Adaptive output regulation via nonlinear Luenberger observer-based internal models and continuous-time identifiers

About Robustness of Control Systems Embedding an Internal Model

“…The importance of keeping s nominal is motivated by the fact that, in a linear setting, knowing s is the main necessary [1] and sufficient [2] condition for the design of a robust regulator (in particular, necessity is known as the internal model principle [1]). When perturbations affect the map s of the exosystem, instead, the problem is typically referred to as adaptive regulation [6]- [10]. For the special role played by s in the design of robust linear regulators, this distinction between robust and adaptive regulation makes considerable sense in the context of linear output regulation.…”

“…This inherent difficulty has recently motivated a shift towards new approaches seeking approximate, rather than asymptotic, results, as they trade weaker claims for stronger robustness properties (see, e.g. [8]- [10], [12]- [15]). Nevertheless, such robustness properties are usually treated in ad hoc manners, poorly characterized, and sometimes not formally proved and just left to intuition.…”

“…1) The existence of a non-empty steady state Ω Σ cl (F,X) (X × X c ) for (10) satisfying the properties of Proposition 1.…”

“…Definition 6 (Robust Steady-State Property): The regulator ( 9) is said to achieve the steady-state property P robustly at (F • , X • ) ∈ F × X and with respect to τ (or to be (P, τ )robust) if it is robustly stabilizing at (F • , X • ) with respect to τ and, by letting N the robust stability neighborhood of (F • , X • ), then the closed-loop system (Σ cl (F, X), X × X c ) given by (10) enjoys P in the sense of Definition 2 for each (F, X) ∈ N . ⊳ Remark 1: One may be concerned with the lack of uniformity on the ultimate bound in Definition 5, in the sense that, in this form, the definition admits the possibility of "horizonescaping" phenomena of the limit sets as (F, X) approaches the frontier of N .…”

“…⊳ Example 13 (Total Stability): Suppose that (F • , X • ) is such that (10) has an equilibrium which is locally exponentially stable with a domain of attraction D including…”