R Ev ans in 19481954 is to accurately sk etc the ro ots of the olynomial as aries from This is to done under the follo wing assumptions and are olynomials with realv alued co e64259cien ts with leading co e64259 cien ts equal to The ro ots of and are ID: 27084 Download Pdf

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R Ev ans in 19481954 is to accurately sk etc the ro ots of the olynomial as aries from This is to done under the follo wing assumptions and are olynomials with realv alued co e64259cien ts with leading co e64259 cien ts equal to The ro ots of and are

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29 Ro ot Lo cus Giv en olynomials and the goal of the Ro ot Lo cus metho (dev elop ed W.R. Ev ans, in 1948-1954) is to accurately sk etc the ro ots of the olynomial as aries from This is to done under the follo wing assumptions: and are olynomials with real-v alued co eﬃcien ts with leading co eﬃ- cien ts equal to The ro ots of and are individually kno wn. Moreo er, since and ha real co eﬃcien ts, kno that the ro ots of and are real and/or come in complex-conjugate pairs. and ha no common ro ots := ord( ord =: Note, if and do ha common ro ot (or common ro

ots), at sa then there exist olynomials and suc that Hence, for ev ery ha Therefore, the ro ots of are simply (regardless of along with the ro ots of 0. In this manner, ou can pre-factor out an common ro ots, un til ou obtain an and that ha no common ro ots, and pro ceed using the metho describ ed elo w. 29.1 Motiv ation Supp ose that is linear system, with transfer function 221

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Consider the feedbac system, with prop ortional feedbac The closed-lo op haracteristic equation is 0. Hence, in order to ho ose an appropriate alue for it ould useful to kno ho the ro ots of are

functions of the parameter This is the canonical example. or this reason, when explaining the Ro ot Lo cus metho d, it is common to refer to the ro ots of as the “zeros, while the ro ots of are referred to as the “p oles. The canonical example seems to of ery limited usefulness. It only concerns the stabilit of system with prop ortional gain feedbac k, and only considers the ariations in the closed-lo op oles as functions of the prop ortional gain alue. But, in fact, man problems whic app ear to more complicated can ultimately cast in this manner. Recall for linear system, stability is prop

ert of the system, and that had essen tially equiv alen deﬁnitions either that all ounded inputs pro duce ounded outputs, or that under no input, all initial conditions deca exp onen tially to zero. Hence in blo diagram, the stabilit is determined the feedbac lo ops, not the exogenous inputs that en ter through summing junctions. act: An blo diagram connection of linear systems, summing junctions, and 222

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single real parameter can massaged in to the follo wing diagram where all of the external inputs are ignored (ie., set to zero). Here is simply the erall transfer

function that the parameter in teracts with in feedbac k. The stabilit of the original diagram can determined from studying the stabilit of is negativ feedbac with the parameter 29.1.1 Examples or or in PID con troller parameter in the plan state equations The stiﬀness, of the ﬂexible cord in the 2-mass exp erimen tal setup. 29.2 Rules for Constructing Ro ot Lo cus Giv en and the main theorem is as follo ws: Theorem: Supp ose is real, and Then is ro ot of the equation if and only if and either and 0, or 223

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0, and 0, the quotien is real, and Pro of: Supp ose 0.

Then, since and ha no common ro ots, it us that 0, and hence 0. If 0, then since 0. If 0, then since 0, it follo ws that either. Dividing out giv es that as desired. If and 0, then since is just some ﬁnite um er, ha 0. If and and then ultiplying out giv es as desired. So, in order to dra the ro ot lo cus, without directly calculating an ro ots of ou “simply do the follo wing: 1. or ev ery complex um er hec to see if either 0, or if is real. 2. If 0, then is ro ot of for 0. In this case, is said to “b on the ro ot lo cus of the pair ). 3. If 0, but is real, then is ro ot of for real-v

alued namely := In this case: If 0, then is said to on the Positive ot cus of the air If 0, then is on the Ne gative ot cus of the air 4. If is not real, then is not ro ot of for an real alue of In this case, is not on the ro ot lo cus of the pair ). Of course, ou can’t actually do this at ev ery complex um er “so man complex um ers, so little time. But, there are few basic rules that can used to accurately get go idea of what the ro ots, as functions of lo ok lik e. The ﬁrst rule follo ws from the reasoning ab e: Basic Rule 1: or an complex um er it is either not on the ro ot lo cus of

), or it is on the ro ot lo cus for one, and only one alue of Put another if and are diﬀeren real um ers, then it is imp ossible for complex um er to ro ot of oth and 0. This 224

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fact gr atly limits the complexit of the ro ot lo cus diagram. This simple fact is not explicitly stated in most textb oks, and as suc h, causes studen ts uc grief the ﬁrst few times sk etc hing ro ot lo cus plots. Next, recall few basic facts from arithmetic: 1. if and are in tegers, then is dd if and only if is dd. Ob viously then, is ev en if and only if is ev en. 2. If and are

complex, then AB 3. If is complex, 0, then 4. If is complex, 0, then is real if and only if is an in teger ultiple of Moreo er, i i is an ev en ultiple of i i is an dd ultiple of 5. The ro ots of are (2 +1) /n for 1. 6. The ro ots of are /n for 1. No w, let the ro ots of 0. Since assume that the leading co eﬃcien of is one, it follo ws tha can factored as Similarly let the ro ots of 0. Hence, can factored as This yields the 2nd basic rule: Basic Rule 2: complex um er is on the ro ot lo cus of if and only if for some in teger Dep ending on whether is ev en or dd determines whether the

oin is on the ositiv or Negativ Ro ot Lo cus. Note that )] )] Moreo er, since for an oin on the ro ot lo cus, the corresp onding alue of is it is easy to see that is on the ositiv Ro ot Lo cus if and only if (2 1) )] )] 225

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for some in teger Similarly is on the Negativ Ro ot Lo cus if and only if )] )] for some in teger 29.2.1 Num er of Ro ots Since for 0, the olynomial is alw ys ’th order, hence there ust alw ys ro ots. Sp cial Case: When and 1, the olynomial is no longer ’th order. What happ ens is that as (from either side), at least one of the ro ots go es to or example,

try the case and 1. 29.2.2 Ro ots when With 0, the olynomial ecomes simply 0. Hence, the ro ots are lo cated at the ro ots of 29.2.3 Ro ots on real axis ak an oin with not ro ot of or Let the um er of real zeros (from the list of that are to the righ of Also, let pR the um er of real oles (from the list of that are to the righ of 226

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rom the picture elo w, it follo ws that )] )] (# pR This is always an inte ger multiple of hence ev ery oin on the real axis is part of the ro ot lo cus of the pair ). can determine whether it is on the ositiv or Negativ ro ot lo cus as follo ws:

is on the ositiv Ro ot Lo cus i pR is dd i pR is dd is on the Negativ Ro ot Lo cus i pR is ev en i pR is ev en 29.2.4 Beha vior as Here, it is est to consider ﬁrst simple example: 2, 1. No w, quadratic form ula, ha that the ro ots of are (5 (5 4(1 Completing the square inside the square ro ot giv es that the ro ots are at (5 (1 20 or ery large alues of the ro ots are near (5 (1 (5 (1 227

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whic are at and Hence, as one ro ot go es to 2, while the other go es to This can generalized to higher order and First note that for large alues of but ﬁnite alues of Hence, as

± of the ro ots of approac the ro ots of 0. The remaining ro ots get large as gets large, in predictable fashion. First, ultiply out and and write them as and Note that in doing so, it is easy to see that =1 =1 =1 =1 (y ou can pro this induction...). Also, since the collections =1 and =1 app ear in complex-conjugate pairs, only the real parts matter in the sums, =1 Re =1 Re( (95) No w, supp ose 0, dividing giv es Therefore, (to see these, note that ﬁrst-order ylor’s series of (1 around is ). ak the ositiv Ro ot Lo cus (so 0). Using the expression for the ro ots of giv es that (2 +1) 228

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for 1. Solving for giv es (2 +1) Note that for giv en and real um er the quan tit cos sin and as go es from lo oks lik Hence, as of the ro ots of go to in the complex plane, along lines radiating from common oin t, called the cen troid, at angles (2 +1) for 1. similar argumen sho ws that as of the ro ots of go to in the complex plane, along lines radiating from the same cen troid, at angles (2 for 1. The last thing to notice is that the cen troid can also written in terms of the zeros of and Using equation 95, see that the cen troid is at cen troid =1 Re =1 Re Summarizing: As

± of the ro ot lo ci tend to ard the zeros of namely the The other ro ots go to along determinable asymptotes. There are asymptotes for There are asymptotes for The cen troid of all of the asymptotes is at =1 Re =1 Re 229

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The angles of the asymptotes are ositiv eRo otLo cus (2 1) and Negativ eRo otLo cus (2 29.2.5 Angle of Departure from ole all of the ro ots of are at the ro ots of namely As increases, the ro ots migrate from eac can hec the oin ts ery close to to see at what angle the ro ot departs from ak ²e where is small, and an to determine so that is on the ro ot lo

ci. Note that )( ²e )( ²e ²e ²e )( ²e ²e )( ²e In order for to on the ositiv Ro ot Lo ci, need this to an dd ultiple of giving (2 1) )] )] Solv for as )] )] (2 1) for an in teger Note that regardless of what in teger hose, just get additiv factor of whic as an angle of departure, pla ys no role. Hence, for simple to remem er the form ula, hose 1, giving )] )] 230

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This holds for the ositiv Ro ot Lo cus or the Negativ Ro ot Lo cus, use an ev en ultiple of whic yields similar form ula, without the last term, )] )] This is for the Negativ Ro ot Lo cus 29.2.6 Angle of Arriv al to

Zero As ± of the lo ci approac the zeros of namely The angle at whic they arriv to the zero can calculated in an iden tical manner as used to determine the angle of departure from ole. can hec the oin ts ery close to to see at what angle the ro ot lo ci arriv es at ak ²e where is small, and an to determine so that is on the ro ot lo ci. Mimicing the tec hnique for departure, get or the ositiv Ro ot Lo cus, the angle of arriv al at zero (in this case, is )] )] or the Negativ Ro ot Lo cus, the angle of arriv al at the zero is )] )] 29.2.7 Imaginary Axis Crossings If plot Bo de plot of ersus can

easily determine alues of for whic /D is real. The alues of suc that /D is negativ corresp ond to oin ts on the ositiv ro ot lo cus, ac hiev ed with Note: This is really the brute-force metho dology describ ed in section 29.2, but applied only to purely imaginary alues of On problems with small (sa 5), direct approac is sometimes successful. Separate the equation 231

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in to real and imaginary parts, giving real equations in real unkno wns and ). Since the co eﬃcien ts of and are real, the real-part equation will in olv ev en ers of while the imaginary-part will in olv

dd er of Exploit the fact that not all ers of app ear in eac equation, and use the quadratic form ula to get expressions of in terms of In some cases, these can solv ed analytically hand. 29.2.8 Symmetry of Ro ot Lo ci Since is olynomial with real co eﬃcien ts, the ro ots are real and/or come in complex-conjugate pairs. This implies that the ro ot lo cus plot is symmet- rical ab out the real axis. 29.2.9 Break oin ts/Multiple Ro ots As aries, it is ossible that real-v alued ro ot turns in to complex-v alued ro ot. or example, consider the ro ots of or 1, the ro ots are real, but for the

ro ots are complex. The olynomial has real co eﬃcien ts, so the ro ots, when complex, come in complex-conjugate pairs. Since complex-conjugate ro ots ust ha equal real parts, and since the ro ots hange con tin uously with the parameter it ust that there are iden tical ro ots at 1, the transition oin t. Indeed, at 1, the olynomial has real ro ots, oth at 2. Also, from Basic Rule kno that an ﬁxed alue can on the ro ot lo cus for atmost one alue of Hence, if real ro ots approac eac other as aries, they ust split in to complex-conjugate pairs at the alue of whic mak es the ro ots

equal. This ccurance is called br akaway oint More generally since the olynomial has real co eﬃcien ts, the ro ots are real and/or come in complex-conjugate pairs. This means that if real-ro ot ecomes complex, it’s complex conjugate ust also app ear at the same real alue (but with an opp ositely signed imaginary part). Hence at the alue of where the real ro ot ecomes complex, there ust in fact iden tical real ro ots. olynomial with ro ot at actually has more than one ro ot at if and only if and dp ds Hence, for to on the ro ot lo cus of ), it ust that (96) 232

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is real.

Denote := or to ultiple ro ot, also need that ds )] Carrying out the diﬀeren tiation, and using what is, ha that at ultiple ro ot, in addition to equation (96), it ust that dN ds dD ds (97) So, to ﬁnd ultiple-ro ots (and ro ot crossings, often called “break oin ts”) simply need to determine if an of the ro ots of dN ds dD ds also satisfy (ie., are actually on the ro ot lo cus). 29.3 Mo dern Ro ot Lo cus Metho ds Most go computer-aided con trol system design pac ages ha ro ot lo cus com- mand (in the MatLab con trol to olb x, the command is rlocus ). These ork purely brute force,

direct calculation of the ro ots of the as aries. The computer uses sp eed, and ell-dev elop ed, olynomial ro ot ﬁnding algorithms to quic kly mak accurate sk etc hes of the dep endencies of the ro ots on 233

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