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I have to design a feedback linearization for the model of a car with a single trailer. This picture shows the system:

And this is the kinematic model I got, where q_dot = [x_dot; y_dot; thA_dot; thP_dot; thR_dot].

I chose y1 = x and y2 = y as outputs. I proved that the system is controllable and observable. To complete the exercise, then, I calculated the time derivatives of y1 and y2. The classic iteration - if I'm not wrong - says to continue the derivatives until a dependance from both inputs comes out. So I calculated the second derivatives with respect to time, the third ones and the fourth ones, until I got two functions (the fourth derivatives of y1 and y2) depending on u1, du1/dt, d²u1/dt², d³u1/dt³ and u2 (the second input comes out only at this step). With these results, I thought to make a system extension: ξ1 = u1, ξ2 = du1/dt and ξ3 = d²u1/dt². The new inputs are w1 = d³u1/dt³ and w2 = u2. Finally I can write:

where v is a new control variable. This was to explain the problem and how I got to this point. Now, my question is: how do I choose v? It should depend on 8 variables, I suppose: q1, q2, q3, q4, q5, ξ1, ξ2, ξ3. But I don't know how.

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You used feedback linearization, so you could use techniques for linear (time invariant) systems, such as pole placement or LQR. Namely you have to have LTI state space model which is two quadruple integrators

$$ \dot{x} = \begin{bmatrix} 0 & I & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & 0 & 0 & I \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} x + \begin{bmatrix} 0 \\ 0 \\ 0 \\ I \\ \end{bmatrix}v, $$

with $x = \begin{bmatrix}y_1 & y_2 & \dot{y}_1 & \dot{y}_2 & \ddot{y}_1 & \ddot{y}_2 & \dddot{y}_1 & \dddot{y}_2\end{bmatrix}^\top$. You could try to find an expression for the derivatives of $y_1$ and $y_2$ as a function of $q$. If this does not have an explicit solution you could also use an observer, since if $y_1$ and $y_2$ are known the system is observable.

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