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# PerformanceEnhancementofDynamicSystems MEM355

Drexel

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This 42 page Class Notes was uploaded by Jada Daniel on Wednesday September 23, 2015. The Class Notes belongs to MEM355 at Drexel University taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/212414/mem355-drexel-university in Mechanical Engineering at Drexel University.

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Date Created: 09/23/15

MEM 355 Performance Enhancement of Dynamical Systems Root Locus Analysis Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline The root locus method was introduced by Evans in the 1950 s It remains a popular tool or 3139 l 39 mp e SISO control deszgn What is a root locus Poles amp Transient Response The Root Locus Method I Problem De nition I The Two Key Formulas I Root Locus Rules Examples I Flexible Spacecraft I Robotic Arm I General Aviation Aircraft I Helicopter Pitch Control What is a Root Locus On the right is a negative feedback loop We wish to examine the closed loop poles as the gain K varies As K increases from zero the 4 H3 poles move from the open loop G H K12s4 values amp trace 4 loci At any particular value of K there are 4 closed loop poles In this example there is a critical value of K at which the system becomes unstable quot at Kmamt stability is lost M Example 1GS71K liszK SK s s S sZKsK02siKi K274K 7 H 8 7 H 8 M V Transient Response Consider a system described by a transfer function GsK ds There are three ways to assess system transient behavior 1 time domain output time trajectories 2 pole or eigenvalue location 3 frequency response Bode or Nyquist plots Here we consider pole location The poles are the roots of dss22p1m1sm12 s22ppmpscos21 s q0 39 Ideal Pole Locations 0 Im M degree of stability decay rate 10 8 1 ideal region for damp39quot9 ram 9 51 P closed loop poles L Re 7 Our goal is to design a compensator so that the closed loop poles lie in the shaded region We get to choose the form of the compensator and select its parameters Problem De nition HS The closed loop input response transfer function is G Gy S 2 1 GH The error response transfer function is recall 6 y y 1 GH G 1 GH The poles of the closed loop system are the roots of 1 GH s 0 G2 s1 Gy s Problem De nition Cont d Suppose ns S Zls Zm SerbmflsmilHIb0 ds 5 P1quot395 Pn Saanilsmlma0 n S d s are completely known but K is a parameter that we can adjust GsHsK Root Locus Problem Generate a sketch in the complex plane of the closed loop poles with varying gain K Solution Strategies 0 We will do this two ways I The easy way Have MATLAB solve for the roots for each of a specified list of values for K and plot them I The hard old way Generate a sketch by hand 0 Why do it the hard way at all I We need to know how to interpret the plot I We obtain insight concerning the choice of compensator I We learn how to set the compensator parameters other than the gain K Root Locus Method 1 1GsHs 0 lt2 K 1 ej2k1 k 012 ds This means K 1 and A K j 2k17r 615 615 for K 2 0 K 1 and 2k17r S Amg s Root Locus Method 2 Our goal is to find values of s that satisfy both of these equations Note that for any given s the magnitude equation is satisfied for some value ofK ie d s n s Note that the angle equation does not depend on K at all Strategy First find values of s that satisfy the angle equation Second calibrate the plot using the magnitude equation Root Locus 3 Using the Angle Formula 63 64 6162 7055 2 2kl7z for any integer k s 2 j3 is not a point on the root locus The points 2 ixE2 is 63 64 6162180 Basic Rules 1 1 Number of branches The number of branches of the root locus equals the number of open loop poles the order of the polynomial d sKns is the order of ds 2 Symmetry The root locus is symmetric about the real aX1s Poles occur in complex conjugate pairs U3 Starting amp ending points The root locus begins at the open loop poles and ends at the nite and in nite open loop zeros dsKns0 gt ds0 asK gt 0 dsns 0 gt ns 0 asK gt oo ifs is bounded s3 Basic Rules 2 4 Realaxis segments ForK gt 0 real axis segments to the left of an odd number of finite real axis poles and0r zeros are part of the root locus Im 16 Test point SI Basic Rules 3 5 Behavior at infinity The root locus approaches infinity along asymptotes with angles 2k 1 2 k 0ili2i3 nite poles nite zeros FuIthermore these asymptotes intersect the real aXis at a common point given by Z nite poles Z nite zeros o nite poles nite zeros Basic Rules 4 Angle part is easy ns m n 4mZilzS Zi ZilzS pi Takeszpe Forp gtoozs L gtzs6 Then 214s Zi Zlzs pi gtm6 n6 ns Sold 2k17rgtm n62k17z Basic Rules 4 6 Real aXis breakaway and breakin points The root locus breaks away from the real aXis Where the gain is a local maximum on the real aXis and breaks into the real aXis Where it is a local minimum To locate candidate break points solve d 1 al 0 7 jmaXis crossings Use Routh test to determine values ofK for Which loci cross imaginary aXis Using MATLAB The basic MATLAB functions are rlocus rlocussys calculates and plow the root locus of the openloop SISO model sys rlocfind KPOLES RLOCFTNDSY S is used for interactive gain selection from the root locus plot of the SISO system SYS generated by RLOCUS RLOCFIND uts up a crosshair cursor in the graphics window which is used to select a pole location on an existing root locus The root locus gain associated with is point is returned in K an all the system poles for this gain are returned in POLES s1s0tool When invoked without input argumenm sisotool opens a SISO Design GUI for interactive compensator design T 39s 39 5 you to desrgn a singleinputsingle output SISO compensator usrng root locus and Bode diagram techniques Flexible Spacecraft Flexible Spacecraft s4sz2s2 sZ szs1 s4s1ij szs05rj0866 GsHsK With One Flex Mode Rigid IIIgt GsHsKs4 2 S Spacecraft Rigid 1723K26 KAZ dc s s 1Ks3 161ltsZ 10Ks8K 1 16K 8K 1K 10K 0 Z 173K6K 8K 0 1K 2K1723K26KZ 173K6KZ 8K 1K20173K6KZ 201723K26KZ 20 One Fex Mode Example Robotic Arm Motor Compensator Ampli er Speed Controller GS20K Z s0l ZOK s0l Z s s4szlls3025 sZS4S55 Robotic Arm 2 Asymptotes 45 135 225 315 Centroid 3725 4b Robotic Arm 3 Plot K1 Gs Imaglnaly Ms Rout anus Real Am nagnarv A 5 m Rea m 20Ks01 5 G51 SS 1554 742533 12132 ZOKS 2K SS 1 74 20K 5 15 121 2K s3 659 1986K s2 121 452K 2K 1 2271Ilt 8976K2 S W s0 2K 121 452K gt0 2271K 8976K2 gt0K gt0 K lt 26769 452 Example Piper Dakota Pitch Control 9 error elevator angle 52 gt pitch angle 6 s25s07 s2 5s40s2 003s006 s3 s20 Plant GP s 160 Lead compensator Gc s K Piper Dakota X A kIm What do we expect Re Re Piper Dakota lmaghary Axis Piper Dakota RED Locus quotquotLuw 11125 Una was 0025 Imaginary Axis A 45 72 r1 5 71 D 5 Real Axis Piper Dakota shep sspnnse 1n swwg 1 2 2 A 5 DB AURA M v V as V EM Open loop response to elevator D 2 5 Zquot m w Closed 100p response with K 15 1 2 7 4 5 6 Time sec Example Helicopter Pitch Control 0 K1 25s003 sl 123 s04s2 036s0l6 sl 2s9 disturbance pilot I Pick the inner loop pitch control gain K2 so that the dominant inner loop poles have a damping ratio of 0707 I Select an outer loop gain sensitivity to place the poles I Determine ultimate error in response to unit step disturbance LIN EIISI39IT Inner Stabilization Loop Gm 255EE3 521 5 141 5 9 527D 5EI16 The inner loop root locus is shown quot on the right Choose a gain K2 15 The inner loop resolves to 5 G 25 SltSgt115PGG 39 c p Ms U 39 25s003s9 i V39 725 s381ij353sl37s00458 25 s003 59 521251 549O4 s337876 5238833 sl70l Outer L009 4 On the basis of the root locus on the right choose a gain of 1 3 The closed loop poles are 1191 I t O 392ij350 0633ij0646 4 I 7 0 03 14 712 710 78 6 4 2 O Disturbance Response Error Gd 2 sztust 25soo3s9 s1191s392ij350s0633ij0646s00314 lim sGdg l 07987 sgt0 s Summary 0 How poles characterize transient response 0 Observing the in uence of gain on closed loop poles using root locus plots Sketching the root locus I The magnitude and gain formulas I Basic rules of root locus sketching I Using MATLAB Examples MEM 355 Performance Enhancement of Dynamical Systems Stale Space Design Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outl 1ne State space techniques emerged around 1960 They are direct and exploit the e icient computations of linear algebra State space models amp similarity transformations Controllability amp Observability Special forms of state equations State feedback amp pole placement Observers Design via separation principle Design examples Similari Transformations 5c Ax bu 5c Ax Bu n m xeR ueR yeRF yCxDu ycxdu Now consider the transformation to new states Z defined by xTzcgt ZT71x 2T 1ATZT IBu T2ATZBu yCTZDu yCTZDu sothat 2AZBquot Aquot T 1ATB T IBC CTD D yCZDu IMagonalFonn 1 z keeigenvalues 71 eigensystern of A hZ hn independent eigenvectors T m hi ha M mFAM m 21 0 0 7 0 z E 7 39 39 0 0 0

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