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engineering
modern control systems

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

P8.2 Sketch the Bode plot representation of the frequency response for the transfer functions given in Problem P8.1.
P8.1 Sketch the polar plot for the following loop transfer functions: (a) L(s)G(s)G(s) = (b) L(s) G(s)G(s) = (c) L(s)G(s)G(s) = 1 (1+0.5s)(1+2s) 3(s2 +1.5s+1) (s-2) 8-6 s2+5s+6 10(s+6) (d) L(s) = G(s)G(s) = = s(s+ 1)(s+3)
E8.15 Consider the single-input, single-output system described by x(t) = Ax(t) + Bu(t) y(t)=Cx(t) where - 0 1 -5-K- C=[24]. Compute the bandwidth of the system for K = 1, 2, and 10. As K increases, does the bandwidth increase or decrease?
E8.14 Consider the nonunity 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 j ( ) ω = 0 dB. Recall that the loop transfer function is L s( ) = G
E8.13 Determine the bandwidth of the feedback control system in Figure E8.13. Controller Process 100 1 R(s) s+1 s+10x+10 Y(s) FIGURE E8.13 Third-order feedback system.
E8.12 Consider the system represented in state variable form -1 2 1 x(t) * (() = |. 3 2 | x() + | 2 | (1) y(t) [12]x(1) (a) Determine the transfer function representation of the system. (b) Sketch the Bode plot.
E8.11 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 log 0 10 G j ( ) ω = dB. FIGURE E8.11 Unity feedback system. Controller 1000 Process 1 s+ 10s + 100 R(s) 5+2 Y(s)
E8.10 The dynamic analyzer shown in Figure E8.10(a)can be used to display the frequency response of a system. Figure E8.10(b) shows the actual frequency response of a system. Estimate the poles and zeros of the device. Note X = 1.37 kHz at the first cursor, and∆ = X 1.257 kHz to the second
E8.9 The Bode plot of a system is shown in Figure E8.9.Estimate the transfer function G s( ). FIGURE E8.9 Bode plot. Magnitude (dB) 20 0 990 40 -20 -40 0.1 1 10 Frequency (rad/s) 90 0 -90 100 1,000 Phase (deg)
E8.8 Two feedback systems with their respective loop transfer function are represented as: (i) T(s)= 100(S-1) (s + 25s+ 100)
E8.7 Consider a system with a closed-loop transfer function 100 G(s) (s+2s+16)(s2+s+64)* This system will have a 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
E8.6 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 function K L(s) = G(s)G(s) = s(s/8+1)(s/100+1)1 Sketch the Bode plot for
E8.5 The magnitude plot of a transfer function G(s)= K' (1+0.125s)(1+as)(1 + bs) s(s+c)(1 + s/16) is shown in Figure E8.5. Estimate K', a, b, and c from the plot.
E8.4 The frequency response for the systemis shown in Figure E8.4. Estimate K and a by examining the frequency response curves. G(s)= Ks (s+a)(s+5s+6.25)
E8.3 A robotic arm has a joint-control loop transfer function L(s) G(s)G(s)= 100(s + 0.1) s(s+1)(s+10) Show that the frequency equals = 11.7 rad/s when the phase angle of L(jw) is -135. Find the magnitude of L(jw) at w11.7 rad/s.
E8.2 A tendon-operated robotic hand can be implemented using a pneumatic actuator [8]. The actuator can be represented by 5000 G(s)= (s+70)(s+500) Plot the frequency response of G(jw). Show that the magnitude of G(jw) is -17 dB at w = 10 and -27.1 dB at w=200. Show also that the phase is -138.7 at
E8.1 Increased track densities for computer disk drives necessitate careful design of the head positioning control [1]. The loop transfer function is L(s) = G(s)G(s) = K (s+1) Plot the frequency response for this system when K 10. Calculate the phase and magnitude at w=0, 0.5, 1, 2, 4, and oo.
15. The transfer function corresponding to the Bode plot in Figure 8.56 is: Phase (deg) Magnitude (dB) 20 0 -20 -40 -60 0 -45 -90 -135
14. Determine the system type (that is, the number of integrators, N):a. N = 0b. N = 1c. N = 2d. N > 2 Phase (deg) Magnitude (dB) 20 0 -20 -40 -60 0 -45 -90 -135
13. Consider the feedback control system in Figure 8.53 with loop transfer functionThe resonant frequency, ωr, and the bandwidth, ωb, are:a. ωr = 1.59 rad/s, ωb = 1.86 rad/sb. ωr = 3.26 rad/s, ωb = 16.64 rad/sc. ωr = 12.52 rad/s, ωb = 3.25 rad/sd. ωr = 5.51 rad/s, ωb = 11.6
12. Suppose that one design specification for a feedback control system requires that the percent overshoot to a step input be P O. . ≤ 10%. The corresponding specification in the frequency domain isa. Mpω ≤ 0.55b. Mpω ≤ 0.59c. Mpω ≤ 1.05d. Mpω ≤ 1.27
11. Consider the Bode plot in Figure 8.55. Phase (deg) Magnitude (dB) -50 -100 -150 -90 -135 -180 -225 -270 10-2 100 50 50 0 10-1 10 10 10 103 Frequency (rad/s) FIGURE 8.55 Bode plot for unknown system.
10. The slope of the asymptotic plot at very low ( ) ω 1 and high ( ) ω 10 frequencies are,respectively:a. At low frequency: slope = / 20 dB decade and at high frequency: slope = / 20 dB decadeb. At low frequency: slope = / 0 dB decade and at high frequency: slope = −20 dB d/ ecadec. At
9. The break frequency on the Bode plot isa. ω = 1 rad/sb. ω = 4.47 rad/sc. ω = 8.94 rad/sd. ω = 10 rad/s L(s) = G(s)G(s) = $2 50 s + 12s + 20
8. Determine the frequency at which the gain has unit magnitude and compute the phase angle at that frequency:a. ω = 1 rad/s, φ = −82°b. ω = 1.26 rad/s, φ = −133°c. ω = 1.26 rad/s, φ = 133°d. ω = 4.2 rad/s, φ = −160° Phase (deg) Magnitude (dB) 50 0 -50 -100 -90 -135
7. The Bode plot of this system corresponds to which plot in Figure 8.54? 8(s+1) L(s) = G(s)G(s) == s(2+s)(2+3s))
6. Consider the stable system represented by the differential equationx() () () t x + = 3 . t u t Determine the phase of this system at the frequency ω = 3 rad/s.a. φ = 0°b. φ = −45°c. φ = −60°d. φ = −180°
5. One advantage of frequency response methods is the ready availability of sinusoidal test signals for various ranges of frequencies and amplitudes. True or False
4. The resonant frequency and bandwidth can be related to the speed of the transient response.True or False
3. A transfer function is termed minimum phase if all its zeros lie in the right-hand s-plane.True or False
2. A plot of the real part of G j ( ) ω versus the imaginary part of G j ( ) ω is called a Bode plot.True or False
1. The frequency response represents the steady-state response of a stable system to a sinusoidal input signal at various frequencies.True or False
CP7.10 Consider the system represented in state variable form(a) Determine the characteristic equation. (b) Using the Routh–Hurwitz criterion, determine the values ofk for which the system is stable. (c) Develop an m-file to plot the root locus, and compare the results to those obtained in (b).
CP7.9 Consider the feedback control system in Figure CP7.9. Develop an m-file to plot the root locus for 0 . K ∞ Find the value of K resulting in a damping ratio of the closed-loop poles equal to ζ = 0.707. FIGURE CP7.9 Unity feedback system with parameter K. R(s) K(s+2) s3+6s+14s+8 Y(s)
CP7.8 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 K Kp D / = 12.Then, develop an m-file using root locus methods find the values of K J D/ and K J p/
CP7.7 Consider the feedback control system in Figure CP7.7. We have three potential controllers for our system:1. G s c( ) = K (proportional controller)2. G s c( ) = K s (integral controller)3. G s c( ) = + K s ( ) 1 1/ (proportional, integral (PI)controller).The design specifications are Ts
CP7.6 A 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
CP7.5 Consider the unity feedback system with the loop transfer functionFor what value of K is the step response to a unit step such that the percent overshoot, P.O. ification is satisfied. L(s) = K s(s+10)
CP7.4 A unity negative feedback system has the loop transfer functionDevelop an m-file to obtain the root locus as p varies;0 . p ∞ For what values of p is the closed-loop stable? p(s-1) L(s) = Ge(s)G(s) = s3+ 4s2+5s+4
CP7.3 Compute the partial fraction expansion ofand verify the result using the residue function. Y(s) = 8+6 s(s+68+5)
CP7.2 A unity negative feedback system has the loop transfer function KG(s) K- S+4 s(s+2)(s+6s+27) Develop 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 = 92.7.
CP7.1 Using the rlocus function, obtain the root locus for the following transfer functions of the system shown in Figure CP7.1 when 0 : 25 a. G(s)=3+10s +40s +25 s+10 b. G(s): = s+2s+10 c. G(s) = c. G(s): = R(s) s+2s+4 s(s + 5s +10) $5 +64 +683 +12s2 + 6s+4 s6 +45 +54 +$3+s + 12s+1 + KG(s) Y(s)
DP7.14 Consider the feedback system shown in Figure DP7.14. The process transfer function is marginally stable. The controller is the proportional-derivative (PD)controller.a. Determine the characteristic equation of the closed-loop system.b. Let τ = / K K P D. Write the characteristic
DP7.13 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 DP7.13(a). By manipulating these surfaces, a pilot
DP7.12 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). (a) Sketch the root locus as K varies from 0 to 10000. Find the roots for K equal to 1000, 1500, and 2500. Predict the
DP7.11 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 φ( )t [13]. The model of the pilot crane control is shown in Figure DP7.11(b).Design a controller that will achieve zero steady-state error for ramp inputs, and maximize
DP7.10 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
DP7.9 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 G s p( ) so that the steady-state error for a step input is equal to zero. Using the G s p( )selected, plot
DP7.8 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
DP7.7 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). Design the controller so that (a) the percent overshoot for a step input is P O. . ≤ 5%; (b) the settling time (with a 2% criterion) is Ts ≤ 6
DP7.6 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 is shown in Figure DP7.6. Using the root locus, select K for the maximum achievable damping ratio of the complex roots. Predict the step response
DP7.5 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
DP7.4 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 T s d ( ) = 0, select K1
DP7.3 A rover vehicle 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], the rover must act independently and reliably. Resembling a cross between a small
DP7.2 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.(a) Sketch the root locus of the system, and determineK when ζ of the complex
DP7.1 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).(a) Sketch
CDP7.1 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 v K T = 1θ. Use the velocity feedback with the adjustable gain K1. Select the best values for the
AP7.14 Consider the unity feedback control system shown in Figure AP7.14. Design 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
AP7.13 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 , ≤ K K ∞.1 2 What is the fastest settling time that you would expect of the closed-loop system in response to a unit step input R s( )
AP7.12 A control system with PI control is shown in Figure AP7.12. (a) Let K KI P / = 0.2 and determineKP 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 AP7.12 A control system with a Pl
AP7.11 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 a percent overshoot ofP O. . ≤ 5%, and the settling time (with a 2% criterion)is Ts ≤ 10 s. FIGURE AP7.11 A control system with parameter K. Controller K
AP7.10 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 a percent overshoot of P O. . ≤ 5% and a settling time (with a 2% criterion) of Ts ≤ 2.5 s. 10 Y(s) R(s) (s+2)(s+5) K s+K FIGURE
AP7.9 A control system is shown in Figure AP7.9.Sketch the root loci for the following transfer functions G s c( ):a. G s c( ) = K b. G s c( ) = + K s( ) K(s+1) $ +20 K(s+ 1)(s+4) = C. Ge(s)= d. G(s) s+10
AP7.8 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.
AP7.7 A feedback system with positive feedback is shown in Figure AP7.7. The root locus for K > 0 must meet the condition KG(s)=1+/k360 for k 0, 1, 2,... = Sketch the root locus for 0 < K < . G(s) 1 R(s) K Y(s) (s+4)(s+8) +
AP7.6 A unity feedback system has a loop transfer functionSketch the root locus for K > 0, and select a value for K that will maximize the damping ratio of the complex roots. L(s) = Ge(s)G(s) = K(s +58 +10) s3+6s2+12s
AP7.5 A unity feedback system has a loop transfer function K L(s) = G(s)G(s) = s3+10s2 +8s-15 a. Sketch the root locus and determine K for a sta- ble system with complex roots with =1/2. b. Determine the root sensitivity of the complex roots of part (a). c. Determine the percent change in K
AP7.4 A remote manipulator control system has unity feedback and a loop transfer function L(s) = Ge(s)G(s) s+1+a $3 +3as2 +2s We want the percent overshoot for a step input to be less than or equal to 30%. Sketch the root locus as a function of the parameter a. Determine the range of a required for
AP7.3 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
AP7.2 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
AP7.1 The top view of a high-performance jet aircraft is shown in Figure AP7.1(a) [20]. Using the block diagram in Figure AP7.1(b), sketch the root locus and determine the gain K so that the damping ratio of the complex poles near the jω-axis is the maximum achievable. Evaluate the roots at this
P7.39 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
P7.38 A unity feedback system has the loop transfer function(a) Determine the range of K so that the closedloop system is stable. (b) Sketch the root locus.(c) Determine the roots for K = 100. (d) For K = 100, predict the percent overshoot for a step input.(e) Determine the actual percent
P7.37 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 a percent overshoot but ultimately attains the final value of 1.When the closed-loop system is subjected to a ramp input, the output response
P7.36 A microrobot with a high-performance manipulator has been designed for testing very small particles, such as simple living cells [6]. The unity feedback system has a loop transfer function(a) Sketch the root locus for K > 0. (b) Find the gain and roots when the characteristic equation has
P7.35 A powerful electrohydraulic forklift can be used to lift pallets weighing several tons on top of 35-foot scaffolds at a construction site. The unity feedback system has a loop transfer function(a) Sketch the root locus for K > 0. (b) Find the gainK when two complex roots have a ζ =
P7.34 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
P7.33 The Bell-Boeing V-22 Osprey Tiltrotor 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
P7.32 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
P7.31 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
P7.30 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. to Z(s) R FIGURE P7.30 RLC network. C
P7.29 A unity feedback control system has the loop transfer function L(s) = Gc(s)G(s) = K(s + 6s+10) s(s+2s+10) We desire the dominant roots to have a damping ratio =0.707. Find the gain K when this condi- tion is satisfied. Show that the complex roots area s-11.1j11.1 at this gain.
P7.28 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 (NO ) x emissions are met mainly by exhaust-gas recirculation
P7.27 A unity feedback system has a loop transfer function L(s) = G(s)G(s) = K(s +0.05) s($+2) Sketch the root locus as a function of K. Determine the values of K where the root locus enters and leaves the real axis.
P7.26 A unity feedback system has a loop transfer function L(s) = Ge(s)G(s) = K(s+ 2) s(s+4)(s+5) (a) Sketch the root locus for 0K
P7.25 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
P7.24 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
P7.23 Repeat Problem P7.22 for the loop transfer function L s( ) = G s c( )G s( ) of Problem P7.1(c).Data from in 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 ζ = 0.707.Evaluate and compare the
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 ζ = 0.707.Evaluate and compare the sensitivity as a function of the poles and zeros of the loop transfer functionL s( ) = G s c( )G s( ).
P7.21 Determine the root sensitivity of the dominant roots of the power system of Problem P7.7. Evaluate the sensitivity for variations of (a) the poles at s = −4, and (b) the feedback gain, 1/R.
P7.20 Determine the root sensitivity for the dominant roots of the design for Problem P7.18 for the gainK = 4 / α β and the pole s = −2.
P7.19 In recent years, many automatic control systems for guided vehicles in factories have been installed.One system uses a magnetic tape applied to the floor to guide the vehicle along the desired lane [10, 15].Using transponder tags on the floor, the automatically guided vehicles can be tasked
P7.18 A feedback control system is shown in Figure P7.18. The filter G s c( ) is often called a compensator, and the design problem involves selecting the parameters α and β. Using the root locus method, determine the effect of varying the parameters. Select a suitable filter so that the time
P7.17 Consider the vibration absorber in Figure P7.17. Using the root locus method, determine the effect of the parameters M2 and k12. Determine the specific values of the parameters M2 and k12 so that the massM1 does not vibrate when F t( ) = a t sin( ) ω0 . Assume that M k 1 1 = = 1, 1, and
P7.16 Control systems for maintaining constant tension on strip steel in a hot strip finishing mill are called“loopers.” A typical system is shown in Figure P7.16.The looper is an arm 2 to 3 feet long with a roller on the end; it is raised and pressed against the strip by a motor [18]. The
P7.15 Suppose that the dynamics of a transport vehicle can be represented by the loop transfer function Ge(s)G(s) = K(s2 +40s+800)(s+40) s(s2+100s+1000)(s2 +250s + 4500) Sketch the root locus for the system. Determine the damping ratio of the dominant roots when K = 1000.
P7.14 The loop transfer function of a unity feedback system is - K(s2 2s+4) L(s)=G(s)G(s)= s(s+4)(s+6) This system is called conditionally stable because it is stable only for a range of the gain K such that k < K < k2. Using the Routh-Hurwitz criteria and the root locus method, determine the range
P7.13 A unity feedback system has the loop transfer function(a) Find the breakaway point on the real axis and the gain for this point. (b) Find the gain to provide two complex roots nearest the jω-axis with a damping ratioζ = 0.6. (c) Are the two roots of part (b) dominant?(d) Determine the
P7.12 A precision speed control system (Figure P7.12) is required for a platform used in gyroscope and inertial system testing where a variety of closely controlled speeds is necessary. A direct-drive DC torque motor system was utilized to provide (1) a speed range of 0.01°/s to 600°/s, and (2)
P7.11 A computer system requires a high-performance magnetic tape transport system [17]. The environmental conditions imposed on the system result in a severe test of control engineering design. A direct-drive DC motor system for the magnetic tape reel system is shown in Figure P7.11, where r

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