第六章习题
E6.1 A system has a characteristic equation :
s3+3Ks2+(2+K)s+5=0.
Determine the range of K for a stable system.
E6.7 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 0.707.
(b)Find the value of the gain when the closed system has two roots on the imaginary axis.
E6.11 A system with a transfer function Y(s)/R(s) is
Determine the steady-state error to a unit step input.is the system stable?
E6.14 A system has a characteristic equation
q(s)=s4+9s3+45s2+87s+50=0.
(a) Determine whether the system is stable,using the Routh-Hurwitz criterion.(b)Determine the roots of the characteristic equation.
E6.23 A closed-loop feedback system is shown in Figure E6.23.For what range of values of the parameters K and p is the system stable?
P6.3 Arc welding is one of the most important areas of application for instustrial robots.In most manufacturing welding situation,uncertainties in dimensions of part,geometry of the joint,and the weiding process itself require the use of sensors for maintaining weld quality.Several systems use a vision system to measure the qeometry of the puddle of melted metal,as shown in Figure P6.3. This system uses a constant rate of feeding the wire to be melted.
(a) Calculate the maximum value for K for the system that will result in a stable sytem
(b) For half of the maximum value of K found in part(a). Determine the roots of the characteristic equation.
(c) Estimate the overshoot of the system of part(b) when it is subjected to step input.
P6.4 A feedback control system is shown in Figure P6.4. The process transfer function is
And the feedback transfer function is H(s)=1/(s+20).
(a)Determine the limiting value of gain K for a stable system
(b)For the gain that results in marginal stability,determine the magnitude of the imaginary roots
(c)Reduce the gain to half the magnitude of the marginal value and determine the relative stability of the system (1)by shifting the axis and using Routh-Hurwitz criterion and (2)by determining the root locations.Show the roots are between -1 and -2.
P6.7 The linear model of a phase detector(phase-lock loop) can be represented by Figure P6.7.The phase-lock systems are designed to maintain zero difference in phase between the input carrier signal and a local voltage-controlled oscillator.Phase-lock loops find application in color television,missile tracking,and space telemetry.The filter for a particular application is chosen as
We want to minimize the steady-state error of the system for a ramp change in
the phase information signal
(a)Determine the limiting value of the gain Kak=kv in order to maintain a stable system
(b)A steady-state error equal to 1o,Is acceptable for a ramp signal of 100% rad/s.For that value of Kv,determine the location of the roots of the system.
P6.8 A very interesting and useful velocity control system has been designed for a wheelchair control system.We want to enable people paralyzed from the neck down to drive themselves about in motorized wheelchairs.A proposed system utilizing velocity sensors mounted in a headgear is shown in Figure P6.8.The headgear sensor provides an output proportional to magnitude of the head movement.There is a sensor mounted at 90o intervals so that forward,left right,or reverse can be commaded.Typical values for the time constant are τ1=0.5s τ3=1s,and τ4=1/4s.
(a) Determine the limiting gain K=K1K2K3 for a stable system
(b) When the gain K is set equal to one-third of the limiting value,determine whether the setting time(to within 2% of the final value of the system) is less than 4s.
(c) Determine the value of gain that results in s system with a settling time of 4s.Also,obtain the value of the roots of the characteristic equation when the settling time is equal to 4s.
P6.9 A cassette tape storage device has been designed for mass-storage.It is necessary to control the velocity of the tape accurately.The speed control of the tape is represented by the system shown in FigureP6.9.
(a) Determine the limiting gain for a stable system
(b) Determine a suitable gain so that the overshoot to a step command is approximately 5%.
P6.10 Robots can be used in manufacturing and assembly operation that require accurate,fast,and versatile manipulation.The open-loop transfer function of a direct-drive arm may be approximateed by
(a)Determine the value of the gain K when the system oscillates.
(b)Calculate the roots of the closed-loop system for the K determined in part(a).
P6.15 On July 16,1993,the elevator in Yokohama's 70-story Landmark Tower,operating at a peak speed of 45km/hr(28 mph),was inaugurated as the fastest
super-fast
elevator.To
reach
such
a
speed
without
leaving
passengers'stomachs on the ground floor,the lift accelerates for longer periods,rather than more than more precipitously.Going up,it reaches full speed only at 27th floor;it begins decelerating 15 floor later.The result is a peak acceleration similar to that of other skyscraper elevator-a bit less than a thenth of force of gravity.
Admirable ingenuity has gone into making this safe and comfortable.Special ceramic brakess had to be developed;iron ones would melt.Computer-controlled
system damp out vibrations.The lift has been streamlined to reduce the wind noise as it hurtles up and down.One proposed control system for the elevator'vertical position is shown in Figure P6.15.Determine the range of K for a stable system.
AP6.4 A bottle-fillinf lines uses a feeder screw mechannism,as shown in Figure AP6.4.The tachometer feedback is used to maitain accurate speed control.Determine and plot the range of K and p that permits stable operation.
FIGURE AP6.4 Speed control of a bottle-filling line
(a) System layout
(b) Bloack diagram
CDP6.1 The capstan drive system uses the amplifier Gc(s)=Ka as the controller.Dertermine the maximum value of the gain Ka before the system becomees unstable.
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.
(a)Determine the value of K required for stablity.
(b)The value of K when one root of the characteristic equation is equal to s=-5.
(c)The value of the two remaining roots for the gain selected in part(b).
(d)Find the response of the system to a step command for the gain selected in part(b).
DP6.6 Consider the potential for a robot steering a motorcycle,as shown in Figure DP6.6(a).The blockdiagram of the system model is shown in Figure DP6.6(b).Determine the range of K for stable operation of the cycle when α1=g/h=9.α2=V2/hc=2.7,and α3=VL/hc=1.35.We assume the motorcycle is moving with a constant velocity V=2m/s.The time constant of the controller is τ=0.2,and K>0.
DP6.7 Consider the feedback control system in Figure Dp6.7. The system has an inner loop and an outer 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 Y(s)/U(s) must be stable.
(b)Select the value of K1 in the sstable range leading to the fastest step response.
(c)For the value of K1 selected in (b),determine the range of K2 such that the closed-loop system,T(s)=Y(s)/R(s),is stable.
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