Csdrhrt

Consider a system of the form
$$dot{x}(t)=Ax(t)+Bu(t)+phi(t)+D(t)$$
I have
$$dot{x}(t)=begin{bmatrix}
-p_1 &G_b & 0 & 0 &0 \
0& -p_2 & p_3 & 0 & 0\
0& 0 & -p_4 & p_5 &0 \
0& 0 & 0 & -p_6 &p_6 \
0& 0& 0 & 0 & -p_6
end{bmatrix}x(t)+begin{bmatrix}
0\
0\
0\
0\
1\
end{bmatrix}u(t)+begin{bmatrix}
-x_1(t)x_2(t)\
0\
0\
0\
0\
end{bmatrix}+begin{bmatrix}
1\
0\
0\
0\
0\
end{bmatrix}D(t)$$
Where $phi(t)$ is a lumped nonlinearity of the system and $D(t)$ is a disturbance acting from outside. I want to convert the system of the form
$$dot{Z}_{i}=Z_{i+1}+text{maybe nonlinearities and disturbances}, i=1,2,...,r-1 \dot{Z}_{r}=u+text{maybe some function oif states}$$
i.e
$$dot{Z_1}=Z_2 \ dot{Z_2}=Z_3 \ cdots \dot{Z_r}=f(Z_1,...,Z_r,t,)+u$$
How to find a transformation matrix to do this?

For a single input system the similarity transformation which transforms the system into its controllable canonical form is given by

$$
vec{v}^top = begin{bmatrix}0 & cdots & 0 & 1end{bmatrix}
begin{bmatrix}
B & B, A & B, A^2 & cdots & B, A^{n-1}
end{bmatrix}^{-1}, tag{1}
$$

$$
T = begin{bmatrix}
vec{v}^top \
vec{v}^top A \
vec{v}^top A^2 \
vdots \
vec{v}^top A^{n-1}
end{bmatrix}. tag{2}
$$

So using the transformation $z(t) = T,x(t)$ gives

$$
dot{z} = T,A,T^{-1} z(t) + T,B,u(t) + T,phi(t) + T,D(t), tag{3}
$$

where $(T,A,T^{-1}, T,B)$ will be in the controllable canonical form.

If you would like to know more about how to derive this then you can look at this related question.