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==Linear System== | ==Linear System== | ||
I'll choose the system | I'll choose the system | ||
| − | <math> | + | <math>x(t) = y(t^2)</math> |
Let's choose <math>A=2</math>, <math>B=3</math>, <math>y1(t)=1+t </math>, and <math> y2(t)=e^t</math> | Let's choose <math>A=2</math>, <math>B=3</math>, <math>y1(t)=1+t </math>, and <math> y2(t)=e^t</math> | ||
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which is the same output as before, so the system is linear. | which is the same output as before, so the system is linear. | ||
| + | |||
| + | == Nonlinear System == | ||
| + | |||
| + | For an example of a nonlinear system, I will use one that APPEARS very close to the one I used for a linear system. | ||
| + | |||
| + | The system <math>x(t)=y(t)^3</math> | ||
| + | |||
| + | We will use the same A, B, y1(t) and y2(t). | ||
| + | |||
| + | <math>y1(t)</math> -> [SYSTEM] -> A*result -> <math>A*(1+t)^3</math> | ||
| + | |||
| + | <math>y2(t)</math> -> [SYSTEM] -> B*result -> <math>B*(e^t)^3</math> | ||
| + | |||
| + | Sum these and you get | ||
| + | |||
| + | <math>A*(1+t)^3+B*(e^t)^3</math> | ||
| + | |||
| + | Now, | ||
| + | |||
| + | <math>A*y1(t)</math> -> [SYSTEM] -> <math>(A(1+t))^3</math> | ||
| + | |||
| + | <math>B*y2(t)</math> -> [SYSTEM] -> <math>(B*e^t)^3</math> | ||
| + | |||
| + | Sum these and you get | ||
| + | |||
| + | <math>A^3(1+t)^3 + B^3(e^t)^3</math> | ||
| + | |||
| + | These two results are clearly not equal, thus the system is not linear (even though it looked very similar to the linear system that I demonstrated before) | ||
Latest revision as of 14:07, 11 September 2008
Definition
A system is linear if two specific inputs yield two specific outputs, and if you multiply the inputs by constants and then run them through the system and sum the result it yields the same result as running them through the system, multiplying the outputs by those same constants, then adding the result together.
Linear System
I'll choose the system $ x(t) = y(t^2) $ Let's choose $ A=2 $, $ B=3 $, $ y1(t)=1+t $, and $ y2(t)=e^t $
$ A*y1(t) $ -> [SYSTEM] -> $ 2(1+t^2) $
$ B*y2(t) $ -> [SYSTEM] -> $ 3(e^{t^2}) $
sum these and you get
$ 2+2t^2 + 3e^{t^2} $
Now,
$ y1(t) $ -> [SYSTEM] -> A*result -> $ (1+t^2)*2 $
$ y2(t) $ -> [SYSTEM] -> B*result -> $ (e^{t^2})*3 $
sum these and you get
$ 2+2t^2+3e^{t^2} $
which is the same output as before, so the system is linear.
Nonlinear System
For an example of a nonlinear system, I will use one that APPEARS very close to the one I used for a linear system.
The system $ x(t)=y(t)^3 $
We will use the same A, B, y1(t) and y2(t).
$ y1(t) $ -> [SYSTEM] -> A*result -> $ A*(1+t)^3 $
$ y2(t) $ -> [SYSTEM] -> B*result -> $ B*(e^t)^3 $
Sum these and you get
$ A*(1+t)^3+B*(e^t)^3 $
Now,
$ A*y1(t) $ -> [SYSTEM] -> $ (A(1+t))^3 $
$ B*y2(t) $ -> [SYSTEM] -> $ (B*e^t)^3 $
Sum these and you get
$ A^3(1+t)^3 + B^3(e^t)^3 $
These two results are clearly not equal, thus the system is not linear (even though it looked very similar to the linear system that I demonstrated before)
