Linearization

[indigo oxide]

"The dynamic evolution of the liquid level of a deposit, represented in the figure, is
define by the equations,
C dh(t)/dt − qe(t) + K * (h(t)^1/2)= 0
Yo
qs(t) = K · (h(t)^1/2)

amb qe(t) cabal d'entrada d'aigua, qs(t) cabal de sortida d'aigua, h(t) level of
liquid to the deposit, C constant of capacity of the dip`osit i K gruix of canonada of
sorted.
If we want to regulate the level of liquid in the deposit at a working point that we will define
com a (qe, h) = (1, 1), linearize the system to the voltant d'aquest punt de treball i
described the resulting linear dynamic system.

[quartz wing]

To linearize the system around the working point (qe, h) = (1, 1), we first need to find the partial derivatives of the equations with respect to h and t:

∂qe(t)/∂h = K/(2h(t)^1/2)
∂^2h(t)/∂t∂h = K/(4C*(h(t)^3/2))

Using these partial derivatives, we can linearize the system by taking the Taylor series expansion around the working point (qe, h) = (1, 1):

C dh(t)/dt − qe(t) + K * (h(t)^1/2) ≈
(C dh(t)/dt − qe(1) − ∂qe(t)/∂h|(qe,h)=(1,1) * (h(t)-1) + K * (h(t)^1/2) + ∂^2h(t)/∂t∂h|(qe,h)=(1,1) * (t-t0) * (h(t)-1))

qs(t) = K · (h(t)^1/2) ≈
(qs(1) + ∂qs(t)/∂h|_(qe,h)=(1,1) * (h(t)-1))

Simplifying and expressing in matrix form, the resulting linear dynamic system is:

[dh(t)/dt; dh(t)/dt-h(t)] =
[0 1; -K/(2C(1^3/2)) -K/(2C(1^1/2))] * [h(t)-1; dh(t)/dt-1] +
[0; K/(2*(1^1/2))] * [qe(t)-1]

[qs(t)] = [K/(2*(1^1/2))] * [h(t)-1] + [qs(1)]