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engineering
modern control systems
Modern Control Systems 14th Global Edition Richard Dorf, Robert Bishop - Solutions
P7.10 New concepts in passenger airliner design will have the range to cross the Pacific in a single flight and the efficiency to make it economical [16, 29]. These new designs will require the use of temperature-resistant, lightweight materials and advanced control systems.Noise control is an
P7.9 The achievement of safe, efficient control of the spacing of automatically controlled guided vehicles is an important part of the future use of the vehicles in a manufacturing plant [14, 15]. It is important that the system eliminates the effects of disturbances (such as oil on the floor) as
P7.8 Consider again the power control system of Problem P7.7 when the steam turbine is replaced by a hydroturbine. For hydroturbines, the large inertia of the water used as a source of energy causes a considerably larger time constant. The transfer function of a hydroturbine may be approximated by
P7.7 The speed control system for an isolated power system is shown in Figure P7.7. The valve controls the steam flow input to the turbine in order to account for load changes ∆L s( ) within the power distribution network. The equilibrium speed desired results in a generator frequency equal to 60
P7.6 An attitude control system for a satellite vehicle within the earth’s atmosphere is shown in Figure P7.6.(a) Draw the root locus of the system as K varies from 0 . ≤ K ∞ (b) Determine the range of K for closedloop stability. (c) Determine the gain K that results in a system with a
P7.5 Automatic control of helicopters is necessary because, unlike fixed-wing aircraft which possess a fair degree of inherent stability, the helicopter is quite unstable. A helicopter control system that utilizes an automatic control loop plus a pilot stick control is shown in Figure P7.5. When
P7.4 Suppose that the loop transfer function of a large antenna is given by ka L(s) G(s)G(s) = TS+1 s (s + 25ws + w )" where t=0.2, 0.707, and co = 1 rad/s. Sketch the root locus of the system as 0
P7.3 A unity feedback system has the loop transfer function K L(s) = Ge(s)G(s) = (s+2)(s+4)(s+6) Find (a) the breakaway point on the real axis and the gain K for this point, (b) the gain and the roots when two roots lie on the imaginary axis, and (c) the roots when K = 10. (d) Sketch the root locus.
P7.2 Consider the loop transfer function of a phase-lock loop system = L(s) Ge(s)G(s) = KaK- 10(s+10) s(s+1)(s+100) Sketch the root locus as a function of the gain K = KK. Determine the value of K, attained if the complex roots have a damping ratio equal to 0.60 [13].
P7.1 Sketch the root locus for the following loop transfer functions of the system shown in Figure P7.1 when 0 K < : a. L(s) = G(s)G(s) = = b. L(s) = G(s)G(s) = . L(s) =G(s)G(s)= d. L(s) Ge(s)G(s) = = K s(s + 5)(s+20) K (s+2s+2)(s+2) K(s+10) s(s+1)(s+20) K (s + 4s +8) s (s+1)
E7.28 Consider the feedback system in Figure E7.28. Obtain the negative gain root locus as −∞ K 0.For what values of K is the system stable? Controller Process Ed(s) R(s) Ge(s) G(s) Y(s) FIGURE E7.28 Feedback system for negative gain root locus.
E7.27 Consider the unity feedback system in Figure E7.27. Sketch the root locus as 0 . ≤ p ∞ For what values of p is the closed loop system stable? FIGURE E7.27 Unity feedback system with parameter p. R(s) Ea(s) Controller S Process 2 Y(s) s+p
E7.26 Consider the single-input, single-output system is described by where A =| 3- 0 3-K x(t)=Ax(t)+ Bu(t) y(t) = Cx(t) 1 B =| | | C = [1 1 ] K | B -2- K Compute the characteristic polynomial and plot the root locus as 0 < K < oo. For what values of K is the system stable?
E7.25 A closed-loop feedback system is shown in Figure E7.25. For what range of K is the system stable? Sketch the root locus as 0 . < < K ∞
E7.24 Consider the system represented in state variable form where x(t) = Ax(t) + Bu(t) y(t)=Cx(t) + Du(t), 0 -246-8 -k-1 C=[10], D=[0]. Determine the characteristic equation and then sketch the root locus as 0 < k < .
E7.23 A unity feedback system has a loop transfer function L(s) = Ge(s)G(s) = 10(s+5) s (s+a) Sketch the root locus for 0 < a < .
E7.22 A high-performance missile for launching a satellite has a unity feedback system with a loop transfer function L(s) = G(s)G(s) = K(s+18)(s+2) ($2-2)(s+12) Sketch the root locus as K varies from 0 < K < 0.
E7.21 A unity feedback system has a loop transfer function L(s) = Ge(s)G(s) = Ks $3+5s+10 Sketch the root locus. Determine the gain K when the complex roots of the characteristic equation have a approximately equal to 0.66. damping ratio
E7.20 A unity feedback system has a loop transfer function L(s) = Ge(s)G(s) = K(s+1) s(s-2)(s+6) of the (a) Determine the range of K for stability. (b) Sketch the root locus. (c) Determine the maximum stable complex roots.
E7.19 A unity feedback system has a loop transfer function K L(s) = G(s)G(s) = s(s+3)(s + 6s+64) (a) Determine the angle of departure of the root locus at the complex poles. (b) Sketch the root locus. (c) Determine the gain K when the roots are on the jw-axis and determine the location of these
E7.18 A closed-loop negative unity feedback system is used to control the yaw of an aircraft. When the loop transfer function is K L(s) = G(s)G(s) = s(s+3)(s +2s+2)* determine (a) the root locus breakaway point and (b) the value of the roots on the jw-axis and the gain required for those roots.
E7.17 A control system, as shown in Figure E7.17, has a process R(s) Ge(s) G(s) Y(s) FIGURE E7.17 Feedback system. (a) When Ge(s) = K, show that the system is always unstable by sketching the root locus. (b) When Ge(s) = K(s+2) $+20 sketch the root locus, and determine the range of K for which the
E7.16 A negative unity feedback system has a loop transfer function = L(s) G(s)G(s) = Ke-sT s+1 where T = 0.1 s. Show that an approximation for the time delay is e-sT 2727 + S Using e-0.1s 20-s 20+s obtain the root locus for the system for K > 0. Determine the range of K for which the system is
E7.15 (a) Plot the root locus for a unity feedback system with loop transfer function L(s) = Ge(s)G(s) = K(s +10)(s+2) $3 (b) Calculate the range of K for which the system is stable. (c) Predict the steady-state error of the system for a ramp input.
E7.14 A unity feedback system has the loop transfer function L(s) = Ge(s)G(s) = K(s+15) s(s+3) (a) Determine the breakaway and entry points of the root locus, and sketch the root locus for K > 0. (b) Determine the gain K when the two characteristic roots have a of 1/2. (c) Calculate the roots.
E7.13 A unity feedback system has a loop transfer function L(s) = G(s)G(s) = (s+4) s(s+2)(s+ z) (a) Draw the root locus as z varies from 0 to 100. (b) Using the root locus, estimate the percent over- shoot and settling time (with a 2% criterion) of the sys- tem at z = 1, 2, and 3 for a step input.
E7.12 A unity feedback system has a loop transfer function L(s) = KG(s) = K(s+1) s(s+68+18) (a) Sketch the root locus for K > 0. (b) Find the roots when K = 10 and 20. (c) Compute the rise time, per- cent overshoot, and settling time (with a 2% criterion) of the system for a unit step input when K
E7.11 A robot force control system with unity feedback has a loop transfer function [6]a. Find the gain K that results in dominant roots with a damping ratio of 0.707. Sketch the root locus.b. Find the actual percent overshoot and peak time for the gain K of part (a). L(s) = KG(s) = K(s+1) s(s +
E7.10 A unity feedback system has the loop transfer function L(s) = KG(s) = K(s+6) s(s+4) a. Find the breakaway and entry points on the real axis. b. Find the gain and the roots when the real part of the complex roots is located at -3. c. Sketch the root locus.
E7.9 The primary mirror of a large telescope can have a diameter of 10 m and a mosaic of 36 hexagonal segments with the orientation of each segment actively controlled. Suppose this unity feedback system for the mirror segments has the loop transfer function K L(s) = G. (s)G(s) = s($+2s+5)* a. Find
E7.8 Sketch the root locus for a unity feedback system with K(s+3) L(s)=G(s)G(s) = = (s+1) (s+19) (a) Find the gain when all three roots are real and equal. (b) Find the roots when all the roots are equal as in part (a).
E7.7 The elevator in a modern office building can travel at a speed of 25 feet per second and still stop within one-eighth of an inch of the floor outside. The loop transfer function of the unity feedback elevator position control is L(s) = Ge(s)G(s) = K(s+3) s(s+1)(s+5)(s+10) Determine the gain K
E7.6 One version of a space station is shown in Figure E7.6 [28]. It is critical to keep this station in the proper orientation toward the Sun and the Earth for generating power and communications. The orientation controller may be represented by a unity feedback system with an actuator and
E7.5 Consider a unity feedback system with a loop transfer function $+4 L(s) = G(s)G(s)=3+10s + 25s +85 (a) Find the breakaway point on the real axis. (b) Find the asymptote centroid. (c) Find the value of K at the breakaway point.
E7.4 Consider a unity feedback system with the loop transfer function(a) Find the angle of departure of the root locus from the complex poles. (b) Find the entry point for the root locus as it enters the real axis. K(s+5) = L(s) G(s)G(s) = $2+2s+8
E7.3 A unity feedback control system for an automobile suspension tester has the loop transfer function [12] L(s) Ge(s)G(s) = K(s+3) s2 (s+10) We desire the dominant roots to have the maximum imaginary part value. Using the root locus, show that K=8.67 is required, and the dominant roots are s =
E7.2 A tape recorder with a unity feedback speed control system has the loop transfer function L(s) = Ge(s)G(s) = K s(s+1)(s2 +10s+24) a. Sketch a root locus for K, and show that the domi- nant roots are s = -0.41j0.384, when K = 8. b. For the dominant roots of part (a), calculate the settling time
E7.1 Consider a device that consists of a ball rolling on the inside rim of a hoop [11]. This model is similar to the problem of liquid fuel sloshing in a rocket. The hoop is free to rotate about its horizontal principal axis as shown in Figure E7.1. The angular position of the hoop may be
15. The departure angles from the complex poles and the arrival angles at the complex zeros are:a. φ φ D A = ±180°, = 0°b. φ φ D A =± =± 116.6°, 198.4°c. φ φ D A =± =± 45.8°, 116.6°d. None of the above
14. Which of the following is the associated root locus? Imaginary Axis (seconds") Imaginary Axis (seconds) TTTT + 3 2 -6 Root Locus Imaginary Axis (seconds") 1.5 25 I 0.5 0 -0.5 -I -1.5 -2 -5-4-3 -2-1 0 1 Real Axis (seconds") (a) Root Locus Root Locus -2 -15 -1 -0.50 -0.5 Real Axis (seconds") (b)
13. Consider the unity feedback system in Figure 7.62 where K L(s) =G(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
12. Suppose that a simple proportional controller is utilized, that is, G s c( ) = K. Using the root locus method, determine the maximum controller gain K for closed-loop stability.a. K = 0.50b. K = 1.49c. K = 4.48d. Unstable for K > 0
11. Suppose that the controller is Ge(s) K(1+0.2s) 1+0.025s Using the root locus method, determine the maximum value of the gain K for closed- loop stability. a. K = 2.13 b. K = 3.88 c. K = 14.49 d. Stable for all K > 0
10. Consider the unity feedback control system in Figure 7.62 where L(s) =G(s)G(s)= 10(s+ z) 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
9. 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 system has a damping ratio (= 2/2. a. K = 25 b. K = 1250 c. K = 2025 d. K = 10500
8. The root locus of this system is given by which of the following: Imaginary Axis 2 -4 -6 -4 -2 0 Real Axis (a) + Imaginary Axis 2 0 4 Imaginary Axis 2 + Imaginay Axis 2 -2 Real Axis (b) 0 2 -10-8 Real Axis -6-4-20 2 -6. -2 0 Real Axis (c) (d)
7. The approximate angles of departure of the root locus from the complex poles area. φd = ±180°b. φd = ±115°c. φd = ±205°d. None of the above
6. Consider the control system in Figure 7.62, where the loop transfer function is Controller Process R(s) Ge(s) G(s) Y(s) FIGURE 7.62 Block diagram for the Skills Check.
5. The root locus provides valuable insight into the response of a system to various test inputs.True or False
4. 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
3. The root locus always starts at the zeros and ends at the poles of G s( ).True or False
2. On the root locus plot, the number of separate loci is equal to the number of poles of G s( ).True or False
1. The root locus is the path the roots of the characteristic equation (given by 1 + KG(s) = 0) trace out on the s-plane as the system parameter 0 ≤ < K ∞ varies.True or False
CP6.9 Consider a system represented in state variable form where x(t) = Ax(t) + Bu(t) y(t)=Cx(t) + Du(t), 0 1 0 A= 0 0 1 B -k-15-3 2 0 C [401], D=[0]. (a) For what values of k is the system stable? (b) Develop an m-file to plot the pole locations as a function of 0 < k < 50, and comment on the
CP6.8 Consider the feedback control system in Figure CP6.8. (a) Using the Routh–Hurwitz method, determine the range of K1 resulting in closed-loop stability. (b) Develop an m-file to plot the pole locations as a function of 0 3 K1 0 and comment on the results. FIGURE CP6.8 Nonunity feedback
CP6.7 Consider a system in state variable form:(a) Compute the characteristic equation using the poly function. (b) Compute the roots of the characteristic equation, and determine whether the system is stable. (c) Obtain the response plot of y t( ) whenu t( ) is a unit step and when the system has
CP6.6 Consider the feedback control system in Figure CP6.6. Using the for function, develop an m-file script to compute the closed-loop transfer function poles for 0 5 ≤ ≤ K and plot the results denoting the poles with the " " × symbol. Determine the maximum range of K for stability with the
CP6.5 A “paper-pilot” model is sometimes utilized in aircraft control design and analysis to represent the pilot in the loop. A block diagram of an aircraft with a pilot“in the loop” is shown in Figure CP6.5. The variableτ represents the pilot’s time delay. Assume that we have a fast
CP6.4 Consider the closed-loop transfer function many poles are in the right half-plane? (b) Compute the poles of T s( ), and verify the result in part (a).(c) Plot the unit step response, and discuss the results. $+6 T(s)= == $3+4s2+15s+42 (a) Using the Routh-Hurwitz method, determine whether the
CP6.3 A unity negative feedback system has the loop transfer function L(s) = G(s)G(s) = s+4 s3+10s2+4s +25 Develop an m-file to determine the closed-loop trans- fer function, and show that the roots of the charac- teristic equation are s = -9.79 and $2,3 = -0.104 j1.7178.
CP6.2 Consider a unity negative feedback system with Ge(s) = K and G(s) = $2-s+2 s+28+1 Develop an m-file to compute the roots of the closed- loop transfer function characteristic polynomial for K=1, 2, and 5. For which values of K is the closed- loop system stable?
CP6.1 Determine the roots of the following characteristic equations:(a) q s( ) = + s s 3 2 3 1 + + 0 1 s 4 0 = .(b) q s( ) = + s s 4 3 8 2 + + 4 3 s s 2 2 1 + =6 0.(c) q s( ) = + s s 4 2 2 1 + = 0.
DP6.8 Consider the feedback system shown in Figure DP6.8. The process transfer function is marginally stable. The controller is the proportional-derivative(PD) controller FIGURE DP6.8 A marginally stable plant with a PD controller in the loop. Controller Process Ea(s) 2 R(s) Kp+ Kps Y(s) 2+2
DP6.7 Consider the feedback control system in Figure DP6.7. The system has an inner loop and an outer loop. The inner loop must be stable and have a quick speed of response. (a) Consider the inner loop first.Determine the range of K1 resulting in a stable inner loop. That is, the transfer function
DP6.6 Consider the single-input, single-output system as described by so that the percent overshoot to a unit step input,R s( ) = /1 ,s is P O. . T s s 4 K, so that the system step response meets the specifications P O. . T s s 4 where 0 x(t) = Ax(t) + Bu(t) y(t) = Cx(t) 1-2 | B=| | | C = [10] A
DP6.5 A traffic control system is designed to control the distance between vehicles, as shown in Figure DP6.5[15]. (a) Determine the range of gain K for which the system is stable. (b) If Km is the maximum value ofK so that the characteristic roots are on the jω-axis, then let K K = / m N, where
DP6.4 The attitude control system of a rocket is shown in Figure DP6.4 [17]. (a) Determine the range of gainK and parameter m so that the system is stable, and plot the region of stability. (b) Select the gain and parameter values so that the steady-state error to a ramp input is less than or equal
DP6.3 A unity negative feedback system with L(s) Ge(s)G(s)= K(1+2s) s(1+Ts)(1 + 5s) has two parameters to be selected. (a) Determine and plot the regions of stability for this system. (b) Select and K so that the steady-state error to a unit ramp input is less than or equal to 0.1. (c) Determine
DP6.2 An automatically guided vehicle on Mars is represented by the system in Figure DP6.2. The system has a steerable wheel in both the front and back of the vehicle, and the design requires that H s( ) = + Ks 1.Determine (a) the value of K required for stability,(b) the value of K when one root
DP6.1 The control of the spark ignition of an automotive engine requires constant performance over a wide range of parameters [15]. The control system is shown in Figure DP6.1, with a controller gain K to be selected. The parameter p is equal to 2 for many autos but can equal zero for those with
AP6.7 A human’s ability to perform physical tasks is limited not by intellect but by physical strength. If, in an appropriate environment, a machine’s mechanical power is closely integrated with a human arm’s mechanical strength under the control of the human intellect, the resulting system
AP6.6 A spacecraft with a camera is shown in Figure AP6.6(a). The camera slews about 16° in a canted plane relative to the base. Reaction jets stabilize the base against the reaction torques from the slewing motors. Suppose that the rotational speed control for the camera slewing has a plant
AP6.5 Consider the closed-loop system in Figure AP6.5.Suppose that all gains are positive, that is,K K 1 2 > > 0, 0, 0 K K 3 4 > > , 0, and K5 > 0.(a) Determine the closed-loop transfer functionT s( ) = Y s( ) / . R s( )(b) Obtain the conditions on selecting the gainsK K 1 2 , , K
AP6.4 A bottle-filling line uses a feeder screw mechanism, as shown in Figure AP6.4. The tachometer feedback is used to maintain accurate speed control.Determine and plot the range of K and p that permits stable operation. Controller FIGURE AP6.4 Speed control of a bottle-filling line. (a) System
AP6.3 A control system is shown in Figure AP6.3. We want the system to be stable and the steady-state error for a unit step input to be less than or equal to 0.1. (a) Determine the range of α that satisfies the error requirement. (b) Determine the range of α that satisfies the stability
AP6.2 Consider the case of a navy pilot landing an aircraft on an aircraft carrier. The pilot has three basic tasks. The first task is guiding the aircraft’s approach to the ship along the extended centerline of the runway. The second task is maintaining the aircraft on the correct glideslope.
AP6.1 A teleoperated control system incorporates both a person (operator) and a remote machine. The normal teleoperation system is based on a one-way link to the machine and limited feedback to the operator.However, two-way coupling using bilateral information exchange enables better operation
P6.21 Consider the system described in state variable form by * (1) = Ax(t) + Bu(t) y(t) = Cx (1) where 0 1 A: -k -k2 B = | | ], and C = [1 1 ], and where k = k2 and both k and k are real numbers. (a) Compute the state transition matrix (1, 0). (b) Compute the eigenvalues of the system matrix A.
P6.20 A personal vertical take-off and landing (VTOL)aircraft is shown in Figure P6.20(a). A possible control system for aircraft altitude is shown in Figure P6.20(b).(a) For K = 17, determine whether the system is stable.(b) Determine a range of stability, if any, for K > 0. FIGURE P6.20 (a)
P6.19 The goal of vertical takeoff and landing (VTOL)aircraft is to achieve operation from relatively small airports and yet operate as a normal aircraft in level flight [16]. An aircraft taking off in a form similar to a rocket is inherently unstable. A control system using adjustable jets can
P6.18 Consider the case of rabbits and foxes. The number of rabbits is x1 and, if left alone, it would grow indefinitely (until the food supply was exhausted) so thatx k 1 1 = x .However, with foxes present, we havex k 1 1 = − x ax2, where x2 is the number of foxes. Now, if the foxes must
P6.17 The elevator in Yokohama’s 70-story Landmark Tower operates at a peak speed of 45 km/hr. To reach such a speed without inducing discomfort in passengers, the elevator accelerates for longer periods, rather than more precipitously. Going up, it reaches full speed only at the 27th floor; it
P6.16 A system has a closed-loop transfer function 1 T(s)= s4+2s3+16s2 + 20s +4 (a) Determine whether the system is stable. (b) Deter- mine the roots of the characteristic equation. (c) Plot the response of the system to a unit step input.
P6.15 The stability of a motorcycle and rider is an important area for study [12, 13]. The handling characteristics of a motorcycle must include a model of the rider as well as one of the vehicle. The dynamics of one motorcycle and rider can be represented by the loop transfer function L(s)= K(s +
P6.14 A feedback control system has a characteristic equations s 6 5 + + 2 12 4 s s 4 3 + + 21s s 2 + + 2 10 0 = .Determine whether the system is stable, and determine the values of the roots.
P6.13 Consider the system in Figure P6.13. Determine the conditions on K, p, and z that must be satisfied for closed-loop stability. Assume that K > 0, ζ > 0, and ωn > 0 . Controller s+z FIGURE P6.13 R(s) Control system with s+p controller with three parameters K, p. and z. Process W
P6.12 A system has the third-order characteristic equations a 3 2 + + s bs c + = 0, where a, b, and c are constant parameters. Determine the necessary and sufficient conditions for the system to be stable. Is it possible to determine stability of the system by just inspecting the coefficients of
P6.11 A feedback control system has a characteristic equations K 3 2 + + ( ) 1 1 s s + + 0 5( ) + = 15K 0.The parameter K must be positive. What is the maximum value K can assume before the system becomes unstable? When K is equal to the maximum value, the system oscillates. Determine the
P6.10 Robots can be used in manufacturing and assembly operations that require accurate, fast, and versatile manipulation [10, 11]. The loop transfer function of a direct-drive arm is Ge(s)G(s) = K(s+4) s(s3+5s2+17s+10) (a) Determine the value of gain K when the system oscillates. (b) Calculate the
P6.9 A cassette tape storage device has been designed for mass-storage [1]. It is necessary to control the velocity of the tape accurately. The speed control of the tape drive is represented by the system shown in Figure P6.9.(a) Determine the limiting gain for a stable system.(b) Determine a
P6.8 A very interesting and useful velocity control system has been designed for a wheelchair control system. A proposed system utilizing velocity sensors mounted in a headgear is shown in Figure P6.8. The headgear sensor provides an output proportional to the magnitude of the head movement. There
P6.7 The linear model of a phase detector (phase-lock loop) can be represented by Figure P6.7 [9]. The phase-lock systems are designed to maintain zero difference in phase between the input carrier signal and a local voltage-controlled oscillator. The filter for a particular application is chosen
P6.6 A unity-feedback control system is shown in Figure P6.6. Determine the stability of the system with the following loop transfer functions using the Routh–Hurwitz criterion: (a) Ge(s)G(s) = (b) Ge(s)G(s) = (c) G(s)G(s) = (10s+30)(s+1) s (s-2) 10 s($3+2s2+5s+2) $2+s+3 s(s + 1)(s+2)
P6.5 Determine the relative stability of the systems with the following characteristic equations (1) by shifting the axis in the s-plane and using the Routh–Hurwitz criterion, and (2) by determining the location of the complex roots in the s-plane:(a) s s 3 2 + + 3 4s + =2 0.(b) s s 4 3 + + 9
P6.4 A feedback control system is shown in Figure P6.4. The controller and process transfer functions are given by G(s) K and G(s) = = s+100 s(s+25) and the feedback transfer function is H(s) = 1/ (s + 50). (a) Determine the limiting value of gain K for a stable system. (b) For the gain that
P6.3 Arc welding is one of the most important areas of application for industrial robots [11]. In most manufacturing welding situations, uncertainties in dimensions of the part, geometry of the joint, and the welding process itself require the use of sensors for maintaining weld quality. Several
P6.2 An antenna control system was analyzed in Problem P4.5, and it was determined that, to reduce the effect of wind disturbances, the gain of the magnetic amplifier, ka, should be as large as possible. (a)Determine the limiting value of gain for maintaining a stable system. (b) We want to have a
P6.1 Utilizing the Routh–Hurwitz criterion, determine the stability of the following polynomials:(a) s s 2 + + 5 2 = 0(b) s s 3 2 + + 4 8s + =4 0(c) s s 3 2 + − 2 6s + = 20 0(d) s s 4 3 + + 2 1 s s 2 + + 2 10 0(e) s s 4 3 + + 3 2 s s 2 + + K = 0(f) s s 5 4 + + 2 6 s s 3 + + = 0(g) s
E6.26 Consider the closed-loop system in Figure E6.26,where 10 1 G(s) = and Ge(s) = s-10 2s+ K (a) Determine the characteristic equation associated with the closed-loop system. (b) Determine the values of K for which the closed- loop system is stable.
E6.25 A closed-loop feedback system is shown in Figure E6.25. For what range of values of the parameters K and p is the system stable? Ks +1 Y(s) R(s) s(s+p) FIGURE E6.25 Closed-loop system with parameters K and p.
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