A unity feedback system has a loop transfer function K(s + 1) L(s) = KG(s) = -- . sis2 +

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 < oo varies.

On the root locus plot, the number of separate loci is equal to the number of poles of G(s).

The root locus always starts at the zeros and ends at the poles of G(s).

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.

The root locus provides valuable insight into the response of a system to various test inputs.

Consider the control system in Figure 7.74, where the loop transfer function is K(s2 + 5s + 9) L(s) = Gc(s)G(s) = s 2 {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 approximate angles of departure of the root locus from the complex poles are a. 4>d = 180 b. 4>d = 115 c (f>d = 205 d. None of the above

The root locus of this system is given by which of the following

A unity feedback system has the closed-loop transfer function given by K T(s) = (s + 45)2 + K Using the root locus method, determine the value of the gain K so that the closed-loop system has a damping ratio = V7/2. a. K = 25 b. K = 1250 c K = 2025 d. K = 10500

Consider the unity feedback control system in Figure 7.74 where 10(s + z) L(s) = Gc(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 In Problems 11 and 12, consider the control system in Figure 7.74 where the model of the process is r( ) _ 750 {S) (s + l)(s + 10)(s + 50)"

Suppose that the controller is K(l + 0.2s) Gc(s) = 1 + 0.025* 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 > 0

Suppose that a simple proportional controller is utilized, that is, Gc(s) = K. Using the root locus method, determine the maximum controller gain K for closed-loop stability. a. K = 0.50 b. K = 1.49 c K = 4.49 d. Unstable for K > 0

Consider the unity feedback system in Figure 7.74 where L(s) = Gc(s)G(s) = , +^,2 ^ .un^ - s(s + 5)(sl + 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, iC = 58.75 d. None of the above In Problems 14 and 15, consider the feedback system in Figure 7.74, where K(s + 1 + j)(s + l-y ) L(s) = Gc(s)G(s) = s{s + 2j)(s - 2/)

Which of the following is the associated root locus?

The departure angles from the complex poles and the arrival angles at the complex zeros are: a. D = 180, 4>A = 0 b. D = 116.6, 4 u = 198.4 c. 4>D= 45.8, A = 116.6 d. None of the above

Let us 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 controlled via the torque T applied to the hoop from a torque motor attached to the hoop drive shaft. If negative feedback is used, the system characteristic equation is 1 + Ks(s + 4) = 0. s- + 2s + 2 (a) Sketch the root locus, (b) Find the gain when the roots are both equal, (c) Find these two equal roots. Torque Hoop FIGURE E7.1 Hoop rotated by motor. (d) Find the settling time of the system when the roots are equal.

A tape recorder has a speed control system so that H(s) = 1 with negative feedback and L(s) = Gc(s)G(s) = K s(s + 2)(52 + 4.v + 5) (a) Sketch a root locus for K, and show that the dominant roots are s = -0.35 /0.80 when K = 6.5. (b) For the dominant roots of part (a), calculate the settling time and overshoot for a step input.

A control system for an automobile suspension tester has negative unity feedback and a process [12] L{s) = Gc(s)G(s) = K(s2 + 4s + 8) s 2 (s + 4) We desire the dominant roots to have a equal to 0.5. Using the root locus, show that K = 7.35 is required and the dominant roots are s = 1.3 /2.2.

Consider a unity feedback system with K(s + 1) L(s) = Ge(s)G(s) = s 2 + 4s + 5

Consider a unity feedback system with a loop transfer function s 2 + 2s + 10 Gc(s)G(s) = - s 4 + 38r1 + 515*' + 2950* + 6000 (a) Find the breakaway points on the real axis, (b) Find the asymptote centroid. (c) Find the values of AT at the breakaway points.

One version of a space sta tion is shown in Figure E7.6 [28]. It is critical to keep this station in the proper orientation toward the Sim and the Earth for generating power and communications. The orientation controller may be represented by a unity feedback system with an actuator and controller, such as Gc{s)G(s) 15 A s^s2 + 15J + 75)' Sketch the root locus of the system as K increases. Find the value of K that results in an unstable system.

The elevator in a modern office building travels at a top speed of 25 feet per second and is still able to stop within one-eighth of an inch of the floor outside. The loop transfer function of the unity feedback elevator position control is K(s + 8) L W = C W = ,(, + 4)(, + 6)(, + 9)- Determine the gain K when the complex roots have a t equal to 0.8.

Sketch the root locus for a unity feedback system with K{s + 1) L{s) = Gc(s)G(s) = s\s + 9)' (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).

The world's largest telescope is located in Hawaii. The primary mirror has a diameter of 10 m and consists of a mosaic of 36 hexagonal segments with the orientation of each segment actively controlled. This unity feedback system for the mirror segments has the loop transfer function L(s) = Gc(s)G(s) K s(s2 + 2s + 5) (a) Find the asymptotes and draw them in the s-plane, (b) Find the angle of departure from the complex poles. (c) Determine the gain when two roots lie on the imaginary axis. (d) Sketch the root locus.

A unity feedback system has the loop transfer function L(x) = KG(s) = K(s + 2) ~s(s + 1)' (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 -2 . (c) Sketch the locus.

A robot force control system with unity feedback has a loop transfer function [6] Us) = KG(s) K(s + 2.5) (s2 + 2s + 2){s2 + 4s + 5) (a) Find the gain A' 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).

A unity feedback system has a loop transfer function K(s + 1) L(s) = KG(s) = -- . sis2 + 6s + 18) (a) Sketch the root locus for K > 0. (b) Find the roots when K - 10 and 20. (c) Compute the rise time, percent overshoot, and settling time (with a 2% criterion) of the system for a unit step input when K = 10 and 20.

A unity feedback system has a loop transfer function 4{s + z) L(s) = Gc(s)G(s) s(s + l)(.y + 3)' (a) Draw the root locus as z varies from 0 to 100. (b) Using the root locus, estimate the percent overshoot and settling time (with a 2% criterion) of the system at z = 0.6, 2, and 4 for a step input, (c) Determine the actual overshoot and settling time at z = 0.6, 2, and 4.

A unity feedback system has the loop transfer function L(s) = Gc(s)G(s) = Kjs + 10) s(s + 5) ' (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/v2. (c) Calculate the roots.

(a) Plot the root locus for a unity feedback system with loop transfer function L(s) = Gc(s)G(s) K(s + 10)0? + 2) (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.

A negative unity feedback system has a loop transfer function L(s) = Gc(s)G(s) = Ke' s+ V where T = 0.1 s. Show that an approximation for the time delay is Using ,-0.1 j2 T 20 - s 20 + s' obtain the root locus for the system for K > 0. Determine the range of K for which the system is stable.

A control system, as shown in Figure E7.17, has a process G(s) 1 s(s - I)-

A closed-loop negative unity feedback system is used to control the yaw of the A-6 Intruder attack jet. When the loop transfer function is L(s) = Gc(s)G{s) = K s(s + 3)(s2 + 2sr + 2) determine (a) the root locus breakaway point and (b) the value of the roots on the /w-axis and the gain required for those roots. Sketch the root locus. Answers: (a) Breakaway: .v = 2.29 (b) jco-axis: s = jl.Q9tK = 8

A unity feedback system has a loop transfer function L(.v) = Gc(s)G(s) = - . W s(s + 3)(s2 + 6i- + 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 /tt-axis and determine the location of these roots.

A unity feedback system has a loop transfer function L(s) = G( .is)Gis) = *tv + 1) sis - 2)is + 6) (a) Determine the range of K for stability, (b) Sketch the root locus, (c) Determine the maximum of the stable complex roots. Answers: (a) K > 16; (b) t = 0.25

A unity feedback system has a loop transfer function Lis) = Gcis)Gis) - -. ^ . w 53 + 5s2 + 10 Sketch the root locus. Determine the gain K when the complex roots of the characteristic equation have a C approximately equal to 0.66.

A high-performance missile for launching a satellite has a unity feedback system with a loop transfer function Gc(s)G(s) = K(s2 + I8)(s + 2) (s2 - 2)(5 + 12) ' Sketch the root locus as K varies from 0 < K < oo.

A unity feedback system has a loop transfer function L(s) = Gc(s)G(s) = 4(s? + 1) Sketch the root locus for 0 s(s + a) a < oo.

Consider the system represented in state variable form x = Ax + Bw y = Cx + DM, where A = c = [ r - 4 1 0], 1 n -k an ,B = d D ri i =[( Determine the characteristic equation and then sketch the root locus as 0 < k < oo.

A closed-loop feedback system is shown in Figure E7.25. For what range of values of the parameters K is the system stable? Sketch the root locus as 0 < K < oo.

Consider the signle-input, single-output system is described by x(/) = Ax(/) + Bu(t) y{t) = Cx(0 where A = 0 3 - K 1 - 2 - K ,B = 0 1 ,C = [1 -1]. Compute the characteristic polynomial and plot the root locus as 0 ^ K < oo. For what values of K is the system stable?

Consider the unity feedback system in Figure E7.27. Sketch the root locus as 0 < p < oo.

Consider the feedback system in Figure E7.28. Obtain the negative gain root locus as -oo < K ^ 0. For what values of K is the system stable?

Sketch the root locus for the following loop transfer functions of the system shown in Figure P7.1 when 0 < K < oo: (a) Gc(s)G{s) = (b) Gc(s)G(s) = (c) Gc(s)G(s) = (d) Gc(s)G(s) = K s(s + 10)(5 + 8) K (s2 + 2s + 2)(5 + 2) K(s + 5) s(s +1)(5+ 10) K(s2 + 45 + 8) s z (s + 1)

The linear model of a phase detector was presented in Problem P6.7. Sketch the root locus as a function of the gain Kv = KaK. Determine the value of Kv attained if the complex roots have a damping ratio equal to 0.60 [13]

A unity feedback system has the loop transfer function Gc(s)G(s) = K s(s + 2)(5 + 5)' 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 = 6. (d) Sketch the root locus.

The analysis of a large antenna was presented in Problem P4.5. Sketch the root locus of the system as 0 < ka < oo. Determine the maximum allowable gain of the amplifier for a stable system.

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 the pilot is not using the control stick, the switch may be considered to be open. The dynamics of the helicopter are represented by the transfer function G(s) = 25(5 + 0.03) (s + 0.4)(52 - 0.365 + 0.16) (a) With the pilot control loop open (hands-off control), sketch the root locus for the automatic stabilization loop. Determine the gain K2 that results in a damping for the complex roots equal to t, = 0.707. (b) For the gain K2 obtained in part (a), determine the steady-state error due to a wind gust Td(s) = 1/5. (c) With the pilot loop added, draw the root locus as K] varies from zero to 00 when K2 is set at the value calculated in part (a), (d) Recalculate the steady-state error of part (b) when K\ is equal to a suitable value based on the root locus.

An attitude control system for a satellite vehicle within the earth's atmosphere is shown in Figure P7.6. The transfer functions of the system are G(5) = K(s + 0.20) (5 + 0.90)(5 - 0.60)(5 - 0.10)

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 AL(s) within the power distribution network. The equilibrium speed desired results in a generator frequency equal to 60 cps. The effective rotary inertia J is equal to 4000 and the friction constant b is equal to 0.75. The steady-state speed regulation factor R is represented by the equation R ~ (Q equals the speed at no load. We want to obtain a very small R, usually less than 0.10. (a) Using root locus techniques, determine the regulation R attainable when the damping ratio of the roots of the system must be greater than 0.60. (b) Verify that the steady-state speed deviation for a load torque change AL(s) = AL/s is, in fact, approximately equal to RMwhen R < 0.1.

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 Gr(s) = -TS + 1 (T/2)S + r where T = 1 second. With the rest of the system remaining as given in Problem P7.7, repeat parts (a) and (b) of Problem P7.7.

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 eliminate the effects of disturbances (such as oil on the floor) as well as maintain accurate spacing between vehicles on a guideway. The system can be represented by the block diagram of Figure P7.9. The vehicle dynamics can be represented by G(s) = (s + 0.1)(.v2 + 2s + 289) s(s - 0.4)(.9 + 0.8)(4-2 + 1.45J + 361)'

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 important issue in modern aircraft designs since most airports have strict noise level requirements. One interesting concept is the Boeing Sonic Cruiser depicted in Figure P7.10(a). It would seat 200 to 250 passengers and cruise at just below the speed of sound. The flight control system must provide good handling characteristics and comfortable flying conditions. An automatic control system can be designed for the next generation passenger aircraft. The desired characteristics of the dominant roots of the control system shown in Figure P7.10(b) have a t = 0.707. The characteristics of the aircraft are io = 25,1 = 0-30, and T = 0.1. The gain factor Kh however, will vary over the range 0.02 at mediumweight cruise conditions to 0.20 at lightweight descent conditions, (a) Sketch the root locus as a function of the loop gain KiK2. (b) Determine the gain K2 necessary to yield roots with f = 0.707 when the aircraft is in the medium-cruise condition, (c) With the gain K2 as found in part (b), determine the f of the roots when the gain K\ results from the condition of light descent.

A computer system requires a high-performance magnetic tape transport system [17].The environmental conditions imposed on the system result in a severe test of control engineering design. A direct-drive DC motor system for the magnetic tape reel system is shown in Figure P7.ll, where r equals the reel radius, and J equals the reel and rotor inertia. A complete reversal of the tape reel direction is required in 6 ms, and the tape reel must follow a step command in 3 ms or less. The tape is normally operating at a speed of

A precision speed control system (Figure P7.12) is required for a platform used in gyroscope and inertial system testing where a variety of closely controlled speeds is necessary. A direct-drive DC torque motor system was utilized to provide (1) a speed range of 0.017s to 6007s, and (2) 0.1% steady-state error maximum for a step input. The direct-drive DC torque motor avoids the use of a gear train with its attendant backlash and friction. Also, the direct-drive motor has a high-torque capability, high efficiency, and low motor time constants. The motor gain constant is nominally K, = 1.8, but is subject to variations up to 50%. The amplifier gain Ka is normally greater than 10 and subject to a variation of 10%. (a) Determine the minimum loop gain necessary lo satisfy the steady-state error requirement, (b) Determine the limiting value of gain for stability, (c) Sketch the rool locus as K varies from 0 to co. (d) Determine the roots when Ka = 40, and estimate the response to a step input.

A unity feedback system has the loop transfer function Us) = GAs)C(s) K s(s + 3)(s2 + 4.v + 7.84)

The loop transfer function of a single-loop negative feedback system is Us) = G,(s)G{s) = K(s + 2.5)(.5 + 3.2) s*($ + !)(.? + 10)(.5+ 30)' This system is called conditionally stable because it is stable only for a range of the gain R such that ky < K < k2. Using the Routh-Hurwitz criteria and the root locus method, determine the range of the gain for which the system is stable. Sketch the root locus fort) < K < oo.

Let us again consider the stability and ride of a rider and high performance motorcycle as outlined in Problem P6.13. The dynamics of the motorcycle and rider can be represented by the loop transfer function G,(s)G(s) = K(s2 + 30s + 625) s(s + 20)(.r + 20s + 200)(.r + 60s + 3400)' Sketch the root locus for the system. Determine the f of the dominant roots when A! = 3 X 104

Control systems for maintaining constant tension on strip steel in a hot strip finishing mill are called "loopers." A typical system is shown in Figure P7.16. The looper is an arm 2 to 3 feet long with a roller on the end; it is raised and pressed against the strip by a motor [18], The typical speed of the strip passing the looper is 2000 ft/min. A voltage proportional to the looper

Consider again the vibration absorber discussed in Problems 2.2 and 2.10 as a design problem. Using the root locus method, determine the effect of the parameters M2 and kn- Determine the specific values of the parameters M2 and kn so that the mass Wj does not vibrate when F(t) = a sin(woT). Assume that Mi = l.jfcj = 1. and 6=1 . Also assume that kn < 1 and that the term k]2~may be neglected.

A feedback control system is shown in Figure P7.18. The filter Gc(s) is often called a compensator. and the design problem involves selecting the parameters a and /3. Using the root locus method, determine the effect of varying the parameters. Select a suitable filter so that the time to settle (to within 2% of the final value) is less than 4 seconds and the damping ratio of the dominant roots is greater than 0.60.

In recent years, many automatic control systems for guided vehicles in factories have been installed. One system uses a magnetic tape applied to the floor to guide the vehicle along the desired lane [10, 15]. Using transponder tags on the floor, the automatically guided vehicles can be tasked (for example, to speed up or slow down) at key locations. An example of a guided vehicle in a factory is shown in Figure P7.19(a). We have G(s) s 2 + As + 100 s(s + 2)(s + 6) and Ka is the amplifier gain. Sketch a root locus and determine a suitable gain K so that the damping ratio of the complex roots is 0.707.

Determine the root sensitivity for the dominant roots of the design for Problem P7.18 for the gain K = 4a//3 and the pole s = -2.

Determine the root sensitivity of the dominant roots of the power system of Problem P7.7. Evaluate the sensitivity for variations of (a) the poles at s = -4 , and (b) the feedback gain, 1/7?.

Determine the root sensitivity of the dominant roots of Problem P7.1(a) when K is set so that the damping ratio of the unperturbed roots is 0.707. Evaluate and compare the sensitivity as a function of the poles and zeros of Gc(s)G(s).

Repeat Problem P7.22 for the loop transfer function Gc(s)G(s) of Problem P7.1(c).

For systems of relatively high degree, the form of the root locus can often assume an unexpected pattern. The root loci of four different feedback systems of third order or higher are shown in Figure P7.24. The open-loop poles and zeros of KG(s) are shown, and the form of the root loci as K varies from zero to infinity is presented. Verify the diagrams of Figure P7.24 by constructing the root loci.

Solid-state integrated electronic circuits are composed of distributed R and C elements. Therefore, feedback electronic circuits in integrated circuit form must be investigated by obtaining the transfer function of the distributed RC networks. It has been shown that the slope of the attenuation curve of a distributed RC network is 10 dB/decade, where n is the order of the RC filter [13]. This attenuation is in contrast with the normal 20n dB/decade for the lumped parameter circuits. (The concept of the slope of an attenuation curve is considered in Chapter 8. If it is unfamiliar,

A single-loop negative feedback system has a loop transfer function L(s) = Gc(s)G(s) = K(s + 2)2 5(.92 +1)( 5 + 8) (a) Sketch the root locus for 0 < K < oo to indicate the significant features of the locus, (b) Determine the range of the gain K for which the system is stable. (c) For what value of K in the range K 2: 0 do purely imaginary roots exist? What are the values of these roots? (d) Would the use of the dominant roots approximation for an estimate of settling time be justified in this case for a large magnitude of gain (K > 50)?

A unity negative feedback system has a loop transfer function L(s) = Gc(s)G(s) K(s2 + 0.1) sis2 + 2) K(s + /0.3162)(5 /0.3162) s(s2 + 1) Sketch the root locus as a function of K. Carefully calculate where the segments of the locus enter and leave the real axis.

To meet current U.S. emissions standards for automobiles, hydrocarbon (HC) and carbon monoxide (CO) emissions are usually controlled by a catalytic converter in the automobile exhaust. Federal standards for nitrogen oxides (NOx) emissions are met mainly by exhaust-gas recirculation (EGR) techniques. However, as NOx emissions standards were tightened from the

A unity feedback control system has a transfer function L(s) = Gc(s)G(s) = K(s2 + 105 + 30) s 2 (s + 10) We desire the dominant roots to have a damping ratio equal to 0.707. Find the gain K when this condition is satisfied. Show that the complex roots are 5 = -3.56 y'3.56 at this gain.

An RLC network is shown in Figure P7.30. The nominal values (normalized) of the network elements are L C = 1 and R 2.5. Show that the root sensitivity of the two roots of the input impedance Z{s) to a change in R is different by a factor of 4.

The development of high-speed aircraft and missiles requires information about aerodynamic parameters prevailing at very high speeds. Wind tunnels are used to test these parameters. These wind tunnels are constructed by compressing air to very high pressures and releasing it through a valve to create a wind. Since the air pressure drops as the air escapes, it is necessary to open the valve wider to maintain a constant wind speed. Thus, a control system is needed to adjust the valve to maintain a constant wind speed. The loop transfer function for a unity feedback system is L(S) = GAs)G(s) K(s + 4) s(s + 0.16)(s + p)(s - p)' where p = 7.3 + 9.7831/.Sketch the root locus and show the location of the roots for K = 326 and K = 1350.

A mobile robot suitable for nighttime guard duty is available. This guard never sleeps and can tirelessly patrol large warehouses and outdoor yards. The steering control system for the mobile robot has a unity feedback with the loop transfer function L(s) = Gc(s)G(s) K(s + 1)(5 + 5) 5(5 + 1.5)(5 + 2)' (a) Find K for all breakaway and entry points on the real axis, (b) Find K when the damping ratio of the complex roots is 0.707. (c) Find the minimum value of the damping ratio for the complex roots and the associated gain K. (d) Find the overshoot and the time to settle (to within 2% of the final value) for a unit step input for the gain, K, determined in parts (b) and (c).

The Bell-Boeing V-22 Osprey Tiltrotor is both an airplane and a helicopter. Its advantage is the ability to rotate its engines to 90 from a vertical position for takeoffs and landings as shown in Figure P7.33(a), and then to switch the engines to a horizontal position for cruising as an airplane [20].The altitude control system in the helicopter mode is shown in Figure P7.33(b). (a) Determine the root locus as K varies and determine the range of K for a stable system, (b) For K = 280, find the actual y{t) for a unit step input r(i) and the percentage overshoot and settling time (with a 2% criterion), (c) When K = 280 and r(t) = 0, find y(i) for a unit step disturbance, Td(s) = \/s. (d) Add a prefilter between R(s) and the summing node so that GJs) and repeat pari (b).

The fuel control for an automobile uses a diesel pump that is subject to parameter variations. A unity negative feedback has a loop transfer function Gf(5)G(s) K(s + 2) (s + 1)(5 + 2.5)(s + 4)(5 + 10)' (a) Sketch the root locus as K varies from 0 to 2000. (b) Find the roots for K equal to 400, 500, and 600. (c) Predict how the percent overshoot to a step will vary for the gain K, assuming dominant roots, (d) Find the actual time response for a step input for all three gains and compare the actual overshoot with the predicted overshoot.

A powerful electrohydraulic forklift can be used to lift pallets weighing several tons on top of 35-foot scaffolds at a construction site. The negative unity feedback system has a loop transfer function L(s) = Gc(s)G(s) K(s + 1)2 s(s2 + 1)' (a) Sketch the root locus for K > 0. (b) Find the gain K when two complex roots have a of 0.707, and calculate all three roots, (c) Find the entry point of the root locus at the real axis, (d) Estimate the expected overshoot to a step input, and compare it with the actual overshoot determined from a computer program.

A microrobot with a high-performance manipulator has been designed for testing very small particles, such as simple living cells [6]. The single-loop unity negative feedback system has a loop transfer function L(s) = Gc(s)G(s) = K(s + l)(.v + 2)(s + 3) s\s - 1) ' (a) Sketch the root locus for K > 0. (b) Find the gain and roots when the characteristic equation has two imaginary roots, (c) Determine the characteristic roots when K = 20 and K = 100. (d) For K = 20, estimate the percent overshoot to a step input, and compare the estimate to the actual overshoot determined from a computer program.

Identify the parameters K, a, and b of the system shown in Figure P7.37. The system is subject to a unit step input, and the output response has an overshoot but ultimately attains the final value of 1. When the closed-loop system is subjected to a ramp input, the output response follows the ramp input with a finite steadystate error. When the gain is doubled to 2K, the output response to an impulse input is a pure sinusoid with a period of 0.314 second. Determine K, a, and b.

A unity feedback system has the loop transfer function L{s) = Ge{s)G(s) = K(s + 1) s(s ~ 3) This system is open-loop unstable, (a) Determine the range of K so that the closed-loop system is stable. (b) Sketch the root locus, (c) Determine the roots for K = 10. (d) For K = 10, predict the percent overshoot for a step input using Figure 5.13. (e) Determine the actual overshoot by plotting the response.

High-speed trains for U.S. railroad tracks must traverse twists and turns. In conventional trains, the axles are fixed in steel frames called trucks.The trucks pivot as the train goes into a curve, but the fixed axles stay parallel to each other, even though the front axle tends to go in a different direction from the rear axle [24]. If the train is going fast, it may jump the tracks. One solution uses axles that pivot independently. To counterbalance the strong centrifugal forces in a curve, the train also has a computerized hydraulic system that tilts each car as it rounds a turn. On-board sensors calculate the train's speed and the sharpness of the curve and feed this information to hydraulic pumps under the floor of each car. The pumps tilt the car up to eight degrees, causing it to lean into the curve like a race car on a banked track. The tilt control system is shown in Figure P7.39. Sketch the root locus, and determine the value of K when the complex roots have maximum damping. Predict the response of this system to a step input R(s).

The top view of a high-performance jet aircraft is shown in Figure AP7.1(a) [20]. Sketch the root locus and determine the gain K so that the of the complex poles near the y'w-axis is the maximum achievable. Evaluate the roots at this K and predict the response to a step input. Determine the actual response and compare it to the predicted re.sponse.

A magnetically levitated high-speed train "flies" on an air gap above its rail system, as shown in Figure AP7.2(a) [24], The air gap control system has a unity feedback system with a loop transfer function Gc(s)G(s) K(s + l)(s + 3) s(s - i)(s + 4)(.? + sy The feedback control system is illustrated in Figure AP7.2(b). The goal is to select K so that the response for a unit step input is reasonably damped and the settling time is less than 3 seconds. Sketch the root locus, and select K so that all of the complex roots have a f greater than 0.6. Determine the actual response for the selected K and the percent overshoot.

A compact disc player for portable use requires a good rejection of disturbances and an accurate position of the optical reader sensor. The position control system uses unity feedback and a loop transfer function L(s) = Gc(s)G(s) = 10 s(s + 1)0 + Py The parameter p can be chosen by selecting the appropriate DC motor. Sketch the root locus as a function of p . Select/? so that the of the complex roots of the characteristic equation is approximately 1/V2.

A remote manipulator control system has unity feedback and a loop transfer function Gc(s)G(s) = (s + a) .r3 + (1 + a)s2 + (a - \)s + 1 - a We want the steady-state position error for a step input to be less than or equal to 10% of the magnitude of the input. Sketch the root locus as a function of the parameter a. Determine the range of a required for the desired steady-state error. Locate the roots for the allowable value of a to achieve the required steady-state error, and estimate the step response of the system.

A unity feedback system has a loop transfer function L(s) = Gc(s)G(s) K 5 3 + 10s2 + 75 - 18' (a) Sketch the root locus and determine K for a stable system with complex roots with equal to l/v2 . (b) Determine the root sensitivity of the complex roots of part (a). (c) Determine the percent change in K (increase or decrease) so that the roots lie on the /w-axis.

A unity feedback system has a loop transfer function K(s2 + 3.? + 6) L(s) = Gc(s)G(s) = s 3 + 2^2 + 3s + 1 Sketch the root locus for K > 0, and select a value for K that will provide a closed step response with settling time less than 1 second.

A feedback system with positive feedback is shown in Figure AP7.7. The root locus for K > 0 must meet the condition KG(s) = l/*360 for k = 0,1,2,... . Sketch the root locus for 0 < K < oo.

A position control system for a DC motor is shown in Figure AP7.8. Obtain the root locus for the velocity feedback constant K, and select K so that all the roots of the characteristic equation are real (two are equal and real). Estimate the step response of the system for the K selected. Compare the estimate with the actual response.

A control system is shown in Figure AP7.9. Sketch the root loci for the following transfer functions Gc(sY (a) Gc(s) = K (b) G( .(.v) = K(s + 3)

A feedback system is shown in Figure AP7.10. Sketch the root locus as K varies when K > 0. Determine a value for K that will provide a step response with an overshoot less than 5% and a settling time (with a 2% criterion) less than 2.5 seconds.

A control system is shown in Figure AP7.11. Sketch the root locus, and select a gain K so that the step response of the system has an overshoot of less than 10% and the settling time (with a 2% criterion) is less than 4 seconds.

A control system with PI control is shown in Figure AP7.12. (a) Let K,/Kp = 0.2 and determine KP so that the complex roots have maximum damping ratio, (b) Predict the step response of the system with KP set to the value determined in part (a).

The feedback system shown in Figure AP7.13 has two unknown parameters K\ and K2. The process transfer function is unstable. Sketch the root locus for 0 < KUK2 < oo. What is the fastest settling time that you would expect of the closed-loop system in response to a unit step input R(s) = 1/s? Explain.

A unity feedback control system shown in Figure AP7.14 has the process C{s) 10 s(s + 10)0 + 7.5)' Design a PID controller using Ziegler-Nichols methods. Determine the unit step response and the unit disturbance response. What is the maximum percent overshoot and settling time for the unit step input?

The drive motor and slide system uses the output of a tachometer mounted on the shaft of the motor as shown in Figure CDP4.1 (switch-closed option). The output voltage of the tachometer is vT = K\B. Use the velocity feedback with the adjustable gain K\, Select the best values for the gain K^ and the amplifier gain K so that the transient response to a step input has an overshoot less than 5% and a settling time (to within 2% of the final value) less than 300 ms.

A high-performance aircraft, shown in Figure DP7.1(a). uses the ailerons, rudder, and elevator to steer through a three-dimensional flight path [20]. The pitch rate control system for a fighter aircraft at 10.000 m and Mach 0.9 can be represented by the system in Figure DP7.1(b), where C(5) = -18Q + 0.015)0 + 0.45) (s2 + 1.2s + 12)(52 + 0.01s + 0.0025)' (a) Sketch the root locus when the controller is a gain. so that Gc(s) - K, and determine K when I for the roots with co > 2 is larger than 0.15 (seek a maximum I). (b) Plot the response q(t) for a step input r(i) with K as in (a), (c) A designer suggests an anticipatory controller with Gc(s) = K, + K2s = K(s + 2). Sketch the root locus for this system as K varies and determine a K so that the f of all the closed-loop roots is >0.8. (d) Plot the response q{t) for a step input r(t) with Kasin (c).

A large helicopter uses two tandem rotors rotating in opposite directions, as shown in Figure P7.33(a). The controller adjusts the tilt angle of the main rotor and thus the forward motion as shown in Figure DP7.2.The helicopter dynamics are represented by G(s) H) i'2 + 4.5s + 9'

The vehicle Rover has been designed for maneuvering at 0.25 mph over Martian terrain. Because Mars is 189 million miles from Earth and it would take up to 40 minutes each way to communicate with Earth [22,27], Rover must act independently and reliably- Resembling a cross between a small flatbed truck and an elevated jeep. Rover is constructed of three articulated sections, each with its own two independent, axle-bearing, one-meter conical wheels. A pair of sampling armsone for chipping and drilling, the other for manipulating fine objectsextend from its front end like pincers. The control of the arms can Y(s) be represented by the system shown in Figure DP7.3. (a) Sketch the root locus for K and identify the roots for K = 4.1 and 41. (b) Determine the gain K that results in an overshoot to a step of approximately 1%. (c) Determine the gain that minimizes the settling time (with a 2% criterion) while maintaining an overshoot of less than 1 %.

A welding torch is remotely controlled to achieve high accuracy while operating in changing and hazardous environments [21]. A model of the welding arm position control is shown in Figure DP7.4, with the disturbance representing the environmental changes. (a) With Ttl(s) = 0, select K\ and K to provide high-quality performance of the position control system. Select a set of performance criteria, and examine the results of your design, (b) For the system in part (a), let R(s) = 0 and determine the effect of a unit steprrf(s) - 1 /sby obtaining y{t).

A high-performance jet aircraft with an autopilot control system has a unity feedback and control system, as shown in Figure DP7.5. Sketch the root locus and select a gain K that leads to dominant poles. With this gain K, predict the step response of the system. Determine the actual response of the system, and compare it to the predicted response.

A system to aid and control the walk of a partially disabled person could use automatic control of the walking motion [25]. One model of a system that is open-loop unstable is shown in Figure DP7.6. Using the root locus, select K for the maximum achievable of the complex roots. Predict the step response of the system, and compare it with the actual step response.

A mobile robot using a vision system as the measurement device is shown in Figure DP7.7(a) [36].The control system is shown in Figure DP7.7(b) where G(s) 1 (v + 1)((),.% + 1)" and Gt .(.f) is selected as a PI controller so that the steady-state error for a step input is equal to zero. We then have Gc(s) KP + K, Design the PI controller so that (a) the percent overshoot for a step input is P.O. s 5%; (b) the settling time (with a 2% criterion) is Ts s 6 seconds; (c) the system velocity error constant Kv > 0.9; and (d) the peak time, Tp. for a step input is minimized.

Most commercial op-amps are designed to be unity-gain stable [26]. That is, they are stable when

A robotic arm actuated at the elbow joint is shown in Figure DP7.9(a), and the control system for the actuator is shown in Figure DP7.9(b). Plot the root locus for K > 0. Select Gp(s) so that the steady-state error for a step input is equal to zero. Using the Gp(s) selected, plot y(t) for K equal to 1, 1.5, and 2.85. Record the rise time, settling time (with a 2% criterion), and percent overshoot for the three gains. We wish to limit the overshoot to less than 6% while achieving the shortest rise time possible. Select the best system fori

The four-wheel-steering automobile has several benefits. The system gives the driver a greater degree of control over the automobile. The driver gets a more forgiving vehicle over a wide variety of conditions.

A pilot crane control is shown in Figure DP7.11(a). The trolley is moved by an input F(i) in order to control x(t) and (t) [13]. The model of the pilot crane control is shown in Figure DP7.11(b). Design a controller that will achieve control of the desired variables when Gc(s) = K.

A rover vehicle designed for use on other planets and moons is shown in Figure DP7.12(a) [21]. The block diagram of the steering control is shown in Figure DP7.12(b), where G(s) = 1.5 (s + 1)(.5 + 2)0$ + 4)(.s + 10)'

The automatic control of an airplane is one example that requires multiple-variable feedback methods. In this system, the attitude of an aircraft is controlled by three sets of surfaces: elevators, a rudder, and ailerons, as shown in Figure DP7.I3(a). By manipulating these surfaces, a pilot can set the aircraft on a desired flight path [20]. An autopilot, which will be considered here, is an automatic control system that controls the roll angle (j> by adjusting aileron surfaces. The deflection of the aileron surfaces by an angle 8 generates a torque due to air pressure on these surfaces. This causes a rolling motion of the aircraft. The aileron surfaces are controlled by a hydraulic actuator with a transfer function lis. The actual roll angle is measured and compared with the input. The difference between the desired roll angle d and the actual angle will drive the hydraulic actuator, which in turn adjusts the deflection of the aileron surface. A simplified model where the rolling motion can be considered independent of other motions is assumed, and its block diagram is shown in Figure DP7.13(b). Assume that Kx = 1 and that the roll rate 4> is fed back using a rate gyro. The step response desired has an overshoot less than 10% and a settling time (with a 2% criterion) less than 9 seconds. Select the parameters Ka and K2-

Consider the feedback system shown in Figure DP7.14. The process transfer function is marginally stable. The controller is the proportional-derivative (PD) controller Gc(s) = Kp + KDs. (a) Determine the characteristic equation of the closed-loop system. (b) Let T = KF/KD.Write the characteristic equation in the form Ms) 1 + K, n(s) d(sY

Using the riocus function, obtain the root locus for the following transfer functions of the system shown in Figure CP7.1 when 0 < K < oo: 30 (a) G(.v) = -% 7 ^ , s 3 + 1%2 + 43i + 30 (b) G(s) = -z s 2 + As + 20 (c) G(.v) = (d) G(s) = s 2 + s + 2 s(s2 + 6s + 10) v5 + 4 / + 653 + 10s2 -I- 65 + 4 s 6 + 4.v5 + 4s4 + s3 + s2 + 10s + 1

A unity negative feedback system has the loop transfer function

Compute the partial fraction expansion of rw = ,' + sis2 + 5s + 4) and verify the result using the residue function.

A unity negative feedback system has the loop transfer function Gc{s)G(s) = (1 +p)s- p s 2 + 4s + 10"

Consider the feedback system shown in Figure CP7.1, where

A large antenna, as shown in Figure CP7.6(a), is used to receive satellite signals and must accurately track the satellite as it moves across the sky. The control system uses an armature-controlled motor and a controller to be selected, as shown in Figure CP7.6(b). The system specifications require a steady-state error for a ramp input r(r) = Bi, less than or equal to 0.012?, where B is a constant. We also seek a percent overshoot to a step input of P.O. < 5% with a settling time (with a 2% criterion) of Ts < 2 seconds, (a) Using root locus methods, create an m-file to assist in designing the controller, (b) Plot the resulting unit step response and compute the percent overshoot and the settling time and label the plot accordingly, (c) Determine the effect of the disturbance T^s) = Q/s (where Q is a constant) on the output Y(s).

Consider the feedback control system in Figure CP7.7. We have three potential controllers for our system: 1. Gc(s) = K (proportional controller) 2. Gc(s) = K/s (integral controller) 3. Gc(s) = K(\ + 1/s) (proportional, integral (PI) controller) The design specifications are 7, s 10 seconds and P.O. & 10% for a unit step input. (a) For the proportional controller, develop an m-file to sketch the root locus for 0 < K < 00, and

Consider the spacecraft single-axis attitude control system shown in Figure CP7.8. The controller is known as a proportional-derivative (PD) controller. Suppose that we require the ratio of Kp/KD = 5.Then, develop

Consider the feedback control system in Figure CP7.9. Develop an m-file to plot the root locus for 0 < K < DO . Find the value of K resulting in a damping ratio of the closed-loop poles equal to 0.707.

Consider the system represented in state variable form x = Ax + Bu y = Cx + DH, where 0 0 -1 1 0 - 5 0 1 - 2 - k_ ,B = ~r 0 _4_ A = C = [1 - 9 12], and D = [0]. (a) Determine the characteristic equation, (b) Using the Routh-Hurwitz criterion, determine the values of k for which the system is stable, (c) Develop an m-file to plot the root locus and compare the results to those obtained in (b).