I have this state-space model:
Is this state space model nonlinear? If it is, why is that?
This is a linear differential equation . It stems from the following electric circuit :
Were the resistor R
, the capacitor C
and the inductor L
are ideal linear components. x1
is the current from the generator of voltage d.u1
and x2
is the voltage of the resistor-capacitor.
To check linearity, let u1_1(t)
and u1_2(t)
be two input signals. The corresponding output signals are x1_1(t),x2_1(t)
and x1_2(t),x2_2(t)
. Then it is easy to see that the output signal corresponding to u1_1(t)+u1_2(t)
is x1_1(t)+x1_2(t),x2_1(t)+x2_2(t)
.
The Laplace transform and the Fourier transform can be used to study this system and compute output signals whenever possible.
d
and u1
are both considered as inputs, the model is non-linear. Think about (d(t),0)
and (0,u1(t))
as sets of inputs. In both cases, the output is 0. But the output corresponding to (d(t)+0,0+u1(t))
is not necessarily 0 ! However, if both d
and u1
are known, d.u1
can be computed and used as input of the linear part. Hence, the analysis of the linear part is not useless.
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u1_1
and u1_2
holds true. If R(t) is a new variable like x1 or x2, then the problem is not linear anymore.
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