Chapter 7 Root Locu Analyi jw + KGH ( ) GH ( ) - K 0 z O 4 p 2 p 3 p
Root Locu Analyi The root of the cloed-loop characteritic equation define the ytem characteritic repone. Their location in the complex -plane lead to prediction of the characteritic of the time domain repone in term of: damping ratio, z natural frequency, w n econd-order mode damping contant, firt-order mode Conider how thee root change a the loop gain i varied from 0 to. Frequency Repone Deign
Root Locu Example R() + E() K ( + 2) C() The cloed-loop tranfer function i The characteritic equation i 2 + + K C( ) K 2 0 R( ) ( + 2) + K Conider the characteritic root a K 0. Frequency Repone Deign
Root Locu Example -± - K For K 0 the cloed-loop pole are at the open-loop pole. For 0 K the cloed-loop pole are on the real axi. For K the cloed-loop pole are complex, with a real value of and an imaginary value increaing with gain K. loci of cloed-loop root K K 2 jw K 0 Frequency Repone Deign
Amplitude Root Locu Example: Step Repone.6 K 50.0 Step Repone K jw.4 K 5.0.2 0.8 0.6 0.4 0.2 K 2.0 K.0 K 0.5 K 2 K 0 0 0 2 4 6 8 0 Time (ec.) Frequency Repone Deign
Root Locu Example: Some Obervation Thi i a econd-order ytem and there two loci. The root loci tart at the open-loop pole. The root loci tend toward the open-loop zero at infinity a K. (Note: the number of zero i equal to the number of pole, when the zero at infinity are included.) The relationhip between the characteritic repone and the increaing gain i een through the root loci. Frequency Repone Deign
The General Root Locu Method Conider the general ytem R() C() G() + where C( ) R( ) H() G( ) + GH ( ) The characteritic equation i + GH( ) 0 or GH( ) - or GH( ) GH( ) (2k k + ) p 0, ±, ± 2 L Frequency Repone Deign
The General Root Locu Method All value of which atify ; ; GH ( ) GH( ) (2k + ) p are root of the cloed-loop characteritic equation. Conider the following general form GH ( ) Note : pi K( + z)( + z2) L ( + z ( + p )( + p ) L ( + p may be zero. 2 m n ) ) k 0, ±, ± 2 L Frequency Repone Deign
Frequency Repone Deign Then The General Root Locu Method i ) ( + + p z K GH n i m i i 2 L,, 0 ± ± k ) (2 ) ( ) ( ) ( + + - + k p z GH n i i m i i p
Root Locu Method: Geometric Interpretation Conider the example K( + z) GH ( ) ( + p )( + p Then the value of which atify K + + p 2 + z + 3 2 p 3 ) z ) - ( + ( + p2) + ( + p )) (2k + )p ( 3 are on the loci and are root of the characteritic equation. q p3 O -z -p 3 A -p 2 q z B D q p2 C jw q p -p Frequency Repone Deign
Root Locu Method: Geometric Interpretation In term of the vector, the condition for to be on the root loci are K A A or BCD BCD K and q p3 A O -z -p 3 q z B D C jw q p -p q k z - ( q + q + q 3) p 0, ±, ± p2 2, L p (2k + ) p -p 2 q p2 Frequency Repone Deign
Root Locu Method When plotting the loci of the root a K 0, the magnitude condition i alway atified. Therefore, a value of that atifie the angle condition, i a point of the root loci. The magnitude condition may then be ued to determine the gain K correponding to that value. How can we eaily determine if the angle condition i atified? Frequency Repone Deign
Root Locu Contruction Rule. The loci tart (K 0) at the pole of the open-loop ytem. There are n loci. 2. The loci terminate (K ) at the zeroe of the open-loop ytem (include zeroe at infinity). For our example ytem + z GH( ) + p + p Therefore, a K 0, GH(), the pole of the loop tranfer function. A K, GH() 0, the zeroe of the loop tranfer function. 2 3 K Frequency Repone Deign
Root Locu Contruction Rule 3. The root loci are ymmetrical about the real axi. 4. A K the loci approach aymptote. There are q n m aymptote and they interect the real axi at angle defined by + p (2k ) q, k 0, ±, ± 2, L The root with imaginary part alway occur in conjugate complex pair. When the loci approach infinity, the angle from all the pole and zeroe are equal. The angle condition then i mq nq (2k + )p Frequency Repone Deign
Root Locu Contruction Rule 5. The aymptote interection point on the real i defined by a poleof GH( ) - 6. Real axi ection of the root loci exit only where there i an odd number of pole and zeroe to the right. q zeroe of GH( ) The angle from pole and zeroe to the left of are zero. The angle from pole and zeroe to the right are p. An odd number are required to atify the angle condition. Frequency Repone Deign
Root Locu Contruction Rule Example Conider our example with z 4, p 2 2j jw aymptote GH ( ) ( Aymptote: a angle K( + 4) + + 2 j)( + - 2 j) [- - + - - ]-[ - ] 0 (2k + ) p p ± 3-2 ( 2 j) ( 3-2 j) ( 4) real axi locu O 4 + 2 2j + 2j Frequency Repone Deign
Root Locu Contruction Rule 7. The angle of departure, q d from pole and arrival, q a to zeroe may be found by applying the angle condition to a point very near the pole or zero. The angle of arrival at the zero, -z i obtained from q + az ( -z + z ) i 2 i - n ( -z + p ) (2k + i )p i m Frequency Repone Deign
Root Locu Contruction Rule Example Departure angle from p 2. q z tan - (2/3) 33.7 q p tan - ( 2/) 6.6 q p3 90 Then -z O 4 33.7 (90 + 6.6 + q p2 ) 80 q p2 352.9 + 7. 33.7 2 q p2 -p 2 -p 3 90 jw 2j 6.6 -p 2j + Frequency Repone Deign
Root Locu Contruction Rule 8. The imaginary axi croing i obtained by applying the Routh- Hurwitz criterion and checking for the gain that reult in marginal tability. The imaginary component are found from the olution of the reulting auxiliary equation. Marginal tability refer to the point where the root of the cloed-loop ytem are on the tability boundary, i.e. the imaginary axi. Frequency Repone Deign
Root Locu Contruction Rule Example Imaginary axi croing: ( Characteritic equation + + 2 2 j)( (5 + + - K) 2 j) 3 2 + + + Routh table 3 5+K 0 2 2 4K 0 5 K 0 0 0 4K 0 4K + K( + 0 4) 0 For marginal tability, K 5 and the auxiliary equation i Therefore, the imaginary axi interection i ± 3.6 j 2 2 + 20 0 ± 0 j ± 3.6 Frequency Repone Deign j
Root Locu Contruction Rule Example Summary: There are three root loci. One on the real axi from -p to -z, and a pair of loci from -p 2 and -p 3 to zeroe at infinity along the aymptote. The departure angle from thee pole i 7. and an imaginary axi croing at 3.6j. -z O 4 2 7. -p 2 -p 3 jw 3.6 j 2j -p 2j + Frequency Repone Deign
Root Locu Contruction Rule Breakaway Point: Some example When two or more loci meet, they will breakaway from thi point at particular angle. The point i known a a breakaway point. It correpond to multiple root. x x x x x x o x x x o x 45 Frequency Repone Deign
Root Locu Contruction Rule 9. The angle of breakaway i 80 /k where k i the number of converging loci. The location of the breakaway point i found from dk d [ ] d GH( ) 0 or d 0 Note: K -2 [ ] d[ GH ( ) ] 0 dk GH ( ) d Alo, - [ ] GH ( ) d D( ) N( ) - N( ) D( ) - [ ] [ ] d GH( ) d N ( ) - D( ) d N( ) D( ) d N( ) D( ) 2 0 D( ) 0 Frequency Repone Deign
Root Locu Plot: Breakaway Point Example Conider the following loop tranfer function. ( ) K GH 2 ( + 3) Real axi loci exit for the full negative axi. Aymptote: angle (2k+)p p/3, p, 5p/3 a 3 (-3-3 - 0) - (0) 3-2 4 2 60 jw 2j + 2j Frequency Repone Deign aymptote
Root Locu Plot: Breakaway Point Example Determine the breakaway point from jw d d K ( + 3) 2 - K(3 + 2 + 9) 3 2 2 ( + 6 + 9) then 2 d d 3 + 0 K 6 2 + 9 4 2 2j 2j + 2 + 4 + 3 -, - 3 ( + )( + 3) 0 Frequency Repone Deign
Root Locu Contruction Rule 0. For a point on the root locu, calculate the gain, K from K + + p z + + p z 2 2 L L Alternately, K may be determined graphically from the root locu plot O A B D C jw K BCD A Frequency Repone Deign
Summary of Root Locu Plot Contruction Plot the pole and zero of the open-loop ytem. Find the ection of the loci on the real axi (odd number of pole an zeroe to the right). Determine the aymptote angle and intercept. + p (2k ) angle, q a pole- q q zeroe n - m, k 0, ±, ± 2, L Frequency Repone Deign
Summary of Root Locu Plot Contruction Determine departure angle. For a pole -p -p+ z ) +( -p + z ) + L-q -(-p + p ) - p L (2k )p ( + 2 2 Check for imaginary axi croing uing the Routh-Hurwitz criterion. Determine breakaway point. angle p / k location from Complete the plot., k d # of [ ] GH( ) d converging 0 loci Frequency Repone Deign
Root Locu Plot Example 3 Loop Tranfer function: ( K GH ) ( + 4)( 2 + 4 + Root: 0, 4, 2 4j Real axi egment: between 0 and 4. Aymptote: angle a 20) (2k + ) p p 3p 5p 7p,,, 4-0 4 4 4 4 (-4-2 -2-0) -2 4 aymptote 4 2 jw 45 4j 2j 2j 4j Frequency Repone Deign +
Root Locu Plot Example 3 Breakaway point: jw 4j d K 4 3 2 d + 8 + 36 + 80 3 2 K(4 + 24 + 72 + - 80) 4 3 2 2 ( + 8 + 36 + 80) 3 2 or + 6 + 8 + 20 0 olving, b -2, - 2 ± 2. 45 j 0 4 2 2j 45 2j + Three point that breakaway at 90. 4j Frequency Repone Deign
Root Locu Plot Example 3 The imaginary axi croing: Characteritic equation 3 2 + 8 + 36 + 80 + K Routh table 4 4 36 K 3 8 80 0 2 26 K 0 80-8K/26 0 0 0 K 0 0 0 Condition for critical tability 80-8K/26 > 0 or K<260 The auxiliary equation 26 2 + 260 0 olving ± 0 j ± 3. 6 Frequency Repone Deign j
Root Locu Plot Example 3 The final plot i hown on the right. jw 4j 3.6j 2j What i the value of the gain K correponding to the breakaway point at b 2 ± 2.45j? 4 2 2j 4j Frequency Repone Deign
Root Locu Plot Example 3 Gain Calculation From the general magnitude condition the gain correponding to the point on the loci i K n i i For the point 2 + 2.45j + p i K 2 + 2.45j 2 + 2.45j + 4 2 + 2.45j + 2 + 4j 2 + 2.45j + 2 4j /.0 3.63 3.63 6.45.55 00.0 m + z i Frequency Repone Deign
Root Locu Plot Example 3 I there a gain correponding to a damping ratio of 0.707 or more for all ytem mode? z 0.707 co(q ) q 45 4 2 45 jw 4j 3.6j 2j 2j 4j Frequency Repone Deign
Root Locu Plot Example 3 Time Repone 4 2 K260 K64 K00 jw 4j 3.6j 2j 2j 4j Examine the repone for the variou gain and relate them to the location of the cloed-loop root. K 64, root are 2, 2, 2±3.46j K 00, root are 2±2.45j, 2±2.45j K 260, root are ±3.6j, 4±3.6j Frequency Repone Deign
Amplitude Root Locu Plot Example 3 Time Repone Step Repone, K 64 2±3.46j 0.8 2, 2 0.6 whole repone 0.4 0.2 0 0 2 3 4 5 Time (ec.) Frequency Repone Deign
Amplitude Root Locu Plot Example 3 Time Repone Step Repone, K 00 0.8 2±2.45j (repeated) 0.6 0.4 whole repone 0.2 0 0 2 3 4 5 Time (ec.) Frequency Repone Deign
Amplitude Root Locu Plot Example 3 Time Repone 2.8.6.4 Step Repone K 260 ±3.6j whole repone.2 0.8 0.6 4±3.6j 0.4 0.2 0 0 0.5.5 2 2.5 3 3.5 4 4.5 5 Time (ec.) Frequency Repone Deign
Root Locu with Other Parameter Can we plot the root locu in term of the variation of a parameter other than gain, for example a time contant? To achieve thi we apply a mathematical trick. Conider the loop tranfer function ( + ) GH ( ) 0 ( + 2) characteritic equation t + 2 + 0 t + 0 2 regrouping the term t ( 2 + 2 + 0) + 0 equivalent loop tranfer function t 0 0 GH ( ) 2 + 2 + 0 0 Frequency Repone Deign
Root Locu with Other Parameter The open-loop pole are t 0 ± 3j and the zero i t 0.294 x 0. Aymptote: 80 t 0.4325 t 0.9 Angle of departure: x x 08 (90 + q d ) 80 4 2 q d 98 Break-in point b 3.6 d d t 0 0 2 + 2 + 0 ( -0) 0 2 x jw O 3j 3j + Frequency Repone Deign
Amplitude Root Locu with Other Parameter Step Repone t 0, ± 3j.2 0.8 0.6 0.4 0.2 0 0 2 3 4 5 6 Time (ec.) t 0.293, 2.5 ± 2j t 0.4325, 3.6, 3.6 t 0.90,, 0 Frequency Repone Deign
Root Locu with Two Parameter The root locu method focue on the root of the characteritic equation. There can be everal different loop tranfer function that have the ame cloed-loop characteritic equation. To contruct the root locu for a characteritic equation which ha two parameter, we contruct fictitiou loop tranfer function and apply the normal method. Frequency Repone Deign
Root Locu with Two Parameter Conider the following characteritic equation. 3 + 2 + b + a 0. Write thi in the general form of + GH() 0 with b a a multiplying gain. b + 0 3 2 + + a GH () Thi will allow the plotting of the root locu with repect to the gain b for any given value of a. The root of the characteritic equation of GH () define the tarting point for the root loci. Conider the loci of thee root. Frequency Repone Deign
Root Locu with Two Parameter The characteritic equation of GH () i 3 + 2 + a 0, which may be written a a + 0 2 ( + ) GH () Now contruct the root locu of GH () in term of the gain a. jw a 2 a a 2 a 2 a 0.3.2, 0.±0.5j a 2.8.66, 0.33±.0j Frequency Repone Deign
Root Locu with Two Parameter Now contruct the root locu for b GH () 3 2 + + a where the open-loop pole correpond to the previou root locu for varying a. Aymptote: ± 90 a 0.3 a (-.2 +0. +0.)/2-0.5 a.8 a (-.66 +0.33 +0.33)/2-0.5 Frequency Repone Deign
Root Locu with Two Parameter Imaginary axi croing: 3 + 2 + b + a 0 3 b 0 2 a 0 b-a 0 0 0 a 0 0 a b, 2 + a 0 - j a 2 jw a 2 a a 2 a O Frequency Repone Deign