Identifiability and Informative Experiments in Open and Closed-Loop Identification

Abstract

This chapter takes a new look at the concept of identifiability and of informative experiments for linear time-invariant systems, both in open-loop and in closed-loop identification. Some readers might think that everything has been said and written about these concepts, which were much studied all through the 1970’s. We shared the same view ... until recently. The motivation for our renewed interest into these very fundamental questions is the recent surge of interest in the question of experiment design, itself triggered by the new concept of least costly identification experiment for robust control [3], [4], [5], [6]. Briefly speaking, least costly experiment design for robust control refers to achieving a prescribed accuracy at the lowest possible price, which is typically measured in terms of the duration of the identification experiment, the perturbation induced by the excitation signal, or any combination of these. In this context, questions like the following become relevant:

• 1. what is the smallest amount of external excitation that is required to achieve identifiability (or to achieve a given accuracy level)?

• 2. assuming that the system operates in closed-loop, is noise excitation sufficient to guarantee identifiability?

• 3. if noise excitation is not sufficient to guarantee identifiability in a closed-loop experiment, then how much additional reference excitation is required?

• 4. assuming that excitation can be applied at different entry points of a multi-input system operating in closed loop, is it necessary to excite each input to achieve identifiability (or to achieve a given accuracy level)?

To Giorgio, who to our great surprise has not solved the rather fundamental problems raised in this chapter, we hope to offer some food for thought.

This chapter presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The second author is also partially supported by the Brazilian Ministry of Education through CAPES.