1) To understand and control complex systems, mathematical models are obtained by deriving differential equations that describe the dynamic behavior using physical laws. 2) For electrical systems, Kirchhoff's laws are used to derive differential equations relating voltage and current. For mechanical systems, Newton's laws relate forces and motions. 3) Most physical systems are nonlinear, so the equations are linearized around an operating point to obtain an approximate linear model for analysis and design of control systems. Linearization involves taking the derivative of the nonlinear equations to determine the slope at the operating point.

1. Mathematical Modeling of
Systems
Riza Muhida

2. • Introduction
• Differential Equations of Physical
Systems
• Linear Approximation of Physical Systems
Outlines

3. Introduction
Why do we need mathematical model
of a system to be controlled?

4. Introduction
Control Systems in Transportation
How can we control our car performances:
• Car speed
• Car position

5. Introduction
We should know the characteristic or behavior (input-
output characteristics) of the controlled object (plant or
system to be controlled) so that we can control the
plant successfully!
Modeling is the process of representing the behavior of
real system (physical system) by collection of
mathematical equation.

6. To understand and control complex systems, we must obtain
mathematical models of these systems. The term
mathematical model, in the control engineering perspective,
implies a set of differential equations that describe the
dynamic behavior of a process.
The set of differential equations that describe the
behavior of physical systems are typically obtained by
utilizing the physical laws of the process.
Introduction

7. Introduction
• The equations of the mathematical model may be
solved using mathematical tools such as the Laplace
Transform.
• Naturally the real system is nonlinear therefore before
solving the equations we usually need to linearize them.

8. • Introduction
• Differential Equations of Physical
Systems
• Linear Approximation of Physical Systems
Outlines

9. How to obtain the mathematical models?
• Physical laws of the process  Differential equations
Electrical system  Kirchhoff's laws
Mechanical system  Newton's laws
Fluid system  ?
Thermal System  ?
Differential Equation of a
Physical System

10. Electrical Systems
Differential Equation of a
Physical System
Electrical system elements are:
1. Resistor
2. Capacitor
3. Inductor
Kirchoff’s laws are used to model the electrical
system:
• Kirchoff’s Voltage Law (KVL)
• Kirchoff’s Current Law (KCL).
Electrical systems are concerned with the behavior of
three fundamental quantities: charge, current and
voltage.

11. Basic Electrical Element of Electrical Systems
Differential Equation of a
Physical System
Voltage current characteristics:
Resistor:
Capacitor:
Inductor:
Ri
v 
dt
dv
C
i 
dt
di
L
v 
R: resistance in Ohm
C: capacitance in F
L: inductance in H

12. Kirchoff’s Law
Differential Equation of a
Physical System
• Kirchoff’s Voltage Law (KVL)
0
1



N
i
i
v
The sum of voltage around a closed loop or path is
zero.

13. Kirchoff’s Law
Differential Equation of a
Physical System
• Kirchoff’s Voltage Law (KVL)
Approach:
• Assume the current direction for each element
• Assign the appropriate polarity for each element
• Assume the KVL loop either CW or CCW
• Start at any point and assign positive sign if the
assumed current direction is the same with KVL loop
and vice versa
• If a voltage result is negative, the voltage actually
drops in the opposite direction.

14. Kirchoff’s Law
Differential Equation of a
Physical System
• Kirchoff’s Voltage Law (KVL)

15. Kirchoff’s Law
Differential Equation of a
Physical System
• Kirchoff’s Current Law (KCL)
0
1



N
n
n
i
The sum of current flowing into a closed surface or
node is zero.

16. Kirchoff’s Law
Differential Equation of a
Physical System
• Kirchoff’s Current Law (KCL)
Approach:
• Assume the current directions arbitrary
• If a current is leaving a node, assign negative sign
• If the calculated current is negative sign, the current
actually flows in the opposite direction.

17. Example:
Differential Equation of a
Physical System
It consists of a source of current r(t), a resistor
characterized by it’s resistance R, a capacitor characterized
by it’s capacitance C, and an inductor with inductance L.
Differential equation of the RLC circuit can be obtained by
utilizing Kirchhoff’s laws, and the voltage-current
relationships for R, L, and C. Indeed, Kirchhoff’s current law
implies that
r(t) = ir + ic + il

18. Example:
Differential Equation of a
Physical System
where ir, ic, and il are currents through the resistor, the capacitor,
and the inductor respectively. On the other hand, Kirchhoff’s
voltage law implies in this particular case that the voltage v across
any of the elements R, L, or C is the same. We have the following
voltage-current relationships for R, L, and C:

19. Example:
Differential Equation of a
Physical System
Thus, we get the following equation of RLC circuit

20. Mechanical Systems
Differential Equation of a
Physical System
Mechanical systems are concerned with the behavior
of matter under the action of forces/torques.
Mechanical system elements are:
1. Mass element
2. Spring element
3. Damper element
Newton’s laws are used to model the Mechanical
system.
Forces cause translational motions and torques
causes rotational motions.

21. Mechanical Systems
-Newton’s Law-
Differential Equation of a
Physical System
• Force-motion relation
• Torque-motion relation
2
2
1 dt
x
d
m
F
N
i
i 


2
2
1 dt
d
J
T
N
i
i





22. Elements of Mechanical System
Translational
systems )
(
)
( t
bv
t
f 
dt
t
dx
b
t
f
)
(
)
( 
Differential Equation of a
Physical System

23. Mechanical System
Example: Mass-Spring System with a Damper
Differential Equation of a
Physical System
Consider the block of mass m attached to a fixed,
vertical support by a spring and a viscous damper.
Drive the model of the system

24. Mechanical System
Differential Equation of a
Physical System
The damping force Fd is assumed to be proportional to the
velocity ý and acts against the motion of the body, i.e.
Such a damping is called viscous damping. Using Newton’s
second law, we can write the equations of motion as follows:

25. Mechanical System
Assumption: wall friction is a viscous force
Example:
Differential Equation of a
Physical System

26. Mechanical System
Exercise:
Differential Equation of a
Physical System
Consider the block of mass m attached to a fixed,
vertical support by two springs and a damper. Drive the
model of the system
m
k1
k2
c F
x

27. Rotational Mechanical System
Rotational
systems
Differential Equation of a
Physical System

28. Rotational Mechanical System
Example:
Differential Equation of a
Physical System
Equations of some rotational mechanical systems can also be obtained
by direct application of Newton’s second law.

29. Rotational Mechanical System
Differential Equation of a
Physical System
T
K
D
J 



 )
( 2
1
1
1
1
1 


 


)
( 1
2
2
2
2
2 


 


 K
D
J 


In this system the two rotors with moments of inertia J1 and J2
are connected by a shaft having a rotational stiffness of K.
The viscous friction coefficients for the two rotors are D1 and
D2 respectively. The system driving torque T is applied to rotor
1. The system equations are obtained by summing the
torques on rotor 1 and rotor 2 respectively.
Applying Newton’s second law, we get

30. • Introduction
• Differential Equations of Physical
Systems
• Linear Approximation of Physical Systems
Outlines

31. • Control method discussed in this course is assumed
that the plant is a linear system.
• Definition. A system is linear, if and only if
– It obey the superposition principle,
– It obey the homogeneity principle
Linear Approximation of
Physical Systems
Linear System
)
(
)
(
)
( 2
1
2
1 u
G
u
G
u
u
G 


)
(
)
( u
G
u
G 
 

32. Linear Approximation of
Physical Systems
Linear System
kx
y 
Example
Examine the linearity of the following system:
2
x
y 

33. • All real systems are nonlinear.
• Almost all physical systems can be closely
approximately by linear models within some range of
the variables
• Linear models make the analysis and design much
simpler.
• Linear model can be obtained using Taylor’s series
about the operating point xo as follows.
    








 !
2
!
1
)
(
)
(
2
0
2
2
0
0
0
0
x
x
dx
g
d
x
x
dx
dg
x
g
x
g
y
x
x
x
x
Linear Approximation of
Physical Systems

34. • The slope at operation point
is a good approximation to the curve over small range
of (x-x0), the deviation from the operation point.
• Then, the reasonable approximation of nonlinear
system becomes:
   
0
0
0
)
( x
x
m
y
x
x
dx
dg
x
g
y o
x
x
o 






Linear Approximation of
Physical Systems
0
x
x
dx
dg

   
0
0 x
x
m
y
y 

 x
m
y 


or

35. Linear Approximation of
Physical Systems
Example: Consider the function: 2
)
( x
x
f 
Suppose we need to find a linear function which approximate f(x) near
the point x0 = 1. Clearly (see the figure), this linear approximation is
given as follows:
Figure: Linearization of
the function y = x2 near
the point x0 = 1

36. Linear Approximation of
Physical Systems
Example: Pendulum system
Derive the exact model, then linearize it for operation
point o = 0

37. Linear Approximation of
Physical Systems
Example: Pendulum system
Example: linearization of the pendulum equations. The pendulum is
described by the following equation
Therefore, the linearized equations of the pendulum about the point q = 0 is
as follows

38. Linear Approximation of
Physical Systems
The linear approximation is reasonable accurate for only
region near the operation point (-/4    /4).
Or if sin 1

39. • Dorf, Modern Control System
– Chapter 2, 2.1 – 2.3
• Nise, Control Systems Engineering
– Chapter 2, 2.4 – 2.6
Further Reading