Modern Control Systems. Richard C. Dorf Robert H. Bishop Twelfth Edition

Transcription

1 Modern Control Systems Richard C. Dorf Robert H. Bishop Twelfth Edition

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3 Video camera Pulley (a) t s s(t 2 s ) FIGURE DP8 Remote-controlled TV camera. (b) represented by the system in Figure DP8(b), where the nominal values are t = 20 ms and t 2 = 2 ms. (a) Compute the sensitivity S T t and the sensitivity. S T t 2 (b) Design the controller gain such that the steadystate tracking error to a unit step disturbance is less than COMPUTER PROBLEMS CP Consider a unity feedback system with 2 s 2 + 2s + 0. Obtain the step response and determine the percent overshoot. What is the steady-state error? CP2 Consider the transfer function (without feedback) 4 s 2 + 2s When the input is a unit step, the desired steady-state value of the output is one. Using the step function, show that the steady-state error to a unit step input is 0.8. CP3 Consider the closed-loop transfer function T(s) = 5 s 2 + 5s +. Obtain the family of step responses for = 0, 200, and 500. Co-plot the responses and develop a table of results that includes the percent overshoot, settling time, and steady-state error. <CP Consider the feedback system in Figure CP Suppose that the controller is G c (s) = = 0. FIGURE CP4 Unity feedback system with controller gain. E a (s) Plant s(s.9) 30

4 (a) Develop an m-file to compute the closed-loop transfer function T(s) = > and plot the unit step response. (b) In the same m-file, compute the transfer function from the disturbance to the output and plot the unit step disturbance response. (c) From the plots in (a) and (b) above, estimate the steady-state tracking error to the unit step input and the steady-state tracking error to the unit step disturbance input. (d) From the plots in (a) and (b) above, estimate the maximum tracking error to the unit step input and the maximum tracking error to the unit step disturbance input. At approximately what times do the maximum errors occur? CP5 Consider the closed-loop control system shown in Figure CP5. Develop an m-file script to assist in the search for a value of k so that the percent overshoot to a unit step input is greater than %, but less than 0%. The script should compute the closed-loop transfer function T(s) = > and generate the step response. Verify graphically that the steady-state error to a unit step input is zero. CP6 Consider the closed-loop control system shown in Figure CP6. The controller gain is = 2. The nominal value of the plant parameter is a =. The nominal value is used for design purposes only, since in reality the value is not precisely known. The objective of our analysis is to investigate the sensitivity of the closedloop system to the parameter a. (a) When a =, show analytically that the steadystate value of y(t) is equal to 2 when r(t) is a unit step. Verify that the unit step response is within 2% of the final value after 4 seconds. (b) The sensitivity of the system to changes in the parameter a can be investigated by studying the effects of parameter changes on the transient response. Plot the unit step response for a = 0.5, 2, and 5. Discuss the results. CP7 Consider the torsional mechanical system in Figure CP7(a). The torque due to the twisting of the shaft is - ku; the damping torque due to the braking device is - bu; the disturbance torque is t d (t); the input torque is r(t); and the moment of inertia of the mechanical system is J. The transfer function of the torsional mechanical system is >J s 2 + (b>j)s + k>j. FIGURE CP5 A closed-loop negative feedback control system. 0 s s k FIGURE CP6 A closed-loop control system with uncertain parameter a. s a FIGURE CP7 (a) A torsional mechanical system. (b) The torsional mechanical system feedback control system. u r(t), Input torque Elastic shaft (a) t d (t), Disturbance torque Braking device u d (s) 0 r(t) (b) Mechanical system J s 2 b J s k J u(s) 3

5 A closed-loop control system for the system is shown in Figure CP7(b). Suppose the desired angle u d = 0, k = 5, b = 0.9, and J =. (a) Determine the open-loop response u(t) of the system for a unit step disturbance (set r(t) = 0). (b) With the controller gain 0 = 50, determine the closed-loop response, u(t) to a unit step disturbance. (c) Plot the open-loop versus the closed-loop response to the disturbance input. Discuss your results and make an argument for using closed-loop feedback control to improve the disturbance rejection properties of the system. CP8 A negative feedback control system is depicted in Figure CP8. Suppose that our design objective is to find a controller G c (s) of minimal complexity such that our closed-loop system can track a unit step input with a steady-state error of zero. (a) As a first try, consider a simple proportional controller G c (s) =, where is a fixed gain. Let = 2. Plot the unit step response and determine the steady-state error from the plot. (b) Now consider a more complex controller G c (s) = 0 + s, where 0 = 2 and = 20. This controller is known as a proportional, integral (PI) controller. Plot the unit step response, and determine the steady-state error from the plot. (c) Compare the results from parts (a) and (b), and discuss the trade-off between controller complexity and steady-state tracking error performance. CP9 Consider the closed-loop system in Figure CP9, whose transfer function is 0s s + 00 and H(s) = 5 s (a) Obtain the closed-loop transfer function T(s) = > and the unit step response; that is, let = >s and assume that N(s) = 0. G(s) H(s) FIGURE CP9 Closed-loop system with nonunity feedback and measurement noise. (b) Obtain the disturbance response when N(s) = 00 s N(s) is a sinusoidal input of frequency v = 0 rad>s. Assume that = 0. (c) In the steady-state, what is the frequency and peak magnitude of the disturbance response from part (b)? CP0 Consider the closed-loop system is depicted in Figure CP0. The controller gain can be modified to meet the design specifications. (a) Determine the closed-loop transfer function T(s) = >. (b) Plot the response of the closed-loop system for = 5, 0, and 50. (c) When the controller gain is = 0, determine the steady-state value of y(t) when the disturbance is a unit step, that is, when = >s and = 0. CP Consider the non-unity feedback system is depicted in Figure CP. (a) Determine the closed-loop transfer function T(s) = >. (b) For = 0, 2, and 5, plot the unit step responses. Determine the steady-state errors and the settling times from the plots. For parts (a) and (b), develop an m-file that computes the closed-loop transfer function and generates the plots for varying. FIGURE CP8 A simple singleloop feedback control system. G c (s) 0 s 0 32

6 FIGURE CP0 Closed-loop feedback system with external disturbances. s s5 s 2 s6.5 FIGURE CP Closed-loop system with a sensor in the feedback loop. Sensor s 20 s 2 4.5s64 ANSWERS TO SILLS CHEC True or False: () True; (2) True; (3) False; (4) False; (5) True Multiple Choice: (6) a; (7) b; (8) a; (9) b; (0) c; () a; (2) b; (3) b; (4) c; (5) c Word Match (in order, top to bottom): e, h, k, b, c, f, i, g, d, a, j TERMS AND CONCEPTS Closed-loop system A system with a measurement of the output signal and a comparison with the desired output to generate an error signal that is applied to the actuator. Complexity A measure of the structure, intricateness, or behavior of a system that characterizes the relationships and interactions between various components. Components The parts, subsystems, or subassemblies that comprise a total system. Disturbance signal An unwanted input signal that affects the system s output signal. Error signal The difference between the desired output and the actual output. Therefore, E(s) = -. Instability An attribute of a system that describes a tendency of the system to depart from the equilibrium condition when initially displaced. Loop gain The ratio of the feedback signal to the controller actuating signal. For a unity feedback system we have L(s) = G c (s)g(s). Loss of gain A reduction in the amplitude of the ratio of the output signal to the input signal through a system, usually measured in decibels. Open-loop system A system without feedback that directly generates the output in response to an input signal. Steady-state error The error when the time period is large and the transient response has decayed, leaving the continuous response. System sensitivity The ratio of the change in the system transfer function to the change of a process transfer function (or parameter) for a small incremental change. Tracking error See error signal. Transient response The response of a system as a function of time. 33

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