EE Control Systems LECTURE 14

Transcription

1 Updated: Tueday, March 3, 999 EE Control Sytem LECTURE 4 Copyright FL Lewi 999 All right reerved ROOT LOCUS DESIGN TECHNIQUE Suppoe the cloed-loop tranfer function depend on a deign parameter k We would like to know how the cloed-loop pole vary with k The next example how that the cloed-loop pole vary continuouly with k and trace out a path or locu in the -plane Thu, by varying k one actually move the pole around in the -plane Example - Variation of Cloed-Loop Pole v a Deign Gain r(t) k (+a) y(t) Let u find how the pole vary a the feedback gain k i increaed from zero The cloed-loop tranfer function i k Y ( ( + a) k T ( R( k ( + a) + k + a ( + ) Compare thi to the tandard form ( β ) ( +α ) + to ee that, if k>, the pole are at real part of -a and imaginary part of β k Thu, a k increae from zero, the pole imply move out along the line a Thi i illutrated in the figure

2 jω j k -a σ Root Locu (RL) Deign Procedure The root locu technique wa developed by Evan working at North American Aviation during World War The importance of thi method lie in the fact that it allow one to ee how the pole vary with k without ever finding the actual root Note that in the previou example, we actually computed the root a a function of k Thi i not neceary with root locu deign The root locu impart a great deal of deign inight about the tructure of the control ytem A baic feedback control ytem i the TRACKING CONTROLLER given in the figure The plant i H( and the compenator K(; the feedback gain i k The function of the tracker i to make the output y(t) follow the command or reference input r(t) by making the tracking error e(t)r(t)-y(t) mall The diturbance d(t) play no part here and i aumed equal to zero r(t) e(t) kk( d(t) H( y(t) The cloed-loop tranfer function i Y ( kk( H ( T ( R( + kk( H ( In root locu deign one focue on the denominator ( ) + kk( H ( + kg( where G( i the open-loop gain Note that we ue the ame ymbol for the denominator of T( a for the tate-variable characteritic polynomial ( I A However, +kg( i actually a polynomial fraction, whoe numerator i the ytem characteritic polynomial

3 Many deign technique rely on trying to determine cloed-loop propertie from open-loop propertie In root locu deign, one ue the all-important open-loop gain G( to etimate the location of the cloed-loop pole, which are the root of the numerator of ( ) + kg( The key point of RL deign i that it i eaier to plot the location of the cloed-loop pole veru the feedback gain parameter k than it i to find the actual cloed-loop pole themelve Thi wa extremely important in day before digital computer when finding root of high-order polynomial wa difficult, and it alo give great inight into the propertie of the cloed-loop ytem In thi dicuion, one mut ditinguih between the cloed-loop pole, which are the root of the numerator of ( +kg(, which depend on the feedback gain k, and the (open-loop) pole and zero of G(, which do not depend on k One alo ue the RELATIVE DEGREE of G( The relative degree of a polynomial fraction i the degree of the denominator minu the degree of the numerator Since the total number of pole and zero i the ame, the number of zero at infinity i equal to the relative degree We conider the cae when the polynomial fraction i PROPER, that i when the relative degree i greater than or equal to zero The fraction i aid to be STRICTLY PROPER if the relative degree i greater than or equal to Define G( p(/q( and denote the relative degree of G( a m deg(q)- deg(p) Then G( ha m zero at infinity The RL i a plot in the -plane of the cloed-loop pole a k varie from zero to infinity To find the cloed-loop pole we are intereted in point where + kg (, which can be written in everal equivalent form, including G( k q( + kp( All point in the -plane that atify thi equation are cloed-loop pole for that value of k Thu one note that on the root locu, G( i real and negative In the textbook it i hown that everal baic rule for plotting the root locu follow directly from thee equation Thee rule aume a poitive gain k The rule for negative k can likewie be derived Baic Rule for Plotting the Root Locu A k goe from zero to infinity, the cloed-loop pole move from the pole of G( to the zero of G( The loci are ymmetric with repect to the real axi 3 The real-axi to the left of an odd number of pole + zero i on the RL (Recall the dicuion on the phae of a rational function on the real axi!) 3

4 4 A k become large, m loci approach infinity aymptotically The aymptote are at angle of o o 8 + i36 ± ; i,,, m and they meet on the real axi at the centroid given by Σ( finite pole of G) Σ( finite zero of G) c m 5 On the RL, G( i real and negative o that angle(g() -8 o Thi i ueful for determining the 'angle of departure' of the RL from a pecific open-loop pole of G There are other rule, including thoe for determining breakaway point from the real axi However, MATLAB ha a very good RL plotting routine which may be ued to determine the RL in more detail than that given by thee imple rule It i ueful to note that the aymptote in Rule 4 take on the following pattern for the firt few value of the relative degree m m m3 6 o m m4 Example - Redo Example Uing RL In Example we had H ( ( + a) K( The cloed-loop tranfer function i 4

5 k Y ( ( + a) T ( R( k + ( + a) o that G( ( + a) kk( H ( + kg( The pole of G( are plotted in the figure Note that there are two zero at infinity jω () -a σ Now we apply the RL rule: Rule The cloed-loop pole move from the x' to the zero at infinity Rule 3 No part of the real axi i on the RL Rule 4 The relative degree i m, o the aymptote are at figure above o ± 9 a depicted in the The RL i therefore determined to be a hown below jω -a σ 5

6 Note that we did not find the cloed-loop root to draw the RL, in contrat to the procedure in Example On the other hand, following the RL rule, we do not know that the imaginary part of the cloed-loop pole i imply equal to j k The RL how it real power in more complex example Example 3 Suppoe the plant ha tranfer function + 5 H ( and one ue the lag compenator + 8 K ( + Then the loop gain i ( + 5)( + 8) G ( K( H ( ( + ) Now we apply the RL rule: The RL move from the x' to o' and The real axi between - and -5 i on the RL The real axi between -8 i on the RL The relative degree i m, o there i one zero at infinity The aymptote i at - 8 o, o the centroid need not be found The RL ketch found from thee rule i given below 6

7 () To plot the RL uing MATLAB, one imply ue the command line: num [ 3 4] ; den [ ] ; rlocu (num,den) The reult i hown Note that the arrowhead were added uing Powerpoint Note that the RL actually croe the jω axi before coming to the left Thi could be determined without MATLAB by uing the angle of departure rule, which we will how in a later example 7

8 The reaon for thi behavior i a looe rule baed on experience that 'the root locu i repelled from pole' Thu, the RL trie to 'get away from' the pole at - Once it doe o, it 'ee' the zero further to the left and head that way The value of k for which the RL croe the imaginary axi i eaily found uing the Routh tet The characteritic polynomial i found from +kg( a 3 ( + ) + k( + 5)( + 8) + ( k + ) + 3k + 4k The Routh Array i given by 3 3k k + 7k 4k 3k 4k where one ha multiplied the third row by k+ To enure that all coefficient in the firt column are poitive, one require that k > 7 Thu, the RL croe the imaginary axi 3 at k 7/3 The root at thi point are determined by ubtituting k 7/3 into the characteritic polynomial and uing the MATLAB root-finder 'root' The MATLAb command line are: k7/3 ; del[ k+ 3*k 4*k] ; root(del) an + 596i - 596i