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Unformatted text preview: is similar to of 30°. Then y? a closed—loop 0.01. :closed—loop 0.01. nputation of ition control :closed-loop that of the mputation of ls and sensor Figure 6.103 Control system for Problem 6.55 Problems 405 (b) Use Bode plot sketches to design a compensator for G(s) so that the closed-loop system satisfies the following specifications: (i) The steady—state error to a unit—ramp input is less than 0.01. (ii) PM 2 45°. (iii) The steady—state error for sinusoidal inputs with a) < 0.2 rad/sec is less than 1/250- (iv) Noise components introduced with the sensor signal at frequencies greater than 200 rad/sec are to be attenuated at the output by at least a factor of 100. (c) Verify and/or refine your design using MATLAB, including a computation of the closed-loop frequency response to verify (iv). ' 6.53 Consider a Type 1 unity—feedback system with K G“) = s(s+1)' Use Bode plot sketches to design a lead compensator so that K, = 20 sec“1 and PM > 40°. Use MATLAB to verify and/or refine your design so that it meets the specifications. 6.54 Consider a satellite attitude-control system with the transfer function 0.05(s + 25) G =——————«.—. (S) s2(s2+0.1s+4) Amplitude-stabilize the system using lead compensation so that GM 3 2 (6 db)7 and PM 2 45°, keeping the bandwidth as high as possible with a single lead. 6.55 In one mode of operation, the autopilot of a jet transport is used to control altitude. For the purpose of designing the altitude portion of the autopilot loop, only the long- period airplane dynamics are important. The linearized relationship between altitude and elevator angle for the long—period dynamics is MS) 20(s + 0.01) ft/ see so) s(s2 + 0.01s + 0.0025) deg ' G(s) - The autopilot receives from the altimeter an electrical signal proportional to altitude. This signal is compared with a command signal (proportional to the altitude selected by the pilot), and the difference provides an error signal. The error signal is processed throngh compensation, and the result is used to command the elevator actuators. A block diagram of this system is shown in Fig. 6.103. You have been given the task of designing the compensation. Begin by considering a proportional control law D(s) = K. (21) Use MATLAB to draw a Bode plot of the open—loop system for D(s) = K = l. (b) What value of K would provide a crossover frequency (i.e., where |G| = 1) of 0.16 rad/sec? (c) For this value of K, would the system be stable if the loop were closed? (d) What is the PM for this value of K? 6.56 6.57 6.58 406 Chapter 6 The Frequency-Response Design Method (e) Sketch the Nyquist plot of the system, and locate carefully any points where the phase angle is 180° or the magnitude is unity. (f) Use MATLAB to plot the root locus with respect to K, and locate the roots for your value of K from part (b). (g) What steady—state error would result if the command was a step change in altitude of 1000 ft? For parts (h) and (i), assume a compensator of the form Ts+l D =K . (S) orTs—I—l (h) Choose the parameters K, T, and or so that the crossover frequency is 0.16 rad/sec and the PM is greater than 50°. Verify your design by superimposing a Bode plot of D(s)G(s)/K on top of the Bode plot you obtained for part (a), and measure the PM directly. (i) Use MATLAB to plot the root locus with respect to K for the system, including the compensator you designed in part (h). Locate the roots for your Value of K from part (h). (j) Altitude autopilots also have a mode in which the rate of climb is sensed directly and commanded by the pilot. (i) Sketch the block diagram for this mode. (ii) Define the pertinent G(s). (iii) Design D(s) so that the system has the same crossover frequency as the altitude hold mode and the PM is greater than 50°. For a system with open-loop transfer function 10 G“) : s[(s/1.4)+1][(s/3)+1]’ design a lag compensator with unity DC gain so that PM 2 40°. What is the approximate bandwidth of this system? For the ship—steering system in Problem 6.39, (a) Design a compensator that meets the following specifications: (i) Velocity constant Kv = 2, (ii) PM 3 50°, (iii) Unconditional stability (PM > 0 for all to S we, the crossover frequency). (b) For your final design, draw a root locus with respect to K, and indicate the location of the closed-loop poles. Consider a unity-feedback system with 1 = s We +1>(s2/1oo2 + 0.55/100 +1)‘ (678) G(s) (a) A lead compensator is introduced with or = 1/5 and a zero at 1/T = 20. How must the gain be changed to obtain crossover at wc : 31.6 rad/sec, and What is the resulting value of Kv? (b) With the lead compensator in place, what is the required value of K for a lag compensator that will readjust the gain to a K, value of 100? Problems 409 Y K G and E l R:1+KG R=1+KG' (d) Explain Why introducing a lead network alone cannot meet the design specifications. (e) Explain why a lag network alone cannot meet the design specifications. (f) Develop a full design using a lead—lag compensator that meets all the design specifications without altering the previously chosen low-frequency open-loop gain. Problems for Section 6.8: Time Delay 6.63 Assume that the system 6.64 6.65 6.66 e—TdS Gm : s+10 has a 0.2—sec time delay (Td = 0.2 sec). While maintaining a phase margin 240°, find the maximum possible bandwidth by using the following: (a) One lead-compensator section s+a D =K , (s) 3+1) where 19/61 = 100; (b) Two lead-compensator sections 2 s—l—a D :K , (s) (M) where b/a = 10. (c) Comment on the statement in the text about the limitations on the bandwidth imposed by a delay. Determine the range of K for which the following systems are stable: (a) G(s) = K8145 e—S s(s + 2) In Chapter 5, we used various approximations for the time delay, one of which is the first order Padé: (1)) Go) 2 K 1 — Tags/2 1 + Tds/z' Using frequency response methods, the exact time delay [TN 3 H1(s) = Has) = e4“ can be obtained. Plot the phase of H1 (s) and H2 (5), and discuss the implications. Consider die heat exchanger of Example 2.15 with the open—loop transfer function ~-5.v E G“) = (10s +1)(60s +1)‘ (3) Design a lead compensator that yields PM 2 45° and the maximum possible closed- loop bandwidth. (b) Design a PI compensator that yields PM 3 45° and the maximum possible closed- loop bandwidth. Problems 593 For a system with a 1 rad/sec bandwidth. describe the consequences of various sample rates. 5. Give two advantages for Selecting a digital processor rather than analog circuitry to implement a controller. 6. Give two disadvantages for selecting a digital processor rather than analog circuitry to implement a controller. 7. Describe how to arrive at a D(z) if the sample rate is 5 x wBW. PROBLEMS Problems for Section 8.2: Dynamic Analysis of Discrete Systems 8.1 The z—transform of a discrete—time filter h(k) at a l Hertz sample rate is 1+ (1/2)z_1 Hz — . O‘ [1—<1/2>z*11[1+<1/3>z‘11 (a) Let u(k) and y(k) be the discrete input and output of this filter. Find a difference equation relating Mk) and y(k). (b) Find the natural frequency and damping coefficient of the filter’s poles. (c) Is the filter stable? 8.2 Use the z—transform to solve the difference equation y(k) — 3y(k — l) + 2y(k — 2): 214(k —- 1) —— 2140c — 2), Where k, k__O, “(k):{ 0 k:0 y(k) = 0, k < O. 8.3 The onersided zitransform is defined as 00 F<z> = Zflez—k. 0 (a) Show that the one—sided transform of f (k + 1) is Z {f (k + 1)} = zF(z) — zf(0). (b) Use the one—sided transform to solve for the transforms of the Fibonacci num— bers generated by the difference equation n(k + 2) : n(k + 1) + u(k). Let 14(0) : 14(1) 2 l. [H int: You will need to find a general expression for the transform of f (k + 2) in terms of the transform of f (k).] (c) Compute the pole locations of the transform of the Fibonacci numbers. (d) Compute the inverse transform of the Fibonacci numbers. (e) Show that, if u(k) represents the kth Fibonacci number, then the ratio n(k + 1) /u(k) will approach (Hg/3) . This is the golden ratio valued so highly by the Greeks. 8.4 Prove the seven properties of the s—plane—to-z—plane mapping listed in Section 8.2.3. W,W..._. _.__.._..._.._...— .__._t_.._...t__..«.._._____ __._.._..._—_ . .t Figure 8.21 Control system for Problem 8.9 Figure 8.2.2 Satellite control schematic for Problem 8.10 Problems 595 Problems for Section 8. 6: Discrete Design 8.9 Consider the system configuration shown in Fig. 8.21, where 4007+ 2) Gm = (s+10)(s2 —1.4)' (a) Find the transfer function C(z) for T = 1 assuming the system is preceded by a ZOH. (b) Use MATLAB to draw the root locus of the system with respect to K. (c) What is the range of K for which the closed—loop system is stable? ((1) Compare your results of part (c) with the case in which an analog controller is used (that is, Where the sampling switch is always closed). Which system has a larger allowable value of K ? (e) Use MATLAB to compute the step response of both the continuous and discrete systems with K chosen to yield a damping factor of r = 0.5 for the continuous case. + Re a)“: > K G(s) OY 8.10 Single—Axis Satellite Attitude Control: Satellites often require attitude control for proper orientation of antennas and sensors with respect to Earth. Figure 2.7 shows a com- munication satellite with a three—axis attitudeacontrol system. To gain insight into the three-axis problem, we often consider one axis at a time. Figure 8.22 depicts this case, where motion is allowed only about an axis perpendicular to the page. The equations of motion of the system are given by I = M C + M D, where I = moment of inertia of the satellite about its mass center, MC 2 control torque applied by the thrusters, M D = disturbance torques, ‘ 0 Inertial 7 reference l Problems 597 ’l Figure 8.23 Schematic of magnetic levitation device for Problems 8.11 l (b) Assume that the input is passed through a ZOH, and let the sampling period he . 0.02 sec. Compute the transfer function of the equivalent discrete-time plant. i i i 5 (c) Design a digital control for the magnetic levitation device so that the closed—loop system meets the following specifications: tr 5 0.1 sec, ts 5 0.4 sec, and overshoot E 20%. ((1) Plot a root locus with respect to k1 for your design, and discuss the possibility of using your closed-loop system to balance balls of various masses. (e) Plot the step response of your design to an initial disturbance displacement on the ball, and show both x and the control current i. If the sensor can measure x only over a range of 131/4 cm and the amplifier can provide a current of only 1 A, what 2 is the maximum displacement possible for control, neglecting the nonlinear terms : inf(x,I)? 8.12 Repeat Problem 5.27 in Chapter 5 by constructing discrete root loci and performing the ‘ . designs directly in the z-plane. Assume that the output y is sampled, the input u is passed “ ': through a ZOH as it enters the plant, and the sample rate is 15 Hz. ' 8.13 Design a digital controller for the antenna servo system shown in Figs. 3.61 and 3.62 and described in Problem 3.31. The design should provide a step response with an overshoot . of less than 10% and a rise time of less than 80 sec. l (a) What should the sample rate be? E (b) Use discrete equivalent design with the matched pole—zero method. i i l (c) Use discrete design and the z—plane root locus. 8.14 The system 1 (s+0.1)(s+3) ‘e G(S) = is to be controlled with a digital controller having a sampling period of T = 0.1 sec. Using a z—plane root locus, design compensation that will respond to a step with a rise time tr 5 1 sec and an overshoot MI; 5 5%. What can be done to reduce the steady-state 1 error? ...
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