Chapter 13
Root Locus
13.1
Introduction
In the previous chapter we had a glimpse of controller design issues through some simple
examples. Obviously when we have higher order systems, such simple design techniques
will not be sufficient. In this chapter we will look at a method that helps us to determine
the position of the roots of the characteristic equation as some design parameter varies.
Root locus is a fairly general graphical technique used to determine the migration of the
closed-loop poles as some parameters, such as controller gain, is varied.
Consider a simple P-controller.
Figure 13.1: A simple proportional controller
the closed loop transfer function is given by
Gc (s) =
K(s)G(s)
kG(s)
=
1 + K(s)G(s)
1 + kG(s)
The closed loop poles are given as the roots of,
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115
1 + kG(s) = 0
We want to select k in order to achieve speed, accuracy, and stability by placing the
poles in certain regions of the s-plane. This region was also shown in the last section.
An approximate sketch is given below.
Figure 13.2: Desirable region of the s-plane
Let us consider the example of an aircraft. From the notion of static stability of an
aircraft we know that the CG (center of gravity) of the aircraft must lie ahead of the
AC (aerodynamic center) of the aircraft for stability.
Figure 13.3: An aircraft example
Let us consider a linearized model of the aircraft dynamics obtained by perturbing the
system about a steady state or ”trim” condition.
α̇ = Zα α + Zq q + Ze δe
q̇ = Mα α + Mq q + Me δe
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116
where, α is the angle of attack, q is the pitch rate, and δe is the elevator deflection (which
is also the input to the system). The coefficients Z and M are the stability derivatives.
The open-loop transfer function can be obtained by taking Laplace transforms on both
sides and assuming initial conditions to be zero (this is true since the ”trim” condition
is taken as the reference).
sα(s) = Zα α(s) + Zq q(s) + Ze δe (s)
sq(s) = Mα α(s) + Mq q(s) + Me δe (s)
Solving for q(s), which the variable of interest to us,
q(s) =
s2
Me s + (Ze Mα − Zα Me )
δe (s)
− (Zα + Mq )s + (Zα Mq − Zq Mα )
Let us consider an aircraft in the unstable configuration at Mach number M = 0.9 and
altitude 20,000 ft. The corresponding values of the coefficients are,
Zα = −1.62 sec−1
Zq = 1.00 sec−1
Ze = −0.17 sec−1
Mα = 2.96 sec−1
Mq = −0.77 sec−1
Me = −22.53 sec−1
Using these values,
q(s) =
−22.53s − 36.55
−22.53(s + 1.62)
=
2
s + 2.39s − 1.71
(s + 2.97)(s − 0.575)
which shows that the open loop system is unstable as it has a pole on the RHS of the
s-plane. So, small deviations in the elevator deflection will cause the pitch rate to grow
without bound.
Let us design a P-control autopilot that stabilizes this system. The feedback system will
be somewhat like that shown in the figure below.
which has the following denominator polynomial,
D(s) = (s + 2.97)(s − 0.575) − 22.53k(s + 1.62)
= s2 + 2.39s − 1.71 − 22.53ks − 36.55k
= s2 + (2.39 − 22.53k)s − (1.71 + 36.55k)
Let us form the Routh array,
s2
1
−(1.71 + 36.55k)
1
s
(2.39 − 22.53k)
0
0
s0 −(1.71 + 36.55k)
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117
Figure 13.4: P-control autopilot
For the system to be stable,
2.39 − 22.53k > 0 ⇒ k < 0.106
−1.71 − 36.55k > 0 ⇒ k < −0.047
which means that a negative k in the order of −1 will stabilize the system.
However, this example is fairly simple and when the aircraft dynamics are realistic we
will need a much higher order system where it will not be very easy to obtain the effect
of k or other gains on the roots. Root locus technique gives us a way by which we can
plot approximate locus of the roots on the s-plane without actually solving for the roots.
It is easy to find the root locus of first and second order systems. But how about those
of higher orders? Let us consider an example of a third order system.
Let the open-loop system be given by,
G(s) =
1
s[(s + 4)2 + 16]
Let us use P-control. Then the closed loop transfer function is,
Gc (s) =
k
k
kG(s)
=
= 3
2
2
1 + kG(s)
s[(s + 4) + 16] + k
s + 8s + 32s + k
Let us check for stability using the Routh array,
1
s3
s2
8
1
s 32 −
k
s0
k
8
32
k
0
0
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118
Figure 13.5: Root locus for the example
For stability,
k < 32 × 8 = 256
If we actually plot the root locus for this system, it will look as above. From the figure
we can see that as k increases beyond the value of 256, the complex conjugate roots
migrate into the RHS of the s-plane. With increasing k the root at the origin migrates
farther into the LHS of the s-plane along the real axis.
13.2
Evan’s Form
Before we can apply the root locus technique, we need to express the system in what is
known as Evan’s form. The Evan’s form is given as,
G(s) =
G1 (s)
1 + kG2 (s)
where k is the parameter to be varied, and we want to plot the roots of
1 + kG2 (s) = 0
as k varies from 0 to ∞.
Lecture Notes on Control Systems/D. Ghose/2012
Example:
A P-control system given below has the closed loop representation as,
Figure 13.6: A P-control system
Gc (s) =
kG(s)
1 + kG(s)
which is already in Evan’s form.
Another Example:
Consider the following system.
Figure 13.7: Another system:PD-control
The closed-loop transfer function is given as,
Gc (s) =
=
(k1 + k2 s)G(s)
1 + k2 sG(s) + k1 G(s)
1
(k1 +k2 s)G(s)
1+k1 G(s)
sG(s)
+ k2 1+k
1 G(s)
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120
Suppose the value of k1 is already fixed, then the system is in Evan’s form and the poles
of this system are the roots of,
1 + k2
sG(s)
=0
1 + k1 G(s)
Another Example:
Consider integral control. Then the closed loop transfer function would be,
k
s
k
· G(s)
· G(s)
s
=
k
1 + s · G(s)
1 + k · G(s)
s
which is in Evan’s form.
Yet Another Example:
Consider PI-control. The closed loop transfer function would be,
k1 +
k2
s
1 + k1 +
G(s)
k2
s
G(s)
=
which is now in Evan’s form for a fixed k1 .
1+
N (s)
1+k1 G(s)
G(s)
k2 s{1+k
1 G(s)}