Question: how to get kalman form

The goal is to get Kalman Form

https://docs.google.com/file/d/0Bxs_ao6uuBDUT25BeVhtWjZwRUU/edit?usp=sharing

https://docs.google.com/file/d/0Bxs_ao6uuBDUVXU5dXh6Rl9zUUU/edit?usp=sharing

I am doing the example in page 707 of Partial Differential control theory Volume 2 by J.F.

when using Robertz Daniel's Ore and Involutive packages

with(Involutive):

with(OreModules):

Alg := DefineOreAlgebra(diff=[D,t], polynom=[t], comm=[a1,a2,a3,u]):

R := evalm([[D, -1, 0, 0],[0, D, -1, 0],[D + 3, 4, 1, 2*D^2 + 5*D + 7]]);

AdjR := Involution(R,Alg);

the adjoint operator is not the same in the example

i do not know how to get

Diff(x1(t),t) = x2(t) + 2*u;

Diff(x2(t),t) = x3(t) - u;

Diff(x3(t),t) = -x1(t) - 4*x2(t) - 3*x3(t) + 2*u;

in the middle step i do not know how to do generically when eliminate all derivatives of lambda

after read

x1 = x1bar

x2 = x2bar + 2*u

x3 = x3bar + 2*Diff(u(t),t) - u(t)

how to get kalman form

using the 5 steps in page 615, i do not know when to use Integrability maple function