Switching Among Feedback-Linearizing Output Sets (Melds): Dwell-Time and Compatibility Guarantees

The gist Switching among feedback-linearizing output sets (melds) involves studying switching among multiple square selections of output functions for nonlinear systems to derive dwell-time and compatibility guarantees for state boundedness and tracking consistency.

How it works

1. The core idea is to switch between a finite set of feedback-linearizing output choices, where each choice induces a distinct feedback-linearizing coordinate chart defined on a common neighborhood Switching between melds therefore produces state-dependent coordinate mismatches that are not captured by classical switched-system analyses.

2. The effect of switching is quantified through uniform Lipschitz bounds on the inverse chart maps and on the associated cross-chart mappings. These bounds are used to derive an explicit dwell-time condition that guarantees exponential decay of active-output tracking errors between switches, seamless tracking of outputs shared by consecutive melds, and uniform boundedness of the state error within the safe set.

3. The analysis introduces a reference-compatibility constant that measures mismatch among reference families across melds. This constant makes explicit that dwell time alone cannot compensate for incompatible references.

Coordinate Transformations and Meld Mappings

** For each output yi = hi(x) of relative degree ri, a map Φi is defined which lifts the state to the (ri − 1)-th jet of the i-th output trajectory. Stacking these maps defines Φσ: R n → R n, which is a local diffeomorphism on Bσ by standard exact linearization results. Assumption 1 states that for each σ ∈ Mx◦, there exists an open set Uσ with K ⊂ Uσ such that the map Φσ: Uσ → Φσ(Uσ) is a diffeomorphism with inverse Ψσ2, and Ψσ is Lipschitz on Φσ(K). The cross-map Θi, σ:= Φi ◦ Ψσ is Lipschitz on Φσ(K). These bounds are key for switching as they allow comparison between inactive references and the trajectories implied by the currently active meld. The uniform bounds LΨ and LΘ are defined based on these Lipschitz constants. **

Reference Trajectories and Tracking Errors

** Desired trajectories y d(t) are defined, and the desired jet for output i is y d i(t):= y d i(t) ˙y d i(t)... y d(ri−1)(t)⊤ ∈ R ri. The tracking error for output yi is defined as the difference between the desired and actual jets: ξi(t):= y d i(t) − yi(t), which represents the compatibility error (or reference mismatch) when yi is inactive. When output yi is part of the active meld, ξi is the standard tracking error. The control law construction involves a virtual input w where each entry wi is of the form: wi = y d,(ri)i + k 1i(y d i − yi) +... + k(ri−1)i(y d,(ri−1)i − y(ri−1)i), z = k⊤ i ξi. **

Seamless Output Tracking of Switching Melds

** Proposition 1 states that under the control law (14), the subsystem associated with ξi evolves on [tk, tk+l+1) according to the same dynamics as in each individual meld interval, i.e., it is not affected by the switching. This ensures that any output shared by consecutive melds retains its exponential convergence across the switch. The analysis shows that for any j = 1,..., Ms and any t ≥ tk such that σ(t′) = σj ∀t′ ∈ [tk, t), the tracking error satisfies ξi(t) ≤ LΘCe−α(t−tk) y d σj(tk) − yσj(tk) + N. **

Guarantees on the Boundedness of the State

** Assumption 3 requires that the reference-induced state χ(t) is well-defined and satisfies χ(t) ∈ K for all t ≥ t0, and there exists a margin condition such that x ∈ R n: x − χ(t) ≤ δ ⊂ K, ∀t ≥ t0. Theorem 2 establishes uniform boundedness by deriving dwell-time conditions (23),(24) that guarantee the tracking error is bounded by S = max(S1, S2) for all t ∈ [t0 + T, t**

i) For all t ∈ [t0 + T, t∗), x(t) − χ(t) ≤ S, where one admissible choice of S is p 2LΨLΘC(ε+N)+pN.

ii) If S ≤ δ, then t∗ = ∞, meaning x(t) ∈ K for all t ≥ t0 + T.

Numerical Example with a 3R Robot and a Multi-dimensional Deck of Outputs

** The example considers a 3R planar manipulator where the deck of candidate outputs is ∆ = (y1 = q1, y2 = q2, y3 = q3, y4 = xG1, y5 = yG1, y6 = xG2, y7 = yG2). Five compatible melds are defined: σ1 to σ5. The task demonstrates that between 7s and 13s, the output q3 continues its exponential convergence to the desired value seamlessly, despite the switching at t = 9s. Similarly, between 13s and 18s, the output q1 maintains its convergence even during the switching at t = 18s. This shows that outputs shared by consecutive melds keep their exponential tracking across the switch. **

The resulting conditions are constructive and apply to general nonlinear systems admitting exact feedback linearization. The paper concludes that dwell time must be chosen together with the reference trajectories so that consecutive meld references are mutually compatible where their charts overlap. Furthermore, the analysis is inherently local and assumes exact feedback linearization with full vector relative degree, possibly after dynamic input extension. Operation near singularities and configurations admitting nontrivial zero dynamics are excluded by construction. Robustness to uncertainty can be handled via standard robust feedback-linearization arguments, but explicit uncertainty-dependent bounds for LΨ, LΘ, and N are left for future work.

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1. textbfSeamless Tracking During Mode Switching: The system can maintain exponential convergence of shared outputs across switching instants because any output shared by consecutive melds retains its exponential convergence across the switch. This ensures that critical, persistent objectives are not disrupted by mode changes, as stated in Proposition 1 and Section E.

2. textbfGuaranteed State Invariance Within a Safe Set: The system's state is guaranteed to remain within a compact safe set K after an initial transient time T, provided switching instants satisfy specific dwell-time conditions (23) and (24). This is achieved by showing that for all subsequent time, x(t) − χ(t) ≤ S, and if S ≤ δ, then x(t) ∈ K.

3. textbfExplicit Dwell-Time Scheduling: The switching frequency can be precisely calculated based on the system's decay rates and reference compatibility constant. The required minimum dwell time τk is derived as τk = 1/α ln LΘC p(ϵ + N), which directly links stability guarantees to the inherent complexity (cross-chart mismatch) of the output selection strategy.

4. textbf Enhanced Reference Management: The system can robustly handle changes in desired objectives by quantifying the incompatibility between them via a reference-compatibility constant. This constant makes it explicit that dwell time alone cannot compensate for incompatible references, guiding designers to ensure consecutive melds have overlapping or compatible coordinate charts.

5. textbf Predictable Tracking Error Bounds: The tracking error for any output during a switching interval is bounded by an expression that combines the exponential decay within the active meld and the reference mismatch: ξi(t) ≤ LΘCe−α(t−tk) Xa∈Iσ(t) ξa(tk) + N. This allows for precise estimation of tracking performance even during dynamic transitions.