Module-6:Root Locus Analysis
Akshya Swain
Department of Electrical, Computer & Software Engineering, The University of Auckland,
Auckland, New Zealand.
Akshya Swain
Module-6:Root Locus Analysis
1/46
Learning Outcome of This Module
I After completion of this module, the student must be able to do the following:
1. Sketch the Root Locus
2. Relative stability analysis
Akshya Swain
Module-6:Root Locus Analysis
2/46
Root Locus: Introduction-1
I While designing any control system, it is often necessary to investigate the performance of the system when one or more parameters of the system varies over a given range.
I Further, it is known that the dynamic behavior (e.g. transient response) of a closed loop system is closely related to the location of the closed-loop poles. ( i.e. location of roots of closed loop characteristic equation).
I Therefore, it is important for the designer to know how the closed-loop poles (i.e. roots of characteristic equation) move in the s plane as one or more parameters of the system varies over a given range.
I A simple method for finding the roots of the characteristic equation has been developed by W. R. Evans
I This method, called the root-locus method, is one in which the roots of the characteristic equation are plotted for all values of a system parameter.
I Note that the root locus technique is not confined to inclusive study of control systems. The equation under investigation does not necessarily have to be the characteristic equation.
I The technique can also be used to assist in the determination of roots of high-order algebraic equations.
Akshya Swain
Module-6:Root Locus Analysis
3/46
Root Locus: Introduction-2
I The root locus problem for one variable parameter can be defined by referring to equations of the form:
F(s) = sn +a1sn−1 +…+an−1s+an
+K(sm +b1sm−1 +…+bm−1s+bm = 0 (1)
where K is the the parameter considered to vary between −∞ and ∞.
I The coefficients a1,…,an,b1,…,bm−1,bm are assumed to be fixed. I The various categories of root loci are defined as follows:
1. Root Loci: The portion of the root loci when K assumes positive values; that is 0 ≤ K < ∞.
2. Complementary Root Loci: The portion of the root loci when K assumes negative values; that is −∞ ≤ K ≤ 0.
3. Root Contours: The loci of roots when more than one parameter varies.
I The complete root loci refers to the combination of the root loci and the complementary root loci.
Akshya Swain
Module-6:Root Locus Analysis
4/46
What is Root Locus And What are its Usefulness ?
I The root locus is the locus of roots of the characteristic equation of the closed-loop system as a specific parameter (usually, gain K) is varied from zero to infinity.
I Such a plot clearly shows the contributions of each open-loop pole or zero to the locations of the closed-loop poles.
Is it Useful in Linear Control Systems Design ?
I It indicates the manner in which the open-loop poles and zeros should be modified so that the response meets system performance specifications.
I For example, by using the root-locus method, it is possible to determine the value of the loop gain K that will make the damping ratio of the dominant closed-loop poles as prescribed.
I If the location of an open-loop pole or zero is a system variable, then the root-locus method suggests the way to choose the location of an open-loop pole or zero.
Akshya Swain
Module-6:Root Locus Analysis
5/46
Basic Conditions of the Root Loci-1
I Consider the system shown in Figure.
The closed-loop transfer function is given by
T(s) = C(s) = KG(s) (2) R(s) 1 + KG(s)H(s)
I The closed loop characteristic equation of the system is
1 + KG(s)H(s) = 0 (3)
I Observe that the closed loop transfer function T(s), as well as the open loop transfer function KG(s)H(s), involves a gain parameter K.
Akshya Swain
Module-6:Root Locus Analysis
6/46
Definition and Concept of Root Locus
Definition 1. The root locus is the path of the roots of the characteristic equation traced out in the complex plane as a system parameter is changed. Example: Consider a video
camera control system shown.
I The closed-loop transfer function of this system is as follows
C(s) = K1K2
R(s)
s2 + 10s + K1K2 =K
s2 +10s+K where K = K1K2
Akshya Swain
Module-6:Root Locus Analysis
7/46
Example-1: Concept of Root Locus
I The closed loop characteristic equation is given by s2 + 10s + K = 0
I The location of poles as the open loop gain K is varied is shown in the Table. K Pole-1 Pole-2
0 -10 0
5 −9.47
10 −8.87
15 −8.16
20 −7.24
25 −5 −5
30 −5 + j2.24
35 −5 + j3.16
40 −5 + j3.87
45 −5 + j4.47
50 −5+j5 −5−j5
−0.53 −1.13 −1.84 −2.76
−5 − j2.24 −5 − j3.16 −5 − j3.87 −5 − j4.47
Akshya Swain
Module-6:Root Locus Analysis
8/46
Example-1: Concept of Root Locus
I Fromtheplot,itisseenthatforK=0,thepolesareatp1 =−10,p2 =0.
I As K increases,p1 moves toward the right, while p2 moves toward the left.
I For K = 25, the poles p1 and p2 meet at −5, break away from the real axis, and
move into the complex plane.
I Further, if 0 < K < 25, the poles are real and distinct, and the system is
overdamped.
I For K = 25, the poles are real and multiple, and the system is critically damped. I For K > 25, the poles are complex conjugate, and the system is underdamped.
Akshya Swain
Module-6:Root Locus Analysis
9/46
Example-2: Concept of Root Locus
I Consider a unity feedback system with open loop transfer function G(s) = K
s(s+2)
I The closed loop transfer function of the system is given by
C(s) = G(s) = K R(s) 1+G(s) s2 +2s+K
I The closed loop characteristic equation is given by s2 + 2s + K = 0
I The roots of the closed loop characteristic equation i.e. the closed loop poles are located at
√ s1,2=−1± 1−K
Akshya Swain
Module-6:Root Locus Analysis
10/46
Example-2: Concept of Root Locus
I The location of poles as the open loop gain K is varied is shown in the Table.
K Pole-1 Pole-2
0 0 -2.0
0.5
0.75
1
2
3
50
−0.293 −0.5 −1.0 −1.0 + j1.0 −1.0 + j1.414 −1.0 + j7.0
−1.707 −1.5 −1.0 −1.0 − j1.0 −1.0 − j1.414 −1.0 − j7.0
Akshya Swain
Module-6:Root Locus Analysis
11/46
Example-2: Concept of Root Locus
I The root locus is shown in the figure
I From the plot, it is seen that for K = 0, the poles are at
p1 = 0,p2 = −2.
I For0