,OBLEMS

PS.1 An important problem for television systems isthe jumping or wobbling of the picture due to themovement of the camera. This effect occurs whenthe camera is mounted in a moving truck or air-plane. The Dynalens system has been designed toreduce the effect of rapid scanning motion; see

CameraB----y-

oX---_ j--

CameraRate gyrospeed Kg---+ ---

STg + I

I ~

Motor BellowsK", speed

STIll + I

Figure P5.1. A maximum scanning motion of 25/s isexpected. Let Kg = K( = 1 and assume that Tg isnegligible. (a) Determine the error of the system(s). (b) Determine the necessary loop gainKaKmK( when a l/s steady-state error is allowable.(c) The motor time constant is 0.40 s. Determine thenecessary loop gain so that the settling time (towithin 2% of the final value of Vb) is less than orequal to 0.03 s.

PS.2 A specific closed-loop control system is to beV designed for an underdamped response to a step

input. The specifications for the system are asfollows:

10% < percent overshoot < 20%,Settling time < 0.6 s.

(a) Identify the desired area for the dominant rootsof the system. (b) Determine the smallest value of a

third root r3 if the complex conjugate roots are torepresent the dominant response. (c) The closed-loop system transfer function yes) is third-order, andthe feedback has a unity gain. Determine the for-ward transfer function G(s) = Y(s)jE(s) when thesettling time to within 2% of the final value is 0.6 sand the percent overshoot is 20%.

PS.3 A laser beam can be used to weld, drill, etch, cut, andmark metals, as shown in Figure P5.3(a) [16]. Assumewe have a work requirement for an accurate laser tomark a parabolic path with a closed-loop control sys-tem, as shown in Figure P5.3(b). Calculate the neces-sary gain to result in a steady-state error of 5 mm forr(t) = t2 cm.

PS.4 The final value of step response of a II order systemis unity and percentage overshoot is 9%. If the stepresponse settled in 2 s, find the transfer function ofthe system. Also find the poles of the system.

PS.S A space telescope is to be launched to carry outastronomical experiments [9]. The pointing controlsystem is desired to achieve 0.01 minute of arc andtrack solar objects with apparent motion up to 0.21arc minute per second. The system is illustrated inFigure P5.5(a). The control system is shown inFigure P5.5(b). Assume that Tl = 1 second and

Lasercavity

FIGURE PS.S(a) The spacetelescope. (b) Thespace telescopepointing controlsystem.

R(s)

Input

72 = 0 (an approximation). (a) Determine the gainK = K tK2 required so that the response to a stepcommand is as rapid as reasonable with an over-shoot of less than 5%. (b) Determine the steady-state error of the system for a step and a rampinput. (c) Determine the value of K1K2 for an ITAEoptimal system for (1) a step input and (2) a rampinput.

P5.6 A robot is programmed to have a tool or welding'v torch follow a prescribed path [8, 13]. Consider a

robot tool that is to follow a sawtooth path, asshown in Figure P5.6(a). The transfer function ofthe plant is

75(s + 1)G(s)------

s(s + 5)(s + 20)

for the closed-loop system shown in Figure 5.6(b).Calculate the steady-state error.

Controller

K2(71S + 1)725 + 1

Yes)

Pointingangle

Yes)Path

trajectory

(b)

FIGURE PS.6 Robot path control.

Dlsturoance1~/(s)

Slope ofpower curveat maximum

power

P(s)

Poweroutput

FIGURE P5.8Solar cell control.

FIGURE 5.9 A model of the antenna for the TelstarSystem at Andover, Maine. (Photo courtesy of BellTelephone Laboratories, Inc.)

Select a feedback gain for the back emf signal to yield astep response with an overshoot of 15%.

P5.11 A simple unity feedback control system has a\j process transfer functionyes) = G(s) = K(s) s

The system input is a step function with an amplitudeA. The initial condition of the system at time to isy(to) = Q, where yet) is the output of the system. Theperformance index is defined as

/ =100e2(t) dt.(a) Show that [ = (A - Q)2j(2K). (b) Determinethe gain K that will minimize the performance index f.Is this gain a practical value? (c) Select a practicalvalue of gain and determine the resulting value of theperformance index.

P5.12 Train travel between cities will increase as trainsaredeveloped that travel at high speeds, making the ttaldtime from city center to city center equivalent to airlinetravel time. The Japanese National Railway has attaincalled the Bullet Express that travels between Tokyoand Osaka on the Tokaido line.This train travels the3211miles in 3 hours and 10 minutes, an average speed of101mph [20]. This speed will be increased as new systemsare used, such as magnetically levitated systems to floatvehicles above an aluminum guideway. To maintainadesired speed, a speed control system is proposed thatyields a zero steady-state error to a ramp input. A thirdorder system is sufficient. Determine the optimum s)~tern transfer function T(s) for an ITAE performancecriterion. Estimate the settling time (with a 2% criterion) and overshoot for a step input when Wn = 10.

P5.13 We want to approximate a fourth-order systembya lower-order model.1l1e transfer function of the original system is

S3 + 7S2 + 24s + 24S4 + lOs3 + 35s2 + 50s + 24

s3 + 7s2 + 245 + 24(s + 1)(5 + 2)(5 + 3)(s + 4)'

Show that if we obtain a second-order model by themethod of Section 5.8, and we do not specify the polesand the zero of L(s), we have

0.2917s + 1L(s) = 0.399s2 + 1375s + 1

0.731(s + 3.428)(s + 1,043)(s + 2.4)'

P5.14 For the original system of Problem P5.13, we wantto find the lower-order model when the poles of thesecond-order model are specified as -1 and - 2 andthe model has one unspecified zero. Show that thislow-order model is

L(s) = 0.986s + 2S2 + 35 + 2

0.986(s + 2.028)(s + l)(s + 2) .

K(5 + 10)(5 + 12)

(a) If T = 2.43, determine the value of K suchthe steady-state error of the closed-loop sresponse to a unit step input, R(s) = lis, is

(b) Determine the percent overshoot Po. andtime to peak T" of the unit step response whetis as in part (a).

FIGURE P5.20 Closed-loop system withparameters k and a.

FIGURE P5.21 Nonunity closed-loop feedback controlsystem.

P5.21 Consider the closed-loop system in Figure PS.2l,where

2 2GcCs)G(s) = 0 and H(s) = --.

s + .2K 2s + T

AP5.1 A closed-loop transfer function is

,~y (s) 96(s + 3)

Y(s) = - = -------R(s) (s + 8)(s2 + 8s + 36)'

(a) Determine the steady-state error for a unit stepinput R(s) = lis.

(b) Assume that the complex poles dominate, anddetermine the overshoot and settling time towithin 2% of the final value.

(c) Plot the actual system response, and compare itwith the estimates of part (b).

AP5.2 A closed-loop system is shown in Figure APS.2.Plot the response to a unit step input for the systemfor T l = 0,0.05,0.1, and 0.5. Record the percent over-shoot, rise time, and settling time (with a 2% criterion)as T z varies. Describe the effect of varying T z' Com-pare the location of the zero -l/T z with the locationof the closed-loop poles.

5440( TZS + I)S(52 + 285 + 432)

AP5.3 A closed-loop system is shown in Figure APS.3.Plot the response to a unit step input for the systemwith Tp = 0, 0.5, 2, and S. Record the percent overshoot, rise time, and settling time (with a 2% criterion)as Tp varies. Describe the effect of varying Tp' Compare the location of the open-loop pole -1/7p withthe location of the closed-loop poles.

Is(s + 2)(7pS + I)