Formula Sheets: Mathematical Modelling of Systems | Control Systems - Electrical Engineering (EE) PDF Download

 Page 1


Control S ystems F ormula Sheet:
Mathematical Modelling of S ystems
1. Basics of Mathematical Modelling
• S ystem Model : Mathematica l representation of a ph ysical system using dif-
ferential equations, tr ansfer functions, or state-space models.
• Linear Time-Invariant (L TI) S ystem : S ystem described b y linear differential
equations with constant coefficients.
• Input-Output Relationship : Relates input u(t) to output y(t) through system
dynamics.
2. Differential Equation Modelling
• Gener al F orm of Linear Differential Equation :
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)
wherey(t) is output,u(t) is input,a
i
andb
i
are constants,n=m .
• Char acteristic Equation : From homogeneous part:
a
n
s
n
+a
n-1
s
n-1
+···+a
0
= 0
Roots determine system poles.
3. Tr ansfer Function Modelling
• Tr ansfer Function : Ratio of Laplace tr ansform of output to 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
• Poles : Roots of denominator (a
n
s
n
+···+a
0
= 0 ).
• Zeros : Roots of numer ator (b
m
s
m
+···+b
0
= 0 ).
• S ystem Order : Highest power ofs in denominator (n ).
1
Page 2


Control S ystems F ormula Sheet:
Mathematical Modelling of S ystems
1. Basics of Mathematical Modelling
• S ystem Model : Mathematica l representation of a ph ysical system using dif-
ferential equations, tr ansfer functions, or state-space models.
• Linear Time-Invariant (L TI) S ystem : S ystem described b y linear differential
equations with constant coefficients.
• Input-Output Relationship : Relates input u(t) to output y(t) through system
dynamics.
2. Differential Equation Modelling
• Gener al F orm of Linear Differential Equation :
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)
wherey(t) is output,u(t) is input,a
i
andb
i
are constants,n=m .
• Char acteristic Equation : From homogeneous part:
a
n
s
n
+a
n-1
s
n-1
+···+a
0
= 0
Roots determine system poles.
3. Tr ansfer Function Modelling
• Tr ansfer Function : Ratio of Laplace tr ansform of output to 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
• Poles : Roots of denominator (a
n
s
n
+···+a
0
= 0 ).
• Zeros : Roots of numer ator (b
m
s
m
+···+b
0
= 0 ).
• S ystem Order : Highest power ofs in denominator (n ).
1
4. State-Space Modelling
• State-Space Model :
? x(t) =Ax(t)+Bu(t)
y(t) =Cx(t)+Du(t)
where:
– x(t) : State vector (n×1 ).
– u(t) : Input vector (m×1 ).
– y(t) : Output vector (p×1 ).
– A : State matrix (n×n ).
– B : Input matrix (n×m ).
– C : Output matrix (p×n ).
– D : F eedthrough matrix (p×m ).
• Tr ansfer Function from State-Space :
G(s) =C(sI-A)
-1
B +D
5. Mechanical S ystems Modelling
• Tr anslational S ystem (Mass-Spring-Damper):
M
d
2
x(t)
dt
2
+B
dx(t)
dt
+Kx(t) =f(t)
whereM is mass,B is damping coefficient, K is spring constant,f(t) is external
force.
• Tr ansfer Function (displacementx(t) to forcef(t) ):
G(s) =
X(s)
F(s)
=
1
Ms
2
+Bs+K
• Rotational S ystem (Moment of Inertia-Damper-Spring):
J
d
2
?(t)
dt
2
+B
d?(t)
dt
+K?(t) =T(t)
where J is moment of inertia, B is damping coefficient, K is torsional spring
constant,T(t) is torque.
• Tr ansfer Function (angle?(t) to torqueT(t) ):
G(s) =
T(s)
T(s)
=
1
Js
2
+Bs+K
2
Page 3


Control S ystems F ormula Sheet:
Mathematical Modelling of S ystems
1. Basics of Mathematical Modelling
• S ystem Model : Mathematica l representation of a ph ysical system using dif-
ferential equations, tr ansfer functions, or state-space models.
• Linear Time-Invariant (L TI) S ystem : S ystem described b y linear differential
equations with constant coefficients.
• Input-Output Relationship : Relates input u(t) to output y(t) through system
dynamics.
2. Differential Equation Modelling
• Gener al F orm of Linear Differential Equation :
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)
wherey(t) is output,u(t) is input,a
i
andb
i
are constants,n=m .
• Char acteristic Equation : From homogeneous part:
a
n
s
n
+a
n-1
s
n-1
+···+a
0
= 0
Roots determine system poles.
3. Tr ansfer Function Modelling
• Tr ansfer Function : Ratio of Laplace tr ansform of output to 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
• Poles : Roots of denominator (a
n
s
n
+···+a
0
= 0 ).
• Zeros : Roots of numer ator (b
m
s
m
+···+b
0
= 0 ).
• S ystem Order : Highest power ofs in denominator (n ).
1
4. State-Space Modelling
• State-Space Model :
? x(t) =Ax(t)+Bu(t)
y(t) =Cx(t)+Du(t)
where:
– x(t) : State vector (n×1 ).
– u(t) : Input vector (m×1 ).
– y(t) : Output vector (p×1 ).
– A : State matrix (n×n ).
– B : Input matrix (n×m ).
– C : Output matrix (p×n ).
– D : F eedthrough matrix (p×m ).
• Tr ansfer Function from State-Space :
G(s) =C(sI-A)
-1
B +D
5. Mechanical S ystems Modelling
• Tr anslational S ystem (Mass-Spring-Damper):
M
d
2
x(t)
dt
2
+B
dx(t)
dt
+Kx(t) =f(t)
whereM is mass,B is damping coefficient, K is spring constant,f(t) is external
force.
• Tr ansfer Function (displacementx(t) to forcef(t) ):
G(s) =
X(s)
F(s)
=
1
Ms
2
+Bs+K
• Rotational S ystem (Moment of Inertia-Damper-Spring):
J
d
2
?(t)
dt
2
+B
d?(t)
dt
+K?(t) =T(t)
where J is moment of inertia, B is damping coefficient, K is torsional spring
constant,T(t) is torque.
• Tr ansfer Function (angle?(t) to torqueT(t) ):
G(s) =
T(s)
T(s)
=
1
Js
2
+Bs+K
2
6. Electrical S ystems Modelling
• RLC Series Circuit :
L
di(t)
dt
+Ri(t)+
1
C
?
i(t)dt =v(t)
where L is inductance, R is resistance, C is capacitance, v(t) is voltage, i(t) is
current.
• Tr ansfer Function (currentI(s) to voltageV(s) ):
G(s) =
I(s)
V(s)
=
1
Ls+R+
1
Cs
• Tr ansfer Function (output voltage across capacitor to input voltage):
G(s) =
V
C
(s)
V(s)
=
1
Cs
Ls+R+
1
Cs
=
1
LCs
2
+RCs+1
7. Block Diagr am Representation
• Series Connection : G
1
(s)G
2
(s) .
• Par allel Connection : G
1
(s)+G
2
(s) .
• Negative F eedback S ystem :
T(s) =
G(s)
1+G(s)H(s)
whereG(s) is forward path,H(s) is feedback path.
3