Root locus Analysis P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: Mr Chaitanya, SYSCON 07
Recap R(t) + _ k p + k s d 1 s( s+ a) C(t) For the above system the closed loop transfer function is given by: ks + d p 2 s + ( kd + a) s+ kp We can see that the closed loop poles location in s-plane s depends on the open loop system gain, poles (i.e( a), zeros (if exist), controller gains (kp( kp, ki and kd). k
Why Root locus? Varying any one or all of these parameters changes the closed loop poles and hence the impulse response and stability of the system changes. And it is laborious each time to find the closed loop poles by varying these parameter values. Root locus is a graphical method which can easily find the closed loop poles by changing any one of these parameters (if others kept constant).
Root locus This method may not give the exact results but a fast approximate result can be achieved. Root locus is defined as the locus of the roots of the characteristic equation of the closed loop system as a specific parameter is varied from zero to infinity.
Root locus approach R(s) - + KG(s) C(s) H(s) Consider a system shown above. The closed loop transfer function of this system is given by C(s) KG(s) = R(s) 1+ KG(s)H(s) So the characteristic equation is 1+ KG(s)H(s) = 0
Root locus approach contd.. Which is same as KG(s)H(s) = -1 Now this can be written as: Angle condition: 0 G(s)H(s) = ±180 (2k +1) (k=0,1..) Magnitude condition: KG(s)H(s) =1 A locus of all the points in the complex plane satisfying the angle condition alone is called root locus. The closed loop poles for a give parameter value are determined from the magnitude condition.
Root locus approach contd.. Suppose KG(s)H(s) ) is of the form: K(s + z )(s + z )...(s + z ) s p s p s p n KG(s)H(s) = 1 2 m ( + )( )...( ) 1 + 2 + Now to sketch the root locus with K as the parameter first we must know the location of poles and zeros of G(s)H(s). Later we test a point in complex plane for the angle condition. If condition is satisfied the test point lies on the root locus.
For a system with 4 poles and 1 zero Assuming two of these poles are complex conjugates. The angle of G(s)H(s) at this test point is: G(s)H(s) = φ1 θ1 θ2 θ3 θ4 Is this value is equal to 0 ±180 (2k +1) then the test point lies on the root locus.
Rules to plot Root locus It is difficult to test each and every point on the complex plane for angle condition. There are 8 rules which states the procedure to plot the root locus of any system. To draw a root locus first we must locate the open loop poles and zeros on the s-plane. s One fact is that the root locus plots are symmetric about the real axis as the complex roots occur in conjugate form.
Set of Rules to plot Root-Locus 1. Locate the poles and zeros of GsHs () () on the S-plane. S The root locus branches start at open loop poles and terminate at zeros. If no. of (poles-zeros) > 0, those (n-m) branches will end at infinite. 2. Determining the root loci on real axis. Taking any test point, if the sum of no. of (poles+zeros) right to it, is an odd number, then that point will be on Root-Loci. 3. Determining the asymptotes of the Root-Loci. These asymptotes show the way through which the (n-m) branches should end at infinite.
Rules contd 4. Finding the break-away away and break-in points. These are the points at which the root locus branch divides or combines. 5. If there are any complex poles or zeros, we have to find the angle of departure or arrival of root loci at that point. 6. Finding the points where the root loci crosses the imaginary axis. This can be found from Routh s s stability criterion. 7. Taking a series of test points in the neighborhood of origin and jw axis at the intersection points. 8. Determining the closed loop poles at desired k value
Rule1 As K increases from 0 to infinity, each branch of the root locus originates from an open-loop pole with K=0 and terminates either on open loop zero or on infinity with K= The number of branches terminating on infinity equals the number of open-loop poles minus zeros. The proof of this statement is as follows:
Rule1 contd.. The general characteristic equation can be rewritten as: When K=0, this equation has roots at - open loop poles. The equation can also be written as: 1 n j j= 1 i= 1 n ( s+ p ) + K ( s+ z ) = 0 ( s+ p ) + ( s+ z ) = 0 j K j = 1 i = 1 When K, this equation has roots at - open loop zeros of the system. m m i i p j which are the z i which are
Rule1 contd.. Therefore m branches terminate at open-loop zeros, the other (n-m) branches terminate at infinity. Examining the magnitude condition m i= 1 n j= 1 ( s+ z ) ( s+ p ) i j s We find that this is satisfied by as K Hence (n-m) branches terminate at infinity as K = 1 K
Rule 2 A point on the real axis lies on the locus if the number of open-loop poles plus zeros on the real axis to the right of this point is odd. This can be easily verified by checking the angle criterion at that point. [G(s)H(s)] = (m - n )180 r Where m r are number of open loop zeros to the right of the point and n r are the number of open-loop poles to the right of the point. r =± + = 0 0 (2q 1)180 ; q 0,1, 2...
Rule 3 (determining the asymptotes of root loci) The (n-m) branches of the root locus which tend to infinity, do so along straight line asymptotes whose angles are given by φ A 0 (2q + 1)180 ; q 0,1, 2...,( n m 1) = = n m Proof: Consider a point in s-plane s which is far away from the open-loop poles and zeros, then the angle made by all the phasors at this point is almost same. Hence, [G(s)H(s)] = -(n - m)φ
Rule 3 (determining the asymptotes of root loci) contd.. If this point has to lie on the root locus then it must satisfy Hence -(n - m) φ =± (2q + 1)180 0 φ A 0 (2q + 1)180 ; q 0,1, 2...,( n m 1) = = n m The asymptotes cross the real axis at a point known as centroid,, determined by the relationship: (sum of real parts of poles sum of real parts of zeros)/(number of poles number of zeros)
Rule 3 (determining the asymptotes of root loci) contd.. The open loop transfer function can also be m m written as m m-1 G(s)H(s) = This can be written as G(s)H(s) = The denominator is like a expansion of which is of the form K[s + ( z )s +...+ ( z )] i i=1 i=1 n n n n-1 pj j=1 j=1 [s + ( )s +...+ ( p )] K n m n-m n-m-1 [s + ( pj- z i)s +...] j=1 i=1 n-m n m 1 [s + ( n m) σ As +...] j i ( ) n s + σ m A
Rule 3 (determining the asymptotes of root loci) contd.. Comparing both as s we get σ = A n m ( p )- (-z ) j j=1 i=1 n-m Because all the complex poles and zeros (if exist) are conjugate pairs, σ A is always a real quantity. Hence σ A is (sum of real parts of poles sum of real parts of zeros)/(number of poles number of zeros) i