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engineering
modern control systems

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

E6.24 Consider the system represented in state variable form(a) What is the system transfer function? (b) For what values of k is the system stable? where A x(t) = Ax(t)+ Bu(t) y(t) = Cx(t) + Du(t), 0 0 0 0 0 B= 0 +111-1 -k -k -k C = [1 0 0]. D=[0].
E6.23 The matrix differential equation of a state variable model of a system is 0 1 k x(t)= 0 0 1 x(t). -1-2-1 Find the range of k where the system is stable.
E6.22 A system has the characteristic equationq s( ) = + s s 3 2 15 + + 30s K = 0.Shift the vertical axis to the right by 1 by usings s 1 = −n , and determine the value of gain K so that the complex roots are s j = −1 3 ± .
E6.21 A system has a transfer function Y s( )/ = R s( )T s( ) = /1 1 s s( ) + . (a) Is this system stable? (b) If r t( )is a unit step input, determine the response y t( ).
E6.20 Find the roots of the following polynomials:(a) s s 3 2 + + 5 8s + =4 0 and(b) s s 3 2 + + 9 27 2 s + =7 0.
E6.19 Determine whether the systems with the following characteristic equations are stable or unstable:(a) s s 3 2 + + 4 6s + = 100 0,(b) s s 4 3 + + 6 10 1 s s 2 + + 7 6 = 0, and(c) s s 2 + + 6 3 = 0.
E6.18 A system has a characteristic equationq s( ) = + s s 3 2 + + 9 9 s = 0.(a) Determine whether the system is stable, using the Routh–Hurwitz criterion. (b) Determine the roots of the characteristic equation.
E6.17 The matrix differential equation of a state variable model of a system is -6 1 -3 x(t)=4-3 3 x(t). -4-1-7 (a) Determine the characteristic equation. (b) Deter- mine whether the system is stable. (c) Determine the roots of the characteristic equation.
E6.16 A system has a characteristic equation q s( ) = + s s 5 4 5 1 + + 2 6 s s 3 2 + + 42s 10 = 0.(a) Determine whether the system is stable, using the Routh–Hurwitz criterion. (b) Determine the roots of the characteristic equation.
E6.15 A system has a characteristic equationq s( ) = + s s 6 5 9 + + 31.25s s 4 3 61.25+ + 67.75s s 2 14.75 + = 15 0.(a) Determine whether the system is stable, using the Routh–Hurwitz criterion. (b) Determine the roots of the characteristic equation.
E6.14 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 is 0 5 -1 *(1) 2 -4 7 x(t), -13 4 where
E6.13 Consider the feedback system in Figure E6.13.Determine the range of KP and KD for stability of the closed-loop system.
E6.12 A system has the second-order characteristic equations a 2 + +s 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
E6.11 A system with a transfer function Y(s)/R(s) is Y(s) 24(s+1) R(s) s+683 +2s+s+3 Determine the steady-state error to a unit step input. Is the system stable?
E6.10 Consider a feedback system with closed-loop transfer function 4 T(s)= 5+4s4+8s3+8s+7s+4" Is the system stable?
E6.9 A system has a characteristic equations s 3 2 + + 2 1 ( ) K s + + 8 0 = .Find the range of K for a stable system.
E6.8 Designers have developed small, fast, vertical-takeoff fighter aircraft that are invisible to radar (stealth aircraft). This aircraft concept uses quickly turning jet nozzles to steer the airplane [16]. The control system for the heading or direction control is shown in Figure E6.8.
E6.7 For the feedback system of Exercise E6.5, find the value of K when two roots lie on the imaginary axis.Determine the value of the roots.
E6.6 A negative feedback system has a loop transfer function(a) Find the value of the gain when the ζ of the closed-loop roots is equal to 0.5. (b) Find the value of the gain when the closed-loop system has two roots on the imaginary axis. L(s) = G(s)G(s) = K(s+1) s(s-2)
E6.5 A unity feedback system has a loop transfer function K L(s)= s(s+2)(s+5)(s+12) where K=15. Find the roots of the closed-loop system's characteristic equation.
E6.4 A control system has the structure shown in Figure E6.4. Determine the gain at which the system will become unstable.
E6.3 A system has the characteristic equation s 4 + 10s 3 +32 37 20 0. s s 2 + + = Using the Routh–Hurwitz criterion, determine if the system is stable.
E6.2 A system has a characteristic equation s 3 + 10s 2 +2 3 s + =0 0. Using the Routh–Hurwitz criterion, show that the system is unstable.
E6.1 A system has a characteristic equation s 3 + 5Ks2 +( ) 2 1 + + K s 5 0 = . Determine the range of K for a stable system.
15. A system has the block diagram representation as shown in Figure 6.27, where 10 K G(s) = and Ge(s)= = s+80 where K is always positive. The limiting gain (s+15) for a stable system is: a. 0 < K
14. Using the Routh–Hurwitz criterion, determine whether the system is stable, unstable, or marginally stable.a. Stableb. Unstablec. Marginally stabled. None of the above In Problems 13 and 14, consider the system represented in a state-space form 0 0 1 0 x = 0 0 1 x+ 0 u -5-10-5 20 y= [101] x
13. The characteristic equation is:a. q s( ) = + s s 3 2 5 1 − − 0 6 s b. q s( ) = + s s 3 2 5 1 + + 0 5 s c. q s( ) = − s s 3 2 5 1 + − 0 5 s d. q s( ) = − s s 2 5 1 + 0 In Problems 13 and 14, consider the system represented in a state-space form 0 0 1 0 x = 0 0 1 x+ 0 u
12. Consider the following unity feedback control system in Figure 6.27 where 1 G(s)= and G(s)= K(s +0.3) (s-2)(s+10x+45) The range of K for stability is a. K < 260.68 b. 50.06 < K < 123.98 c. 100.12 < K < 260.68 d. The system is unstable for all K > 0
11. Use the Routh array to assist in computing the roots of the polynomial a. s = 1; 823 = j b. $ = 1; $2,3 = j c. s = -1; $2,3 = 1j- =1 d. s = -1; 823 = 1 2 = q(s) 283+2s2+s+1=0.1
10. A system is represented by x A = x , where 0 1 0 A = 1 00 -5-K -10 The values of K for a stable system are a. K < 1/2 b. K > 1/2 c. K = 1/2 d. The system is stable for all K
9. Consider a unity negative feedback system in Figure 6.27 with loop transfer function where K L(s) =G(s)G(s) = (1+0.5s)(1+0.5s+ 0.25s Determine the value of K for which the closed-loop system is marginally stable. a. K = 10 b. K = 3 c. The system is unstable for all K d. The system is stable for
8. Consider the feedback control system block diagram in Figure 6.27. Investigate closed 1 for the two cases where loop stability for G(s) = K(s+1) and G(s)= K = 1 and K = 3. a. Unstable for K = 1 and stable for K = 3 b. Unstable for K = 1 and unstable for K = 3 c. Stable for K = 1 and unstable for
7. Utilizing the Routh–Hurwitz criterion, determine whether Systems 1 and 2 with the following polynomials are stable or unstable:p s 1( ) = + s s 2 10 + 5,p s 2( ) = + s s 4 3 + + 5 2 s s 2 0 10. +a. System 1 is stable, System 2 is stableb. System 1 is unstable, System 2 is
6. A system has the characteristic equationq s( ) = + s K 3 2 4 5 s K + + ( )s + = 10 0.The range of K for a stable system is:a. K ≥ 0.46b. K < 0.46c. 0 0.46 < < K d. Unstable for all K
5. Relative stability characterizes the degree of stability.True or False
4. The Routh–Hurwitz criterion is a necessary and sufficient criterion for determining the stability of linear systems.True or False
3. A system is stable if all poles lie in the right half-plane.True or False
2. A marginally stable system has poles on the jω-axis.True or False
1. A stable system is a dynamic system with a bounded output response for any input.True or False
CP5.12 A closed-loop transfer function is given by Y(s) T(s)= 12(s+3) R(s) (s+10)(s2+ 6s+45)* (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 T, (use a 2% criterion) and percent overshoot P.O.? (b) Neglecting the real pole
CP5.11 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.(b) By consecutively adding zeros at 0, − − 0.5, 1.5, and −2.5, 0, − − 0.5, 1.5, and −2.5, determine the step response. Compare the
CP5.10 Develop an m-file to simulate the response of the system in Figure CP5.10 to a parabolic inputR s( ) = /1 . s3 What is the steady-state error? Display the output on an x-y graph. R(s) (s+1)(s+2)(s+3)(s+4)(s+5) s2(s+6)(s+7)(s+8) Y(s) FIGURE CP5.10 Closed-loop system for m-file.
CP5.9 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? 30 R(s) Y(s) s+30 5 0.2s+5 FIGURE CP5.9 Nonunity feedback system.
CP5.8 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 M T pt, p, and Ts for the closed-loop system due to a unit step input. Compare the predicted results with the actual unit step response obtained
CP5.7 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 G s c ( ) = 5. Using the Isim function, compute and plot the ramp response forθd ( )t a = t, where a = / 1.5° s. Determine
CP5.6 The closed-loop transfer function of a simple second-order system is W 3 + 2wns +w Consider the following cases: 1. w1, 0.5, 2. wn=2, 0.5,
CP5.5 Consider the feedback system in Figure CP5.5.Develop an m-file to design a controller and prefilter G(s) = K+2 and Gp(s): S+T s+P such that the ITAE performance criterion is minimized. For w=0.45 and 0.59, plot the unit step response and determine the percent overshoot and settling time.
CP5.4 Consider the unit step response of the simple second-order closed-loop control system shown in Figure CP5.4.(a) Determine analytically the damping ratio ξ and natural frequency ωn of the closed-loop system response to a unit step input, and the corresponding closed-loop system transfer
CP5.3 A working knowledge of the relationship between the pole locations of a second-order system and the transient response is important in control design. With that in mind, consider the following five pole location cases:Using the impulse and subplot functions, create a plot containing two
CP5.2 A unity negative feedback system has the loop transfer function 2s+8 L(s) = Ge(s)G(s)= $ ($+5s+20) Using Isim, obtain the response of the closed-loop sys- tem to a unit ramp input, R(s) = 1/s. Consider the time interval 0 <
CP5.1 Consider the closed-loop transfer function 35 T(s)= s+12s+35 Obtain the impulse response analytically, and compare the result to one obtained using the impulse function.
DP5.8 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
DP5.7 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
DP5.6 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 ≤ 0.7 s, and the percent overshoot for a step input is≤ 5%. FIGURE DP5.6 Position control robot. R(s) 10 K s(s+4) (1+2Ks) Y(s)
DP5.5 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
DP5.4 The space satellite, as 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 closedloop system.(b) Using the second-order model, select a gainK so that the percent overshoot is ≤ 10%, and the
DP5.3 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]. The
DP5.2 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.The damping ratio ζ , the gain K, and the natural frequency ωn can be selected. (a) Determine K, andωn so that the response to a unit
DP5.1 The roll control autopilot of an aircraft is shown in Figure DP5.1. The goal is to select a suitable K so that the response to a step command φd ( )t A = ≥ , t 0, will provide a response φ( )t that is a fast response and has an percent overshoot of P O. . ≤ 20%. (a) Determine the
CDP5.1 The capstan drive system of the previous problems (see CDP1.1–CDP4.1) has a disturbance due to changes in the part that is being machined as material is removed. The controller is an amplifierG s c a ( ) = K . Evaluate the effect of a unit step disturbance, and determine the best
AP5.9 The unity negative feedback system in Figure AP5.9 has the process given byThe 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 equal to 0.707. Determine the
AP5.8 A unity negative feedback system has an open loop transfer function K G(s) = $2+8s Determine the gain K that results in the fastest re- sponse without overshoot. What are the correspond- ing poles?
AP5.7 Consider the closed-loop system in Figure AP5.7 with transfer functions(a) Assume that the complex poles dominate, and estimate the settling time and percent overshoot to a unit step input for K = 1, 10, 25, and 50.(b) Determine the actual settling time and percent overshoot to a unit step
AP5.6 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 in terms of K, Kb, and Km.(b) Let Km = 12 and Kb = 0.01, and select K so that steady-state tracking error is equal to 1.(c) Plot the
AP5.5 A system with a controller is shown in Figure AP5.5. The zero of the controller may be varied.(a) Determine the steady-state error for a unit step input for α = 0 and α ≠ 0.(b) Let α = 1, 15, and 75. Plot the response of the system to a unit step input disturbance for the three values
AP5.4 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. Consider the three values for K equal to 1, 10, and 100.(a) Determine the steady-state error.(b) Determine and plot the
AP5.3 A closed-loop system is shown in Figure AP5.3.Plot the response to a unit step input for the system with τp = 0, 0.2, 1, and 4. Record the percent overshoot, rise time, and settling time (with a 2% criterion) as τp varies. Describe the effect of varying τp.Compare the location of the
AP5.2 A closed-loop system is shown in Figure AP5.2.Plot the response to a unit step input for the system for τz = 0, 0.05, 0.1, and 0.5. Record the percent overshoot, rise time, and settling time (with a 2%criterion) as τz varies. Describe the effect of varyingτz. Compare the location of the
AP5.1 Consider the following closed-loop transfer functions T(s) = 10(s+1) (s+5)(s +2s+2) and s+10 T (s) = (s+5)(s+2s+2) (a) Determine the steady-state error for a unit step input. (b) Assume that the complex poles dominate, and determine the percent overshoot and settling time to within 2% of the
P5.22 Consider the closed-loop system in Figure P5.22, where 2 2 G(s)G(s) = and H(s)= S+0.2K 2s+T (a) If = 2.43, determine the value of K such that the steady-state error of the closed-loop system response to a unit step input, is zero. (b) Determine the percent overshoot and the time to peak of
P5.21 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 P O. . ≤ 5%. R(s) 1 s+2k 1 s+a FIGURE
P5.20 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, whereE s( ) = − R s( ) Y s( ).(b) Select K1, so that the steady-state error is zero. K R(s) K Y(s) (s+5)(s +11) FIGURE P5.20 System with pregain, K.
P5.19 A closed-loop control system with negative unity feedback has a loop transfer function L(s) = G(s)G(s) = 8 s(s+68+12) (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 ap- proximation
P5.18 Consider the third-order system 1 G(s) = s3+5s2+10s+1' Determine a first-order model with one pole unspecified and no zeros that will represent the third-order system.
P5.17 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
P5.16 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
P5.15 Consider a unity feedback system with loop transfer function K(s+3) L(s) = G(s)G(s) = (s+5)(s +48 +10) Determine the value of the gain K such that the per- cent overshoot to a unit step is minimized.
P5.14 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 GL(s)= = 0.986s+2 $2+3s+2 0.986(s +2.028) (s+1)(s+2)
P5.13 We want to approximate a fourth-order system by a lower-order model. The transfer function of the original system is GH(S)= s3+7s2+24s+24 s4+10s3+35s + 50s +24 3+7s2+248 + 24 (s+1)(s+2)(s+3)(s+4)* Show that if we obtain a second-order model by the method of Section 5.8, and we do not specify
P5.12 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 Shinkansen train that travels an average speed of 320
P5.11 A unity feedback control system has a process transfer function Y(s) K =G(s): E(s) S The system input is a step function with an amplitude A. The initial condition of the system at time to is y(to) Q, where y(t) is the output of the system. The performance index is defined as 1 = e (1) dt. 0
P5.10 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 (see Example 2.5). (b) Calculate the steady-state error of this system to a step input command setting the speed to a new level.
P5.9 Antennas that receive and transmit signals to communication satellites generally include an extremely large horn antenna. The microwave antenna can be 175 ft long and weigh 340 tons. A photo of an antenna is shown in Figure P5.9. Suppose that the communication satellite is 3 ft in diameter and
P5.8 Photovoltaic arrays 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 such
P5.7 Astronaut Bruce McCandless II took the first untethered 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 astronaut is
P5.6 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 loop transfer function of the plant is for the closed-loop system shown in Figure P5.6(b).Calculate the steady-state
P5.5 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 arc minute per second. The system is illustrated in Figure P5.5(a). The control system is
P5.4 The loop transfer function of a unity negative feedback systemA system response to a step input is specified as follows:Tp = 0.25 s,P O. . = 10%.(a) Determine whether both specifications can be met simultaneously. (b) If the specifications cannot be met simultaneously, determine a compromise
P5.3 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 linear path with a closed-loop control system, as shown in Figure P5.3(b). Calculate the necessary gains K and K1 to result in
P5.2 A specific closed-loop control system is to be designed for an underdamped response to a step input.The specifications for the system are as follows:10% < < P O. . 20%,Ts < 0.6 s.(a) Identify the desired area for the dominant roots of the system. (b) Determine the smallest value of a third
P5.1 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
E5.20 Consider the closed-loop system in Figure E5.20,where (s+2) L(s)= -Ka- (s2+5s) (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. (c) Select a value for K, so that the steady-state
E5.19 A second-order system has the closed-loop transfer function Y(s) T(s) = w R(s) s+2ws + w/1 20 s+5.38s+20 (a) Estimate the percent overshoot P.O., the time to peak Tp, and the settling time T, of the unit step response. (b) Obtain the system response to a unit step, and verify the results in
E5.18 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. R(s) K y(1) (a) 1.0 0.8 0 0 G(s) Y(s) (b) FIGURE E5.18 Feedback system with prefilter.
E5.17 A closed-loop control system transfer functionT s( ) has two dominant complex conjugate poles.Sketch the region in the left-hand s-plane where the complex poles should be located to meet the given specifications. (a) 0.6 0.8, W 10 (b) 0.5 0.707, Wn 10 (c) ( 0.5, 5
E5.16 A second-order system isConsider the case where 1 8 z . Obtain the partial fraction expansion, and plot the output for a unit step input for z = 2, 4, and 6. Y(s) (10/2)(s+z) T(s) = R(s) (s+1)(s+8)]
E5.15 A closed-loop control system has a transfer func T s( ) as follows:Plot y t( ) for a unit step input when (a) the actualT s( ) is used, and (b) using the dominant complex poles. Compare the results. T(s)= Y(s) 2500 R(s) (s+50)(s2+10s +50)*
E5.14 A feedback system is shown in Figure E5.14.(a) Determine the steady-state error for a unit step when K = 0.6 and G s p( ) = 1.(b) Select an appropriate value for G s p( ) so that the steady-state error is equal to zero for the unit step input. K R(5) Gp(s) Y(s) s(s+5) s+1 (s+ 0.5) FIGURE
E5.13 For the system with unity feedback shown in Figure E5.11, determine the steady-state error for a step and a ramp input when 20 G(s) = s +14s +50
E5.12 The Ferris wheel is often 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 bridge

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