Control Theory and Self-Regulation (TCA - 22134)

Lecture 5: M3.2. Control design using root-locus

Outline
o Module M3: Feedback control design
• M3.1: Analysis of feedback
• M3.2: Control design using root-locus
§ M3.2.1: Root-locus and guidelines for determining it
§ M3.2.2: Dynamic compensation design using root-locus

2
Module M3

Module M3: Feedback control design

Contents
• M3.1: Analysis of feedback
• M3.2: Control design using root-locus
• M3.3: Control design using frequency response
• M3.4: Control design using state-space
• M3.5: Digital control
• M3.6: Introduction to non-linear systems and adaptive control 3
Module M3.2

Module M3.2: Control design using root-locus

Contents
• M3.2.1: Root-locus and guidelines for determining it
• M3.2.2: Dynamic compensation design using root-locus

4
M3.2.1 Root-locus and
guidelines for determining it
o Root-locus of a feedback system
• The objective is to know how the poles (roots of denominator) locations of a
feedback system change as a parameter K (real number) changes
• Root-locus (Evans): graph of all possible roots of 1 + KL(s) = 0 when K varies
from zero to infinity
Transfer Function (not including
disturbances W or V)

Characteristic equation of the TF

Root-locus form : It’s fundamental to convert the

𝑏 𝑠 characteristic equation into its
, 𝑤ℎ𝑒𝑟𝑒 𝐿 𝑠 = root-locus form in order to
𝑎 𝑠
perform root locus analysis.

We have several ways of expressing the root-locus equations.

Each one has exactly the same roots and is equivalent. From
now on, we refer to them as the root-locus form:
5
M3.2.1 Root-locus and
guidelines for determining it
𝑏 𝑠
, 𝑤ℎ𝑒𝑟𝑒 𝐿 𝑠 =
𝑎 𝑠
n (poles of L) ≥ m (zeros of L)
!
Example: Given 𝐺 𝑠 =
"("$%)
, 𝐷' 𝑠 = 𝐻 𝑠 = 1 , obtain the characteristic equation and put it in root locus form with A = K

6
M3.2.1 Root-locus and
guidelines for determining it
!
Example: Given 𝐺 𝑠 = "("$%) , 𝐷' 𝑠 = 𝐻 𝑠 = 1 , obtain the characteristic equation and put it in root locus form with A = K
Sol: 𝑏 𝑠 = 1
!
1 + 𝐾 · "("$!) = 0 𝑎 𝑠 = 𝑠 𝑠+1
! "
𝐿 𝑠 =# "

In order to study the behaviour of the feedback transfer function as gain K that we apply grows from 0 -> ∞, it’s usually useful
to keep track of the zeros and poles of the system from the beginning, as they’re going to be used repeatedly for some of the
equations involved in the root-locus analysis.

Thus, for the example provided we have:

• 𝑛( = 2 “nº of poles” • 𝑝* = {0, −1} “Location of poles”

• 𝑚) = 0 “nº of zeros” • 𝑧* = { } “Location of zeros” (none in this case)

If we plot out zeros and poles in the s-plane we obtain the following:

The Root-locus method is a graph of the roots of the quadratic equation:

a s +K·b s s(𝑠 + 1) + 𝐾·(1) = 0
Which we can solve for K
7
M3.2.1 Root-locus and
guidelines for determining it
There are some rules involved in the plotting of root-locus, which are going to be covered later.

In this specific case, as gain K grows the two poles come closer together

• Poles come together at s = -0.5 and bifurcate at ±90º from the real axis, becoming a conjugate pair when K>0.25

Probing the point:

8
M3.2.1 Root-locus and
guidelines for determining it
o Root-locus of a feedback system - 2nd example:
Root-locus graph example:
"$%
In this other specific case, if we plot the root locus of a feedback system with transfer function L(s) = "("$!)
:

Real roots become a

conjugate complex pair as K
Roots become real again as increases
K continues increasing

Roots starts (K = 0) at
the locations of the
poles of L (a(s) = 0)

The other root stops at the

zero (of L) location as K
tends to +∞
One root tends to −∞ as
9
K becomes huge
M3.2.1 Root-locus and
guidelines for determining it
o Formal definitions for a positive (180º) root-locus
• In the previous example, the gain K varies from 0 to +∞ (K is positive).
!
• In this case, L(s) is always real and negative since 𝐿 𝑠 = −
'

In other words, L(s) has a phase of 180º. This is called positive or 180º root-locus, which is
defined next:

• DEFINITION I: Test point

The root locus is the set of values of s for which 1 + 𝐾𝐿(𝑠) = 0 is satisfied, as K varies
from 0 all the way to +∞.
In most cases 1 + 𝐾 · 𝐿(𝑠) is the characteristic equation of the system, and in this case
the roots on the locus are the closed-loop poles of that system.

• DEFINITION II: Phase condition

The root locus of L(s) is the set of points in the s-plane where the phase of L(s) is 180º.
To test whether any chosen point in the s-plane is on the locus, define:
𝝍𝒊 as the angle to the test point from a zero; and 𝝓𝒊 as the angle to the test point from a
pole.
Then, the locus is expressed as those points in the s-plane where, ∀l:

10
M3.2.1 Root-locus and
guidelines for determining it
o Rules for determining a positive (180º) root-locus
• Evans defined 5 rules for determining a root-locus, which are the following:
§ RULE 1:
𝑛( branches of the locus start at the poles of L(s), and
𝑚( of these same branches end on the zeros of L(s).

§ RULE 2: The loci are on the real axis to the left of an odd
number of poles and zeros of L(s)

§ RULE 3: For large 𝑠 and 𝐾, (𝑛 − 𝑚) branches of the loci

are asymptotic to lines at angles 𝝓𝒍 radiating out from
the point s = 𝛼 on the real axis, where:

** For doing a root-locus quickly for seminar and lab session exercises, use rlocus(tf) function of MATLAB 11
M3.2.1 Root-locus and
guidelines for determining it
o Rules for determining a positive (180º) root-locus
§ RULE 4:
• The angle of departure of a branch of the locus • For poles repeated q times:
from a single pole is given by:

Where : 𝛴 𝝓𝒊 is sum of the angles to remaining poles Where l is a value in [1, 2 ,..., q-1 ,q ]
𝛴 𝝍𝒊 is sum of the angles to all the zeros

• Similarly, the angle(s) of arrival at a zero with

multiplicity “q” is given by:

Departure and arrival angles are found by selecting a test point (very) near to a
pole or zero

12
M3.2.1 Root-locus and
guidelines for determining it
o Rules for determining a positive (180º) root-locus
§ RULE 5:
The locus can have multiple roots at points on the locus of multiplicity q: The branches will approach a point of q
roots at angles separated by:

These same branches will depart at angles with same separation, forming an array of 2q rays.
If the point is on the real axis, then the orientation of this array is given by the real-axis rule. If the point is in the
complex plane, then the angle of departure rule must be applied.

• These 5 rules are defined here for sake of formality, however with only the 3 first
rules we are able to draw most root locus plots.
• Rule 4 is useful to understand how segments of the locus will depart
• Rule 5 is useful to interpret plots that come from computers, and to explain
“qualitative” changes in some loci when a zero or pole is moved
13
M3.2.1 Root-locus and guidelines
for determining it: Example H(s) = 1

Assuming that we have Dc(s) and G(s) in a system in a feedback loop such that:

What is the root-locus form of the equation with respect to c?

𝑠 + 2𝑐
Closed-loop transfer 𝐻𝐷%& 𝐺(𝑠) 𝑠(𝑠 + 𝑐) 𝑠 + 2𝑐 Characteristic equation: s(s+c) +s + 2c = 0
= =
function 1 + 𝐻𝐷%& 𝐺(𝑠) 1 + 𝑠 + 2𝑐 𝑠 𝑠 + 𝑐 + 𝑠 + 2𝑐
𝑠(𝑠 + 𝑐)
Characteristic equation
grouped by “c terms”

Divide everything by s2+s = s(s+1)

Characteristic equation
in root-locus form

14
M3.2.1 Root-locus and
guidelines for determining it
o Example of root-locus
• Plot the resulting root-locus
graph (MATLAB)

Point info

15
M3.2.1 Root-locus and
guidelines for determining it
o Example of root-locus
• Reminder: H(s) and poles location of
second-order system

Damping ratio Poles location

Undamped natural
frequency 16
M3.2.1 Root-locus and
guidelines for determining it
o Other examples of root-locus: L(s) has a particular graph
depending on locations of poles and zeros

17
Module M3.2

Module M3.2: Control design using root-locus

Contents
• M3.2.1: Root-locus and guidelines for determining it
• M3.2.2: Dynamic compensation design using root-locus

18
 M3.2.2 Dynamic compensation
design using root-locus
o Reminder: parameters of step response and relationships
with poles of first and second-order systems

Second order

First order

Damping ratio Poles location

Undamped natural
frequency 19
M3.2.2 Dynamic compensation
design using root-locus
o Lead compensation
• Approximates the function of PD control à speeds up time response by lowering rise
time and decreasing transient overshoot
, (Eq. 5.70)

20
M3.2.2 Dynamic compensation
design using root-locus
o Lead compensation: Example
𝟏
• Find a compensation for 𝑮 𝒔 = that will provide a step response with an overshot of no more than 20%
𝒔 𝒔)𝟏
and a rise time of no more than 0.3 s.
STEP 1: Find the closed-loop roots (conjugate complex pair) to meet the specification limits for
both ωn and ζ.
1.8 1.8 1.8
𝑡𝑟 ≤ 0.3 𝑠 𝑡* = → 𝜔+ = = = 6 𝑟𝑎𝑑/𝑠 → 𝜔+ ≥ 6 𝑟𝑎𝑑/𝑠
𝜔+ 𝑡* 0.3

2
! log 𝑀, log 0.2 2
𝑀𝑝 ≤ 0.2 𝑀, = 𝑒 ./0⁄ 1.0 →𝜁= 2 = ≈ 0.46 𝑟𝑎𝑑/𝑠 → 𝜁 ≥ 0.46
𝜋 2 + log 𝑀, 𝜋 2 + log 0.2 2

STEP 2: Create the root-locus without compensation (Dc(s) = K, proportional control)

𝐾
𝐷% (𝑠)𝐺(𝑠) 𝑠(𝑠 + 1) 𝐾 1
= = →𝐾+𝑠 𝑠+1 =0→1+𝐾 =0
1 + 𝐷% (𝑠)𝐺(𝑠) 1 + 𝐾 𝐾+𝑠 𝑠+1 𝑠 𝑠+1
𝑠(𝑠 + 1)
Extract the 21
characteristic Equation
M3.2.2 Dynamic compensation
design using root-locus
o Lead compensation example:
STEP 2: Create the root-locus without compensation: We apply proportional control: 𝐷𝑐 (𝑠) = 𝐾

As we increase K to reach 𝝎𝒏 ≥ 𝟔 𝒓𝒂𝒅/𝒔,

ζ decreases.

Then, there is no feasible solution that

meets both 𝝎𝒏 ≥ 6 rad/s and ζ ≥ 0.46

We cannot meet the

specifications with only
Proportional control

22
M3.2.2 Dynamic compensation
design using root-locus
o Lead compensation example
' "$+
STEP 3: More damping is required à Lead compensation (𝐷𝑐 𝑠 = "$(
, 𝑧 < 𝑝) with
• z will be a value between z = ¼ ωn and z = ωn ; and pick p = 10·z

We choose z = 2 and p = 20. Then the new feedback transfer function with lead
compensation added is:
𝐾(𝑠 + 2)
𝐷% (𝑠)𝐺(𝑠) 𝑠(𝑠 + 1)(𝑠 + 20) 𝐾(𝑠 + 2)
= =
1 + 𝐷% (𝑠)𝐺(𝑠) 1 + 𝐾(𝑠 + 2) 𝐾(𝑠 + 2) + 𝑠 𝑠 + 1 (𝑠 + 20)
𝑠(𝑠 + 1)(𝑠 + 20)

𝑠+2
𝐾 𝑠 + 2 + 𝑠 𝑠 + 1 𝑠 + 20 = 0 → 1+𝐾 =0
𝑠 𝑠 + 1 (𝑠 + 20)

Root Locus form 1+KL(s) = 0

23
M3.2.2 Dynamic compensation
design using root-locus
o Lead compensation example
\
• Find a compensation for G(s) = that will provide a step response with an
] ]^\
overshot of no more than 20% and a rise time of no more than 0.3 s
Steps 4 and 5: Examine the resulting root-locus; adjust p and z if needed.
If root-locus is OK (can meet specifications as is) we just select the value of K that meets the constraints

For K = 140, ωn ≥ 6 rad/s and ζ ≥ 0.46

For K = 140, specifications are met in root-locus,

but we have to check specifications in time-domain
(step response) since relationships between time-
domain parameters and system poles are
estimations for second-order systems,
(but in this case our system is third order)

24
M3.2.2 Dynamic compensation
design using root-locus
o Lead compensation example
• Find a compensation for G(s) = 1/(s(s+1)) that will provide a step response with an
overshot of no more than 20% and a rise time of no more than 0.3 s

Step 6: Verify that time-domain specifications are met

Overshot is 10% approximately

(𝑠 + 2)
𝐷, (𝑠) = 140
(𝑠 + 20) Rise time is 0.2 s approximately

25
M3.2.2 Dynamic compensation
design using root-locus
o Lag compensation
• Approximates the function of PI control à reduces steady-state error

Eq. (5.72)

Equations for system type:

26
M3.2.2 Dynamic compensation
design using root-locus
o Lag compensation example
• Improve steady-state error to polynomial inputs of previous example by a factor of 3

1 1 1
𝐸 𝑠 = 𝑅 𝑠 = <)1
1 + 𝐷% 𝑠 𝐺 𝑠 140 𝑠 + 2 𝑠
1+
𝑠 𝑠 + 1 𝑠 + 20
1 1
𝑘 = 0 → 𝑒"" = lim 𝑠𝐸 𝑠 = lim 𝑠 =0
"→> "→> 140 𝑠 + 2 𝑠 System Type 1
1+
𝑠 𝑠 + 1 𝑠 + 20
1 1 1 1 1
𝑘 = 1 → 𝑒"" = lim 𝑠𝐸 𝑠 = lim 𝑠 = lim = =
"→> "→> 140 𝑠 + 2 𝑠 2 "→> 140 𝑠 + 2 140 0 + 2 14
1+ 𝑠+ 0+
𝑠 𝑠 + 1 𝑠 + 20 𝑠 + 1 𝑠 + 20 0 + 1 0 + 20

1
Goal: ess = ?2

Step 1: Determine the amount of gain amplification to be contributed by the lag compensation at
low frequencies in order to achieve the desired error
𝑠+𝑧 Constant gain is K = 1 because
Gain amplification at low frequencies à z/p = 3 𝐷%2 𝑠 = we don’t want to affect
𝑠+𝑝
previous design 27
M3.2.2 Dynamic compensation
design using root-locus
o Lag compensation example
• Improve steady-state error to polynomial inputs of previous example by a factor of 3

Step 2: Select the value of z so it is approximately a factor of 100 to 200 smaller than the system
dominant natural frequency

Root-locus of lead- This makes the new zero-

compensated system pole pair have no effect at
frequencies near ωn, only
ωn = 10.7 rad/s à We choose z = 0.06 affects low frequencies (e.g.
steady-state)

Step 3: Select the value of p so that z/p is the desired gain at low frequencies

𝑠 + 0.06
Gain amplification at low frequencies à z/p = 3 à p = 0.02 𝐷%2 𝑠 =
𝑠 + 0.02
28
 M3.2.2 Dynamic compensation
design using root-locus
o Lag compensation example
• Improve steady-state error to polynomial inputs of previous example by a factor of 3

Step 4: Examine the resulting root-locus; adjust the lead compensator if needed

New root-locus
(lead and lag
compensation)
Previous root- à no change at
conjugate
locus (lead
complex poles
compensation)
à no change in
rise time and
overshot
expected

29
 M3.2.2 Dynamic compensation
design using root-locus
o Lag compensation example
• Improve steady-state error to polynomial inputs of previous example by a factor of 3

Step 5: Verify that time-domain specifications are met

New (lead
Previous (lead)
and lag)

30
M3.2.2 Dynamic compensation
design using root-locus
o Lag compensation example
• Improve steady-state error to polynomial inputs of previous example by a factor of 3

Now the steady-state error is

1 1 1
𝐸 𝑠 = 𝑅 𝑠 =
1 + 𝐷% 𝑠 𝐷%2 𝑠 𝐺 𝑠 140 𝑠 + 2 (𝑠 + 0.06) 𝑠 <)1
1+
𝑠 𝑠 + 1 𝑠 + 20 (𝑠 + 0.02)
1 1 1
𝑘 = 1 → 𝑒"" = lim 𝑠𝐸 𝑠 = lim 𝑠 2
= lim
"→> "→> 140 𝑠 + 2 (𝑠 + 0.06) 𝑠 "→> 140 𝑠 + 2 (𝑠 + 0.06)
1+ 𝑠+
𝑠 𝑠 + 1 𝑠 + 20 (𝑠 + 0.02) 𝑠 + 1 𝑠 + 20 (𝑠 + 0.02)

1 1 0.02 1
= = =
140 0 + 2 (0 + 0.06) 14 0.06 42
0+
0 + 1 0 + 20 (0 + 0.02)

Improved by a
factor of z/p = 3
31
Module M3.2
o Summary
• A root-locus is a graph of the values of s that are solutions to the equation 1 + KL(s) =
0 with respect to a real parameter K. When K > 0, s is on the locus if the phase of L(s)
is 180º, producing a 180º or positive root-locus. When K < 0, s is on the locus if the
phase of L(s) is 0º, producing a 0º or negative root- locus

• If KL(s) is the loop transfer function of a system with negative feedback, then the
characteristic equation of the closed-loop system is 1 + KL(s) = 0, and the root-locus
method displays the effect of changing the gain K on the closed-loop system roots

• A specific locus for a system in MATLAB notation can be plotted by rlocus function

32
Module M3.2
o Summary
• A working knowledge of how to determine a root-locus is useful for verifying
computer results and for suggesting design alternatives (RULES 1 and 2)

• Lead compensation (z < p) approximates the function of PD control: it speeds up time

response by lowering rise time and decreasing transient overshoot

• Lag compensation (z > p) approximates the function of PI control: it reduces steady-

state error

33