(a) For the system given in Fig. 5.56, plot the root locus of the characteristic

Give two definitions for the root locus.

Define the negative root locus

Where are the sections of the (positive) root locus on the real axis?

What are the angles of departure from two coincident poles at s = a on the real axis? There are no poles or zeros to the right of a.

What are the angles of departure from three coincident poles at s = a on the real axis? There are no poles or zeros to the right of a.

What is the principal effect of a lead compensation on a root locus?

What is the principal effect of a lag compensation on a root locus in the vicinity of the dominant closed-loop roots?

What is the principal effect of a lag compensation on the steady-state error to a polynomial reference input?

Why is the angle of departure from a pole near the imaginary axis especially important?

Define a conditionally stable system.

Show, with a root-locus argument, that a system having three poles at the origin MUST be conditionally stable.

Set up the listed characteristic equations in the form suited to Evanss root-locus method. Give L(s), a(s), and b(s) and the parameter K in terms of the original parameters in each case. Be sure to select K so that a(s) and b(s) are monic in each case and the degree of b(s) is not greater than that of a(s). (a) s + (1/) = 0 versus parameter (b) s 2 + cs + c + 1 = 0 versus parameter c (c) (s + c) 3 + A(Ts + 1) = 0 (i) versus parameter A, (ii) versus parameter T, (iii) versus the parameter c, if possible. Say why you can or cannot. Can a plot of the roots be drawn versus c for given constant values of A and T by any means at all? (d) . Assume that , where c(s) and d(s) are monic polynomials with the degree of d(s) greater than that of c(s). (i) versus kp (ii) versus kI (iii) versus kD (iv) versus

Roughly sketch the root loci for the polezero maps as shown in Fig. 5.51 without the aid of a computer. Show your estimates of the center and angles of the asymptotes, a rough evaluation of arrival and departure angles for complex poles and zeros, and the loci for positive values of the parameter K. Each polezero map is from a characteristic equation of the form where the roots of the numerator b(s) are shown as small circles o and the roots of the denominator a(s) are shown as s on the s-plane. Note that in Fig. 5.51(c) there are two poles at the origin. Figure 5.51 Polezero maps

For the characteristic equation (a) Draw the real-axis segments of the corresponding root locus. (b) Sketch the asymptotes of the locus for K . (c) Sketch the locus? (d) Verify your sketch with a MATLAB plot.

Real poles and zeros. Sketch the root locus with respect to K for the equation 1 + KL(s) = 0 and the listed choices for L(s). Be sure to give the asymptotes, and the arrival and departure angles at any complex zero or pole. After completing each hand sketch, verify your results using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales.

Complex poles and zeros. Sketch the root locus with respect to K for the equation 1 + KL(s) = 0 and the listed choices for L(s). Be sure to give the asymptotes and the arrival and departure angles at any complex zero or pole. After completing each hand sketch, verify your results using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales.

Multiple poles at the origin. Sketch the root locus with respect to K for the equation 1 + KL(s) = 0 and the listed choices for L(s). Be sure to give the asymptotes and the arrival and departure angles at any complex zero or pole. After completing each hand sketch, verify your results using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales.

Mixed real and complex poles. Sketch the root locus with respect to K for the equation 1 + KL(s) = 0 and the listed choices for L(s). Be sure to give the asymptotes and the arrival and departure angles at any complex zero or pole. After completing each hand sketch, verify your results using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales.

RHP and zeros. Sketch the root locus with respect to K for the equation 1 + KL(s) = 0 and the listed choices for L(s). Be sure to give the asymptotes and the arrival and departure angles at any complex zero or pole. After completing each hand sketch, verify your results using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. the model for a case of magnetic levitation with lead compensation. ; the magnetic levitation system with integral control and lead compensation. What is the largest value that can be obtained for the damping ratio of the stable complex roots on this locus?

Put the characteristic equation of the system shown in Fig. 5.52 in root-locus form with respect to the parameter , and identify the corresponding L(s), a(s), and b(s). Sketch Figure 5.52 Control system for Problem 5.9 the root locus with respect to the parameter , estimate the closed-loop pole locations, and sketch the corresponding step responses when = 0, 0.5, and 2. Use MATLAB to check the accuracy of your approximate step responses.

Use Rouths criterion to find the range of the gain K for which the systems in Fig. 5.53 are stable, and use the root locus to confirm your calculations. Figure 5.53 Feedback systems for Problem 5.11

Sketch the root locus for the characteristic equation of the system for which and determine the value of the root-locus gain for which the complex conjugate poles have a damping ratio of 0.5.

For the system in Fig. 5.54, (a) Find the locus of closed-loop roots with respect to K. (b) Is there a value of K that will cause all roots to have a damping ratio greater than 0.5? (c) Find the values of K that yield closed-loop poles with the damping ratio = 0.707. (d) Use MATLAB to plot the response of the resulting design to a reference step. Figure 5.54 Feedback system for Problem 5.13 Figure 5.55 Feedback system for Problem 5.14

For the feedback system shown in Fig. 5.55, find the value of the gain K that results in dominant closed-loop poles with a damping ratio = 0.5.

A simplified model of the longitudinal motion of a certain helicopter near hover has the transfer function and the characteristic equation 1 + D(s)G(s) =0. Let D(s) = kp at first. (a) Compute the departure and arrival angles at the complex poles and zeros. (b) Sketch the root locus for this system for parameter K = 9.8 kp . Use axes 4 x 4; 3 y 3. (c) Verify your answer using MATLAB. Use the command axis([4 4 3 3]) to get the right scales. (d) Suggest a practical (at least as many poles as zeros) alternative compensation D(s) that will at least result in a stable system

(a) For the system given in Fig. 5.56, plot the root locus of the characteristic equation as the parameter K1 is varied from 0 to with = 2. Give the corresponding L(s), a(s), and b(s). (b) Repeat part (a) with = 5. Is there anything special about this value? (c) Repeat part (a) for fixed K1 = 2, with the parameter K = varying from 0 to . Figure 5.56 Control system for Problem 5.16

For the system shown in Fig. 5.57, determine the characteristic equation and sketch the root locus of it with respect to positive values of the parameter c. Give L(s), a(s), and b(s), and be sure to show with arrows the direction in which c increases on the locus. Figure 5.57 Control system for Problem 5.17

Suppose you are given a system with the transfer function where Z and p are real and Z > p. Show that the root locus for 1 + KL(s) = 0 with respect to K is a circle centered at Z with radius given by r = (z p). Hint: Assume s + Z = re j and show that L(s) is real and negative for real under this assumption.

The loop transmission of a system has two poles at s = 1 and a zero at s = 2. There is a third real-axis pole p located somewhere to the left of the zero. Several different root loci are possible, depending on the exact location of the third pole. The extreme cases occur when the pole is located at infinity or when it is located at s = 2. Give values for p and sketch the three distinct types of loci.

Consider the system in Fig. 5.59. (a) Using Rouths stability criterion, determine all values of K for which the system is stable. (b) Use MATLAB to draw the root locus versus K and find the values of K at the imaginary-axis crossings. Figure 5.59 Feedback system for Problem 5.21

Let Using root-locus techniques, find values for the parameters a, b, and K of the compensation D(s) that will produce closed-loop poles at s = 1 j for the system shown in Fig. 5.60. Figure 5.60 Unity feedback system for Problems 5.22 to 5.28 and 5.33

Suppose that in Fig. 5.60 Sketch the root locus with respect to K of the characteristic equation for the closed-loop system, paying particular attention to points that generate multiple roots if KL(s) = D(s)G(s).

Suppose the unity feedback system of Fig. 5.60 has an open-loop plant given by G(s) = 1/s 2 . Design a lead compensation to be added in series with the plant so that the dominant poles of the closed-loop system are located at s = 2 2j.

Assume that the unity feedback system of Fig. 5.60 has the open-loop plant Design a lag compensation to meet the following specifications: The step response settling time is to be less than 5 sec. The step response overshoot is to be less than 17%. The steady-state error to a unit-ramp input must not exceed 10%.

A numerically controlled machine tool positioning servomechanism has a normalized and scaled transfer function given by Performance specifications of the system in the unity feedback configuration of Fig. 5.60 are satisfied if the closed-loop poles are located at . (a) Show that this specification cannot be achieved by choosing proportional control alone, D(s) = kp . Figure 5.61 Elementary magnetic suspension (b) Design a lead compensator that will meet the specification.

A servomechanism position control has the plant transfer function You are to design a series compensation transfer function D(s) in the unity feedback configuration to meet the following closed-loop specifications: The response to a reference step input is to have no more than 16% overshoot. The response to a reference step input is to have a rise time of no more than 0.4 sec. The steady-state error to a unit ramp at the reference input must be less than 0.02. (a) Design a lead compensation that will cause the systemtomeet the dynamic response specifications. (b) If D(s)is proportional control, D(s) = kp , what is the velocity constant Kv? (c) Design a lag compensation to be used in series with the lead you have designed to cause the system to meet the steady-state error specification. (d) Give the MATLAB plot of the root locus of your final design. (e) Give the MATLAB response of your final design to a reference step

Assume that the closed-loop system of Fig. 5.60 has a feed-forward transfer function Design a lag compensation so that the dominant poles of the closed-loop system are located at s = 1 j and the steady-state error to a unit-ramp input is less than 0.2.

An elementary magnetic suspension scheme is depicted in Fig. 5.61. For small motions near the reference position, the voltage e on the photo detector is related to the ball displacement x (in meters) by e = 100x. The upward force (in newtons) on the ball caused by the current i (in amperes) may be approximated by f = 0.5i + 20x. The mass of the ball is 20 g and the gravitational force is 9.8 N/kg. The power amplifier is a voltage-tocurrent device with an output (in amperes) of i = u + V0 . (a) Write the equations of motion for this set-up. (b) Give the value of the bias V0 that results in the ball being in equilibrium at x = 0. (c) What is the transfer function from u to e? (d) Suppose that the control input u is given by u = Ke. Sketch the root locus of the closed-loop system as a function of K. Figure 5.62 Block diagram for rocket-positioning control system (e) Assume that a lead compensation is available in the form . Give values of K, Z, and p that yield improved performance over the one proposed in part (d).

Consider the rocket-positioning system shown in Fig. 5.62. (a) Show that if the sensor that measures x has a unity transfer function, the lead compensator stabilizes the system. (b) Assume that the sensor transfer function is modeled by a single pole with a 0.1 sec time constant and unity DC gain. Using the root-locus procedure, find a value for the gain K that will provide the maximum damping ratio. Figure 5.63 Control system for Problem 5.32

For the system in Fig. 5.63, (a) Find the locus of closed-loop roots with respect to K. (b) Find the maximum value of K for which the system is stable. Assume K = 2 for the remaining parts of this problem. (c) What is the steady-state error (e = r y) for a step change in r? (d) What is the steady-state error in y for a constant disturbance w1? (e) What is the steady-state error in y for a constant disturbance w2? (f) If you wished to have more damping, what changes would you make to the system?

Consider the plant transfer function to be put in the unity feedback loop of Fig. 5.60. This is the transfer function relating the input force u(t) and the position y(t) of mass M in the noncollocated sensor and actuator problem. In this problem we will use rootlocus techniques to design a controller D(s) so that the closed-loop step response has a rise time of less than 0.1 sec and an overshoot of less than 10%. You may use MATLAB for any of the following questions: (a) Approximate G(s) by assuming that m 0, and let M = 1, K = 1, b = 0.1, and D(s) = K. Can K be chosen to satisfy the performance specifications? Why or why not? (b) Repeat part (a) assuming that D (s) = K(s + Z), and show that K and Z can be chosen to meet the specifications. (c) Repeat part (b), but with a practical controller given by the transfer function Pick p so that the values for K and Z computed in part (b) remain more or less valid. (d) Now suppose that the small mass m is not negligible, but is given by m = M/10. Check to see if the controller you designed in part (c) still meets the given specifications. If not, adjust the controller parameters so that the specifications are met.

Consider the Type 1 system drawn in Fig. 5.64. We would like to design the compensation D(s) to meet the following requirements: (1) The steady-state value of y due to a constant unit disturbance w should be less than , and (2) the damping ratio = 0.7. Using root-locus techniques, (a) Show that proportional control alone is not adequate. (b) Show that proportional-derivative control will work. (c) Find values of the gains kp and kD for D (s) = kp + kDs that meet the design specifications. Figure 5.64 Control system for Problem 5.34 Figure 5.65 Positioning servomechanism Source: Reprinted from Clark, 1962, with permission

Consider the positioning servomechanism system shown in Fig. 5.65, where (a) What is the range of the amplifier gain KA for which the system is stable? Estimate the upper limit graphically using a root-locus plot. (b) Choose a gain KA that gives roots at = 0.7. Where are all three closed-loop root locations for this value of KA?

We wish to design a velocity control for a tape-drive servomechanism. The transfer function from current I(s) to tape velocity (s) (in millimeters per millisecond per ampere) is We wish to design a Type 1 feedback system so that the response to a reference step satisfies tr 4 msec, ts 15 msec, Mp 0.05. (a) Use the integral compensator kI/s to achieve Type 1 behavior, and sketch the root locus with respect to kI . Show on the same plot the region of acceptable pole locations corresponding to the specifications. (b) Assume a proportional-integral compensator of the form kp (s + )/s, and select the best possible values of kp and you can find. Sketch the root-locus plot of your design, giving values for kp and , and the velocity constant Kv your design achieves. On your plot, indicate the closed-loop poles with a dot () and include the boundary of the region of acceptable root locations.

The normalized, scaled equations of a cart as drawn in Fig. 5.66 of mass mc holding an inverted uniform pendulum of mass mp and length l with no friction are where is a mass ratio bounded by 0 < < 0.75. Time is measured in terms of = o t where . The cart motion y is measured in units of pendulum length as and the input is force normalized by the system weight . These equations can be used to compute the transfer functions In this problem you are to design a control for the system by first closing a loop around the pendulum, Eq. (5.96), and then, with this loop closed, closing a second loop around the cart plus pendulum, Eq. (5.97). For this problem, let the mass ratio be mc = 5mp . (a) Draw a block diagram for the system with V input and both Y and as outputs. (b) Design a lead compensation for the loop to cancel the pole at s = 1 and place the two remaining poles at 4 j4. The new control is U(s), where the force is V (s) = U(s) + D(s)(s). Draw the root locus of the angle loop. (c) Compute the transfer function of the new plant from U to Y with D(s) in place. (d) Design a controller Dc(s) for the cart position with the pendulum loop closed. Draw the root locus with respect to the gain of Dc(s). (e) Use MATLAB to plot the control, cart position, and pendulum position for a unit step change in cart position. Figure 5.66 Figure of cart pendulum for Problem 5.37

Consider the 270-ft U.S. Coast Guard cutter Tampa (902) shown in Fig. 5.67. Parameter identification based on sea-trials data (Trankle, 1987) was used to estimate the hydro-dynamic coefficients in the equations of motion. The result is that the response of the heading angle of the ship to rudder angle and wind changes w can be described by the second-order transfer functions where = heading angle, rad, r = reference heading angle, rad, r = yaw rate, rad/sec, = rudder angle, rad, w = wind speed, m/sec. (a) Determine the open-loop settling time of r for a step change in . (b) In order to regulate the heading angle , design a compensator that uses and the measurement provided by a yaw-rate gyroscope (that is, by = r). The settling time of to a step change in r is specified to be less than 50 sec, and for a 5 change in heading, the maximum allowable rudder angle deflection is specified to be less than 10. (c) Check the response of the closed-loop system you designed in part (b) to a wind gust disturbance of 10 m/sec. (Model the disturbance as a step input.) If the steady-state value of the heading due to this wind gust is more than 0.5, modify your design so that it meets this specification as well. Figure 5.67 USCG cutter Tampa (902) for Problem 5.38

Golden Nugget Airlines has opened a free bar in the tail of their airplanes in an attempt to lure customers. In order to automatically adjust for the sudden weight shift due to passengers rushing to the bar when it first opens, the airline is mechanizing a pitch-attitude autopilot. Figure 5.68 shows the block diagram of the proposed arrangement. We will model the passenger moment as a step disturbance Mp (s) = M0/s, with a maximum expected value for M0 of 0.6. (a) What value of K is required to keep the steady-state error in to less than 0.02 rad ( 1)? (Assume the system is stable.) (b) Draw a root locus with respect to K. (c) Based on your root locus, what is the value of K when the system becomes unstable? (d) Suppose the value of K required for acceptable steady-state behavior is 600. Show that this value yields an unstable system with roots at s = 2.9,13.5,+1.2 6.6j. (e) You are given a black box with rate gyro written on the side and told that, when installed, it provides a perfect measure of , with output KT . Assume that K = 600 as in part (d) and draw a block diagram indicating how you would incorporate the rate gyro into the autopilot. (Include transfer functions in boxes.) (f) For the rate gyro in part (e), sketch a root locus with respect to KT . (g) What is the maximum damping factor of the complex roots obtainable with the configuration in part (e)? (h) What is the value of KT for part (g)? (i) Suppose you are not satisfied with the steady-state errors and damping ratio of the system with a rate gyro in parts (e) through (h). Discuss the advantages and disadvantages of adding an integral term and extra lead networks in the control law. Support your comments using MATLAB or with rough root-locus sketches.Figure 5.68 Golden Nugget Airlines autopilot

Plot the loci for the 0 locus or negative K for each of the following: (a) The examples given in Problem 5.3 (b) The examples given in Problem 5.4 (c) The examples given in Problem 5.5 (d) The examples given in Problem 5.6 (e) The examples given in Problem 5.7 (f) The examples given in Problem 5.8

Suppose you are given the plant where is a system parameter that is subject to variations. Use both positive and negative root-locus methods to determine what variations in can be tolerated before instability occurs.

Consider the system in Fig. 5.70. (a) Use Rouths criterion to determine the regions in the (K1 , K2 ) plane for which the system is stable. (b) Use RLTOOL to verify your answer to part (a). Figure 5.70 Feedback system for Problem 5.43

The block diagram of a positioning servomechanism is shown in Fig. 5.71. (a) Sketch the root locus with respect to K when no tachometer feedback is present (KT = 0). (b) Indicate the root locations corresponding to K = 16 on the locus of part (a). For these locations, estimate the transient-response parameters tr , Mp , and ts . Compare your estimates to measurements obtained using the step command in MATLAB. (c) For K = 16, draw the root locus with respect to KT . (d) For K = 16 and with KT set so that Mp = 0.05 ( = 0.707), estimate tr and ts . Compare your estimates to the actual values of tr and ts obtained using MATLAB. (e) For the values of K and KT in part (d), what is the velocity constant Kv of this system? Figure 5.71 Control system for Problem 5.44

Consider the mechanical system shown in Fig. 5.72, where g and a0 are gains. The feedback path containing gs controls the amount of rate feedback. For a fixed value of a0 , adjusting g corresponds to varying the location of a zero in the s-plane. (a) With g = 0 and = 1, find a value for a0 such that the poles are complex. (b) Fix a0 at this value, and construct a root locus that demonstrates the effect of varying g. Figure 5.72 Control system for Problem 5.45

Sketch the root locus with respect to K for the system in Fig. 5.73 using the Pad(1,1) approximation and the first-order lag approximation. For both approximations, what is the range of values of K for which the system is unstable? Figure 5.73 Control system for Problem 5.46

Prove that the plant G(s) = 1/s 3 cannot be made unconditionally stable if pole cancellation is forbidden.

For the equation 1 + KG(s), where use MATLAB to examine the root locus as a function of K for p in the range from p = 1 to p = 10, making sure to include the point p = 2.