Feedback Classification of Nonlinear Single-Input Control Systems with Controllable Linearization: Normal Forms, Canonical Forms, and Invariants

Normal Forms of Nonlinear Control Systems

“…For linearly controllable and analytic systems that are not feedback linearizable, the group of stationary symmetries contains at most two elements and the group of non stationary symmetries consist of at most two 1-parameter families. This surprising result follows from the canonical form obtained for single-input systems by Tall-Respondek [35]. Respondek [31] establishes the relationship between flatness and symmetries for two classes of systems: feedback linearizable systems and systems equivalent to the canonical contact system for curves.…”

“…However, a normal form of degree k is not unique under transformations of degree less than k. If a normal form is unique under transformation of arbitrary degree, it is call a canonical form. Tall-Respondek [35] solved the problem of canonical form for single-input and linearly controllable systems.…”

Smooth and Analytic Normal and Canonical Forms for Strict Feedforward Systems

“…In this paper we will show that an analytic special strict feedforward system can be brought to an analytic canonical form. These normal and canonical forms are, respectively, smooth and analytic counterparts of the corresponding formal forms obtained, respectively, by Kang [12] (normal form) and the authors [31] (canonical form).…”

“…The action of Γ ∞ on the system Π ∞ step by step leads to formal normal forms. The following normal form was obtained by Kang [12] (see also [14], [31]) and then completed by the authors who obtained the canonical forms (see [31] for details):…”

“…In this paper, we will use a very fruitful approach of Kang and Krener [14], [12], [13] who have proposed to analyze (following Poincaré), step by step, the action of the Taylor series expansion of the feedback transformation Γ on the Taylor series expansion of the system Π and have obtained for single-input control systems with controllable linearization normal forms for the quadratic terms [14] and then for higher order terms [12]. The results of Kang and Krener [14], [12] have been completed by the authors who obtained canonical forms and dual canonical forms for single-input nonlinear control systems with controllable [31] and then with uncontrollable linearization [32] (see also [17]). Recently those results have been generalized by Tall [29], [30] to multi-input nonlinear control systems.…”