Effective control of insulin injections can result in better lives for diabetic persons

In general, a third-order system can be approximated by a second-order system's dominant roots if the real part of the dominant roots is less than 1/10 of the real part of the third root.

The number of zeros of the forward path transfer function at the origin is called the type number.

The rise time is defined as the time required for the system to settle within a certain percentage of the input amplitude.

For a second-order system with no zeros, the percent overshoot to a unit step is a function of the damping ratio only.

A type-1 system has a zero steady-state tracking error to a ramp input. True or False Consider the closed-loop control system in Figure 5.50 for Problems 6 and 7 with Us) = G.MGM = ^ .

The steady-state error to a unit step input R(s) = \/s is: a. ess = lime(0 = 1 b. ess = \ime(t) = 1/2 c e^ = hme(f) = 1/6 d. ess = lime(f) = oo

The percent overshoot of the output to a unit step input is: a. P.O. = 9% b. P.O. = 1% c. P.O. = 20% d. No overshoot Consider the block diagram of the control system shown in Figure 5.50 in Problems 8 and 9 with the loop transfer function K L(s) = Gc(s)G(s) = s(s + 10)'

Find the value of K so that the system provides an optimum ITAE response. a. K = 1.10 b. K = 12.56 c K = 51.02 d. K = 104.7

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

A system has the closed-loop transfer function T(s) given by Y(s) 2500 T(s) = R(s) (s + 20)(.92 + 105 + 125)'

Consider the unity feedback control system in Figure 5.50 where K L{s) = Gc{s)G{s) = s(s + 5)' The design specifications are: i. Peak time Tn < 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 Tp < 1.0 can be satisfied. c. Only the second specification P.O. ^ 10% can be satisfied. d. Neither specification can be satisfied.

Consider the feedback control system in Figure 5.51 where G(s) = K s + 10 W R(s) O ) " (5> . 1 * Controller 1 s Process G(s) Measurement Y(s) FIGURE 5.51 Feedback system with integral controller and derivative measurement. The nominal value of K 10. Using a 2% criterion, compute the settling time, Ts for a unit step disturbance, TX-s) = l/s. a. Ts = 0.02 s b. Ts = 0.19 s c. Ts = 1.03 s d. Ts = 4.83 s

A plant has the transfer function given by G(5 ) = (1 + 5)(1 + 0.55) and is controlled by a proportional controller Gc(s) - K, as shown in the block diagram in Figure 5.50. 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. K = 49 b. K = 99 Skills Check 367 c. K = 169 d. None of the above In Problems 14 and 15, consider the control system in Figure 5.50, where G{s) = - and Gc(s) = ^rr. w (s + 5)(s + 2) c w s + 50

A second-order approximate model of the loop transfer function is: (3/25) A: a. Gc(s)G(s) = b. Gc(s)G(s) = c. Gc(s)G(s) = d. Gc(s)G(s) = s 2 + Is + 10 (1/25)# s 2 + Is + 10 (3/25)X 5 2 + 7^ + 500 6K 5 2 + Is + 10

Using the second-order system approximation (see Problem 14), estimate the gain K so that the percent overshoot is approximately P.O. 15%. a. K = 10 b. A = 300 c. A = 1000 d. None of the above

A test input consisting of an impulse of infinite amphtude system and zero width, and having an area of unity

A motor control system for a computer disk drive must reduce the effect of disturbances and parameter variations, as well as reduce the steady-state error. We want to have no steady-state error for the head-positioning control system, which is of the form shown in Figure 5.18. (a) What type number is required? (How many integrations?) (b) If the input is a ramp signal, and we want to achieve a zero steady-stale error, what type number is required?

The engine, body, and tires of a racing vehicle affect the acceleration and speed attainable [9]. The speed control of the car is represented by the model shown in Figure E5.2. (a) Calculate the steady-state error of the car to a step command in speed, (b) Calculate overshoot of the speed to a step command. Answer: (a) e5S = /1/11: (b) P.O. = 36%

New passenger rail systems that could profitably compete with air travel are under development. Two of these systems, the French TGV and the Japanese Shinkansen, reach speeds of 160 mph [17]. The Transrapid, a magnetic levitation train, is shown in Figure E5.3(a). The use of magnetic levitation and electromagnetic propulsion to provide contactless vehicle movement makes the Transrapid technology radically different. The underside of the carriage (where the wheel trucks would be on a conventional car) wraps around a guideway. Magnets on the bottom of the guideway attract electromagnets on the "wraparound," pulling it up toward the guideway. This suspends the vehicles about one centimeter above the guideway. The levitation control is represented by Figure E5.3(b). (a) Using Table 5.6 for a step input, select K so

A feedback system with negative unity feedback has a loop transfer function L(s) = Gc(s)G(s) 2(5 + 8) s(s + 4)' (a) Determine the closed-loop transfer function T(s) = Y(s)/R(s). (b) Find the time response, y(t), for a step input r(t) = A for t > 0. (c) Using Figure 5.13(a), determine the overshoot of the response. (d) Using the final-value theorem, determine the steady-state value of y(t).

Consider the feedback system in Figure E5.5. Find K such that the closed-loop system minimizes the ITAE performance criterion for a step input.

Consider the block diagram shown in Figure E5.6 [16]. (a) Calculate the steady-state error for a ramp input. (b) Select a value of K that will result in zero overshoot to a step input. Provide the most rapid response that is attainable. Plot the poles and zeros of this system and discuss the dominance of the complex poles. What overshoot for a step input do you expect?

Effective control of insulin injections can result in better lives for diabetic persons. Automatically controlled insulin injection by means of a pump and a sensor that measures blood sugar can be very effective. A pump and injection system has a feedback control as shown in Figure E5.7. Calculate the suitable gain K so that the overshoot of the step response due to the drug injection is approximately 7%. R(s) is the desired blood-sugar level and Y(s) is the actual bloodsugar level. (Hint: Use Figure 5.13a.)

A control system for positioning the head of a floppy disk drive has the closed-loop transfer function T(s) = 11.1(5 + 18) (s + 20)(52 + 4s + 10) Plot the poles and zeros of this system and discuss the dominance of the complex poles. What overshoot for a step input do you expect?

A unity negative feedback control system has the loop transfer function L(s) = Gc(s)G(s) = K s(s + V2K)' (a) Determine the percent overshoot and settling time (using a 2% settling criterion) due to a unit step input. (b) For what range of K is the settling time less than 1 second?

A second-order control system has the closed-loop transfer function T(s) = Y(s)/R(s). The system specifications for a step input follow: (1) Percent overshoot P.O. < 5%. (2) Settling time Ts < 4s. (3) Peak time Tp < Is. Show the permissible area for the poles of T(s) in order to achieve the desired response. Use a 2% settling criterion to determine settling time.

A system with unity feedback is shown in Figure E5.ll. Determine the steady-state error for a step and a ramp input when G(s) = 5(s + 8) s(s + 1)(5 + 4)(5 + 10)'

We are all familiar with the Ferris wheel 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 construction. He conceived and constructed his famous wheel for the 1893 Columbian Exposition in Chicago [8]. To avoid upsetting passengers, set a requirement that the steady-state speed must be controlled to within 5% of the desired speed for the system shown in Figure E5.12. (a) Determine the required gain K to achieve the steady-state requirement. (b) For the gain of part (a), determine and plot the error e(f) for a disturbance 1}(s) = \/s. Does the speed change more than 5%? (Set R(s) = 0 and recall that E(s) = R(s) - T(s).)

For the system with unity feedback shown in Figure E5.ll, determine the steady-state error for a step and a ramp input when G(s) 20 s 2 + 14^ + 50*

A feedback system is shown in Figure E5.14. (a) Determine the steady-state error for a unit step when K = 0.4 and Gp(s) = 1. /?(.v) Y{s) (b) Select an appropriate value for Gp(s) so that the steady-state error is equal to zero for the unit step input.

A closed-loop control system has a transfer function T(s) as follows: Y(s) R(s) T(s) = 2500 (s + 50)(^2 + 10s + 50) Ploty(*) for a step input R(s) when (a) the actual T(s) is used, and (b) using the relatively dominant complex poles. Compare the results.

A second-order system is R(s) T(S) (10/Z)(5 + Z) (5 + 1)(5 + 8) FIGURE E5.14 Feedback system. Consider the case where 1 < z < 8. Obtain the partial fraction expansion, and plot y(t) for a step input r(t) for z = 2,4, and 6.

A closed-loop control system transfer function T(s) has two dominant complex conjugate poles. Sketch the region in the left-hand 5-plane where the complex poles should be located to meet the given specifications. (a) 0.6 < < 0.8, o)lt < 10 (b) 0.5 < ^ 0.707, o> > 10 (c) C ^ 0.5, 5 < (on < 10 (d) < 0.707, 5 < (o < 10 (e) > 0.6, o> < 6

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 steadystate error is equal to zero.

A second-order system has the closed-loop transfer function i \ & i 5 + ft>2 , 52 + 3.1755 + l'

Consider the closed-loop system in Figure E5.19, where Gc(s)G(s) = ^-^-an d H(s) = Ka. s 2 + 03s (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, R($) = 1/j2 . (c) Select a value for Ka so that the steady-state error of the system response to a unit step input, R(s) = L/i, is zero.

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 P5.1. A maximum scanning motion of 25/s is expected. Let Kg ~ K, = I and assume that Tg is negligible, (a) Determine the error of the system E(s). (b) Determine the necessary loop gain KaKnK, when a l/s steady-state error is allowable, (c) The motor time constant is 0.40 s. Determine the necessary loop gain so that the settling time (to within 2% of the final value of v!:) is less than or equal to 0.03 s

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% < percent overshoot < 20%, Settling time < 0.6 s. (a) Identify the desired area for the dominant roots of the system, (b) Determine the smallest value of a third root /; if the complex conjugate roots are to represent the dominant response, (c) The closed-loop system transfer function T(s) is third-order, and the feedback has a unity gain. Determine the forward transfer function G(s) = Y(s)/E(s) when the settling time to within 2% of the final value is 0.6 s and the percent overshoot is 20%

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 parabolic path with a closed-loop control system, as shown in Figure P5.3(b). Calculate the necessary gain to result in a steady-state error of 5 mm for r(() = t2 cm.

The loop transfer function of a unity negative feedback system (see Figure E5.ll) is Us) = G&ycm = ^YY A system response to a step input is specified as follows: peak time Tp = 1.1 s, percent overshoot P.O. = 5%. (a) Determine whether both specifications can be met simultaneously, (b) If the specifications cannot be met simultaneously, determine a compromise value for K so that the peak tune and percent overshoot specifications are relaxed by the same percentage.

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 shown in

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 transfer function of the plant is G(s) 75(5 + 1) s(s + 5)(s + 25) for the closed-loop system shown in Figure 5.6(b). Calculate the steady-state error.

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). 20 30 Time (s) (a) Xr -> V G(s) mi trajectory A'(.v) (b) FIGURE P5.6 Robot path control. The controller can be represented by a gain K2, as shown in Figure P5.7(b). The moment of inertia of the equipment and man is 25 kg m~. (a) Determine the

Photovoltaic arrays (solar cells) 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 antenna that receives and transmits signals to the Telstar communication satellite is the largest horn antenna ever built. The microwave antenna is 177 ft long, weighs 340 tons, and rolls on a circular track. A photo of the antenna is shown in Figure P5.9. The Telstar satellite is 34 inches in diameter and moves about 16,000 mph at an altitude of 2500 miles. The FIGURE 5.9 A model of the antenna for the Telstar System at Andover, Maine. (Photo courtesy of Bell Telephone Laboratories, inc.) the array at its maximum available as the solar incidence changes during the day. One such closed-loop system is shown in Figure P5.8. The transfer function for the process is where K - 20. Find (a) the time constant of the closed-loop system and (b) the settling time to within 2% of the final value of the system to a unit step disturbance. antenna must be positioned accurately to 1/10 of a degree, because the microwave beam is 0.2 wide and highly attenuated by the large distance. If the antenna is following the moving satellite, determine the K necessary for the system.

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 Equation (2.69)). (b) Calculate the steadystate error of this system to a step input command setting the speed to a new level. Assume that Ra= La = J = b = \, the motor constant is Km = 1, and Kh = 1. (c) Select a feedback gain for the back emf signal to yield a step response with an overshoot of 15%.

A simple unity feedback control system has a process transfer function = 6(5 = . E(s) s The system input is a step function with an amplitude A. The initial condition of the system at time f0 is y(t0) = Q, where y(t) is the output of the system. The performance index is defined as e\t) dt.

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 Bullet Express that travels between Tokyo and Osaka on theTokaido line. This train travels the 320 miles in 3 hours and 10 minutes, an average speed of 101 mph [17]. This speed will be increased as new systems are used, such as magnetically levitated systems to float vehicles above an aluminum guideway. To maintain a desired speed, a speed control system is proposed that yields a zero steady-state error to a ramp input. A third-order system is sufficient. Determine the optimum system transfer function T(s) for an ITAE performance criterion. Estimate the settling time (with a 2% criterion) and overshoot for a step input when &> = 10

We want to approximate a fourth-order system by a lower-order model. The transfer function of the original system is and the zero of GL(s), we have GH(s) = s 3 + Is2 + 24s + 24 s 4 + 10.53 + 35s2 + 50s + 24 s 3 + 7s2 + 24s + 24 (s + l)(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 the poles Gds) = 0.2917s + 1 0.399s2 + 1.375s + 1 _ 0.731 (s + 3.428) ~ (s + 1.043)(s + 2.4)*

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 = 0.986s + 2 = 0.986(s + 2.028) dS) s2 + 3s + 2 (s+ 1)(5 + 2)

Consider a unity feedback system with loop transfer function L(s) = Gc(s)G(s) = K(s + 1) (s + 4)(s2 + s + 10)* Determine the value of the gain K such that the percent overshoot to a unit step is minimized.

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 function V0(s)/Vin(s) has a damping ratio of 1/ v2 . The time constant of the magnetic amplifier is equal to 1 second, and the gain is K = 10. Calculate the settling time (with a 2% criterion) of the resulting system.

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 be G(s) = K s(s/12 + 1) Design the amplifier gain to yield a system with a settling time to a step disturbance of less than 1 second. The overshoot to a step in desired heart rate should be less than 10%. (a) Find a suitable range of K. (b) If the nominal value of K is K = 10, find the sensitivity of the system to small changes in K. (c) Evaluate the sensitivity of part (b) at DC (set s = 0). (d) Evaluate the magnitude of the sensitivity at the normal heart rate of 60 beats/minute.

Consider the original third-order system given in Example 5.9. Determine a first-order model with one pole unspecified and no zeros that will represent the third-order system.

A closed-loop control system with negative unity feedback has a loop transfer function L{s) = Gc(s)G(s) = s(s2 + 6s + 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 secondorder approximation to a unit step input and compare the results.

A system is shown in Figure P5.20. (a) Determine the steady-state error for a unit step input in terms of K and 2^, where E(s) = R(s) - Y(s). (b) Select Ki so that the steady-state error is zero.

Consider the closed-loop system in Figure P5.21. Determine values of the parameters k and a so that the following specifications are satisfied: FIGURE P5.22 Nonunity closed-loop feedback control (a) The steady-state error to a unit step input is zero,

Consider the closed-loop system in Figure P5.22, where Ge(s)G(s) = s + 0.2K and H(s) = 2s + T (a) If r = 2.43, determine the value of K such that the steady-state error of the closed-loop system response to a unit step input, R(s) = l/s, is zero. (b) Determine the percent overshoot P.O. and the time to peak Tp of the unit step response when K is as in part (a).

A closed-loop transfer function is T(s) = Y(s) 108(* + 3) = R(s) (s + 9)(52 + 8.v + 36)' (a) Determine the steady-state error for a unit step input R(s) = \/s. (b) Assume that the complex poles dominate, and determine the overshoot and settling time to within 2% of the final value. (c) Plot the actual system response, and compare it with the estimates of part (b).

A closed-loop system is shown in Figure AP5.2. Plot the response to a unit step input for the system for T, = 0,0.05,0.1, and 0.5. Record the percent overshoot, rise time, and settling time (with a 2% criterion) as TZ varies. Describe the effect of varying TC. Compare the location of the zero -1/r, with the location of the closed-loop poles.

A closed-loop system is shown in Figure AP5.3. Plot the response to a unit step input for the system with Tp = 0, 0.5, 2, and 5. Record the percent overshoot, rise time, and settling time (with a 2% criterion) as Tp varies. Describe the effect of varying rp. Compare the location of the open-loop pole -\/rp with the location of the closed-loop poles.

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 r(t). Consider the three values for K equal to 1,10, and 100. (a) Determine the steady-state error. (b) Determine and plot the response y{t) for (i) a unit step input R(s) - \(s and (ii) a unit step disturbance input Tlt(s) = 1/s. (c) Create a table showing overshoot, settling time (with a 2% criterion), ess for r(t), and ly/tjmax for the three values of K. Select the best compromise value.

A system with a controller is shown in Figure AP5.5. The zero of the controller may be varied. Let a = 0, 10,100. (a) Determine the steady-state error for a step input r(t) for a = 0 and a ^ 0. Y(s) Speed (b) Plot the response of the system to a step input disturbance for the three values of a. Compare the results and select the best value of the three values of a.

The block diagram model of an armature-currentcontrolled DC motor is shown in Figure AP5.6. (a) Determine the steady-state tracking error to a ramp input r(/) = t,t > 0, in terms of K, Kh, and (b) Let Km = 10 and / = 0.05, and select K so that steady-state tracking error is equal to 1. (c) Plot the response to a unit step input and a unit ramp input for 20 seconds. Are the responses acceptable?

Consider the closed-loop system in Figure AP5.7 with transfer functions ^ / ^ 10 0 , ^ / x K G c(s ) = 77^ a " d G( s s ) = + 100 s(s + 50)' where 1000 < K ^ 5000. (a) Assume that the complex poles dominate and estimate the settling time and percent overshoot to a unit step input for K = 1000, 2000, 3000, 4000, and 5000.

A unity negative feedback system (as shown in Figure E5.ll) has the loop transfer function L{s) = Gc(s)G(s) = K(s + 2) i 2 + h + \ Determine the gain K that minimizes the damping ratio of the closed-loop system poles. What is the minimum damping ratio?

The unity negative feedback system in Figure AP5.9 has the process given by G(s) = 1 s(s + 15)(.y + 25) The controller is a proportional plus integral controller with gains Kp and /C/.The objective is to design the controller gains such that the dominant roots have a damping ratio equal to 0.707. Determine the resulting peak time and settling time (with a 2% criterion) of the system to a unit step input.

The capstan drive system of the previous problems r C> (see CDP1.1-CDP4.1) has a disturbance due to changes \^vj in the part that is being machined as material is removed. The controller is an amplifier Gc(s) K. Evaluate the effect of a unit step disturbance, and determine the best value of the amplifier gain so that the overshoot to a step command r(t) = A,t > 0 is less than 5%, while reducing the effect of the disturbance as much as possible.

The roll control autopilot of a jet fighter is shown in Figure DPS. 1. The goal is to select a suitable K so that the response to a unit step command 4>d{t) = A, t 2: 0, will provide a response dj>{t) that is a fast response and has an overshoot of less than 20%. (a) Determine the closed-loop transfer function <(s)/<rf(s)- (b) Determine the roots of the characteristic equation for K = 0.7, 3, and 6. (c) Using the concept of dominant roots, find the expected overshoot and peak time for the approximate second-order system, (d) Plot the actual response and compare with the approximate results of part (c). (e) Select the gain K so that the percentage overshoot is equal to 16%. What is the resulting peak time?

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, where = 0.6, and the gain K and the natural frequency (on can be selected, (a) Determine K and co so that the response to a unit step input achieves a peak l Aileron actuator K s + 7 Aircraft dynamics s Gyro K I A . 12.2 (s + 2.2) (.V) Roll angle time for the first overshoot (above the desired level of I) that is less than or equal to 1 second and the overshoot is less than 5%. (Hint: Try 0.2 < K/a> < 0.4.) (b) Plot the response of the system designed in part (a) to a step input.

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]. Select a design value L Actuator and amplifier K s Arm dynamics co}, x 2 + 2

The space satellite shown in Figure DP5.4(a) uses (b) a control system to readjust its orientation, as shown in Figure DP5.4(b). (a) Determine a second-order model for the closed- (c ) loop system. Using the second-order model, select a gain K so that the percent overshoot is less than 15% and the steady-state error to a step is less than 12%. Verify your design by determining the actual performance of the third-order system.

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 [8,11]. While force control has been able to address the problem of accuracy, it has been more difficult to solve the contact stability problem. In fact, by closing the force loop and introducing a compliant wrist force sensor (the most common type of force control), one can add to the stability problem. A model of a robot deburring system is shown in Figure DP5.5. Determine the region of stability for the system for K\ and K%, Assume both adjustable gains are greater than zero.

The model for a position control system using a DC motor is shown in Figure DPS .6. The goal is to select X| and K2 so that the peak time is T s 0.5

A three-dimensional cam for generating a function

Computer control of a robot to spray-paint an and y may be controlled using a position control system automobile is accomplished by the system shown in

Consider the closed-loop transfer function 15 T(s) = s 2 + 8s + 15 Obtain the impulse response analytically and compare the result to one obtained using the impulse function.

A unity negative feedback system has the loop transfer function L(s) = Ge{s)G(s) = A- + 10 s 2 (s + 15)' Using Isim, obtain the response of the closed-loop system to a unit ramp input, R(s) = 1/52 . Consider the time interval 0 s t < 50. What is the steady-state error?

A working knowledge of the relationship between the pole locations of the second-order system shown in Figure CP5.3 and the transient response is important in control design. With that in mind, consider the following four cases: 1. (on = 2, = 0, 2. a,, = 2, I = 0.1, 3. ns + a>;, Y(s) FIGURE CP5.3 A simple second-order system. Using the impulse and subplot functions, create a plot containing four subplots, with each subplot depicting the impulse response of one of the four cases listed. Compare the plot with Figure 5.17 in Section 5.5, and discuss the results.

Consider the control system shown in Figure CP5.4. (a) Show analytically that the expected percent overshoot of the closed-loop system response to a unit step input is about 50%. (b) Develop an m-file to plot the unit step response of the closed-loop system and estimate the percent overshoot from the plot. Compare the result with part (a).

Consider the feedback system in Figure CP5.5. Develop an m-file to design a controller and prefilter Gc(s) = K s + z s + p and Gp(s) = Kr S + T such that the ITAE performance criterion is minimized. For (on = 0.45 and = 0.59, plot the unit step response and determine the percent overshoot and settling time.

The loop transfer function of a unity negative feedback system is 25 L(s) = Gc(s)G(s) = s(s + 5)' Develop an m-file to plot the unit step response and determine the values of peak overshoot Mp, time to peak Tp, and settling time Ts (with a 2% criterion).

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 Gc(s) = 2. Using the Isim function, compute and plot the ramp response for 8(f(t) = at, where a = 0.5/s. Determine the attitude error after 10 seconds. (b) If we increase the complexity of the controller, we can reduce the steady-state tracking error. With this objective in mind, suppose we replace the constant gain controller with the more sophisticated controller K 1 Gc(s) = Ky + = 2 + -. This type of controller is known as a proportional plus integral (PI) controller. Repeat the simulation of part (a) with the PI controller, and compare the steadystate tracking errors of the constant gain controller versus the PI controller.

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 Mp Tp, and Ts for the closed-loop system due to a unit step input. Compare the predicted results with the actual unit step response obtained with the step function. Explain any differences.

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?

Develop an m-file to simulate the response of the system in Figure CP5.10 to a ramp input R(s) - l/s2 . What is the steady-state error? Display the output on an x-y graph.

Consider the closed-loop system in Figure CP5.11. Develop an m-file to accomplish the following tasks: (a) Determine the closed-loop transfer function T(s) = Y(s)/R(s), (b) Plot the closed-loop system response to an impulse input R(s) = 1, a unit step input R(s) = l/s, and a unit ramp input R(s) = l/s2 . Use the subplot function to display the three system responses.