Term 8 CISE: Linear Control Sytem Dr. Samir Al-Amer Chapter 7: Root locu CISE_ch 7 Al-Amer8 ١ Learning Objective Undertand the concept of root locu and it role in control ytem deign Be able to ketch root locu and ue MATLAB to plot the root locu Recognize the role of root locu in parameter deign and enitivity analyi Be able to deign controller uing root locu CISE_ch 7 Al-Amer8 ٢
Outline The concept of root locu CISE_ch 7 Al-Amer8 ٣ Root locu The location of the root of the characteritic equation determine the ytem repone. Modifying one or more of the ytem parameter caue the root of the characteritic equation to change. Root locu i a graphical method that determine the location of the root a one parameter change. CISE_ch 7 Al-Amer8 ٤
Root locu Root locu wa developed by Evan in 948. Ued to analyze and deign ytem We ue it to enure that the root of the characteritic equation are in a deirable region Can be ued to tudy enitivity of the pole location w.r.t change in the parameter. Deirable region CISE_ch 7 Al-Amer8 ٥ Root Locu Uing MATLAB rlocu(,[ ]) CISE_ch 7 Al-Amer8 ٦
Example the characteritic equation root are : ± 4 4 r, r, r E() R() _ ( ) Y() Root Locu CISE_ch 7 Al-Amer8 ٧ Root locu can be ued to tudy enitivity of ytem w.r.t variation in the parameter :.:; Grlocu(,[ ],); plot(g,. ) CISE_ch 7 Al-Amer8 ٨ 4
Mathematic of root locu c R() E() G() _ Y() G( ) G G( ) j G ; G( ) 8 ± k6 k : integer CISE_ch 7 Al-Amer8 ٩ Quetion c R() E() G() _ Y() How doe the root of the characteritic equation G( ) change a varie between and? CISE_ch 7 Al-Amer8 ١٠ 5
Example G( ) ( ) E() R() Y() _ ( ) When ( ) or The locu of the root of G() tart at the pole of G() CISE_ch 7 Al-Amer8 ١١ Root Locu Procedure Initial Step: Expre the characteritic equation in the form P() Factor P() and rewrite the equation in the form m i ( z ) i i n ( p i ) CISE_ch 7 Al-Amer8 ١٢ 6
Example Apply the initial tep of the root locu procedure for the ytem _ CISE_ch 7 Al-Amer8 ١٣ Example Step : Expre the characteritic a P( ) _ CISE_ch 7 Al-Amer8 ١٤ 7
Example Step : Factor P() P( ) ( )( ) _ n m CISE_ch 7 Al-Amer8 ١٥ Root Locu Procedure Locate the open loop pole and zero of P() Locate the pole and zero of P() on the -plane. Ue for pole and O for zero ( ) ( ) - CISE_ch 7 Al-Amer8 ١٦ 8
The root locu tart at the pole of P() and terminate at the zero of P() or at infinity CISE_ch 7 Al-Amer8 ١٧ Root Locu Procedure # of eparate root loci The number of eparate root loci The number of pole of P() Note that ( # pole # zero) CISE_ch 7 Al-Amer8 ١٨ 9
Example How many eparate root loci doe the root locu of the ytem ha? _ P( ) ( )( ) # of pole Three eparate root loci CISE_ch 7 Al-Amer8 ١٩ Root Locu Procedure Segment of real line that i part of root locu : Segment of real line: A egment of real line i a part of root locu if um of # of pole plu # of zero to right of that egment i odd. Ye No Ye No CISE_ch 7 Al-Amer8 ٢٠
Example A egment of real line i a part of root locu if um of # of pole plu # of zero to right of that egment i odd. Double pole 6 5 No Ye Ye No Ye No CISE_ch 7 Al-Amer8 ٢١ Example A egment of real line i a part of root locu if um of # of pole plu # of zero to right of that egment i odd. Complex pole or zero do not affect the count 5 4 Ye No Ye No Ye No CISE_ch 7 Al-Amer8 ٢٢
Example Step : Segment of real axi that are part of the root locu ( )( ) CISE_ch 7 Al-Amer8 ٢٣ Aymptote If n > m # of pole of P() ># of zero The root locu approache infinity at n-m direction. CISE_ch 7 Al-Amer8 ٢٤
Aymptote Step : # of aymptote n m pole zero { } {} centroid n m q Φ A 8 q,, n m Φ 6, 8, A - - 6 CISE_ch 7 Al-Amer8 ٢٥ Example How many aymptote? ( ) n, m, n-m aymptote Centroid ((-))/ -.5 CISE_ch 7 Al-Amer8 ٢٦
4 CISE_ch 7 Al-Amer8 ٢٧ Interection with Imaginary axi To find interection with imaginary axi ue Routh Hurwitz method 6 Interection 6 Characteritic equation 6 CISE_ch 7 Al-Amer8 ٢٨ Interection with Imaginary axi 6 Equation 6 j Auxiliary ± 6 Interection with imaginary axi
CISE_ch 7 Al-Amer8 ٢٩ Breakaway point Breakaway point are point at which the root loci breakaway from real axi or the root loci return to real axi. At breakaway point dp( ) d Solve the above equation to determine the breakaway point. Select olution that are in egment of real axi that i part of root locu CISE_ch 7 Al-Amer8 ٣٠ 5
Breakaway point To find breakaway point d d - - ( 6 ) ( ) break away point 5774. and Select. 46. 46 CISE_ch 7 Al-Amer8 ٣١ Symmetry of root locu The root locu i ymmetric with repect to the real axi. CISE_ch 7 Al-Amer8 ٣٢ 6
Angle of departure Sum of angle contribution of pole and zero (meaured with tandard reference) 86k In the example all pole/zero are real No need to do thi CISE_ch 7 Al-Amer8 ٣٣ Angle criteria At all point of the root loci : Sum of angle contribution of pole and zero (meaured with tandard reference) 86k CISE_ch 7 Al-Amer8 ٣٤ 7
Example 8 Draw root locu for the following ytem _ ( ) Characteritic Equation : P( ), n m CISE_ch 7 Al-Amer8 ٣٥ Root Locu of ( ) CISE_ch 7 Al-Amer8 ٣٦ 8
Example 9 Draw root locu for the following ytem _ ( 5) ( )( )( ) Characteritic Equation : 5 ( )( )( ) 5 P( ), ( )( )( ) n, m CISE_ch 7 Al-Amer8 ٣٧ Example 9 5 P( ), n, m ( )( )( ) Pole :,,, Zero: 5 # of ymptote Angle 9, 7 ( 5) Centroid CISE_ch 7 Al-Amer8 ٣٨ 9
root locu of 5 ( )( )( ) CISE_ch 7 Al-Amer8 ٣٩ Example How do we draw root locu for thi ytem? _ CISE_ch 7 Al-Amer8 ٤٠
CISE_ch 7 Al-Amer8 ٤١ Example How do we draw root locu for thi ytem? _ Initial Step: Expre the characteritic a )? ( ) ( i P What P CISE_ch 7 Al-Amer8 ٤٢ Example Initial Step: Expre the characteritic a ( ) ) ( Characteritic Equation : P Continue the Root Locu Procedure
CISE_ch 7 Al-Amer8 ٤٣ Example How do we draw root locu for thi ytem? _ ( ) CISE_ch 7 Al-Amer8 ٤٤
CISE_ch 7 Al-Amer8 ٤٥ Example Initial Step: Expre the characteritic a ( ) ) ( ) ( ) ( Characteritic Equation : P Continue the Root Locu Procedure CISE_ch 7 Al-Amer8 ٤٦
Controller Deign Baed on Root Locu The idea here i to elect the parameter o that all the pole in the deired location. Deired region CISE_ch 7 Al-Amer8 ٤٧ The deired region i obtained to atify the given pecification Draw the deired region on the root locu Select the gain uch that all pole of the cloed loop ytem are in the deired region CISE_ch 7 Al-Amer8 ٤٨ 4
Deired region CISE_ch 7 Al-Amer8 ٤٩ What can you tell about behavior of thi ytem? CISE_ch 7 Al-Amer8 ٥٠ 5
CISE_ch 7 Al-Amer8 ٥١ What can you tell about thi ytem? CISE_ch 7 Al-Amer8 ٥٢ 6
eyword Root locu Breakaway point Aymptote Centroid Angle of departure Angle of arrival CISE_ch 7 Al-Amer8 ٥٣ 7