (b) Determine the roots of the characteristic equation.
A system has a characteristic equation s3 + I0s2 + 2s + 30 = 0. Using the Routh-Hurwitz criterion, show that the system is unstable.
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