Linearizing a second order non-linear equation

[zaotron]

Homework Statement

I am modeling a set of equations for a protein network. It is a feedback loop between 2 proteins. I have gotten the differential equations for this model and plan on doing an extended Kalman Filter to estimate the levels of protein in real time. However, I am having trouble trying to linearize the equations so I can put them into a state space equation, which is essential to the Kalman Filter.

Homework Equations

The equations are shown below, the only variables are x, y, and S. All the other variables are just constants.

The Attempt at a Solution

I have attempted to try and linearize this model by discretizing the equations. I represented each equation as a function of time with its initial value at zero (x(0)) added to the value at time n, where n is equal to the number of steps in time. Then, I tried to represent each of the equation by putting them into the s-domain and then solving for the first derivative. However, I don't think that I'm going about the correctly, I'm having trouble linearizing these equations. Can anyone help?

 

[CEL]

Homework Statement

I am modeling a set of equations for a protein network. It is a feedback loop between 2 proteins. I have gotten the differential equations for this model and plan on doing an extended Kalman Filter to estimate the levels of protein in real time. However, I am having trouble trying to linearize the equations so I can put them into a state space equation, which is essential to the Kalman Filter.

Homework Equations

The equations are shown below, the only variables are x, y, and S. All the other variables are just constants.

The Attempt at a Solution

I have attempted to try and linearize this model by discretizing the equations. I represented each equation as a function of time with its initial value at zero (x(0)) added to the value at time n, where n is equal to the number of steps in time. Then, I tried to represent each of the equation by putting them into the s-domain and then solving for the first derivative. However, I don't think that I'm going about the correctly, I'm having trouble linearizing these equations. Can anyone help?

You have $$\frac{dX}{dt} = f(X,t)$$
Where X is your state vector (x, y, S)'.
Now you calculate $$F = \frac{df}{dX}$$ at some known point (x(0), y(0), S(0))'.
Your linearized equation will be $$\frac{dX}{dt} = F \cdot X$$.
After each iteration of the filter F must be recalculated at the new estimate.

 

[zaotron]

Thanks a ton! That helped out a lot! I also used another method called the Runge Kutta method to help with linearization.

 

[CEL]

Thanks a ton! That helped out a lot! I also used another method called the Runge Kutta method to help with linearization.

You can use Runge-Kutta for the propagation of the state. For the propagation of the error covariance matrix you should use the matrix F that I proposed.