Process Control Design for Industrial Applications

Appendix AMatrix Transformation from Any Representation to the Companion Form or Arrow Form

Let us consider the process for which the evolution is described by equation:

[A.1]

with the matrix of change of basis P_c such that

[A.2]

makes it possible to obtain a representation of the system described by equation (S′) in controllable form, with x_c representing the state vector of the process in the new base. We obtain:

[A.3]

[A.4]

Let us note , the characteristic polynomial of matrix A_c:

[A.5]

If the pair (A(x₀), B(x₀)) is controllable, the search for a description of the system in controllable form allows us to characterize the system through the pair (A_c, B_c) , which is defined by:

[A.6]

We obtain:

[A.7]

Let us note P_ci, the ith column of P_c such that:

[A.8]

The P_c columns are deduced from ...