Root Locu Technque ELEC 32 Cloed-Loop Control The control nput u t ynthezed baed on the a pror knowledge of the ytem plant, the reference nput r t, and the error gnal, e t The control ytem meaure the output, and compare t to the dered output reference nput through a feedback path to generate an error gnal
What a Root Locu? A graphcal decrpton of the movement of each of the cloed-loop pole n the complex -plane a open-loop parameter gan, pole value, etc. vare. Obey magntude and phae crtera Pont on the root locu are the only poble cloed-loop pole locaton Ued n predctng the ytem overall performance tranent repone dynamc, teady-tate error, and ytem tablty Ad n controller degn For a gven value of K, a pont n the complex plane can be a cloed-loop pole.e. on the root locu of the above ytem f atfe the followng crtera: Magntude: KG H = Phae: KG H = ±80 2l +, l = 0,,2, 2
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Example For the followng ytem, determne the tandard form of the charactertc equaton whoe cloedloop pole can be ued to contruct the root locu of the ytem. 5
Example For the followng ytem, determne the tandard form of the charactertc equaton whoe cloedloop pole can be ued to contruct the root locu of the ytem. Rule The root locu ha a many branche a there are open-loop pole Each branch repreent the perambulaton of a cloed-loop pole n the -plane a K vared from 0 to Each branch begn at an open-loop pole K = 0 and end at a fnte open-loop zero or at a zero at nfnty K Aumng real ytem, all coeffcent are real. Conequently, the root locu plot wll be ymmetrc about the real ax 6
Rule 2: Real-ax locu Startng at + and movng along the real-ax toward the left, the root locu le on the real ax to the left of an odd number of real-ax open-loop pole or zeroe n any combnaton Multplcte taken nto account Only the cumulatve number of pole or zeroe odd or even mportant Pole or zeroe off the real ax not ncluded Example: Determne the root locu for each of the ytem defned by the followng tranfer functon: G H = + 2 + + 3 2 G 2 H 2 = + 2 + 3 + + 4 7
Rule 3: Break-away and Break-n pont Both pont le on the root locu n between two open-loop pole Break-away: a pont where the root locu leave the real ax Break-n: a pont where the root locu enter the real ax At thee pont, branche form an angle of 80/n wth the real ax, where n = # of CL pole arrvng or departng Determne the root locu for each of the ytem defned by the followng open-loop tranfer functon: G 3 H 3 = + 3 + 4 + + 2 G 4 H 4 = + 22 + 5 + 0.25 + + 3 + 4 8
Rule 4: Aymptote & Centrod Aymptote gve general drecton n whch root locu branche wll radate toward the zeroe at nfnty A centrod a common pont on the real ax where the aymptote come together n-m branche of the root locu go to nfnty a K aumng n > m n = # of open-loop pole m = # of open-loop zeroe Rule 4 The root locu goe to nfnty by radatng from a centrod located at at angle of 9
Determne the root locu for the ytem defned by 6 G 6 H 6 = + + 2 Example Determne the root locu for the ytem defned by 7 G 7 H 7 = + + 2 + 3 0
Example Determne the root locu for the ytem defned by 8 G 8 H 8 = + + 2 + 3 + 4 Example Determne the root locu for the ytem defned by G H = + 2 + + 3 + 4
2 Evaluaton of Complex Functon Ung Vector Notaton...... 2 2 n m n m p p p z z z p z F Evaluatng F at = yeld F = M where p m n m p z p z F M to length from to length from pole angle zero angle n m p z F Example Gven, evaluate F at the pont = 3+j4. 2 F
Graphcal method for determnng the angle and magntude of K + z G H = + p + p 2 + p 3 + p 4 G H KB = A A 2 A 3 A 4 G H = φ θ θ 2 θ 3 θ 4 Rule 5 The root locu make an angle D angle of departure wth repect to the potve real ax a t leave the complex conjugate pole wth potve magnary part, p. Th equaton come from atfyng the phae crteron at a pont near p. 3
Rule 5 Contnued The root locu make an angle A angle of arrval wth repect to the potve real ax a t arrve at the complex conjugate zero wth potve magnary part, z. Th equaton come from atfyng the phae crteron at a pont near z. 4
Determne the root locu for the ytem defned by 5 G 5 H 5 = 2 + 2 + 2 2 + 5 + 0 5
Example Determne the root locu for the ytem defned by 9 G 9 H 9 = 2 + 2 + 2 Example Determne the root locu for the ytem defned by 0 G 0 H 0 = 2 + 2 + 2 6
Example Determne the root locu for the ytem defned by 2 G 2 H 2 = + 2 + + 3 2 + 2 + 2 G H = + 3 + 2 2 + 4 + 5 7
G H = + 3 + 2 2 + 4 + 3 Rule 6 At any pont on the locu, the varable K can be calculated a the product of dtance from the pont to the open-loop pole dvded by the product of dtance from the pont to the open-loop zeroe. Note:. If there are no zeroe, the denomnator. 2. The calculaton aume that GH ha a numerator wth a leadng coeffcent of. 8
Rule 7 Crong of the j ax If branche of the root locu cro the magnary ax, the locaton of the crong and the value of the correpondng gan K can be found by Form the charactertc equaton from the cloed-loop tranfer functon 2 Replace wth j and equate the real and magnary part and olve Example Determne the root locu gan at the three pont ndcated on the plot below. 9
Example Plot a root locu f the four pole are at -, -4, -5 and -8. The ngle zero at = -3. Fnd the gan whch produce ntablty. Alo, calculate the gan for a dampng rato of 0.5. num = poly[-3]; den = poly[- -4-5 -8]; rlocunum,den grd0.5,[]; ax[-0 2-0 0] [k pole] = rlocfndnum,den 20
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