1. Automatic Control Theory CSE 322
Lec. 3
Modeling of Dynamic System

2. Quiz (10 mins)
Find the transfer function of the following block diagrams

3. 1. Moving pickoff point A behind block
Solution:
1. Moving pickoff point A behind block

4. 2. Eliminate loop I and Simplify
feedback
Not feedback

5. 3. Eliminate loop II & IIII

6. Types of Systems
Static System: If a system does not change with time, it is called a static system.
The o/p depends on the i/p at the present time only
Dynamic System: If a system changes with time, it is called a dynamic system.
The o/p depends on the i/p at the present time and the past time. (integrator, differentiator).

7. Dynamic Systems
A system is said to be dynamic if its current output may depend on the past history as well as the present values of the input variables.
Mathematically,
Example: A moving mass M y u
Model: Force=Mass X Acceleration

8. Ways to Study a System System Experiment with actual System
Experiment with a model of the System
Physical Model
Mathematical Model
Analytical Solution
Simulation
Frequency Domain
Time Domain
Hybrid Domain

9. Modeling
Mathematical.
State space.

10. What is Mathematical Model?
A set of mathematical equations (e.g., differential equs.) that describes the input-output behavior of a system.
What is a model used for?
Simulation
Prediction/Forecasting
Prognostics/Diagnostics
Design/Performance Evaluation
Control System Design

11. 1. Electrical Systems

12. Basic Elements of Electrical Systems
The time domain expression relating voltage and current for the resistor is given by Ohm’s law i.e.
The Laplace transform of the above equation is

13. Basic Elements of Electrical Systems
The time domain expression relating voltage and current for the Capacitor is given as:
The Laplace transform of the above equation (assuming there is no charge stored in the capacitor) is

14. Basic Elements of Electrical Systems
The time domain expression relating voltage and current for the inductor is given as:
The Laplace transform of the above equation (assuming there is no energy stored in inductor) is

15. V-I and I-V relations Symbol V-I Relation I-V Relation Resistor
Component
Symbol
V-I Relation
I-V Relation
Resistor
Capacitor
Inductor

16. Example-1
The two-port network shown in the following figure has vi(t) as the input voltage and vo(t) as the output voltage. Find the transfer function Vo(s)/Vi(s) of the network. C
i(t)
vi( t)
vo(t)

17. Example-1
Taking Laplace transform of both equations, considering initial conditions to zero.
Re-arrange both equations as:

18. Example-1
Substitute I(s) in equation on left

19. Example-1
The system has one pole at

20. Example-2
Design an Electrical system that would place a pole at -4 if added to another system.
System has one pole at
Therefore, C
i(t)
vi( t)
vo(t)

21. Transform Impedance (Resistor)
iR(t)
vR(t) + -
IR(S)
VR(S)
ZR = R
Transformation

22. Transform Impedance (Inductor)
iL(t)
vL(t) + -
IL(S)
VL(S)
ZL=LS

23. Transform Impedance (Capacitor)
ic(t)
vc(t) + -
Ic(S)
Vc(S)
ZC(S)=1/CS

24. Equivalent Transform Impedance (Series)
Consider following arrangement, find out equivalent transform impedance. L C R

25. Equivalent Transform Impedance (Parallel)

26. 2. Mechanical Systems

27. Mechanical Systems Part-I: Translational Mechanical System
Part-II: Rotational Mechanical System
Part-III: Mechanical Linkages

28. Basic Types of Mechanical Systems
Translational
Linear Motion
Rotational
Rotational Motion

29. 1. Translational Mechanical Systems

30. Basic Elements of Translational Mechanical Systems
Translational Spring i)
Translational Mass
ii)
Translational Damper
iii)

31. Translational Spring
A translational spring is a mechanical element that can be deformed by an external force such that the deformation is directly proportional to the force applied to it.
Translational Spring i)
Circuit Symbols
Translational Spring

32. Translational Spring If F is the applied force
Then is the deformation if
Or is the deformation.
The equation of motion is given as
Where is stiffness of spring expressed in N/m

33. Translational Mass Translational Mass is an inertia element.
A mechanical system without mass does not exist.
If a force F is applied to a mass and it is displaced to x meters then the relation b/w force and displacements is given by Newton’s law.
Translational Mass
ii) M

34. Common Uses of Dashpots
Door Stoppers
Vehicle Suspension
Bridge Suspension
Flyover Suspension

35. Translational Damper
Where b is damping coefficient (N/ms-1).

36. Modeling a Simple Translational System
Example-1: Consider a simple horizontal spring-mass system on a frictionless surface, as shown in figure below. or

37. Example-2 M Consider the following system (friction is negligible)
Free Body Diagram M
Where and are force applied by the spring and inertial force respectively.

38. Example-2 M Then the differential equation of the system is:
Taking the Laplace Transform of both sides and ignoring initial conditions we get

39. Example-2
The transfer function of the system is if

40. Example-2
The pole-zero map of the system is

41. Example-3
Consider the following system
Mechanical Network ↑ M

42. Example-3
Mechanical Network ↑ M
At node
At node

43. Example-4: Automobile Suspension

44. Automobile Suspension

45. Automobile Suspension
Taking Laplace Transform of the equation (2)

46. Example-5: Train Suspension
Car Body
Bogie-2
Bogie
Frame
Bogie-1
Wheelsets
Primary
Suspension
Secondary

47. Example-5: Train Suspension