1Lag-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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3II-C Compensator Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5II-D Making the Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5II-E Uncompensated Phase Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5II-F Determination of φmaxand αd........................................ 6II-G Determination of zcdand pcd......................................... 7II-H Determination of αg.............................................. 7II-I Determination of zcgand pcg......................................... 8III DESIGN EXAMPLE 9III-A Plant and Specifications............................................ 9III-B Compensator Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10III-C The Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10III-D Uncompensated Phase M argin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11III-E Determination of φmaxand αd........................................ 11III-F Lead Compensator Zero and Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12III-G Determination of αg.............................................. 12III-H Determination of zcgand pcg......................................... 13III-I Implementation of the Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13III-J Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14References 15LIST OF FIGURES1 Bode magnitude and phase plots for a typical lag-lead compensator. . . . . . . . . . . . . . . . . . . . . . . . . 42 Bode plots after the steady-state error specification has been satisfied......................... 63 Bode plots after the lead portion of the lag-lead compensator has been designed. . . . . . . . . . . . . . . . . . 84 Bodeplotsforthelag-leadcompenatedsystem..................................... 95 Bode plots for the plant after the steady-state error specification has been satisfied. ................ 116 Bodeplotsafterthedesignoftheleadpartofthelag-leadcompensator. ...................... 127 Bode plots for the finallag-leadcompensatedsystem. ................................ 148 Closed-loop step responses for the various systems definedintheexample...................... 15These 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 and phase lead compensation, the purpose of lag-lead compensator design in the frequency domaingenerally is to satisfy specifications on steady-state accuracy and phase margin. Typically, there is also a specification (implicitlyor explicitly) on gain crossover frequency or closed-loop bandwidth. A phase margin specification can represent a requirementon relative stability due to pure time delay in the system, or it can represent desired transient response characteristics thathave been translated from the time domain into the frequency domain. A specification on bandwidth or crossover frequencycan represent a requirement on speed of response in the time domain or a frequency-domain requirement on which sinusoidalfrequencies will be passed by the system without significant attenuation.The overall philosophy in the design procedure presented here is for the lead part of the compensator to adjust the system’sBode phase curve to establish the required phase margin at a specified frequency, without reducing the zero-frequency magnitudevalue. The lag part of the compensator is used to drop the magnitude curve down to 0 db at that specified frequency. The lagcompensator must attenuate the magnitude of the series combination of the lead compensator Gc_lead(s) and the plant Gp(s)at the chosen frequency. Thus, in the procedure presented here, the lead compensator is designed first. In order for lag-leadcompensation to work in this context, the following two characteristics are needed:• the uncompensated phase shift at the chosen gain crossover frequency must be more negati ve than the value needed tosatisfy the phase margin specification (otherwise, no lead compensation is needed);• the Bode magnitude curve (after the lead compensator has been designed) must be above 0 db at the frequency chosenfor the gain crossover frequency (otherwise no lag compensation is needed, just additional gain).The basic lag-lead compensator has two stages, one each of lag and lead compensation. If the compensator is to have asingle-stage lead compensator, then the amount that the phase curve needs to be moved up at the gain crossover frequencyin order to satisfy the phase margin specification must be less than 90◦, and is generally restricted to a maximum value inthe range 55◦–65◦. Multiple stages of lead compensation can be used, following the same procedure as shown below, and areneeded when the amount that the Bode phase curve must be moved up exceeds the available phase shift for a single stage ofcompensation.If the compensator is to have a single-stage lag compensator, then it must be possible to drop the magnitude curve down to0 db at the gain crossover frequency without using excessively large component values. Multiple stages of compensation canbe used, following the same procedure as shown below. Multiple stages are needed when the amount that the Bode magnitudecurve must be moved down is too large for a single stage of compensation. More is said about this later.The …