Mechanical Engineering Modeling and Control of Dynamic electro Mechanical System

# Mechanical Engineering Modeling and Control of Dynamic electro Mechanical System

Page 1

NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
State Space Approach in Modelling State Space Approach in Modelling
D Bi h kh Bh tt h Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
Page 2

NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
State Space Approach in Modelling State Space Approach in Modelling
D Bi h kh Bh tt h Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
Answer of the Last Assignment g
Following Mason’s law, there are two forward paths in the SFG:
T
1
=  G
1
G
2
G
3
and
T
2
=  G
4
There are four loops:
L
1
= -G
1
H
1
L
2
=  -G
3
H
2
L
3
= -G
1
G
2
G
3
H
3
L
3
G
1
G
2
G
3
H
3
L
4
=  -G
4
H
3
?= 1 –(L
1
+  L
2
+ L
3
+ L
4
) + L
1
L
2
?
1
= 1
?
2
=  1
h ffi ldb d ( )/
2
Hence, the transfer function could be expressed as (T
1
+ T
2
)/ ?
Joint Initiative of IITs and IISc -Funded by MHRD
Page 3

NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
State Space Approach in Modelling State Space Approach in Modelling
D Bi h kh Bh tt h Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
Answer of the Last Assignment g
Following Mason’s law, there are two forward paths in the SFG:
T
1
=  G
1
G
2
G
3
and
T
2
=  G
4
There are four loops:
L
1
= -G
1
H
1
L
2
=  -G
3
H
2
L
3
= -G
1
G
2
G
3
H
3
L
3
G
1
G
2
G
3
H
3
L
4
=  -G
4
H
3
?= 1 –(L
1
+  L
2
+ L
3
+ L
4
) + L
1
L
2
?
1
= 1
?
2
=  1
h ffi ldb d ( )/
2
Hence, the transfer function could be expressed as (T
1
+ T
2
)/ ?
Joint Initiative of IITs and IISc -Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
The Lecture Contains
? State Space Modeling
?EOM of a SDOF system in State Space Form
?Response of a State Space System
?Examples to Solve
Joint Initiative of IITs and IISc - Funded by MHRD
Page 4

NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
State Space Approach in Modelling State Space Approach in Modelling
D Bi h kh Bh tt h Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
Answer of the Last Assignment g
Following Mason’s law, there are two forward paths in the SFG:
T
1
=  G
1
G
2
G
3
and
T
2
=  G
4
There are four loops:
L
1
= -G
1
H
1
L
2
=  -G
3
H
2
L
3
= -G
1
G
2
G
3
H
3
L
3
G
1
G
2
G
3
H
3
L
4
=  -G
4
H
3
?= 1 –(L
1
+  L
2
+ L
3
+ L
4
) + L
1
L
2
?
1
= 1
?
2
=  1
h ffi ldb d ( )/
2
Hence, the transfer function could be expressed as (T
1
+ T
2
)/ ?
Joint Initiative of IITs and IISc -Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
The Lecture Contains
? State Space Modeling
?EOM of a SDOF system in State Space Form
?Response of a State Space System
?Examples to Solve
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
State-Space Modelling
The stateof a model of a dynamic system is a set of
independent physical quantities, the specification of which (in
the absence of excitation) completely determines the future
positions of the system
Dynamics describes how the state evolves. The dynamicsof a
model is an update rule for the system state that describes
howthestateevolves asafunctiononthecurrentstateand how the state evolves, as a function on the current state and
any external inputs
.
.
1
x
?
?
?
?
?
?
) ( ) (
.
.
2 .
t U B t X A
x
X ? ?
?
?
?
?
?
?
?
?
?
?
?
Joint Initiative of IITs and IISc -Funded by MHRD
.
x
n
?
?
?
?
?
?
Page 5

NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
State Space Approach in Modelling State Space Approach in Modelling
D Bi h kh Bh tt h Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
Answer of the Last Assignment g
Following Mason’s law, there are two forward paths in the SFG:
T
1
=  G
1
G
2
G
3
and
T
2
=  G
4
There are four loops:
L
1
= -G
1
H
1
L
2
=  -G
3
H
2
L
3
= -G
1
G
2
G
3
H
3
L
3
G
1
G
2
G
3
H
3
L
4
=  -G
4
H
3
?= 1 –(L
1
+  L
2
+ L
3
+ L
4
) + L
1
L
2
?
1
= 1
?
2
=  1
h ffi ldb d ( )/
2
Hence, the transfer function could be expressed as (T
1
+ T
2
)/ ?
Joint Initiative of IITs and IISc -Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
The Lecture Contains
? State Space Modeling
?EOM of a SDOF system in State Space Form
?Response of a State Space System
?Examples to Solve
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
State-Space Modelling
The stateof a model of a dynamic system is a set of
independent physical quantities, the specification of which (in
the absence of excitation) completely determines the future
positions of the system
Dynamics describes how the state evolves. The dynamicsof a
model is an update rule for the system state that describes
howthestateevolves asafunctiononthecurrentstateand how the state evolves, as a function on the current state and
any external inputs
.
.
1
x
?
?
?
?
?
?
) ( ) (
.
.
2 .
t U B t X A
x
X ? ?
?
?
?
?
?
?
?
?
?
?
?
Joint Initiative of IITs and IISc -Funded by MHRD
.
x
n
?
?
?
?
?
?
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4
Wh tlkbt lt hi l t dldb diff ti l When we talk about electro-mechanical systems modeled by differential
equations, such as masses and springs, electric circuits or satellites (rigid
bodies) rotating in space, we can attach some additional intuition: the
variables inthestate shouldbeadequate tospecifytheenergyofthe variables in the state should be adequate to specify the energyof the
system.
For example, take a ball free-falling to earth: we can specify the position
of the ball by specifying the height (h) above the ground, but we also
need to include the velocity of the ball (dh/dt) to specify the total energy
(E = 1/2*m*(dh/dt)^2 + mgh). Therefore, the state of the ball is (h,dh/dt).
Joint Initiative of IITs and IISc -Funded by MHRD