Linear control systems Automatic Control

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1. Linear Systems Theory
Ramkrishna Pasumarthy
Department of Electrical Engineering,
Indian Institute of Technology Madras

2. Module 1
Lecture 1
Introduction to Linear Systems
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 1/18

3. Engineering Tools
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 2/18

4. The Importance of Math
On Teaching Mathematics https://www.uni-muenster.de/Physik.TP/ munsteg/arnold.html
▶Mathematics is a part of physics.
▶Physics is an experimental science, a part of natural science.
▶Mathematics is the part of physics where experiments are cheap.
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 3/18

5. The Importance of Math
On Teaching Mathematics https://www.uni-muenster.de/Physik.TP/ munsteg/arnold.html
▶Mathematics is a part of physics.
▶Physics is an experimental science, a part of natural science.
▶Mathematics is the part of physics where experiments are cheap.
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 3/18

6. The Importance of Math
On Teaching Mathematics https://www.uni-muenster.de/Physik.TP/ munsteg/arnold.html
▶Mathematics is a part of physics.
▶Physics is an experimental science, a part of natural science.
▶Mathematics is the part of physics where experiments are cheap.
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 3/18

7. System Modelling

8. What is a Model?
▶Models are mathematical representations to help understand dynamical behaviour
of systems.
▶Can describe behaviour of a physical process. Eg. How long would it take for a fan to
come to a halt, once switched off?
▶Can predict response of a system to certain inputs. Eg.- Effect of GST on economy.
▶This is important because in most cases the system response is not instantaneous,
but evolves with time.
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 4/18

9. What is a Good Model?
▶Depends on the purpose of the model.
▶Should incorporate the factors that govern the dynamics or evolution of a system
with time.
▶The same system can be represented by different models.
Eg.- A lumped and distributed parameter model of a transmission line.
▶The model abstraction must be compact, as opposed to a rule based formulation.
▶How accurate should the model be?
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 5/18

10. Some Basic Modelling Elements
1.Inductance:stores energy in a magnetic field when electric current flows through it.
ϕ/i;v=
_
ϕ;v/
di
dt
The energy stored in an inductorW=
∫
ϕ
0
i(ϕ)dϕ.
In the linear caseW=
1
2
ϕ
2
L
=
1
2
Li
2
2.Mass: An inertia element:Newtons second law
For a point massM>0, moving in thexdirection,p=Mvin the nonrelativistic case.
From the Newtons second lawF=
dp
dt
.
If the mass is moved by a force, workdone is
Fdx=
dp
dt
dx=vdp=
p
M
dp
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 6/18

11. Some Basic Modelling Elements
The Kinetic Energy=
∫
p
0
p
M
dp=
p
2
2M
.
Figure 1:Kinetic energy and co-kinetic energy
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 7/18

12. Examples of Systems and their
Models

13. A Simple Mechanical System
▶One of the most simple models is a mass-spring-damper system.
▶Unforced system - Model equation:
mx+d_x+kx=0:
▶mrepresents the mass,kthe spring constant, anddis the damping coefficient.
▶x2Ris a variable which denotes the position of the mass w.r.t its rest position.
▶_xandxrespectively denoting the velocity and acceleration respectively.
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 8/18

14. A Simple Mechanical System
▶This system can also be written as
x=x1
_x1=x2
_x2=
d
m
x2
k
m
x1
▶Or in a structured way of the form
[
_x1
_x2
]
=
[
0 1

k
m

d
m
]
|{z}
The state matrix A
[
x1
x2
]
|{z}
state vectorx
_x=Ax
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P. 9/18

15. A Simple Mechanical System0 2 4 6 8 10 12 14
-4
-2
0
2
4
t
Position
Velocity
Damped Oscillator: Under Damped Case
Figure 2:Under damped response of the mass
spring damper system-2 -1 0 1 2
-4
-2
0
2
4
x
v
Phase Portrait: Under Damped
Figure 3:Phase portrait of the mass spring damper
system
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.10/18

16. A Simple Mechanical System0 1 2 3 4 5
-2
-1
0
1
2
t
Position
Velocity
Damped Oscillator: Critically Damped Case
Figure 4:Critically damped response of the
mass spring damper system-2 -1 0 1 2
-2
-1
0
1
2
x
v
Phase Portrait: Critically Damped Figure 5:Phase portrait of the mass spring damper
system
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.11/18

17. A Simple Mechanical System0 1 2 3 4 5
-2
-1
0
1
2
t
Position
Velocity
Damped Oscillator: Over Damped Case
Figure 6:Over damped response of the mass
spring damper system-2 -1 0 1 2
-2
-1
0
1
2
x
v
Phase Portrait: Over Damped Figure 7:Phase portrait of the mass spring damper
system
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.12/18

18. A Simple Electrical System
▶Series RLC Circuit with voltage source
▶The governing equations of the system are
V=RiL+L
diL
dt
+vC
iL=C
dvC
dt
▶Written in the state space form as
[
dvC
dt
diL
dt
]
=
[
0
1
C

1
L

R
L
]
|{z}
A
[
vC
iL
]
|{z}
x
+
[
0
1
L
]
|{z}
The input matrix B
V(The Control input u)
_x=Ax+Bu
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.13/18

19. A Simple Electrical System
Impulse Response0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time [s]
-1
0
1
2
3
4
5
Current [A]
10
-5 Current for Impulse Input
Figure 8:Current through the RLC circuit0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time [s]
-0.5
0
0.5
1
1.5
2
2.5
Voltage [V]
10
-3Voltage across Capacitor for Impulse Input Figure 9:Voltage across the capacitor
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.14/18

20. A Simple Electrical System
Step Response0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time [s]
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
Current [A]
10
-3 Current for Step Input
Figure 10:Current through the RLC circuit0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time [s]
0
0.2
0.4
0.6
0.8
1
1.2
Voltage [V]
Voltage across Capacitor for Step Input Figure 11:Voltage across the capacitor
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.15/18

21. A Simple Electrical System
Response to Sinusoidal Input0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time [s]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Current [A]
10
-4 Current for Sinusoidal Input
Figure 12:Current through the RLC circuit0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time [s]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Voltage [V]
Voltage across Capacitor for Sinusoidal Input Figure 13:Voltage across the capacitor
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.16/18

22. Predator-Prey Model
▶An ecological system where one species feeds on the other.
▶Model describes the evolution of population of both predator and prey.
▶Eg.- Prey: Small FishS(k); Predator: Big FishB(k)
Model Equations:
S(k+1) =S(k) +aS(k)cS(k)B(k)
B(k+1) =B(k)bB(k) +dS(k)B(k)
▶k represents time instances anda;b;c;dare constants
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.17/18

23. Predator-Prey Model
▶The populations of both the small and Big fish are oscillatory in nature.0 10 20 30 40 50 60 70 80 90
Years
0
5
10
15
20
25
30
Population in Thousands
Predator Prey Model
Prey
Predator
Figure 14:Population of predator and prey for 90 years
Linear Systems Theory Module 1 Lecture 1 Ramkrishna P.18/18