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computer science
modern database management 13th edition
Modern Control Systems 13th Global Edition Robert Bishop Richard Dorf - Solutions
For physically realizable systems, the loop gain L(s) = Gc(s)G(s)must be large for high frequencies. True or False
A robust control system exhibits the desired performance in the presence of significant plant uncertainty. True or False
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(t) - 2x1(t) + r(t). We define x1(t) = y(t) and r(t) is a refer- ence input. a. s-1+j1 s = -1 - jl b. s = -2+12 32-2-j2 c. s = -1+12 $2 =
A feedback system has a state-space representation *(1) = [ x(t -75 0] x(t) + [0] u(t) y(t) = [0_3600] x(t), 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
Consider the system where -3 0 = A-78-8 B C = [01]. It is desired to place the observer poles at $1.2=-3 13. Determine the appropriate state-variable feedback control gain matrix L. a. L = b. L c. L = d. None of the above
Consider the closed-loop system in Figure 11.37, where A = = [-1 -1] -10] B = [] C = [01]. 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 = [-14]
A system has the transfer function s + a T(s) = s+63 +12s + 12s +6 Determine the values of a that render the system unobservable. a. a 1.30 or a = -1.43 b. a 3.30 or a = 1.43 c. a -3.30 or a = -1.43 d. a -5.7 or a = -2.04
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) 1 X2(5) s+3 Y(s) X(3) 5 S FIGURE 11.38 Two-loop feedback control system.
Consider the closed-loop system in Figure 11.37, where -12 -10 -5 A = 1 0 0 B=0 01 0 C=[35-5]. Determine the state-variable feedback control gain matrix K so that the closed-loop system poles are s = -3, -4, and -6. a. K [1 44 67] = b. K [10 44 67] c. K = [44 1 1] d. K [1 67 44]
A system has the state variable representation -1 0 0 x(t) = 0 -3 0 x(t) + 0 0 -5 y(t) = [121]x(t) Determine the associated transfer function model G(s) 5s + 32s + 35 a. G(s) s39s+23s+ 15 5s + 32s+35 b. G(s) = s4+93 +235 +15 2s + 16s+22 c. G(s)=3+9s + 23s + 15 d. G(s) = 5s + 32 s + 32s+9 1u(t) Y(s)
Consider the systemThis system is:a. Controllable, observableb. Not controllable, not observablec. Controllable, not observabled. Not controllable, observable 10 G(s) = s(s+2)(s+2s+5)*
Consider the systemThe system is:a. Controllable, observableb. Not controllable, not observablec. Controllable, not observabled. Not controllable, observable Lo y(t) = [02]x(t) (1) x(t) = [84]) + [2]4) u(t)
Ackerman’s formula is used to check observability of a system. True or False
Optimal control systems are systems whose parameters are adjusted so that the performance index reaches an extremum value. True or False
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
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
True or False
A system is said to be controllable on the interval 3t0, tf 4 if there exists a continuous input u1t2 such that any initial state x1t02 can be transformed to any arbitrary state x1tf2 in a finite interval tf - t0 7
Using a Nichols chart, determine the gain and phase margin of the system in Figure 10.38 with loop gain transfer function a. G.M. L(s) = Ge(s)G(s) 20.4 dB, P.M. = 58.1 b. G.M. = dB, P.M. = 47 c. G.M. 6 dB, P.M. = 45 d. G.M. = 0 dB, P.M. = 23 8s + 1 s(s+2s+4)
Consider the feedback system depicted in Figure 10.38, where G(s) = s(s + 4) A suitable compensation Ge(s) for this system that satisfies the specifications: (i) P.O. 20%, and (ii) velocity error constant K, 10, is which of the following: a. Ge(s) = S+4 (s + 1) 160(10s + 1) b. G.(s) = 200s 1 24(s
A viable phase-lag compensator for a unity negative feedback system with plant transfer function 1000 G(s) = (s+8)(s+14) (s + 20) that satisfies the design specifications: (i) percent overshoot P.O. 5%; (ii) rise time T, 20 s, and (iii) position error constant K, > 6, is which of the following:
Consider the feedback system in Figure 10.38, where 1 G(s) = s(s+10) (s +15)
Consider a feedback system in which a phase-lead compensator is placed in series with the plant Ge(s) = 1 +0.4s 1 + 0.04s 500 G(s) = (s+1)(s+5)(s + 10)* The feedback system is a negative unity feedback control system shown in Figure 10.38. Compute the gain and phase margin. a. G.M. dB, P.M. = 60 ==
Consider the feedback system in Figure 10.38, where 1 G(s) = s(1s/8)(1 + s/20)* The design specifications are: K, 100, G.M. > 10 dB, P.M. > 45, and the crossover frequency, w 10 rad/s. Which of the following controllers meets these specifications? (1 + s)(1 + 20s) = a. Ge(s) = b. Ge(s) c. Ge(s) d.
Consider the feedback system in Figure 10.38, where the plant model is G(s) = 500 s(s + 50) and the controller is a proportional-plus-integral (PI) controller given by K Ge(s) = Kp + S Selecting K, = 1, determine a suitable value of Kp for a percent overshoot of P.O = 20%. a. Kp = 0.5 b. Kp = 1.5
Consider a unity feedback system in Figure 10.38, where G(s) = 1450 s(s+3)(s +25) A phase-lead compensator is introduced into the feedback loop, where 1 + 0.3s Ge(s) 1 + 0.03s The peak magnitude and the bandwidth of the closed-loop frequency response are: a. Mp = 1.9 dB: w = b. Mp = 12.8 dB; w 12.1
A position control system can be analyzed using the feedback system in Figure 10.38, where the process transfer function is 5 G(s): = s(s + 1)(0.4s + 1)* A phase-lag compensator that provides a phase margin of P.M. 30 is: 1+s a. Ge(s) = 1 + 106s 1 + 26s b. G.(s) = c. Ge(s) = 1 + 115s 1 + 106s
Consider the feedback system in Figure 10.38, where 1000 G(s) = s(s + 400) (s + 20)* A phase-lag compensator is designed for the system to give additional attenuation at higher frequencies. The compensator is 1 + 0.25s Ge(s) 1 + 2s When compared with the uncompensated system (that is, G.(s) = 1),
A phase-lead compensator can be used to increase the system bandwidth. True or False
A deadbeat response of a system is a rapid response with minimal percent overshoot and zero steady-state error to a step input. True or False
The arrangement of the system and the selection of suitable components and parameters is part of the process of control system design. True or False
Generally, a phase-lag compensator speeds up the transient response. True or False
A cascade compensator is a compensator that is placed in parallel with the system process. True or False
Consider a control system with unity feedback as in Figure 9.69 with loop transfer function (s + 4) L(s) = Ge(s)G(s) s(s + 1)(s +5)* The gain and phase margin are: a. G.M. dB, P.M. = 58.1 b. G.M. = 20.4 dB, P.M. = 47.3 c. G.M. = 6.6 dB, P.M. = 60.4 d. Closed-loop system is unstable
A feedback model of human reaction time used in analysis of vehicle control can use the block diagram model in Figure 9.69 with 1 Ge(s) = est and G(s) = = s(0.25 + 1) A typical driver has a reaction time of T = 0.3 s. Determine the bandwidth of the closed- loop system. a. p = 0.5 rad/s b. w 10.6
Consider the control system in Figure 9.69, where the loop transfer function is 1 L(s) = Ge(s)G(s) = s(s + 1) The value of the resonant peak, M, and the damping factor, 5, for the closed-loop system are: a. Mp. = 0.37,5 = 0.707 b. Mp 1.15, = 0.5 = c. Mp = 2.55,5 = 0.5 d. Mp = 0.55,5 = 0.25
Consider the feedback system in Figure 9.69, where -0.2s Ge(s) = K and G(s) = s+5 Notice that the plant contains a time-delay of T = 0.2 seconds. Determine the gain K such that the phase margin of the system is P.M. = 50. What is the gain margin for the same gain K? a. K=8.35, G.M. = 2.6 dB b. K =
Consider the closed-loop system in Figure 9.69, where the loop transfer function is K(s + 4) L(s) = Ge(s)G(s) 52 Determine the value of the gain K such that the phase margin is P.M. = 40. a. K = 1.64 b. K = 2.15 c. K = 2.63 d. Closed-loop system is unstable for all K > 0
Determine whether the closed-loop system in Figure 9.69 is stable or not, given the loop transfer function s+1 = L(s) Ge(s)G(s) == s(4s + 1) In addition, if the closed-loop system is stable, compute the gain and phase margins. a. Stable, G.M. = 24 dB, P.M. = 2.5 b. Stable, G.M. = 3 dB, P.M. = 24 c.
Using the value of K in Problem 8, compute the gain and phase margins when Td = 0.2.a. G.M. = 14 dB, P.M. = 27b. G.M. = 20 dB, P.M. = 64.9c. G.M. = dB, P.M. = 60d. Closed-loop system is unstable For Problems 8 and 9, consider the block diagram in Figure 9.69 where G(s) 9 (s+1)(s+3s+ 9)' and the
When Td = 0, the PD controller reduces to a proportional controller, Gc1s2 = K. In this case, use the Nyquist plot to determine the limiting value of K for closed-loop stability.a. K = 0.5b. K = 1.6c. K = 2.4d. K = 4.3 For Problems 8 and 9, consider the block diagram in Figure 9.69 where G(s) 9
Consider the block diagram in Figure 9.69. The plant transfer function isUtilize the Nyquist stability criterion to characterize the stability of the closed-loop system.a. The closed-loop system is stable.b. The closed-loop system is unstable.c. The closed-loop system is marginally stable.d. None
Consider the closed-loop system in Figure 9.69 where L(s) = Ge(s)G(s) = 3.25(1 + s/6) s(1s/3)(1s/8)* The crossover frequency and the phase margin are: a. w = 2.0 rad/s, P.M. = 37.2 b. w = 2.5 rad/s, P.M. = 54.9 c. w = d. w = 5.3 rad/s, P.M. = 68.1 10.7 rad/s, P.M. = 47.9
The phase margin of a second-order system (with no zeros) is a function of both the damping ratio z and the natural frequency, vn. True or False
A Nichols chart displays curves describing the relationship between the open-loop and closed-loop frequency responses. True or False
The gain and phase margin are readily evaluated on either a Bode plot or a Nyquist plot. True or False
A conformal mapping is a contour mapping that retains the angles on the s-plane on the transformed F1s2-plane. True or False
The gain margin of a system is the increase in the system gain when the phase is -180 that will result in a marginally stable system. True or False
The transfer function corresponding to the Bode plot in Figure 8.56 is: 100(s+10) (s+5000) a. G(s) = s(s + 5)(s+6) b. G(s) = = 100 c. G(s) = d. G(s) = 100 (s + 1)(s +20) (s+1)(s +50) (s + 200) 100(s+20)(s + 5000) (s+ 1)(s+50)(s + 200)
Determine the system type (that is, the number of integrators, N):a. N = 0b. N = 1c. N = 2d. N 7 2
Consider the feedback control system in Figure 8.53 with loop transfer function L(s) G(s)G(s) = 100 s(s + 11.8)
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. Mpv … 0.55b. Mpv … 0.59c. Mpv … 1.05d. Mpv … 1.27
Consider the Bode plot in Figure 8.55. Magnitude (dB) 100 50 0 -50 -100 -150 -90 -135 -180 -225 Phase (deg) -270 10-2 10-1 10 101 102 80 Frequency (rad/s) FIGURE 8.55 Bode plot for unknown system. 103
The slope of the asymptotic plot at very low 1v V 12 and high 1v W 102 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/decadec. At low frequency: slope =
The break frequencies on the Bode plot are a. w b. 1 and w = 12 rad/s 2 and w = 10 rad/s c. w = 20 and w = 1 rad/s d. w = 12 and w = 20 rad/s
Determine the frequency at which the gain has unit magnitude and compute the phaseangle at that frequency: L(s) = G(s)G(s) = 8(s+1) s(2 + s)(2 + 3s)
The Bode plot of this system corresponds to which plot in Figure 8.54? L(s) = G(s)G(s) = 8(s+1) s(2 + s)(2 + 3s)
Consider the stable system represented by the differential equation x(t) + 3x(t) = u(t), where u(t) = sin 31. Determine the phase lag for this system. a. =0 b. = -45 c. --60 d. = -180
One advantage of frequency response methods is the ready availability of sinusoidal test signals for various ranges of frequencies and amplitudes. True or False
The resonant frequency and bandwidth can be related to the speed of the transient response. True or False
A transfer function is termed minimum phase if all its zeros lie in the right-hand s-plane. True or False
A plot of the real part of G1jv2 versus the imaginary part of G1jv2 is called a Bode plot. True or False
The frequency response represents the steady-state response of a stable system to a sinusoidal input signal at various frequencies. True or False
The departure angles from the complex poles and the arrival angles at the complex zeros are: a. p = 180, A = 0 b. op . = +116.6, A +198.4 = +45.8, A = 116.6 d. None of the above
Which of the following is the associated root locus? Imaginary Axis Imaginary Axis -4 -2 Real Axis (a) -4 -3 -2 0 Real Axis (c) 0 Imaginary Axis Imaginary Axis T 40 20 0 2 -1 Real Axis (b) 0 1 -25 -20 -15 -10 -5 Real Axis (d) 0
Consider the unity feedback system in Figure 7.62 where K == L(s) Ge(s)G(s): == s(s + 5)(s + 6s+ 17.76)* Determine the breakaway point on the real axis and the respective gain, K. a. s -1.8, K = 58.75 b. s = -2.5, K = 4.59 c. s 1.4, K = 58.75 d. None of the above
Suppose that a simple proportional controller is utilized, that is, Gc1s2 = K. Using the root locus method, determine the maximum controller gain K for closed-loop stability.a. K = 0.50b. K = 1.49c. K = 4.49d. Unstable for K 7 0 G(s) = 7500 (s+1)(s+10) (s + 50)*
Suppose that the controller isUsing the root locus method, determine the maximum value of the gain K for closed-loop stability.a. K = 2.13b. K = 3.88c. K = 14.49d. Stable for all K 7 0 K(1 + 0.2s) Ge(s) = 1 +0.025s
Consider the unity feedback control system in Figure 7.62 where 10(s + z) L(s) = G(s)G(s) = s(s + 4s +8) Using the root locus method, determine that maximum value of z for closed-loop stability. a. z = 7.2 b. z = 12.8 c. Unstable for all z > 0 d. Stable for all z > 0
A unity feedback system has the closed-loop transfer function given by K T(s) (s + 45) + K Using the root locus method, determine the value of the gain K so that the closed-loop sys- tem has a damping ratio 5 = 2/2. a. K = 25 b. K = 1250 c. K = 2025 d. K = 10500
The root locus of this system is given by which of the following: Imaginary Axis Imaginary Axis 4 -6 -2 Real Axis (a) 0 2 Imaginay Axis Imaginary Axis + -6 -4 -2 Real Axis (b) -10-8 -6 -4 -2 0 2 -6 -4 0 Real Axis Real Axis (c) (d) 0
The approximate angles of departure of the root locus from the complex poles are a. = 180 == b. d = 115 c. d = 205 d. None of the above
Consider the control system in Figure 7.62, where the loop transfer function is K(s + 5s +9) L(s) = Ge(s)G(s) s (s + 3) Using the root locus method, determine the value of K such that the dominant roots have a damping ratio = 0.5. a. K = 1.2 b. K = 4.5 c. K = 9.7 d. K = 37.4
The root locus provides valuable insight into the response of a system to various test inputs. True or False
The root locus provides the control system designer with a measure of the sensitivity of the poles of the system to variations of a parameter of interest. True or False
The root locus always starts at the zeros and ends at the poles of G1s2. True or False
On the root locus plot, the number of separate loci is equal to the number of poles of G1s2. True or False
The root locus is the path the roots of the characteristic equation (given by 1 + KG1s2 = 0) trace out on the s-plane as the system parameter 0 … K 6 ∞ varies. True or False
A system has the block diagram representation as shown in Figure 6.27, where 10 G(s) (s +15)2 system is: and Ge(s) == a. 0 < K < 28875 b. 0 < K < 27075 c. 0 < K < 25050 d. Stable for all K > 0 K S+ 80' where K is always positive. The limiting gain for a stable
Using the Routh–Hurwitz criterion, determine whether the system is stable, unstable, or marginally stable.a. Stableb. Unstablec. Marginally stabled. None of the above
The characteristic equation is: a. q(s)=3+5s - 10s - 6 b. q(s) c. q(s) d. q(s) + 3 5s + 10s - 5 $25s + 10 35s + 10s + 5
Consider the following unity feedback control system in Figure 6.27 where 1 K(s + 0.3) G(s) (s 2) (s2+10s+45) and Ge(s) 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
Use the Routh array to assist in computing the roots of the polynomial a. S = -1; 523 b. $ = 1; $23 = = c. S =-1;5231 + d. s = -1; 523 1 q(s) =23+2s+s+1=0.
A system is represented by x # = Ax, where 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 0 1 0 0 0 1 -K 10
Consider a unity negative feedback system in Figure 6.27 with loop transfer function where K L(s) = Ge(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
Consider the feedback control system block diagram in Figure 6.27. Investigate closedloop stability for Ge(s) = K(s + 1) and G(s) where K = 1 and K = 3. ====== for the two cases. (s + 2)(s1)'
Utilizing the Routh–Hurwitz criterion, determine whether the following polynomials are stable or unstable: Pi(s)+10x+5=0, P2(s) 4+3+5s + 20s + 10 = 0. a. Pi(s) is stable, p2(s) is stable b. pi(s) is unstable, p2(s) is stable c. Pi(s) is stable, p2(s) is unstable d. P(s) is unstable, P(s) is
A system has the characteristic equation q(s)=s3+ 4Ks + (5+ K)s + 10 = 0. The range of K for a stable system is: a. K > 0.46 b. K < 0.46 c. 0 < K < 0.46 d. Unstable for all K
Relative stability characterizes the degree of stability. True or False
The Routh–Hurwitz criterion is a necessary and sufficient criterion for determining the stability of linear systems. True or False
A system is stable if all poles lie in the right half-plane. True or False
A marginally stable system has poles on the jv-axis. True or False
A stable system is a dynamic system with a bounded output response for any input. True or False
Using the second-order system approximation, estimate the gain K so that the percent overshoot is approximately P.O. ≈ 15,.a. K = 10b. K = 300c. K = 1000d. None of the above
A second-order approximate model of the loop transfer function is: a. (s)(s) = b. e(s)(s) == c. e(s)(s) = d. .(s)(s) == (3/25)K 27s+10 (1/25)K +7s+10 (3/25)K +7s+500 6K s+7s+ 10
A plant has the transfer function given by G(s) = (1 + s)(1 + 0.5s) and is controlled by a proportional controller G.(s) = K, as shown in the block diagram in Figure 5.42. The value of K that yields a steady-state error E(s) = Y(s) - R(s) with a magnitude equal to 0.01 for a unit step input is: a.
Consider the feedback control system in Figure 5.43 where G(s) K s + 10
Consider the unity feedback control system in Figure 5.42 where K L(s) = Ge(s)G(s) = s(s +5)* The design specifications are: i. Peak time T, 1.0 ii. Percent overshoot P.O. 10%. With K as the design parameter, it follows that a. Both specifications can be satisfied. b. Only the first specification
A system has the closed-loop transfer function T1s2 given by Y(s) T(s) = R(s) 2500 (s+20) (s + 10s + 125) Using the notion of dominant poles, estimate the expected percent overshoot. a. P.O.5% b. P.O.20% c. P.O. 50% d. No overshoot expected
Compute the expected percent overshoot to a unit step input.a. P.O. = 1.4,b. P.O. = 4.6,c. P.O. = 10.8,d. No overshoot expected K L(s)G(s)G(s) = == s(s + 10)
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