Process Control Design for Industrial Applications

6Ill-Defined and/or Uncertain Systems

To calculate command laws for a process, we use a model. If the model used is ill-defined and/or uncertain with bounded uncertainties or disturbances, we then need to check if the process evolves in accordance with the prescribed law with an error that is consistent with the desired behavior.

A particularly effective approach to this problem is to study the stability of the command process and estimate the maximum error in process behavior relative to that desired. This can be done using the concept of a comparison system and an overvaluing system, which are defined using vector norms [BOR 74].

6.1. Study of the stability of nonlinear systems from vector norms

6.1.1. Vector norms

Let E = be a vector space and E₁, E₂…,E_k be subspaces of E such that E = E₁ E₂…E_k and let x be a vector defined on E with its projection on E_i denoted as x_i.

[6.1]

where P_ri is the projection operator of E in E_i.

Let us note , a scalar norm of vector x_i and

If y is ...