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A little control engineering (for those who are interested)
This summary is no substitute for study the theory of closed-loop control but is intended to
provide information on some relationships and help with practical application.
Time equation and frequency response
Every linear system can be described by one or more coupled linear differential equations.
The solution to a non-homogeneous linear differential equation is composed of a
homogeneous and a non-homogeneous partial solution.
If the input variable of a linear system is a harmonic oscillation in a particular case, the non-
homogeneous partial solution that characterizes the steady state is also a harmonic
oscillation.
Time equation:
Input variable: x
e
= ^x
e
* sin ( t) ^x = peak value of x
Solution method for the output variable: x
a
= ^x
a
* sin ( t + )
Or, in complex notation: x
e
= ^x
e
* e
t
; x
a
= ^x
a
* e
j( t + )
For an nth-order linear differential equation, the following applies:
x
a
+ a
1
* dx
a
/ dt + a
2
* d
2
x
a
/ dt
2
+...+ a
n
* d
n
x
a
/ dt
n
= k * x
e
Where d / dt
j = p, d
2
/ dt
2
(j )
2
= p
2
etc applies:
x
a
* e
j( t + )
*(1 + pa
1
+ p
2
a
2
+ ...+ p
n
a
n
) = k^x
e
e
t
Further substitution and transformation then yields the following:
G(p) = x
a
/ x
e
= k / (1 + pa
1
+ p
2
a
2
+....+p
n
a
n
)
G(p) is termed the frequency response.
The frequency response of a linear system is the relation between the non-homogeneous
partial solution of the output variable to that of the input variable, where the input variable is a
harmonic oscillation.
As can be seen above, the frequency response is obtained from the differential equation by
substituting p.
The advantage of the frequency response is that only the fundamental operation of
arithmetic: addition, subtraction, multiplication and division have to be applied, unlike solving
differential equations.
Frequency response and transfer function
The frequency response equation describes the response of a system for a harmonic
oscillation. It is therefore a special case of an equation, similar to the differential equation,
that describes the response of a system for any input variables. Such an equation can be
obtained by applying the Laplace transform to the system equations.
By means of the integral F(s):
a time-dependent function f(t) is transformed into a function F(s) depending on the complex
variable
s = + j .
In this case, f(t) is the original function and F(s) is the Laplace transform. Applying the
Laplace function to an original function in the form of a linear differential equation results in a
Laplace transform in the form of a linear algebraic equation. By solving this algebraic
equation and re-transformation to the original, this solution can be calculated as a differential
equation. Please refer to the extensive literature on this topic.
Using an example of a second-order linear differential equation:
x
a
+ a
1
*d/dt*x
a
+ a
2
*d
2
/dt
2
*x
a
= k*x
e
and the addition rule:
a
1
f
1
(t) +…+a
n
f
n
(t)
a
1
f
1
(s) +…+ a
n
f
n
(s) and the differential rule:
d
n
/dt
n
*f(t)
s
n
*f(s) + s
n-1
*f(0) + s
n-2
*f’(0) +...+ f
(n-1)
(0)
dt
*
)
t
(
f
)
s
(
F
e
st
0
