Finding the state space model of a nonlinear system

[asd1249jf]

Homework Statement

The state space model of a nonlinear system is
x'_1(t) = 2x^2_2(t) - 50
x'_2(t) = -x_1(t) - 3x_2(t) + u(t)

Where x_1(t) and x_2(t) are the states, and u(t) is the input. The output of the system is x_2(t).

Find the state space model of this system linearized at the equilibrium point (-15, 5)

Homework Equations

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 x'_1(t) = 2x^2_2(t) - 50 contains non linear term, we must first take a derivative of it as denoted in Jacobian matrix. derivative of x_1(t) is

x''_1(t) = 4x_2(t)

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?

 

[pasmith]

The linearization about (-15, 5) is obtained by evaluating the Jacobian matrix at (-15,5). Here the Jacobian matrix is <br /> J(x_1,x_2) = \begin{pmatrix} 0 &amp; 4x_2 \\ -1 &amp; -3 \end{pmatrix}<br /> and the latex code is

J(x_1,x_2) = \begin{pmatrix} 0 & 4x_2 \\ -1 & -3 \end{pmatrix}

 

[asd1249jf]

The linearization about (-15, 5) is obtained by evaluating the Jacobian matrix at (-15,5). Here the Jacobian matrix is <br /> J(x_1,x_2) = \begin{pmatrix} 0 &amp; 4x_2 \\ -1 &amp; -3 \end{pmatrix}<br /> and the latex code is

J(x_1,x_2) = \begin{pmatrix} 0 & 4x_2 \\ -1 & -3 \end{pmatrix}

So the evaluation at (-15,5) becomes

y&#039;= \begin{pmatrix} 0 &amp; 20 \\ -1 &amp; -3 \end{pmatrix}

?