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engineering
modern control systems

Modern Control Systems 14th Global Edition Richard Dorf, Robert Bishop - Solutions

CP13.7 An industrial grinding process is given by the transfer function [15] 10 Gp($)= s(s+5) The objective is to use a digital computer to im- prove the performance, where the transfer function of the computer is represented by D(z). The design specifications are (1) phase margin of P.M.> 45, and
CP13.6 Consider the sampled-data system with the loop transfer function(a) Plot the root locus using the rlocus function.(b) From the root locus, determine the range of K for stability. z-z+1.5 G(z)D(z)=-12z+0.1
CP13.5 Consider the feedback system in Figure CP13.5.Obtain the root locus, and determine the range of K for stability.
CP13.4 Plot the root locus for the system Z G(z) D(z) K- = z-z+0.45 Find the range of K for stability.
CP13.3 The closed-loop transfer function of a sampleddata system is given by Y(z) T(z) = R(z) 0.684 (z -0.4419) z -0.7524z + 0.0552 (a) Compute the unit step response of the system using the dstep function, and assume a sampling period. of 70.1 s. (b) Determine the continu- ous-time transfer
CP13.2 Convert the following continuous-time transfer functions to sampled-data systems using the c2d function. Assume a sample period of 1 second and a zero-order hold G s 0( ). (a) Gp(s)= (b) Gp(s) = (c) Gp(s) = (d) Gp(s) = S s +2 S+4 s+3 1 s(s+8)
CP13.1 Develop an m-file to plot the unit step response of the systemVerify graphically that the steady-state value of the output is 1. G(z) = 0.575z+ 0.025 z2-0.8z+0.4
DP13.6 A sampled-data system closed-loop block diagram is shown in Figure DP13.6. Design D(z) to such that the closed-loop system response to a unit step response has a percent overshoot P O. . ≤ 12% and a settling time Ts ≤ 20 s. FIGURE DP13.6 A closed-loop sampled-data system with sampling
DP13.5 Plastic extrusion is a well-established method widely used in the polymer processing industry [12].Such extruders typically consist of a large barrel divided into several temperature zones, with a hopper at one end and a die at the other. Polymer is fed into the barrel in raw and solid form
DP13.4 A machine-tool system has the form shown in Figure E13.6(b) with [10]The sampling rate is chosen as T = 1 s. We desire the step response to have a percent overshoot ofP O. . ≤ 20% and a settling time (with a 2% criterion) of Ts ≤ 10 s. Design a D(z) to achieve these specifications.
DP13.3 Vehicle traction control, which includes antiskid braking and antispin acceleration, can enhance vehicle performance and handling. The objective of this control is to maximize tire traction by preventing the wheels from locking during braking and from spinning during acceleration.Wheel slip,
DP13.2 A disk drive read-write head-positioning system has a system as shown in Figure P13.10 [11]. The process transfer function is Gp($)= 2 9 +0.85s+788 Accurate control using a digital compensator is required. Let T = 10 ms and design a compensa- tor, D(z), using (a) the G(s) - to - D(z)
DP13.1 A temperature system, as shown in Figure P13.10, has a process transfer function Gp($)= 0.8 3s+1 and a sampling period T of 0.5 s. (a) Using D(z) K, select a gain K so that the system is stable. (b) The system may be slow and overdamped, and thus we seek to design a phase-lead compensator.
CDP13.1 Design a digital controller for the system using the second-order model of the motor-capstan-slide as described in CDP2.1 and CDP4.1. Use a sampling period of T = 1 ms and select a suitable D(z) for the system shown in Figure P13.10. Determine the response of the designed system to a step
AP13.5 Consider the closed-loop sampled-data system shown in Figure AP13.5. Determine the acceptable range of the parameter K for closed-loop stability FIGURE AP13.5 A closed-loop sampled-data system with sampling time T = 0.1s. Zero-order hold Gp(s) K R(s) Go(s) Y(s) s(s+3) T=0.1
AP13.4 A system of the form shown in Figure E13.10 hasDetermine the range of sampling period T for which the system is stable. Select a sampling period T so that the system is stable and provides a rapid response. Gp(s) = 8 s+2
AP13.3 A system of the form shown in Figure P13.10 has D z( ) = K andWhen T = 0.05 s, find a suitable K for a rapid step response with a percent overshoot of P O. . ≤ 10%. Gp(s) = 10 s(s+5)
AP13.2 A manufacturer uses an adhesive to form a seam along the edge of the material, as shown in Figure AP13.2. It is critical that the glue be applied evenly to avoid flaws; however, the speed at which the material passes beneath the dispensing head is not constant.The glue needs to be dispensed
AP13.1 A closed-loop system, as shown in Figure E13.16,has a processwhere a is adjustable to achieve a suitable response.Plot the root locus when a = 6. Determine the range of K for stability when T = 0.5 s. Gp(s)= K(1+ as) $2
P13.18 A unity feedback system, as shown in Figure E13.10, hasIf the system is continuous ( ) T = 0 , then K = 1 yields a step response with a percent overshoot of P O. . = 16%and a settling time (with a 2% criterion) of Ts = 8 s.Plot the response for 0 ≤ ≤ T 1.2, varying T by increments of
P13.17 A closed-loop system, as shown in Figure E13.10, has Gp(s)= K s(s +2.5) and 7=1.5 s. Plot the root locus for K > 0, and de- termine the gain K that results in the two roots of the characteristic equation on the z-circle (at the stability limit).
P13.16 A closed-loop system as shown in Figure E13.10 has 0.5 Gp(s)= s(s+5) Calculate and plot y(kT) for 0 k 10 when T=1 s, and the input is a unit step.
P13.15 A closed-loop system with a sampler and hold, as shown in Figure E13.10, has a process transfer functionDetermine the first 6 samples of y(kT) whenT = 0.1 s. The input signal is a unit step. Gp(s)= 17 8-3
P13.14 A sampled-data system with a sampling periodT = 0.05 s is(a) Plot the root locus. (b) Determine K when the two real poles break away from the real axis. (c) Calculate the maximum K for stability. G(z) = K(z +10.3614z2+ 9.758z+0.8353) z3.7123z3 +5.1644z23.195z + 0.7408
P13.13 The azimuth control system of an aircraft has a transfer function Gp(s) = (s+3) s(s+1) It is implemented with a sampler and hold as shown in Figure E13.10. (a) Find the transfer function of the plant and zero-order hold at a sampling rate 1 Hz. (b) Plot the root locus, and determine the
P13.12 The transfer function of a plant and a zero-order hold is(a) Plot the root locus. (b) Determine the range of gain K for a stable system. G(z) = K(z+0.45) z(z-3)
P13.11 (a) For the system described in Problem P13.10,design a lag compensator G s c ( ) to achieve a percent overshoot P O. . ≤ 30% and a steady-state error of ess = 0.01 for a ramp input. Assume a continuous nonsampled system with G s p( ).(b) Determine a suitable D(z) to satisfy the
P13.10 Consider a system as shown in Figure P13.10 with a zero-order hold, a process Gp(s) = 1 s(s + 10)' and T = 0.1 s. Note that G(z) = Z{G(s)Gp(s)}. (a) Let D(z) = K and determine the transfer function G(z) D(z). (b) Determine the characteris- tic equation of the closed-loop system. (c)
P13.9 A suspended, mobile, remote-controlled system to bring three-dimensional mobility to professional NFL football is shown in Figure P13.9. The camera can be moved over the field as well as up and down.The motor control on each pulley is represented by Figure E13.10 with We wish to achieve a
P13.8 Consider the computer-compensated system shown in Figure E13.6(b) when T = 1 s and K KGp(s) = s(s+10) Select the parameters K and r of D(z) when D(z) = z- 0.3678 z+r Select within the range 1 < K
P13.7 A closed-loop system is shown in Figure E13.10. This system represents the pitch control of an aircraft. The process transfer function is G s p( ) =K s[ ] ( ) 0.5s + 1 . Select a gain K and sampling periodT so that the percent overshoot is limited to 0.3 for a unit step input and the
P13.6 Consider the closed-loop system shown in Figure E13.10, where Given the sampling period T = 0.05 s, find the output Y(z) to a unit step input. Find the initial and final value directly from Y(z), and plot the unit step response. Gp(s)= 1 (0.2s+1)
P13.5 For the system in Problem P13.4, let r(t) be a unit step input and calculate the response of the system by synthetic division for five time steps.Data from in P13.4 A closed-loop system has a hold circuit and process as shown in Figure E13.10. Determine G(z)when T = 0.5 s and Gp(s) = 3 s+3
P13.4 A closed-loop system has a hold circuit and process as shown in Figure E13.10. Determine G(z)when T = 0.5 s and Gp(s) = 3 s+3
P13.3 A unit ramp r t( ) = > t t , 0, is used as an input to a process where G s( ) = + 1 1 ( ) s , as shown in Figure P13.3. Determine the output y(kT) for the first four sampling instants. 1*(1) r(1) G(s) y(t) FIGURE P13.3 Sampling system.
P13.2 The input to a sampler is r t( ) = sin , ( ) ωt whereω π = 2 . The output of the sampler enters a zeroorder hold. Plot the output of the zero-order holdp(t) for the first 2 seconds when T = 0.125 s.
P13.1 The input to a sampler is r t( ) = sin , ( ) ωt whereω π = 1.5 . Plot the input to the sampler and the output r t *( ) for the first 2 seconds when T = 0.25 s.
E13.16 Consider the sampled-data system shown in Figure E13.16. Determine the transfer function G(z)and when the sampling time T = 0.5 s. FIGURE E13.16 An open-loop sampled-data system with Zero-order hold Gp(s) r*(t) 1 sampling time r(t) Go(s) T = 0.5 s. T=0.5 s(s+2) y(t)
E13.15 Consider the sampled-data system shown in Figure E13.15. Determine the transfer function G(z)when the sampling time T = 1 s. FIGURE E13.15 An open-loop sampled-data system with Zero-order hold Gp(s) 1*(1) sampling time T = 1s. r(1) 3 Go(s) T=1 (s+1)(x+2) y(t)
E13.14 A unity feedback system, as shown in Figure E13.10, has a plantwith T = 0.5 s. Determine whether the system is stable when K = 4. Determine the maximum value ofK for stability. Gp(s) = K s(2s+1)
E13.13 The characteristic equation of a sampled system isFind the range of K so that the system is stable. z+(K 3)z+0.7 = 0.
E13.12 Find the z-transform of X(s) = s+1 s+5s +6 when the sampling period is T = 1 s.
E13.11 A system has a process transfer function Gp(s)= 9 $ +9 (a) Determine G(z) for Gp (s) preceded by a zero- order hold with T = 0.15 s. (b) Determine whether the digital system is stable. (c) Plot the impulse re- sponse of G(z) for the first 15 samples. (d) Plot the first 30 samples of the
E13.10 A system haswith T = 0.01 s and τ = 0.008 s. (a) Find K so that the percent overshoot is P O. . ≤ 40%. (b) Determine the steady-state error in response to a unit ramp input.(c) Determine K to minimize the integral squared error. Gp(s) = K s(TS+1)
E13.9 (a) Determine y(kT) for k = 0 to 3 when(b) Obtain a closed form solution for y(kT) as a function of k. Y(z) = = 1.5z + 0.5z z-1
E13.8 Determine whether the closed-loop system withT(z) is stable when Z T(z)= z + 0.2z-1.0
E13.7 Find the response for the first four sampling instants forThen, find y(0), y(1), y(2), and y(3). Y(z) = z3 +222 +1 z3-1.5z2 +0.5z
E13.6 Computer control of a robot to spraypaint an automobile is shown by the system in Figure E13.6(a) [1].The system is of the type shown in Figure E13.6(b), where 1 Gp(s)= s(0.25s + 1) and we want a phase margin of P.M. = 45. Using frequency response methods, a compensator was de- veloped, given
E13.5 The space shuttle, with its robotic arm, is shown in Figure E13.5(a). An astronaut controls the robotic arm and gripper by using a window and the TV cameras[9]. Discuss the use of digital control for this system and sketch a block diagram for the system, including a computer for display
E13.4 We have a function 1 Y(s) = s(s+2)(s+3) Using a partial fraction expansion of Y(s) and a table of z-transforms, find Y(z) when T = 0.2 s.
E13.3 Obtain the z-transform Y(z) for the responsey k( ) T k = T, k ≥ 0, where T is the sampling time, (a) by using the definition X(z) = 0x(KT)zk, (b) by applying the property of differentiation in dX z-domain, Z{tx (1)}=-27 dx (2), given that dz Z{u(t)} = z-1
E13.2 (a) Find the values y(kT) when 2z Y(z) = 22-4z+3 for k = 0 to 3. (b) Obtain a closed form of solution for y (KT) as a function of k.
E13.1 State whether the following signals are discrete or continuous:(a) Elevation contours on a map.(b) Temperature in a room.(c) Digital clock display.(d) The score of a soccer game.(e) The output of a loudspeaker.
15. Consider a continuous-time system with the closed-loop transfer function T(s)= S s+4s+8 Using a zero-order hold on the inputs and a sampling period of 7 = 0.02 s, determine which of the following is the equivalent discrete-time closed-loop transfer function representation: 0.019z -0.019 a.
14. The range of the sampling period T for which the closed-loop system is stable is:a. T ≤ 1.12b. The system is stable for all T > 0.c. 1.12 ≤ ≤ T 10d. T ≤ 4.23
13. The closed-loop transfer function T(z) of this system using a sampling period of T = 0.5s is which of the following: a. T(z) = 1.76z +1.76 z + 2.76 0.87z + 0.23 b. T(z) = z20.14z + 0.24 0.87z + 0.23 c. T(z) = z 1.01z+0.011 0.92z+0.46 d. T(z)=-1.01z
12. The unit step response of the closed-loop system is: 1.76z + 1.76 a. Y(z) = z + 3.279z+2.76 1.76z + 1.76 b. Y(z) = z3+2.279z20.5194z - 2.76 1.76z+1.76z c. Y(z) = z + 2.279z2-0.5194z - 2.76 1.76z + 1.76z d. Y(z) = 2.279z2-0.5194z-2.76
11. The closed-loop transfer function T(z) of this system with sampling at T = 1 s is a. T(z) = b. T(z)= c. T(z) = 1.76z +1.76 z+3.279z+2.76 z+1.76 z +2.76 1.76z + 1.76 z +1.519z+1 Z d. T(z) = 2+1
10. Consider the unity feedback system in Figure 13.47, where Zero-order Process hold e(t) R(s) Go(s) Gp(s) Y(s) FIGURE 13.47 Block diagram for the Skills Check.
9. The characteristic equation of a sampled system is q(z) = z+(2K-1.75) z + 2.5 = 0, where K >0. The range of K for a stable system is: a. 0 < K < 2.63 b. K 2.63 c. The system is stable for all K > 0. d. The system is unstable for all K > 0.
8. Consider a sampled-data system with the closed-loop system transfer function z +2z T(z) = K z + 0.2z-0.5 This system is: a. Stable for all finite K. b. Stable for -0.5 < K < 0. c. Unstable for all finite K. d. Unstable for -0.5 < K < 0.
7. The impulse response of a system is given by z + 2z +2 Y(z) = z3-25z +0.6z Determine the values of y(nT) at the first four sampling instants. a. y(0) =1, y(T) = 27, y(27) = 647, y(37) = 660.05 b. y(0) = 0, y(T) = 27, y(27) = 47, y(37) = 60.05 c. y(0) = 1, y(T) = 27, y(27) = 674.4, y(37) =
6. Consider the function in the s-domain 10 Y(s) = s(s+2)(s+6) Let T be the sampling time. Then, in the z-domain the function Y(s) is 5 N 5 a. Y(z) = Z 5 Z 6 z-1 4 z-e-27 + 5 b. Y(z) = N 5 Z + 6 z-1 5 N c. Y(z) = 6 z-1 4 z-e-67 + 5 12 z-e-67 5 12 z e- z-e-67 12 z-e-27 Z z 1 d. Y(z) N Z 5 Z = + 6 z
5. The z-transform is a conformal mapping from the s-plane to the z-plane by the relation z e = sT.True or False
4. A sampled system is stable if all the poles of the closed-loop transfer function lie outside the unit circle of the z-plane.True or False
3. Root locus methods are not applicable to digital control system design and analysis.True or False
2. The sampled signal is available only with limited precision.True or False
1. A digital control system uses digital signals and a digital computer to control a process.True or False
CP12.8 The Gamma-Ray Imaging Device (GRID) is a NASA experiment to be flown on a long-duration, high-altitude balloon during the coming solar maximum. The GRID on a balloon is an instrument that will qualitatively improve hard X-ray imaging and carry out the first gamma-ray imaging for the study of
CP12.7 A unity feedback control system has the loop transfer function L(s) = G(s)G(s)= a(s+0.5) s + 0.15s We know from the underlying physics of the prob- lem that the parameter a can vary only between 0 < a
CP12.6 The industrial process shown in Figure CP12.6 is known to have a time delay in the loop. In practice, it is often the case that the magnitude of system time delays cannot be precisely determined. The magnitude of the time delay may change in an unpredictable manner depending on the process
CP12.5 A model of a flexible structure is given bywhere ωn is the natural frequency of the flexible mode, and ζ is the corresponding damping ratio. In general, it is difficult to know the structural damping precisely, while the natural frequency can be predicted more accurately using
CP12.4 Consider the feedback control system in Figure CP12.4. The exact value of parameter b is unknown;however, for design purposes, the nominal value is taken to be b = 4. The value of a = 8 is known very precisely. a. Design the proportional controller K so that the closed-loop system
CP12.3 Consider the control system in Figure CP12.3. The value of J is known to change slowly with time, although, for design purposes, the nominal value is chosen to be J = 28.(a) Design a PID controller (denoted by G s c( )) to achieve a phase margin P M. . ≥ 45° and a bandwidthωB ≤ 4
CP12.2 An aircraft aileron can be modeled as a first-order system where p depends on the aircraft. Obtain a family of step responses for the aileron system in the feedback configuration shown in Figure CP12.2.The nominal value of p = 15. Compute reasonable values of Kp and KI so that the step
CP12.1 A closed-loop feedback system is shown in Figure CP12.1. Use an m-file to obtain a plot of S j T K ( ) ω versus ω. Plot T j ( ) ω versus ω, where T s( ) is the closedloop transfer function. Plant 6 R(s) K Y(s) s(s+3) FIGURE CP12.1 Closed-loop feedback system with gain K.
DP12.12 A benchmark problem consists of the mass–spring system shown in Figure DP12.12, which represents a flexible structure. Let m m 1 2 = = 1 and 0.5 ≤ ≤k 2.0 2[ ] 9 . It is possible to measure x t 1 ( ) andx t 2 ( ) and use a controller prior to u t( ). Obtain the system description,
DP12.11 Electromagnetic suspension systems for aircushioned trains are known as magnetic levitation(maglev) trains. One maglev train uses a superconducting magnet system [17]. It uses superconducting coils, and the levitation distance x t( ) is inherently unstable. The model of the levitation is
DP12.10 A photovoltaic system is mounted on a space station in order to develop the power for the station.The photovoltaic panels should follow the Sun with good accuracy in order to maximize the energy from the panels. The unity feedback control system uses a DC motor, so that the transfer
DP12.9 One arm of a space robot is shown in Figure DP12.9(a). The block diagram for the control of the arm is shown in Figure DP12.9(b). (a) If G s c ( ) = K, determine the gain necessary for a percent overshoot of P O. . = 4.5%, and plot the step response. (b) Design a proportional plus
DP12.8 A model of the feedback control system is shown in Figure DP12.8 for an electric ventricular assist device. This problem was introduced in AP9.11. The motor, pump, and blood sac can be modeled by a time delay with T = 1 s. The goal is to achieve a step response with less than 5%
DP12.7 The goal is to design an elevator control system so that the elevator will move from floor to floor rapidly and stop accurately at the selected floor(Figure DP12.7). The elevator will contain from one to three occupants. However, the weight of the elevator should be greater than the weight
DP12.6 The use of control theory to provide insight into neurophysiology has a long history. As early as the beginning of the last century, many investigators described a muscle control phenomenon caused by the feedback action of muscle spindles and by sensors based on a combination of muscle
DP12.5 The system described in DP12.4 is to be designed using the frequency response techniques. Select the coefficients of Gc(s) so that the phase margin isP M. . = 45°. Obtain the step response of the system with and without a prefilter Gp(s).
DP12.4 Objects smaller than the wavelengths of visible light are a staple of contemporary science and technology. Biologists study single molecules of protein or DNA; materials scientists examine atomic-scale flaws in crystals; microelectronics engineers lay out circuit patterns only a few tenths
DP12.3 Many university and government laboratories have constructed robot hands capable of grasping and manipulating objects. But teaching the artificial devices to perform even simple tasks required formidable computer programming. However, a special hand device can be worn over a human hand to
DP12.2 Consider the closed-loop system depicted in Figure DP12.2. The process has a parameter K that is nominally K = 1. Design a controller that results in a percent overshoot P O. . ≤ 20% for a unit step input for all K in the range 1 4 ≤ ≤ K . Controller Process FIGURE DP12.2 A unity
DP12.3 Many university and government laboratories have constructed robot hands capable of grasping and manipulating objects. But teaching the artificial devices to perform even simple tasks required formidable computer programming. However, a special hand device can be worn over a human hand to
DP12.1 A position control system for a large turntable is shown in Figure DP12.1(a), and the block diagram of the system is shown in Figure DP12.1(b) [11, 14].This system uses a large torque motor with Km = 15.The objective is to reduce the steady-state effect of a step change in the load
CDP12.1 Design a PID controller for the capstan-slide system of Figure CDP4.1. The percent overshoot should be P O. . ≤ 3% and the settling time should be(with a 2% criterion) Ts ≤ 250 ms for a step input r t( ).Determine the response to a disturbance for the designed system.
AP12.11 A unity feedback system has a plant G(s) = 1 (s+2)(s+4)(s+6) We want to attain a steady-state error for a step input. Select a compensator G. (s) using the pseudo-QFT method, and determine the performance of the system when all the poles of G(s) change by -50%. Describe the robust nature of
AP12.10 A system of the form shown in Figure 12.1 has str G(s) = (s+p)(s+q) where 3 p 5, 0
AP12.9 The position control of a suspension system can be represented by a unity feedback system with controller Gc(s). The plant has a gain K and viscous friction coefficientThe system has its gain varying in a large range, 4 2 ≤ ≤ K 5, with low damping, 0.5 ≤ ≤b 2. The desired 2%
AP12.8 A machine tool control system is shown in Figure AP12.8. The transfer function of the power amplifier, prime mover, moving carriage, and tool bit is 50 G(s)= s(s+1)(s+4)(s+5)* The goal is to have a percent overshoot of P.O.
AP12.7 Consider a unity feedback system with G(s)= The goal is to select a PI controller using the ITAE de- sign criterion while constraining the control signal as |u(t)|1 for a unit step input. Determine the appro- priate PI controller and the settling time (with a 2% criterion) for a step input.
AP12.6 Consider a unity feedback system with K G(s)= S(TS+1)' where K = 1.5 and 7 = 0.001 s. Select a PID con- troller so that the settling time (with a 2% criterion) for a step input is T,
AP12.5 The plant of a driverless car is modeled as G(s) K where K = 1 under nominal con- s(s+10) ditions. The system has unity feedback with a control- ler G, (s). To increase the system's robustness, a phase margin of P.M. = 50 is required. Design a PID con- troller for G, (s), and determine the
AP12.4 A robot has been designed to aid in hipreplacement surgery. The device, called RoBoDoc, is used to precisely orient and mill the femoral cavity for acceptance of the prosthetic hip implant. Clearly, we want a very robust surgical tool control, because there is no opportunity to redrill a
AP12.3 Antiskid braking systems present a challenging control problem, since brake/automotive system parameter variations can vary significantly (e.g., due to the brake-pad coefficient of friction changes or road slope variations) and environmental influences (e.g., due to adverse road conditions).
AP12.2 One promising solution to traffic gridlock is a magnetic levitation (maglev) system. Vehicles are suspended on a guideway above the highway and guided by magnetic forces instead of relying on wheels or aerodynamic forces. Magnets provide the propulsion for the vehicles [7, 12, 17]. Ideally,

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