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Chapter: Control Systems : Time Response Analysis

Root Locus Technique, Analysis and Application Procedure

The roots of the closed-loop characteristic equation define the system characteristic responses

Root Locus Technique


o   Introduced by W. R. Evans in 1948


o   Graphical method, in which movement of poles in the s-plane is sketched when some parameter is varied The path taken by the roots of the characteristic equation when open loop gain K is varied from 0 to ∞ are called root loci


o   Direct Root Locus = 0 < k < ∞


o   Inverse Root Locus = - ∞ < k < 0


Root Locus Analysis:


o   The  roots  of  the  closed-loop  characteristic  equation  define  the  system  characteristic responses


o   Their location in the complex s-plane lead to prediction of the characteristics of the time domain responses in terms of:


o   damping ratio ζ,


o   natural frequency, wn



o   damping constant ζ, first-order modes


o   Consider how these roots change as the loop gain is varied from 0 to∞


Basics of Root Locus:


o   Symmetrical about real axis


o   RL branch starts from OL poles and terminates at OL zeroes


o   No. of RL branches = No. of poles of OLTF


o   Centroid is common intersection point of all the asymptotes on the real axis


o   Asymptotes are straight lines which are parallel to RL going to ∞ and meet the RL at ∞


o   No. of asymptotes = No. of branches going to ∞


o   At Break Away point , the RL breaks from real axis to enter into the complex plane


o   At BI point, the RL enters the real axis from the complex plane


Constructing Root Locus:


o   Locate the OL poles & zeros in the plot Find the branches on the real axis


o   Find angle of asymptotes & centroid


o   Φa= ±180º(2q+1) / (n-m)


o   ζa = (Σpoles - Σzeroes) / (n-m) Find BA and BI points


o   Find Angle Of departure (AOD) and Angle Of Arrival (AOA)


o   AOD = 180º- (sum of angles of vectors to the complex pole from all other poles) + (Sum of angles of vectors to the complex pole from all zero)


o   AOA = 180º- (sum of angles of vectors to the complex zero from all other zeros) + (sum of angles of vectors to the complex zero from poles)


o   Find the point of intersection of RL with the imaginary axis.


Application of the Root Locus Procedure


Step 1: Write the characteristic equation as


1+ F(s)= 0

Step 2: Rewrite preceding equation into the form of poles and zeros as follows


Step 3:


Locate the poles and zeros with specific symbols, the root locus begins at the open-loop poles and ends at the open loop zeros as K increases from 0 to infinity


If open-loop system has n-m zeros at infinity, there will be n-m branches of the root locus approaching the n-m zeros at infinity


Step 4:


The root locus on the real axis lies in a section of the real axis to the left of an odd number of real poles and zeros


Step 5:


The number of separate loci is equal to the number of open-loop poles


Step 6:


The root loci must be continuous and symmetrical with respect to the horizontal real axis


Step 7:


The loci proceed to zeros at infinity along asymptotes centered at centroid and with angles


Step 8:


The actual point at which the root locus crosses the imaginary axis is readily evaluated by using Routh‗s criterion


Step 9:


 Determine the breakaway point d (usually on the real axis)


Step 10:


Plot the root locus that satisfy the phase criterion


Step 11:

Determine the parameter value K1 at a specific root using the magnitude criterion


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Control Systems : Time Response Analysis : Root Locus Technique, Analysis and Application Procedure |

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