How to create a state space equation from three differential equations where input(u) is multiplied by state (x1)?

My setup is:

I have a 3 states model built out of 3 differential equations. Namely:

Here the parameters F and R are constants, and u is the input with a range of

The system has initial values:

My goal is to put this system in Matlab and analysis this system using bodeplots and stability criteria.

I tried to put the differential equations into state-space format (), to be able to use the ss() command, with no succes. The multiplication of makes this not possible.

I looked into linearizing the problem, but this also didnt work. On top of that I would like to have Matlab do the hard work for me.

My question is: How can I decouple the input from the A matrix in order to put the differential equations into state-space form?

Is ode a possible outcome for my problem? Should I transform the equations into LaPlace somehow?

Thanks in advance!

Answer 1

[Sam Chak]

Hi @Gaston Gabriël

This is a nonlinear system,

which cannot be expressed in the standard linear state-space model

unless you understand how u behaves and can express it using the Takagi-Sugeno Fuzzy System. For example

Rule 1: If Condition 1 is true, then .

Rule 2: If Condition 2 is true, then .

Rule 3: If Condition 3 is true, then .

Laplace transforms cannot be applied on Nonlinear systems.

[Sam Chak]

Another method is through the Carleman Linearization method for Nonlinear System.

But I'm unsure if there is any direct MATLAB command that computes the Carleman Linearization algorithm. Carleman's method uses the Kronecker product of matrices. The method is detailed in Nonlinear Dynamical Systems and Carleman Linearization, by K Kowalski and W.-H. Steeb.