Input Dexterity and Output Negotiation in Feedback-Linearizable Nonlinear Systems

TL;DR

A task-relative taxonomy of actuator inputs for nonlinear systems within the input-output feedback-linearization framework is introduced and it is shown that a subset is dexterity if and only if it can appear as additional output channels (flat-input complement) on a common validity set.

Abstract

We introduce a task-relative taxonomy of actuator inputs for nonlinear systems within the input-output feedback-linearization framework. Given a flat output specifying the task, inputs are classified as essential, redundant, or dexterity: essential inputs are required for exact linearization, redundant inputs can be removed without effect, and dexterity inputs can be deactivated while preserving exact linearization of a reduced task. We show that a subset is dexterity if and only if, under a suitable dynamic prolongation, it can appear as additional output channels (flat-input complement) on a common validity set. Whenever a family of systems obtained by (de)activating dexterity inputs admits a common prolongation, the family can be interpreted as a single prolonged system endowed with different output selections. This enables a unified linearizing controller that negotiates between full and reduced tasks without transients on shared outputs under compatibility and dwell-time conditions. Simulations on a fully actuated aerial platform illustrate graceful task downgrades from six-dimensional pose tracking as lateral-force channels are deactivated.

Figures (5)

• Figure 1: (Motivating example, Section \ref{['sec:motv_exmp']}). Direct shutdown: $\mathcal{K}_\Sigma$ regulates $\mathbf{y}=[x_1\ x_3\ x_4]^\top$ until $t=8s$; then $u_2$ is set to zero and $\mathcal{K}_{\Sigma_{\overline{\{2\}}}}$ regulates $\mathbf{y}_{\overline{\{3\}}}=[x_1\ x_3]^\top$. A transient appears on $x_3$.
• Figure 2: Unified prolonged control: one controller $\mathcal{K}_{\Sigma^{(\ell)}}$ regulates pre‑switch $\mathbf{y}$ and post‑switch $\mathbf{y}_{\overline{\{3\}},\{2\}}=[x_1\ x_3\ u_2]^\top$, and smoothly drives $u_2\to 0$ (no transient on the kept outputs).
• Figure 3: Direct shutdown of $u_1$ and $u_2$ at $t=8s$, then regulation of $\mathbf{y}_{\overline{\{2,3\}}}=x_1$ with $\mathcal{K}_{\Sigma_{\overline{\{1,2\}}}}$. A transient appears in $x_1$ due to a relative‑degree jump.
• Figure 4: Schematic of the families introduced in this work: admissible prolongations $\mathcal{L}^{\mathcal{A}}_{(\Sigma,\mathbf{y})}$; admissible index sets $\Omega^{\mathcal{A}}_{(\Sigma,\mathbf{y})}$; $\ell$‑realizable family $\mathcal{N}^{\ell}_{(\Sigma,\mathbf{y})}$; and realizing pairs $\mathcal{C}^{\mathcal{A}}_{(\Sigma,\mathbf{y})}$.
• Figure 5: Simulation. Switching among three pairwise compatible melds on $\Sigma^{(\ell)}$: DF (#2), FM (#1), and QM (#13). Dark gray: DF; light coral: FM; light gray: QM. No transients on shared outputs, consistent with Lemma \ref{['lem:zero_transient_common_ell']} and Section \ref{['sec:switching-negotiable-common-ell']}.

Theorems & Definitions (29)

• Definition 1: Dexterity subset of inputs
• Definition 2: Family of dexterity subsets
• Definition 3: Essential vs. dexterity inputs; minimum loss
• Remark VI.1
• Example VI.1
• Definition 4: Realizing pairs
• Remark VI.2
• Definition 5: Flat-input complement subset
• Definition 6: Family of flat-input complement subsets
• Definition 7: Realizing pairs for flat-input complement