The state space model of a nonlinear system is
[tex]x'_1(t) = 2x^2_2(t) - 50[/tex]
[tex]x'_2(t) = -x_1(t) - 3x_2(t) + u(t)[/tex]
Where [tex]x_1(t)[/tex] and [tex]x_2(t)[/tex] are the states, and u(t) is the input. The output of the system is [tex]x_2(t)[/tex].
Find the state space model of this system linearized at the equilibrium point (-15, 5)
State space modeling, Jacobian Matrix(?)
The Attempt at a Solution
I think my attempt lies on false assumption, but here's what I did.
Since we're linearizing at a known equilibrium point for a nonlinear system, and since [tex]x'_1(t) = 2x^2_2(t) - 50[/tex] contains non linear term, we must first take a derivative of it as denoted in Jacobian matrix. derivative of x_1(t) is
[tex]x''_1(t) = 4x_2(t)[/tex]
So the state space model we get is
[x'_1] = [ 0 4 ] [x_1] + [ 0 ] u(t)
[x'_2]aa[-1 -3 ] [x_2] a[ 1 ]
(Sorry I have no clue how to put this matrix in Latex).
But then how do we get the output from this state space model?