Linearization of a non-linear model

Question

Context:-Consider you are holding a plate of cookies in your hand at angle $x$ with force $k$(you may change this at your discretion.The plate puts a force $f$ on your hands. Some people take cookies out of the plate at random time thus decreasing $f$ and then accordingly you have to adjust $k$.The plate must be held at constant height(basically $x$ must remain constant). Consider the function as follows $$\frac{d^2x}{dt^2}=f(t)\sin(x(t))-k(t)x(t)...(1)$$ Linearize the given function and rewrite it in terms of $\bar{x},\bar{f},\bar{k}$. Where bar denotes the difference between value at any given point of time and steady state value. $$\text{At steady state there is no change in values of functions wrt time}$$. $$\text{Attempt}$$ At steady state we have $0=f_s(t)\sin(x_s)-k_s(t)x_s(t)$.The general approach is to subtract this from (1) but I can't make substantial process with this. My first thought was using Taylor series for the $\sin$ part but what's bugging me is the presence of $f(t)$ with it. I have come across spring-system models where we have 2nd order derivative but never across a function which has multiplication of both the dependent variables. I couldn't even get a paper which relates to such kind of functions. Any help is appreciated!

Answer 1

Something like section 5.3 in this chapter should help

I would go about it this way, first assign your state variables $x_i$

$$\frac{d^2x}{dt^2} = f(t)\sin(x(t))-k(t)x(t)$$

Let $$x_{1} = x(t)$$ $$x_{2} = \frac{dx}{dt}$$

Then just for the sake of convention,

$$u_{1} = f(t)$$

$$u_{2} = k(t)$$

Then substitute into our original equation

$$\dot x_{2} = u_{1}sin(x_{1}) - u_{2}x_{1}$$

$$\dot x_{1} = x_{2}$$

Now you can calculate the linearization matrices about the nominal point, the following is in traditional state space vector/matrix format.

$$ \dot x = f(x,u) $$

$ A = \frac{\partial f}{\partial x} $ evaluated at $x = \bar x, u = \bar u$

$ B = \frac{\partial f}{\partial u} $ evaluated at $x = \bar x, u = \bar u$

Then the familiar state space equation appears:

$$ \dot{\bar{x}} = A \bar x + B \bar u $$