ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about basic rules of sketching root locus.

1. ME-314 Control Engineering
Dr. Bilal A. Siddiqui
Mechanical Engineering
DHA Suffa University

2. System Stability in terms of Poles
• Shape of response is determined only by poles. Zeros
determine only the magnitude of response
• Poles in the left half of s-plane yield either pure
exponential decay (𝑒 𝜎𝑡) or damped sinusoidal
(𝑒 𝜎𝑡cos𝜔 𝑑 𝑡) natural responses.
• These natural responses decay to zero as time
approaches infinity.
• Stable systems have closed-loop transfer functions
with poles only in the left half-plane, i.e. real part
of all poles is negative (𝝈 < 𝟎)
• Unstable systems have at least one pole in right hand
plane; real part is positive (𝝈 > 𝟎)

3. Another type of instability
• A system is marginally stable if there are complex
pair of poles on imaginary axis (𝜎 = 0)
• Another type of instability can occur if the system has
repeated poles on the imaginary axis. This will cause
natural response to be of the type 𝐴𝑡 𝑛 cos 𝜔 𝑑 𝑡, where
n+1 is the number of repeated roots at imaginary axis.
• Unstable systems have closed loop transfer functions
with at least one pole in the right half-plane and/or
repeated poles on the imaginary axis.

4. Determining Stability of Systems when
Loop is Closed
• Feedback is generally useful. Too much feedback
can also destabilize a system originally stable
without feedback.
• Consider the system 𝐺 𝑠 =
3
𝑠 𝑠+1 𝑠+2
• We know this system is stable in open loop with
non-oscillatory response.
• Let us now close the loop. Is the system still stable
with feedback?

5. Stability of Close Loop Systems
• The closed loop transfer function is
𝐶 𝑠
𝑅 𝑠
=
𝐺
1 + 𝐺
=
3
𝑠 𝑠 + 1 𝑠 + 2 + 3
• So now, we have to find the poles of above
• Turns out the non-oscillatory open loos
response has now become oscillatory

6. Destabilizing Effect of Feedback
• Consider the system 𝐺 𝑠 =
7
𝑠 𝑠+1 𝑠+2
• We know this system is stable in open loop. We
also know the response is non-oscillatory
• Let us now close the loop. Is the system still stable
with feedback?
• Turns out, feedback is destabilizing in this case

7. Root Locus (Kuo)
• In the preceding examples, we saw the importance of the poles and
zeros of the closed-loop transfer function of a linear control system
on the dynamic performance of the system. Roots of the
characteristic equation, which are the poles of the closed-loop
transfer function, determine the absolute and the relative stability of
linear SISO systems.
• Conundrum: we have to solve a polynomial every time a parameter
of the system changes to find new location of poles.
• Root locus, a graphical presentation of the closed-loop poles as a
system parameter is varied, is a powerful method of analysis and
design for stability and transient response
Root locus gives more information than methods we studied so far.

8. Control System Problem (Nise)
• Poles of the open-loop transfer function (KG(s)) are easily
found (they are known by inspection and do not change with
changes in system gain)
• Poles of closed-loop transfer function (T(s)) are more difficult
to find (they cannot be found without factoring the closed-loop
system’s denominator)
• Also, closed-loop poles change with changes in system gain
Since the system’s transient response
and stability are dependent upon the
poles of T(s), we have no knowledge of
the system’s performance unless we
factor the denominator for specific
values of K.
The root locus will be used to give us a
vivid picture of the poles of T(s) as K
varies, without factoring it.

9. Example
• Consider position control of a security camera tracking a subject
• Root locus technique can be used to
analyze and design the effect of loop
gain on the system’s transient response
• Closed loop poles are obviously at −5 ±
𝑗 𝐾 − 25. Let’s plot movement of poles
with values of 0 ≤ 𝐾 ≤ +∞
• We can make several observations about
system out as K changes

10. Characteristic Equation
• We saw that we had to solve the polynomial for poles at different K’s to
get close loop poles.
• Root locus offers a way of getting this pole trajectory without solving for
the poles explicitly. Only open loop poles/zeros should be known.
• Root Locus means the trajectory of pole movements as a system
parameter changes. If several parameters change, we call it Root Contour
(not subject of this course)
• Let the close loop transfer function be (unity feedback)
𝑇 𝑠 =
𝐾𝐺 𝑠
1 + 𝐾𝐺 𝑠
• Poles of the transfer function are roots of 𝟏 + 𝑲𝑮 𝒔 = 𝟎
• This equation is called the characteristic equation
• Gain K can also be negative, but there is no loss of generality in
considering K>0
• Sketching root locus is based on few rules (as K varies from 0 to +∞)
• Let’s see these rules now, without derivation.

11. RL Sketching: Rules 1-3
• Let open loop tf 𝐺 𝑠 =
𝑁 𝑠
𝐷 𝑠
, then we write the characteristic equation
as 𝐷 𝑠 + 𝐾𝑁 𝑠 = 0, or 𝐷 𝑠 /𝐾 + 𝑁 𝑠 = 0
• As 𝐾 → 0, poles of the closed loop system are roots of D(s), i.e. open
loop poles.
• As 𝐾 → ∞, poles of the closed loop system are roots of N(s), i.e. open
loop zeroes.
1) Start and End of Roots: 1st rule of root locus is that all closed loop
poles start at open loop poles (as 𝐾 → 0) and end at open loop zeros
(as 𝐾 → ∞)
2) Number of branches. Each closed-loop pole moves as the gain is
varied. Define a branch as the path that one pole traverses, then there
will be one branch for each closed-loop pole. The number of branches
of the root locus equals the number of closed-loop poles.
3) Symmetry. Since roots are either real (𝜎) or complex conjugate (𝜎 ±
𝑗𝜔), the root locus will always be symmetric about the real axis.

12. RL Sketching: Rule 4
• We saw that root loci start from a pole of G(s), and end at a zero of G(s)
• What if we have more poles than zeros of G(s)? This is often the case
• Let’s say we have ‘m’ zeros and ‘n’ poles of G(s). Let n ≥m
• Then, ‘m’ closed loop poles will start at poles of G(s) and end at zeros of
G(s). The rest of n-m close loop will start at some poles and go where?
• For this case, we consider n-m zeros to be situation at infinity.
• Then the remaining poles of close loop will start at poles of G(s) as K=0
and go towards infinity as 𝐾 → ∞
• In which direction will the poles go? This is given by asymptotes.
4) Asymptotes: The root locus approaches straight lines as asymptotes as
the locus approaches infinity. Equation of the asymptotes is given by the
real-axis intercept, 𝜎 𝑎 and angle, 𝜃 𝑎 as follows
𝝈 𝒂 =
𝒖=𝟏
𝒏
𝒑 𝒖 − 𝒗=𝟏
𝒎
𝒛 𝒗
𝒏 − 𝒎
, 𝜽 𝒂 =
±𝟏𝟖𝟎° 𝟐𝒌 + 𝟏
𝒏 − 𝒎
Where p and z are the finite poles and zeros; k=1,2,…(n-m)

13. Examples of Asymptotes
• Verify this below.
• Just remember the figures above. No need to
remember the formulae!
𝝈 𝒂 =
𝒖=𝟏
𝒏
𝒑 𝒖 − 𝒗=𝟏
𝒎
𝒛 𝒗
𝒏 − 𝒎
,
𝜽 𝒂 =
±𝟏𝟖𝟎° 𝟐𝒌 + 𝟏
𝒏 − 𝒎

14. RL Sketching: Rule 5
• Number the zeros and poles of open loop tf from
right to left.
5) Real Axis Segments: For K > 0 root locus
exists to the left of an odd numbered open-loop pole
or zero on the real axis.

15. Sketching RL: Rule 6
Numerous root loci appear to break away from the real axis
as the system poles move from the real axis to the complex
plane. At other times the loci appear to return to the real axis
as a pair of complex poles becomes real.
6) Break -ins and -outs. Roots moving in opposite
directions along the x-axis from two poles “break out” of the
x-axis at 900. Similarly, roots rejoining the x-axis to enter a
zero also “break into” the x-axis at 90o. There are two ways
of finding break in and break out point locations, but we will
not use them. Let Matlab handle that. With practice, we can
do that by intuition approximately.

16. Revisiting the previous example

17. Example
• Plot Root locus for system shown below by
applying rules studied so far

18. RL Sketching – Rule 7
• In the previous example, we see that the system’s poles are in the left half-
plane up to a particular value of gain. Above this value of gain, two of the
closed-loop system’s poles move into the right half-plane, meaning that the
system is unstable.
• The gain at the imaginary-axis crossing, in this example, yields the
maximum positive gain for system stability. This is known as “gain
margin”: the maximum gain you can put in the system before it becomes
unstable.
• In some cases, the roots move from the right half-plane to left half-plane.
For small values of gain, the system’s poles are in the right half-plane.
Above a particular value of gain, closed-loop poles move into the left half-
plane, meaning that the system is stable. In any case 𝑗𝜔-axis crossing is v.
important
7) Imaginary-axis Crossing: To find the imaginary-axis crossing, we can
use the Routh-Hurwitz criterion
• However, we will not use this rule. Let Matlab handle this as well!

19. Summary of Basic Rules for Sketching
Root Loci
1) Number of branches. The number of branches of the root locus equals the
number of closed-loop poles.
2) Symmetry. Root locus will always be symmetric about the real axis.
3) Start and End of Roots: Closed loop poles start at open loop poles (as 𝐾 → 0)
and end at open loop zeros (as 𝐾 → ∞)
4) Asymptotes: If open loop zeros=m are less than poles=n, then n-m closed loop
poles approach infinity as following straight asymptotic lines when 𝐾 → ∞.
5) Real Axis Segments: For K > 0 root locus exists to the left of an odd numbered
open-loop pole or zero on the real axis.
6) Break-in / Break-out Points: Roots moving in opposite directions along the x-
axis “break out” or “break into” the x-axis at 900.

20. Another Example
Given open loop transfer function 𝐺 𝑠 =
𝐾 𝑠−2 𝑠−4
𝑠2+6𝑠+25
, plot locus
of closed loop poles.

21. Assignment 6
• Problem 3 & 45 in Nise. Use Matlab in both
questions to confirm your hand sketched root
locus plot.
• Submit by 3rd of November, 2017