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engineering
modern control systems

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

P11.5 Consider the system described by x(t) = Ax(t) + Bu(t), where x(t) = (x1(1), x2 (1))", 0 1 and B = 00 The state feedback of the system is u(t) = -kx (1) -k2x2 (1), where k = 3k and k = k, k > 0. Given the initial condition x (0) = (0,1)", and determine the gain k to minimize the performance
P11.4 Consider the system x(t) =[ABK]x(t) = Hx(t), 0 1 where H = Determine the feedback gain -k -k k that minimizes the performance index J = x(t)x(t)dt when x(0) = 2, -1]. Plot the performance index J versus the gain k.
P11.3 An unstable robot system is described by the vector differential equation [9] 10 0 *x() = [ 1 8 ]x(0) + [ 9 ] (1), 00 where x(t) = (x1 (1) = x2 (1)). Both state variables are measurable, and so the control signal is set as u(t) = -k(x1 (1)+ x2 (1)). Design the gain k so that the perfor- mance
P11.2 To account for the expenditure of energy and resources, the control signal is often included in the performance integral. Then the operation will not involve an unlimited control signal u(t). One suitable performance index, which includes the effect of the magnitude of the control signal,
P11.1 A first-order system is represented by the time-domain differential equationx t ( ) = + x t( ) u t( ).A feedback controller is to be designed such thatu t( ) = −2 , kx( )t and the desired equilibrium condition is x t( ) = 0 ast → ∞. The performance integral is defined asand the
E11.12 Consider a single-input, single-output system that is described by x(t) = Ax(t) + Bu(t) y(t) = Cx(t),
E11.11 Consider the system shown in block diagram form in Figure E11.11. Obtain a state variable representation of the system. Determine if the system is controllable and observable. U(s) FIGURE E11.11 State variable block diagram with a feedforward term. + 51 8 9 5 2 Y(s)
E11.10 Consider the block diagram model in Figure E11.10.Write the corresponding state variable model in the form x(t) = Ax(t) + Bu(t) y(t)=Cx(t)+Du(t).
E11.9 Consider the second-order systemFor what values of k1 and k2 is the system completely controllable? 1-3 k x(t)= -4 1 x(1) + u(t) k y(t) =|10|x(t) +[0]u(t).
E11.8 Consider the third-order systemSketch a block diagram model of the system. 0 1 4 x (1) = 1 0 0 x(t) + -7-1-2 -1 u(t) 9 y(t)= [12 4 3x(1).
E11.7 Consider the system represented in state variable formSketch a block diagram model of the system. *(1) = Ax(1) + Bu(1) y(t)= Cx(1)+Du(1), where -5 -8 A= 1 B = C= [-13], and D = [0}-
E11.6 A system is described by the matrix equationsDetermine whether the system is controllable and observable. 0-7 y(t) = | 0 1 ]x(t). (1)x 4 5 1 = x(1) + [
E11.5 A system is described by the matrix equationsDetermine whether the system is controllable and observable. 0 = (1) x 1 -1-2 y(t) = [01] x(t). (1)| = |+ (0)| 1 -2 +(1) x
E11.4 A system is described by the matrix equations Determine whether the system is controllable and observable.
E11.3 A system is described by the matrix equations 300-400+300 y(t) = | 01 |x(t). Determine whether the system is controllable and observable.
E11.2 A magnetically suspended steel ball can be described by the linear equationThe state variables are x t 1 ( ) = position and x t 2 ( ) =velocity. Select a feedback so that the settling time(with a 2% criterion) is Ts = 4 s and P.O. ≤ 10% to a unit step input. Choose the feedback in the
E11.1 The ability to balance actively is a key ingredient in the mobility of a device that hops and runs on one springy leg, as shown in Figure E11.1 [8]. The control of the attitude of the device uses a gyroscope and a feedback such that u t( ) = Kx( )t , whereDetermine a value for k so that the
15. Consider the system Y(s) = G(s)U (s) = |||U(s). Determine the eigenvalues of the closed-loop system when utilizing state variable feed- back, where u(t) = -2x2 (1) -2x1 (1) + r(1). We define x1 (t) = y(t), x2 (t) = x (1), and r(t) is a reference input. a. s-1j1 $ = -1-j1 = b. s = -2+ j2 82 =
14. A feedback system has a state-space representation *(1) -75 === 1 y(t) = 0 3600 |x(1), where the feedback is u(t) = -Kx+r(t). The control system design specifications are: (i) the overshoot to a step input approximately P.O. 6%, and (ii) the settling time T 0.1 s. A state variable feedback gain
13. Consider the system where It is desired to place the observer poles at $12 = -3 13. Determine the appropriate state-variable feedback control gain matrix L. a. L = b. L= c. L = d. None of the above
12. Consider the closed-loop system in Figure 11.37, where -7 -10 Determine the state variable feedback control gain matrix K for a zero steady-state tracking error to a step input. a. K=3-9 b. K = 3-6 c. K = -3 2 d. K = -1 4
11. A system has the transfer function s+a T(s)= 4 +683 +12s2+12s+61 Determine the values of a that render the system unobservable. a. a=1.30 or a = -1.44 b. a = 3.30 or a = 1.44 c. a=-3.30 or a = -1.44 d. a = -5.7 or a = -2.04
10. Consider the system depicted in the block diagram in Figure 11.38. This system is:a. Controllable, observableb. Not controllable, not observablec. Controllable, not observabled. Not controllable, observable R(s) X(5) Y(s) s+3 X(s) 5 FIGURE 11.38 Two-loop feedback control system.
9. Consider the closed-loop system in Figure 11.37, where -12 -10 5 1 0 0 0 1 0 1 C=[355] Determine the state-variable feedback control gain matrix K so that the closed-loop system poles are s=-3, 67 ] a. K = [1 44 b. K = [10 44 67] c. K = 44 1 1 4, and -6. d. K = 1 67 44
8. A system has the state variable representation -1 0 0 x(t) = 0 -3 0 0 x(t) + 1 0 0-5 -1 y(t) = 1 2 1 ]x(1). 1 Determine the associated transfer function model G(s) = 5s + 32s+35 a. G(s)= s3+9s+23s+15 5s2 +32s+35 b. G(s) = s4+983+23s+15 2s2+16s+22 c. G(s)= s3+9s+23s +15 5s +32 d. G(s) = s+32s+9
7. Consider the systemThis system is:a. Controllable, observableb. Not controllable, not observablec. Controllable, not observabled. Not controllable, observable G(s)= 10 s2 (s+2)(s2+2s+5)*
6. Consider the systemThe system is:a. Controllable, observableb. Not controllable, not observablec. Controllable, not observabled. Not controllable, observable x(1) = y(1) 01 0 2 u(t) [ 3 ] + 1 7 8 ] - 0 -4 x(1) =|02|x(t).
5. Ackerman’s formula is used to check observability of a system.True or False
4. Optimal control systems are systems whose parameters are adjusted so that the performance index reaches an extremum value.True or False
3. The problem of designing a compensator that provides asymptotic tracking of a reference input with zero steady-state error is called state-variable feedback.True or False
2. The poles of a system can be arbitrarily assigned through full-state feedback if and only if the system is completely controllable and observable.True or False
1. A system is said to be controllable on the interval    t t 0, f    if there exists a continuous input u t( ) such that any initial state x( ) t0 can be transformed to any arbitrary state( )x tf in a finite interval t t f − >0 0.True or False
CP10.10 Consider the feedback control system shown in Figure CP10.10. The time delay is T = 0.7 s. Plot the phase margin for the system versus the gain in the range 0.1 ≤ ≤ K 10. Determine the gain K that maximizes the phase margin. What is the stability limit for K? R(s) FIGURE CP10.10
CP10.9 Consider a circuit with the transfer functionwhereC C 1 2 = = 1 mF, 0.1 Fµ , 1 R1 = Ω0 , andR2 = Ω 10 k . Plot the frequency response of the circuit. Vo(s) G(s) = 1+ RC2s Vi (s) 1+ RCs
CP10.8 Consider a unity feedback system with the loop transfer functionwhere z = 2 and p = 5. The actual percent overshoot of the compensated system will be P O. . = 59.1%. We want to reduce the percent overshoot to P O. . = 35%.Using an m-file script, determine an appropriate value for the zero
CP10.7 A lateral beam guidance system has an inner loop as shown in Figure CP10.7 [26].(a) Design a control system to meet the following specifications: (1) settling time (with a 2% criterion)to a unit step input of Ts ≤ 1 s, and (2) steady-state tracking error for a unit ramp input of less than
CP10.6 Consider the control system shown in Figure CP10.6. Design a lag compensator using root locus methods to meet the following specifications: (1) steadystate error less than 10% for a step input, (2) phase margin of P M. . ≥ 45°, and (3) settling time (with a 2%criterion) of Ts ≤ 5 s
CP10.5 The pitch attitude motion of a rigid spacecraft is described bywhere J is the principal moment of inertia, and is the input torque on the vehicle [7]. Consider the PD controllerG s c P ( ) = + K KDs.(a) Design a unity feedback control system to meet the following specifications: (1)
CP10.4 Consider the aircraft unity feedback control system in Figure CP10.4, where θ( )t is the pitch rate(rad/s) and δ( )t is the elevator deflection (rad). The four poles represent the phugoid and short-period modes. The phugoid mode has a natural frequency of 0.1 rad/s, and the short period
CP10.3 Consider the system in Figure CP10.1, whereDesign a compensator G s c( ) so that the steady-state tracking error to a ramp input is zero and the settling time (with a 2% criterion) is Ts ≤ 5 s. Obtain the response of the closed-loop system to the inputR s( ) = 1 s2, and verify that the
CP10.2 A unity feedback control system is shown in Figure CP10.2. Design the proportional controllerG s c ( ) = K so that the system has a phase margin ofP M. . = 55°. Develop an m-file to obtain a Bode plot, and verify that the design specification is satisfied.
CP10.1 Consider the control system in Figure CP10.1, whereDevelop an m-file to show that the phase margin is approximately P M. . = 50° and that the percent overshoot to a unit step input is approximately P O. . = 18%. 1 G(s)= $+9.5 and G(s)= 99 S
DP10.11 Modern microanalytical systems used for polymerase chain reaction (PCR) requires fast, damped tracking response [30]. The control of the temperature of the PCR reactor can be represented as shown in Figure DP10.11. The controller is chosen to be PID controller, denoted by G s c( ), with a
DP10.10 A unity feedback system has the process transfer functionDesign the controller G s c ( ) such that the Bode magnitude plot of the loop transfer function L s( ) =G s c( )G s( ) is greater than 20 dB for ω ≤ 0.01 rad s and less than −20 dB for ω ≤ 10 rad s. The desired shape of
DP10.9 Consider the feedback control system shown in Figure DP10.9. Design a PID compensator G s c1( )and a lead-lag compensator G s c2( ) such that, in each case, the closed-loop system is stable in the presence of a time-delay T = 0.1 s. Discuss the capability of each compensator to insure
DP10.8 A simple closed-loop control system has been proposed to demonstrate proportional-integral (PI)control of a windmill radiometer [27]. The windmill radiometer is shown in Figure DP10.8(a) and the control system is shown in Figure DP10.8(b). The variable to be controlled is the angular
DP10.7 A high-performance jet airplane is shown in Figure DP10.7(a), and the roll-angle control system is shown in Figure DP10.7(b). Design a controller G s c ( ) so that the step response is well behaved and the steady-state error is zero. That is, P.O. ≤ 10% and Ts ≤ 2 s. FIGURE DP10.7
DP10.6 The past years have witnessed a significant engine model-building activity in the automotive industry in a category referred to as “control-oriented” or “control design” models. These models contain representations of the throttle body, engine pumping phenomena, induction process
DP10.5 High-performance tape transport systems are designed with a small capstan to pull the tape past the read/write heads and with take-up reels turned by DC motors. The tape is to be controlled at speeds up to 200 inches per second, with start-up as fast as possible, while preventing permanent
DP10.4 A high-speed train is under development in Texas[21] with a design based on the French Train à Grande Vitesse (TGV). Train speeds of 186 miles per hour are foreseen. To achieve these speeds on tight curves, the train may use independent axles combined with the ability to tilt the train.
DP10.3 NASA has identified the need for large deployable space structures, which will be constructed of lightweight materials and will contain large numbers of joints or structural connections. This need is evident for programs such as the space station. These deployable space structures may have
DP10.2 The heading control of the traditional bi-wing aircraft, shown in Figure DP10.2(a), is represented by the block diagram of Figure DP10.2(b).(a) Determine the minimum value of the gain K when G s c( ) = K, so that the steady-state effect of a unit step disturbance T s d ( ) = 1 s is less
DP10.1 In Figure DP10.1, two robots are shown cooperating with each other to manipulate a long shaft to insert it into the hole in the block resting on the table.Long part insertion is a good example of a task that can benefit from cooperative control. The unity feedback control system of one robot
CDP10.1 The capstan-slide system of Figure CDP4.1 uses a PD controller. Determine the necessary values of the gain constants of the PD controller so that the deadbeat response is achieved. Also, we want the settling time (with a 2% criterion) to be Ts ≤ 250 ms.Verify the results.
AP10.9 The plant dynamics of a chemical process are represented byWe desire that the unity feedback system have a small steady-state error for a ramp input so that Kυ = 100.For stability purposes, we desire a gain margin ofG M. . ≥ 10 dB and a phase margin of P M. . ≥ 40°.Determine a
AP10.8 The Manutec robot has large inertia and arm length resulting in a challenging control problem, as shown in Figure AP10.8(a). The block diagram model of the system is shown in Figure AP10.8(b).The percentage overshoot for a step input should be P O. . ≤ 20% with a rise time of Tr ≤ 0.5 s
AP10.7 A unity feedback system has(a) Determine the percent overshoot and rise time for G s p( ) = 1 and for p = 1. (b) Select an appropriate value for p that will give an overshoot of P O. . ≤ 1%, and compare the results. G(s) = 1 s(s+1)(s+5) with a phase-lead compensator Ge(s) = K(s+2) S+15
AP10.6 Consider a unity feedback system with loop transfer functionWe wish to minimize the settling time of the system while requiring that K priate compensator parameters p and z that will minimize the settling time. Plot the system response. S+z K L(s) = G(s)G(s) = s+p s(s+1)
AP10.5 A unity feedback system is shown in Figure AP10.5. We want the step response of the system to have a percent overshoot of P O. . ≤ 10% and a settling time (with a 2% criterion) of Ts ≤ 4 s. (a) Design a phase-lead compensator G s c( ) to achieve the dominant roots desired. (b)
AP10.4 A DC motor control system with unity feedback has the form shown in Figure AP10.4. Select K1 andK2 so that the system response has a settling time(with a 2% criterion) Ts ≤ 0.5 s and a percent overshoot of P O. . ≤ 10% for a step input.
AP10.3 The system of Advanced Problem AP10.1 is required to have a percent overshoot of P O. . ≤ 13%with a steady-state error for a unit ramp input less than 0.125 ( 8 Kυ = ). Design a proportional plus integral (PI) controller to meet the specifications.
AP10.2 The system of Advanced Problem AP10.1 is to have a percent overshoot of P O. . ≤ 13%. In addition, we desire that the steady-state error for a unit ramp input will be less than 0.125 ( 8 Kυ = ) [24]. Design a lag compensator to meet the specifications. Check the resulting percent
AP10.1 A three-axis pick-and-place application requires the precise movement of a robotic arm in three-dimensional space, as shown in Figure AP10.1 for joint 2. The arm has specific linear paths it must follow to avoid other pieces of machinery. The overshoot for a step input should be less than
P10.43 A unity feedback system has a loop transfer functionPlot the percent overshoot of the closed-loop system response to a unit step input for K in the range 0 K 100. Explain the behavior of the system response for K in the range 0.129 K 69.872. K(s+2s+20) = L(s) G(s)G(s)= = s(s+2)(s+28+1)
P10.42 Consider the system shown in Figure P10.42 and let R s( ) = 0 and T s d ( ) = 0. Design the compensator G s c( ) = K such that, in the steady-state, the response of the system is less than −40 dB when the noise N s( ) is a sinusoidal input at a frequency ofω ≥ 100 rad s. FIGURE
P10.41 Repeat Example 10.12 when we want the rise time to be Tr = 1 s.
P10.40 For the system and requirements of Problem P10.39,determine the required compensator when the steadystate error for the ramp input must be equal to 0.02.
P10.39 A unity feedback system has a plantWe desire that the phase margin be P M. . = 30°. For a ramp input r t( ) = t, we want the steady-state error to be equal to 0.05. Design a phase-lag compensator to satisfy the requirements. Verify the results. G(s)= = 40 s(s+2)
P10.38 A unity feedback system has a plantWe desire to have a phase margin of P M. . = 35° and a relatively large bandwidth. Select the crossover frequency ωc = 10 rad s, and design a phase-lead compensator. Verify the results. G(s) = 30 s(s+3)
P10.37 A unity feedback system has the loop transfer functionDesign a compensator G s c( ) so that the percent overshoot for a step input R s( ) is P O. . ≤ 5% and the steady-state error is less than 1%. Determine the bandwidth of the system. 1 L(s) = G(s)G(s) = G(s) (s+2)(s+8)
P10.36 A system transfer function is a pure time delay of 0.5 s, so that G s( ) = e−s/2. Select a compensator G s c ( )so that the steady-state error for a step input is less than 2% of the magnitude of the step and the phase margin is P M. . ≥ 30°. Determine the bandwidth of the compensated
P10.35 A unity feedback system has the loop transfer functionwhere T is a time delay and K is the controller proportional gain. The block diagram is illustrated in Figure P10.35. The nominal value of K = 2. Plot the phase margin of the system for 0 2 ≤ ≤ T s when K = 2.What happens to the
P10.34 A magnetically levitated train operated in Berlin,Germany from 1989–1991. Fully automated trains can run at short intervals and operate with excellent energy efficiency. The control system for the levitation of the car is shown in Figure P10.34. Select a compensator so that the phase
P10.33 Consider the block diagram of the extender robot system shown in Figure P10.33 [14]. The goal is that the compensated system will have a velocity constant Kυ equal to 80, so that the settling time (with a 2% criterion) will be Ts = 1.6 s, and that the percent overshoot will be P O. . =
P10.32 When a motor drives a flexible structure, the structure’s natural frequencies, as compared to the bandwidth of the servodrive, determine the contribution of the structural flexibility to the errors of the resulting motion. In current industrial robots, the drives are often relatively
P10.31 For the system of Problem P10.30, use a phase-lag compensator and attempt to achieve a phase margin of P M. . = 50°. Determine the actual percent overshoot and peak time for the compensated system.
P10.30 An automated guided vehicle (AGV) can be considered as an automated mobile conveyor designed to transport materials. Most AGVs require some type of guide path. The steering stability of the guidance control system has not been fully solved. The slight“snaking” of the AGV about the track
P10.29 A liquid-level control system has a loop transfer function L(s) = Ge(s)G(s), where Ge(s) is a compensator, and the plant is G(s)= 10e-sT s (s +10)' where T = 50 ms. Design a compensator so that Mpw does not exceed 3.5 dB and w, is approxi- mately 1.4 rad/s. Predict the percent overshoot and
P10.28 An adaptive suspension vehicle uses a legged locomotion principle. The control of the leg can be represented by a unity feedback system with [12]We desire to achieve a steady-state error for a ramp input of 10% and a damping ratio of the dominant roots of ζ = 0.707. Determine a suitable
P10.27 An engineering design team is attempting to control a process shown in Figure P10.27. It is agreed that a system with a phase margin of P M. . = 50° is acceptable. Determine G s c( ).First, let G s c( ) = K and find (a) a value of K that yields a phase margin of P M. . = 50° and the
P10.26 A computer uses a printer as a fast output device.We desire to maintain accurate position control while moving the paper rapidly through the printer.Consider a system with unity feedback and a transfer function for the motor and amplifier of 0.2 G(s)= s(s+1)(6s+1) Design a phase-lead
P10.25 The possibility of overcoming wheel friction,wear, and vibration by contactless suspension for passenger-carrying mass-transit vehicles is being investigated throughout the world. One design uses a magnetic suspension with an attraction force between the vehicle and the guideway with an
P10.24 The stability and performance of the rotation of a robot (similar to waist rotation) presents a challenging control problem. The system requires high gains in order to achieve high resolution; yet a large percent overshoot of the transient response cannot be tolerated. The block diagram of
P10.23 A system the loop transfer function with unity feedback hasWe desire the steady-state error to a step input to be approximately 4% and the phase margin to beP M. . = 60°. Design a phase-lag compensator to meet these specifications. L(s) = G(s)G(s) = K (s+4)
P10.22 For the system of Problem P10.20, we wish to achieve the same phase margin and Kυ, but in addition, we wish to limit the bandwidth to 2 rad s ≤ ≤ ωB 10 rad s. Use a lead-lag compensation to compensate the system. The compensator could be of the formwhere a is to be selected for the
P10.21 For the system of Problem P10.20, design a phaselag compensator to yield the desired specifications, with the exception that a bandwidth ωB ≥ 2 rad s will be acceptable.
P10.20 An uncompensated control system with unity feedback has a plant transfer functionWe want to have a velocity error constant of Kυ = 20.We also want to have a phase margin of P M. . = 45°and a closed-loop bandwidth ωB ≥ 4 rad s. Use two identical cascaded phase-lead compensators to
P10.19 There have been significant developments in the application of robotics technology to nuclear power plant maintenance problems. Thus far, robotics technology in the nuclear industry has been used primarily on spent-fuel reprocessing and waste management. The industry is applying the
P10.18 NASA is developing remote manipulators that can be used to extend the hand and the power of humankind through space by means of radio. A concept of a remote manipulator is shown in Figure P10.18(a)[11, 22]. The closed-loop control is shown schematically in Figure P10.18(b). Assuming an
P10.17 A unity feedback control system for a robot submarine has a plant with a third-order transfer function [20]:methods. Let the zero of the compensator be located at s = −15, and determine the compensator pole.Determine the resulting system Kυ. K G(s)= s(s+10)(s+50) We want the percent
P10.16 A driver and car may be represented by the simplified model shown in Figure P10.16 [17]. The goal is to have the speed adjust to a step input with a percent overshoot of P O. . ≤ 10% and a settling time (with a 2% criterion) of Ts = 1 s. Select a proportional plus integral (PI)
P10.15 A robot with an extended arm has a heavy load, whose effect is a disturbance, as shown in Figure P10.15 [22]. Let R s( ) = 0 and design G s c ( ) so that the maximum value of the disturbance response is less than 0.25 and the steady-state error to a unit step disturbarce is zero.
P10.14 For the system described in Problem P10.13, the goal is to achieve a phase margin of P M. . = 45° with the additional requirement that the time to settle (to within 2% of the final value) is Ts ≤ 10 s. Design a phase-lead compensator to meet the specifications.As before, we require
P10.13 Materials testing requires the design of control systems that can faithfully reproduce normal specimen operating environments over a range of specimen parameters [23]. From the control system design viewpoint, a materials-testing machine system can be considered a servomechanism in which
P10.12 A unity feedback control system has a plantSelect a compensator G s c( ) so that the phase margin is P M. . ≥ 80°. Use a two-stage lead compensatorIt is required that the error for a ramp input be 1% of the magnitude of the ramp input ( ) Kυ = 100 . 25 G(s)= s(1+0.2s) (1+0.1s)
P10.11 A unity feedback control system has the loop transfer functionSelect a lead-lag compensator so that the percent overshoot for a step input is P O. . ≤ 5% and the settling time (with a 2% criterion) is Ts ≤ 1 s. It also is desired that the acceleration constant Ka be greater than 7500.
P10.10 A unity feedback system has the loop transfer function(a) Determine the step response when G s c ( ) = 1, and calculate the settling time and steady state for a ramp input r t( ) = > t t , 0. (b) Design a phase-lag compensator using the root locus method so that the velocity constant
P10.9 The Avemar ferry, shown in Figure P10.9(a), is a large 670-ton ferry hydrofoil built for Mediterranean ferry service. It is capable of 45 knots (52 mph) [29].The boat’s appearance, like its performance, derives from the innovative design of the narrow “wavepiercing” hulls which move
P10.8 is an interesting problem in attaining sufficient ac A numerical path-controlled machine turret lathe -curacy [2, 23]. A block diagram of a turret lathe control system is shown in Figure P10.8. The gear ratio isn J = = 0.2, 10−3, and b = × 2.0 10−2. It is necessary to attain an
P10.7 A chemical reactor process whose production rate is a function of catalyst addition is shown in block diagram form in Figure P10.7 [10]. The time delay isT = 50 s, and the time constant τ is approximately 40 s. The gain of the process is K = 1. Design a compensator using Bode plot methods

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