1Compensator Design to Improve TransientPerformance Using Root LocusProf. 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 System Type N ................................................ 4II-D Selecting a Dominant Closed-Loop Pole Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4II-E Determining the Compensator’s Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6II-E.1 Plant Phase Shift at s1...................................... 6II-E.2 Compensator Phase Shift at s1.................................. 7II-E.3 PlacingtheCompensatorZero .................................. 8II-E.4 PlacingtheCompensatorPole .................................. 9II-E.5 DeterminingtheCompensatorGain ............................... 10II-E.6 CompensatorPhaseShiftRevisited................................ 11II-E.7 SimultaneousPlacementofCompensatorPoleandZero .................... 13II-E.8 Multi-StageCompensation .................................... 13III Design Example 16III-A Phase Lead Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16III-A.1 Given System and Specifications................................. 16III-A.2 Selection of the Dominant Closed-Loop Pole . . . . . . . . . . . . . . . . . . . . . . . . . . 16III-A.3 Designing the Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16III-B Phase Lag Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18III-B.1 Given System and Specifications................................. 18III-B.2 Selection of the Dominant Closed-Loop Pole . . . . . . . . . . . . . . . . . . . . . . . . . . 19III-B.3 Designing the Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19References 22LIST OF FIGURES1 Allowable region for s1in order to satisfy specifications on overshoot, settling time, and frequency of oscillation. 62 Calculating the phase shift of G(s) at the chosen point s = s1. ........................... 73 One possible solution for locating the compensator zero and pole. . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Comparison of closed-loop step responses for 5 compensator designs with s1= −4+j5.4575........... 115 Rootlocusplotforfourcompensatordesigns...................................... 126 Comparisonoflagandleadcompensationforaparticularsystem........................... 147 Selecting the compensator zero and pole to maximize α = zc/pc........................... 158 Step response of the uncompensated system for the phase lead design example. . . . . . . . . . . . . . . . . . . 179 Root locus of the uncompensated system and the desired closed-loop pole s1= −0.25 + j0.488.......... 1810 Lead compensated root locus and step response for the design example. . . . . . . . . . . . . . . . . . . . . . . 1911 Comparisonofrootlocusplotswithtwolagcompensatordesigns. ......................... 2112 Comparison of closed-loop step responses for two lag compensators. . . . . . . . . . . . . . . . . . . . . . . . . 22These 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 Univ e rsity, Fairfax, VA. Additional notes can be found at: http://teal.gmu.edu/~ gbeale/examples.html.2I. INTRODUCTIONIn my approach to root locus design, the purpose of compensator design using root locus methods generally is to establish aspecified point in the s-plane, s = s1, as a closed-loop pole. The assumption is that time-domain transient specifications, suchas settling time and overshoot, will be satisfied if s1is a dominant closed-loop pole. In the simplest case, s1is already onthe root locus of the uncompensated system. The compensator is then just a gain Kcthat is chosen to satisfy the magnitudecriterion at the point s1.More often, the point s1is not on the uncompensated root locus, so the compensator must add enough phase shift at thepoint s1to satisfy the phase angle criterion so that the compensated root locus does pass through s1. This is done by choiceof the compensator’s poles and zeros. The compensator gain is then chosen to satisfy the magnitude criterion at the point s1.In many cases, the speed of response and/or the damping of the uncompensated system must be increased in order to satisfythe specifications. This requires moving the dominant branches of the root locus to the left. A phase lead compensator (providingpositive phase shift at s1) is used for this purpose. If the branches need to be moved to the right, a phase lag compensator(providing negative phase shift at s1) is used. The design techniques are identical for the two types of compensator; the roles ofthe compensator’s poles and zeros are just reversed. Because of this similarity in the design methods, phase lead compensationwill be discussed in detail here. An example of each type of compensation will be given after the general design procedure isdescribed.Conceptually, the design procedure presented here is graphical in nature. The process of locating the compensator’s polesand zeros to satisfy the phase requirements can be visualized from the trigonometric relationships that must be satisfied at thedesired dominant closed-loop pole. The computations can be easily done by calculator. If data arrays representing the numeratorand denominator polynomials of the open-loop system are available, then the procedure can be done using a software packagesuch as MATLAB, and in many cases it can be automated. The examples and plots presented here are all done in MATLAB ,and the various measurements that are presented in the examples are obtained from the arrays storing the appropriate variables.The primary references for the procedures described in these notes are [1]–[3]. Other references that contain similar materialare [4]–[11].II. DESIGN PROCEDUREA. Compensator StructureThe basic phase lead or phase lag compensator consists of a gain, one real pole, and one real zero. Based on the usualelectronic implementation of the compensator [3], the