Finding the state space model of a nonlinear system

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1. Feb 5, 2016

l46kok

1. The problem statement, all variables and given/known data
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)

2. Relevant equations

State space modeling, Jacobian Matrix(?)

3. 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?

2. Feb 6, 2016

pasmith

The linearization about (-15, 5) is obtained by evaluating the Jacobian matrix at (-15,5). Here the Jacobian matrix is $$J(x_1,x_2) = \begin{pmatrix} 0 & 4x_2 \\ -1 & -3 \end{pmatrix}$$ and the latex code is
Code (LaTeX):

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

Last edited: Feb 6, 2016
3. Feb 6, 2016

l46kok

So the evaluation at (-15,5) becomes

$$y'= \begin{pmatrix} 0 & 20 \\ -1 & -3 \end{pmatrix}$$

?

Last edited: Feb 6, 2016