How to linearize a kinematic bicycle model?

Question

I have the following system:

$$\begin{aligned} x(k+1) &= x(k) + T_sv\cos(\phi(k) + \beta(k)) \\ y(k+1) &= y(k) + T_sv\sin(\phi(k) + \beta(k)) \\ \phi(k+1) &= \phi(k) + \frac{T_sv}{l}\sin(\beta(k))\end{aligned}$$

I am confused how should I put this in linear state-space representation?

1. Clearly there is no equilibria when $v\neq0$, does this mean that I cannot linearize the system or I can linearize it arbitrarily?
2. I got the following ss form using the standard linearization method. Again, how to pick $x_e/u_e$? does it make sense and is there a better way to linearize this system?

$$ \begin{bmatrix}\delta x(k+1) \\ \delta y(k+1) \\ \delta \phi(k+1) \\\end{bmatrix}= \begin{bmatrix}1&0&-T_sv\sin(\phi_e + \beta_e) \\ 0&1&T_sv\cos(\phi_e + \beta_e) \\ 0&0&1 \\\end{bmatrix} \begin{bmatrix}\delta x(k) \\ \delta y(k) \\ \delta \phi(k) \\\end{bmatrix}+ \begin{bmatrix}-T_sv\sin(\phi_e + \beta_e) \\ T_sv\cos(\phi_e + \beta_e) \\ \frac{T_sv}{l}\cos(\beta_e) \\\end{bmatrix} \begin{bmatrix} \delta \beta(k) \\ \end{bmatrix}$$ where $ \delta x(k) = x(k) - x_e(k) $

Answer 1

1. You should always start any state space modelling effort by clearly defining (a) the state variables, (b) the inputs, and (c) the outputs. It looks like your state variables are $x,y,\phi$. What physical quantities do those represent? It looks like $\beta$ is an input. What physical quantity does that represent? What outputs are there?

2. You can still linearize the system. You just need to linearize about a nonstationary solution. Assume that $v \neq 0$ and $\beta = 0$ (note that it is common to linearize about the "free-response", e.g. with an input of zero.) Then it looks like $\phi$ is constant and $x$ and $y$ have simple solutions. That looks like a good place to start. Remember that you will need to add in this solution when you want to go from the perturbative analysis back into the first-order approximation of the overall solution.

3. That looks reasonable for the $A$ and $B$ matrices. The reason you have uncertainty at this point is that you skipped the crucial step of defining the solution you are linearizing about before linearizing. $\beta_e(t),x_e(t),x_e(t),\phi_e(t)$ can be any solution you want. I suggested the solution consistent with $\beta_e(t) = 0$, since this would be the most common choice, but you might have some other need. The procedure of linearization cannot tell you this--you as an engineer need to decide what is best for your application.