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MASON ECE 421 - Phase Lead Compensator Design Using Bode Plots

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1Phase Lead Compensator DesignUsing Bode PlotsProf. Guy BealeElectrical and Computer En gineering DepartmentGeorge Mason UniversityFairfax, VirginiaCONTENTSI INTRODUCTION 2II DESIGN PROCEDURE 2II-A Compensator Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2II-B Outline of the Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4II-C Compensator Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4II-D Making the Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5II-E Uncompensated Phase Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5II-F Determination of φmaxand α ......................................... 5II-G Compensated Gain Crossover Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8II-H Determination of zcand pc.......................................... 8II-I Evaluating the Design – A Potential Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8III DESIGN EXAMPLE 10III-A Plant and Specifications............................................ 10III-B Compensator Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11III-C The Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11III-D Uncompensated Phase Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11III-E Determination of φmaxand α ......................................... 12III-F Compensated Gain Crossover Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12III-G Compensator Zero and Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12III-H Eva luating the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13III-I Implementation of the Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14III-J Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15References 17LIST OF FIGURES1 Magnitudeandphaseplotsforatypicalleadcompensator............................... 32 BodeplotsforthesysteminExample2. ....................................... 63 Polar plot for phase lead compensator with Kc=1,α=0.16. ........................... 74 Bodeplotsforthelead-compensatedsysteminExample8. ............................. 105 Bode plots for the plant after the steady-state error specification has been satisfied. ................ 126 Bodeplotsforthecompensatedsystem......................................... 147 Closed-loop frequency response magnitudes for the example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Step responses for the closed-loop systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16These notes are lecture notes prepared by Prof. Guy Beale for presentation in ECE 421, Classical Systems and Control Theory, in the Electrical andComputer Engineering Department, George Mason University, Fairfax, VA. Additional notes can be found at: http://teal.gmu.edu/~gbeale/examples.html.2I. INTRODUCTIONAs with phase lag compensation, the purpose of phase lead compensator design in the frequency domain generally is tosatisfy specifications on steady-state accuracy and phase margin. There may also be a specification on gain crossove r frequencyor closed-loop bandwidth. A phase margin specification can represent a requirement on relative stability due to pure time delayin the system, or it can represent desired transient response characteristics that have been translated from the time domain intothe frequency domain.The overall philosophy in the design procedure presented here is for the compensator to adjust the system’s Bode phasecurve to establish the required phase margin at the existing gain-crossover frequency, ideally without disturbing the system’smagnitude curve at that frequency and without reducing the zero-frequency magnitude value. The unavoidable shift in the gaincrossover frequency is a function of the amount of phase shift that must be added to satisfy the phase margin requirement. Inorder for phase lead compensation to work in this context, the following two characteristics are needed:• the Bode magnitude curve (after the steady-state accuracy specification has been satisfied) must pass through 0 db insome acceptable frequency range;• the uncompensated phase shift at the gain crossov er frequency must be more negative than the value needed to satisfythe phase margin specification (otherwise, no compensation is needed).If the compensation is to be performed by a single-stage compensator, then the amount that the phase curve needs to bemoved up at the gain crossov er frequency in order to satisfy the phase marg in specification must be less than 90◦,andisgenerally restricted to a maximum value in the range 55◦–65◦. Multiple stages of compensation can be used, following thesame procedure as shown below, and are needed when the amount that the Bode phase curve must be moved up exceeds theavailable phase shift for a single stage of compensation. More is …


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MASON ECE 421 - Phase Lead Compensator Design Using Bode Plots

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