Short Notes Mathematical Modelling of Systems - Control Systems - Electrical Engineering

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Mathematical Mo delling of Systems
Mathematical mo delling in con trol systems in v olv es represen ting a ph ysical system using mathematical
equations to analyze and design con trol strategies. The goal is to capture system dynamics accurately
for analysis and con troller design.
1. In t ro d uction to Mathematical Mo delling
A mathematical mo del describ es the relationship b et w een system inputs, outputs, and in ternal states
using equations. Mo dels can b e:
• Linear or Nonlinear : Linear mo dels assume prop ortionalit y , while nonlinear mo dels accoun t for
complex b eha viors.
• Time-In v arian t or Time-V arian t : Time-in v arian t systems ha v e constan t parameters.
• Con tin uous or Discrete : Con tin uous mo dels use differen tial equations; discrete mo dels use dif-
ference equations.
2. Mo delling Approac hes
T w o primary approac hes are used in con trol systems:
• T ransfer F unction Approac h : Suitable for linear, time-in v arian t (L TI) systems, using Laplace
transforms.
• State-Space Approac h : General represen tation for b oth linear and nonlinear systems, using
state v ariables.
3. Differen tial Equation Mo dels
Ph y sical systems are often describ ed b y differen tial equations deriv ed from ph ysical la ws (e.g., Newton’s
la ws, Kirc hhoff ’s la ws). F or an L TI system, the general form of a linear differen tial equation is:
a
n
d
n
y(t)
dt
n
+a
n-1
d
n-1
y(t)
dt
n-1
+···+a
0
y(t) = b
m
d
m
u(t)
dt
m
+···+b
0
u(t)
where y(t) is the output, u(t) is the input, and a
i
, b
j
are constan ts.
4. T ransfer F unction Mo del
The transfer function is the ratio of the Laplace transform of the output to the input, assuming zero
initial conditions:
G(s) =
Y(s)
U(s)
=
b
m
s
m
+b
m-1
s
m-1
+···+b
0
a
n
s
n
+a
n-1
s
n-1
+···+a
0
where s is the Laplace v ariable. It describ es system dynamics in the frequenc y domain.
Example : F or a mass-spring-damp er system with massm , damping co e?icien t c , and spring constan t
k , the differen tial equation is:
m
d
2
x(t)
dt
2
+c
dx(t)
dt
+kx(t) = f(t)
T aking the Laplace transform:
G(s) =
X(s)
F(s)
=
1
ms
2
+cs+k
5. Sta te-Space Mo del
State-space represen tation uses a set of first-order differen tial equations. F or an L TI system:
? x(t) =Ax(t)+Bu(t)
y(t) =Cx(t)+Du(t)
where:
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Mathematical Mo delling of Systems
Mathematical mo delling in con trol systems in v olv es represen ting a ph ysical system using mathematical
equations to analyze and design con trol strategies. The goal is to capture system dynamics accurately
for analysis and con troller design.
1. In t ro d uction to Mathematical Mo delling
A mathematical mo del describ es the relationship b et w een system inputs, outputs, and in ternal states
using equations. Mo dels can b e:
• Linear or Nonlinear : Linear mo dels assume prop ortionalit y , while nonlinear mo dels accoun t for
complex b eha viors.
• Time-In v arian t or Time-V arian t : Time-in v arian t systems ha v e constan t parameters.
• Con tin uous or Discrete : Con tin uous mo dels use differen tial equations; discrete mo dels use dif-
ference equations.
2. Mo delling Approac hes
T w o primary approac hes are used in con trol systems:
• T ransfer F unction Approac h : Suitable for linear, time-in v arian t (L TI) systems, using Laplace
transforms.
• State-Space Approac h : General represen tation for b oth linear and nonlinear systems, using
state v ariables.
3. Differen tial Equation Mo dels
Ph y sical systems are often describ ed b y differen tial equations deriv ed from ph ysical la ws (e.g., Newton’s
la ws, Kirc hhoff ’s la ws). F or an L TI system, the general form of a linear differen tial equation is:
a
n
d
n
y(t)
dt
n
+a
n-1
d
n-1
y(t)
dt
n-1
+···+a
0
y(t) = b
m
d
m
u(t)
dt
m
+···+b
0
u(t)
where y(t) is the output, u(t) is the input, and a
i
, b
j
are constan ts.
4. T ransfer F unction Mo del
The transfer function is the ratio of the Laplace transform of the output to the input, assuming zero
initial conditions:
G(s) =
Y(s)
U(s)
=
b
m
s
m
+b
m-1
s
m-1
+···+b
0
a
n
s
n
+a
n-1
s
n-1
+···+a
0
where s is the Laplace v ariable. It describ es system dynamics in the frequenc y domain.
Example : F or a mass-spring-damp er system with massm , damping co e?icien t c , and spring constan t
k , the differen tial equation is:
m
d
2
x(t)
dt
2
+c
dx(t)
dt
+kx(t) = f(t)
T aking the Laplace transform:
G(s) =
X(s)
F(s)
=
1
ms
2
+cs+k
5. Sta te-Space Mo del
State-space represen tation uses a set of first-order differen tial equations. F or an L TI system:
? x(t) =Ax(t)+Bu(t)
y(t) =Cx(t)+Du(t)
where:
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• x(t) : State v ector
• u(t) : Input v ector
• y(t) : Output v ector
• A : System matrix
• B : Input matrix
• C : Output matrix
• D : F eedthrough matrix
Example : F or the mass-sp ring-damp er system, define states x
1
= x (p osition), x
2
= ? x (v elo cit y):
[
? x
1
? x
2
]
=
[
0 1
-
k
m
-
c
m
][
x
1
x
2
]
+
[
0
1
m
]
f(t)
y(t) =
[
1 0
]
[
x
1
x
2
]
6. Blo c k Diagram Represen tation
Systems can b e visualized using blo c k diagrams, where blo c ks represen t comp onen ts (e.g., in tegrators,
gains) and arro ws indicate signal flo w. The transfer function or state-space mo del can b e deriv ed from
the blo c k diagram.
7. Mo del Simplification
T o simplify mo dels:
• Neglect insignifican t dynamics (e.g., high-frequency mo des).
• Linearize nonlinear systems around op erating p oin ts using T a ylor series:
f(x,u)˜ f(x
0
,u
0
)+
?f
?x
(x-x
0
)+
?f
?u
(u-u
0
)
8. Appli cations
Mathematical mo dels are used for:
• System analysis (stabilit y , p erformance).
• Con troller design (PID, state feedbac k).
• Sim ulation and prediction of system b eha vior.
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