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Continuous-time Constrained Funnel Synthesis for Incrementally Quadratic Nonlinear Systems

Taewan Kim, Dayou Luo, Behçet Açıkmeşe

TL;DR

The paper tackles robust trajectory guidance by synthesizing time-varying controlled invariant funnels around a nominal trajectory for nonlinear systems with bounded disturbances. It develops a convex optimization framework using delta-quadratic constraints and a DLMI to guarantee funnel invariance, and reformulates the DLMI as a differential matrix equation to enable numerical optimal control methods. Two continuous-time constraint satisfaction strategies are proposed: intermediate constraint-checking points and a subgradient-based successive convexification approach, with theoretical justification for subgradient existence in the latter. Demonstrations on a unicycle and a 6-DoF quadrotor show that L-smooth nonlinearities are less conservative than Lipschitz, and that CTCS methods effectively enforce continuous-time feasibility, offering a practical path to robust, safety-guaranteed planning and control.

Abstract

This paper presents a convex optimization-based framework for synthesizing time-varying controlled invariant funnels and associated feedback control around a given nominal trajectory for nonlinear systems subject to bounded disturbances. Nonlinearities are modeled using incremental quadratic constraints, including Lipschitz, L-smooth, and sector-bounded nonlinearities. Funnel invariance is ensured via a DLMI. Together with pointwise-in-time LMIs for state and input constraints, we formulate a continuous-time funnel synthesis problem. To solve it using numerical optimal control techniques, the DLMI is reformulated into a differential matrix equality (DME) and an LMI, where the DME acts as a funnel dynamics equation. We explore different formulations of these funnel dynamics. Continuous-time constraint satisfaction is addressed through two convex methods: one based on intermediate constraint-checking points, and another using a successive convexification method with subgradients to handle nondifferentiable maximum eigenvalue functions. Theoretical justification is provided for the existence of a measurable and integrable subgradient for the latter. The method is demonstrated on two numerical examples: the control of a unicycle and a 6-degree-of-freedom quadrotor for obstacle avoidance.

Continuous-time Constrained Funnel Synthesis for Incrementally Quadratic Nonlinear Systems

TL;DR

The paper tackles robust trajectory guidance by synthesizing time-varying controlled invariant funnels around a nominal trajectory for nonlinear systems with bounded disturbances. It develops a convex optimization framework using delta-quadratic constraints and a DLMI to guarantee funnel invariance, and reformulates the DLMI as a differential matrix equation to enable numerical optimal control methods. Two continuous-time constraint satisfaction strategies are proposed: intermediate constraint-checking points and a subgradient-based successive convexification approach, with theoretical justification for subgradient existence in the latter. Demonstrations on a unicycle and a 6-DoF quadrotor show that L-smooth nonlinearities are less conservative than Lipschitz, and that CTCS methods effectively enforce continuous-time feasibility, offering a practical path to robust, safety-guaranteed planning and control.

Abstract

This paper presents a convex optimization-based framework for synthesizing time-varying controlled invariant funnels and associated feedback control around a given nominal trajectory for nonlinear systems subject to bounded disturbances. Nonlinearities are modeled using incremental quadratic constraints, including Lipschitz, L-smooth, and sector-bounded nonlinearities. Funnel invariance is ensured via a DLMI. Together with pointwise-in-time LMIs for state and input constraints, we formulate a continuous-time funnel synthesis problem. To solve it using numerical optimal control techniques, the DLMI is reformulated into a differential matrix equality (DME) and an LMI, where the DME acts as a funnel dynamics equation. We explore different formulations of these funnel dynamics. Continuous-time constraint satisfaction is addressed through two convex methods: one based on intermediate constraint-checking points, and another using a successive convexification method with subgradients to handle nondifferentiable maximum eigenvalue functions. Theoretical justification is provided for the existence of a measurable and integrable subgradient for the latter. The method is demonstrated on two numerical examples: the control of a unicycle and a 6-degree-of-freedom quadrotor for obstacle avoidance.

Paper Structure

This paper contains 36 sections, 11 theorems, 112 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Statement of contributions
  4. Notation
  5. Nonlinear Systems, Funnels, and Invariance Conditions
  6. Nonlinear systems
  7. Funnel definition and invariance conditions
  8. Continuous-time Funnel Synthesis Problem
  9. Structured nonlinearity and mean value theorem
  10. Characterization of nonlinearity using $\delta$QC
  11. Lipschitz and L-smooth nonlinearities
  12. Invariance condition by DLMI
  13. Funnel constraints
  14. Funnel cost functions
  15. Funnel synthesis problem
  16. ...and 21 more sections

Key Result

Lemma 10

Let $Q:[t_{0},t_{f}]\rightarrow\mathbb{S}_{++}^{n_{x}}$, $K:[t_{0},t_{f}]\rightarrow\mathbb{R}^{n_{u}\times n_{x}}$, $\lambda_{w}:[t_{0},t_{f}]\rightarrow\mathbb{R}_{+}$ be piecewise continuous functions. Define the Lyapunov function $V:[t_{0},t_{f}]\times\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}_{+}$ for some decay rate $\alpha\in\mathbb{R}_{++}$. Then, the state funnel $\mathcal{E}_{\eta}(t)$, def

Figures (12)

  • Figure 1: The synthesized state funnel projected on $x$ ($x_1$) and $y$ ($x_2$) position coordinates.
  • Figure 2: The synthesized state funnel with local Lipschiptz and L-smooth constants.
  • Figure 3: (Top): The synthesized state funnel with Lyapunov-type and $\text{Direct}$-type dynamics. (Bottom): The time history of minimum eigenvalues of $Q(t)$.
  • Figure 4: Time evolution of the maximum eigenvalue of each pointwise-in-time LMI constraint $L_l(t)$ associated with the corresponding constraint labeled in each subplot title. “Obstacle 1” denotes the obstacle at (1,2) in the xy-coordinate plane.
  • Figure 5: Mosek solve time versus number of constraints for two approaches: increasing node points and adding intermediate constraint-checking points (proposed).
  • ...and 7 more figures

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 4
  • Definition 5
  • Example 6
  • Example 7
  • Remark 8
  • Example 9
  • Lemma 10
  • ...and 26 more