Scalable Distributed Nonlinear Control Under Flatness-Preserving Coupling

TL;DR

This work tackles the challenge of designing scalable distributed control for networks of nonlinear, differentially flat subsystems under dynamic coupling. It identifies a class of lower-triangular couplings that preserve joint flatness, and shows that the joint flatness diffeomorphism can be constructed from the subsystems’ maps, with a sparsity pattern that mirrors the coupling graph. Leveraging this structure, the authors develop a distributed tracking controller that computes inputs using only local information, with stronger locality under strongly lower-triangular couplings. Validation on planar quadrotors coupled by aerodynamic downwash demonstrates accurate trajectory tracking and a clear trade-off between communication and performance when using approximate coupling models.

Abstract

We study distributed control for a network of nonlinear, differentially flat subsystems subject to dynamic coupling. Although differential flatness simplifies planning and control for isolated subsystems, the presence of coupling can destroy this property for the overall joint system. Focusing on subsystems in pure-feedback form, we identify a class of compatible lower-triangular dynamic couplings that preserve flatness and guarantee that the flat outputs of the subsystems remain the flat outputs of the coupled system. Further, we show that the joint flatness diffeomorphism can be constructed from those of the individual subsystems and, crucially, its sparsity structure reflects that of the coupling. Exploiting this structure, we synthesize a distributed tracking controller that computes control actions from local information only, thereby ensuring scalability. We validate our proposed framework on a simulated example of planar quadrotors dynamically coupled via aerodynamic downwash, and show that the distributed controller achieves accurate trajectory tracking.

Figures (5)

• Figure 1: Illustration of the allowed dependencies for $\Delta^2_2$ on $N=3$ subsystems with relative degree $r=3$. (a) Lower-triangular coupling (Def. \ref{['defn: lower-triangular-coupling']}) (b) Strongly lower-triangular coupling (Def. \ref{['defn: block-lower-triangular']})
• Figure 2: Visualization of the $F_D$ for a pair of vehicles as a function of the horizontal separation, plotted at different vertical separations $\delta_y$.
• Figure 3: Evolution of position error over time, averaged over the vehicles.
• Figure 4: Trajectory of the quadrotors. Dotted grey lines represent the reference trajectories. Flatness-based controllers constructed under both the exact and the approximate downwash coupling achieve good tracking performance.
• Figure 5: Average position errors for $N=10$ vehicles as we vary the approximation cutoff threshold $\bar{\boldsymbol{\delta}}=[\bar{\boldsymbol{\delta}}, \bar{\boldsymbol{\delta}}]$. The position error decreases as we increase the cutoff threshold (communication range), as the average number of vehicles in the communication set increases. The error bars denote the inter-quartile range computed over the vehicles.

Theorems & Definitions (11)

• Remark 1
• Definition 1: Lower-Triangular Coupling
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 3
• proof : Proof
• Definition 2: Strongly Lower-Triangular Coupling
• Corollary 4