Usbrth

I have the following system:

$$beginaligned x(k+1) &= x(k) + T_svcos(phi(k) + beta(k)) \ y(k+1) &= y(k) + T_svsin(phi(k) + beta(k)) \ phi(k+1) &= phi(k) + fracT_svlsin(beta(k))endaligned$$

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

1. Clearly there is no equilibria when $vneq0$, 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?

$$
beginbmatrixdelta x(k+1) \ delta y(k+1) \ delta phi(k+1) \endbmatrix=
beginbmatrix1&0&-T_svsin(phi_e + beta_e) \ 0&1&T_svcos(phi_e + beta_e) \ 0&0&1 \endbmatrix
beginbmatrixdelta x(k) \ delta y(k) \ delta phi(k) \endbmatrix+
beginbmatrix-T_svsin(phi_e + beta_e) \ T_svcos(phi_e + beta_e) \ fracT_svlcos(beta_e) \endbmatrix
beginbmatrix delta beta(k) \ endbmatrix$$
where $ delta x(k) = x(k) - x_e(k) $

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.