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Funnel Synthesis via LMI Copositivity Conditions for Nonlinear Systems

Taewan Kim, Behçet Açıkmeşe

TL;DR

This work addresses the problem of computing time-varying controlled invariant funnels around nominal trajectories for nonlinear systems with bounded disturbances over a finite horizon. It develops an incremental LPV representation of the nonlinear dynamics, treats the linearization/approximation error as structured uncertainty governed by IQCs, and enforces a continuous-time DLMI via copositive LMIs. The resulting convex SDP yields both the funnel shape (via $Q(t)$) and the state-feedback gains (via $K(t)$) with guarantees that the funnel remains invariant for all times in $[t_0,t_f]$ despite approximations and disturbances. Numerical demonstrations on a 2D unicycle and a 6-DoF rocket descent illustrate invariant, feasible funnels and highlight advantages over baseline node-discretized DLMI approaches in terms of feasibility, cost, and computational effort. The method provides a principled, convex framework for invariant funnel synthesis applicable to robust motion planning and constrained control in nonlinear systems.

Abstract

Funnel synthesis refers to a procedure for synthesizing a time-varying controlled invariant set and an associated control law around a nominal trajectory. The computation of the funnel involves solving a continuous-time differential equation or inequality, ensuring the invariance of the funnel. Previous approaches often compromise the invariance property of the funnel; for example, they may enforce the equation or the inequality only at discrete temporal nodes and do not have a formal guarantee of invariance at all times. This paper proposes a computational funnel synthesis method that can satisfy the invariance of the funnel without such compromises. We derive a finite number of linear matrix inequalities (LMIs) that imply the satifaction of a continuous-time differential linear matrix inequality guaranteeing the invariance of the funnel at all times from the initial to the final time. To this end, we utilize LMI conditions ensuring matrix copositivity, which then imply continuous-time invariance. The primary contribution of the paper is to prove that the resulting funnel is indeed invariant over a finite time horizon. We validate the proposed method via a three-dimensional trajectory planning and control problem with obstacle avoidance constraints, and a six-degree-of-freedom powered descent guidance.

Funnel Synthesis via LMI Copositivity Conditions for Nonlinear Systems

TL;DR

This work addresses the problem of computing time-varying controlled invariant funnels around nominal trajectories for nonlinear systems with bounded disturbances over a finite horizon. It develops an incremental LPV representation of the nonlinear dynamics, treats the linearization/approximation error as structured uncertainty governed by IQCs, and enforces a continuous-time DLMI via copositive LMIs. The resulting convex SDP yields both the funnel shape (via ) and the state-feedback gains (via ) with guarantees that the funnel remains invariant for all times in despite approximations and disturbances. Numerical demonstrations on a 2D unicycle and a 6-DoF rocket descent illustrate invariant, feasible funnels and highlight advantages over baseline node-discretized DLMI approaches in terms of feasibility, cost, and computational effort. The method provides a principled, convex framework for invariant funnel synthesis applicable to robust motion planning and constrained control in nonlinear systems.

Abstract

Funnel synthesis refers to a procedure for synthesizing a time-varying controlled invariant set and an associated control law around a nominal trajectory. The computation of the funnel involves solving a continuous-time differential equation or inequality, ensuring the invariance of the funnel. Previous approaches often compromise the invariance property of the funnel; for example, they may enforce the equation or the inequality only at discrete temporal nodes and do not have a formal guarantee of invariance at all times. This paper proposes a computational funnel synthesis method that can satisfy the invariance of the funnel without such compromises. We derive a finite number of linear matrix inequalities (LMIs) that imply the satifaction of a continuous-time differential linear matrix inequality guaranteeing the invariance of the funnel at all times from the initial to the final time. To this end, we utilize LMI conditions ensuring matrix copositivity, which then imply continuous-time invariance. The primary contribution of the paper is to prove that the resulting funnel is indeed invariant over a finite time horizon. We validate the proposed method via a three-dimensional trajectory planning and control problem with obstacle avoidance constraints, and a six-degree-of-freedom powered descent guidance.
Paper Structure (17 sections, 6 theorems, 53 equations, 10 figures, 1 algorithm)

This paper contains 17 sections, 6 theorems, 53 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Invariance of funnel
  4. Nonlinear systems
  5. Incremental dynamical system and its LPV approximation
  6. Invariance condition
  7. Differential linear matrix inequality
  8. Funnel synthesis problem
  9. Transformation of DLMI into a finite number of LMIs
  10. Constraints
  11. Problem formulation
  12. Estimation of the constants via sampling
  13. Algorithm summary
  14. Numerical simulation
  15. Unicycle model
  16. ...and 2 more sections

Key Result

Lemma 1

Suppose there exists a PD matrix-valued continuous function $Q(t)\in\mathbb{S}_{++}^{n_{x}}$ and a continuous matrix-valued function $K(t)\in\mathbb{R}^{n_{u}\times n_{x}}$ such that all trajectories of the system eq:LPV_final satisfy for almost all $t\in[t_{0},t_{f}]$. Then, with every almost everywhere continuous signal $w(\cdot)$ such that eq:bound_on_2, $\mathcal{E}_{Q}(t)$ defined in eq:stat

Figures (10)

  • Figure 1: Uncertain LPV system interconnection with feedback control.
  • Figure 2: (Top) The synthesized state funnel projected on $x$ ($x_1$) and $y$ ($x_2$) position coordinates. (Bottom) Time history of the state funnel projected on yaw angle ($x_3$) coordinate.
  • Figure 3: Time history of the synthesized input funnel projected on velocity command ($u_1$) and angular velocity command ($u_2$) coordinates, shown in the top and bottom figures, respectively.
  • Figure 4: The cost results of \ref{['eq:SDP']} with different values of $\lambda_w$ are presented for both cases: using \ref{['eq:LMI_copositive1']} in Lemma \ref{['lem:first_copositive1']} and \ref{['eq:LMI_copositive2']} in Lemma \ref{['lem:first_copositive2']}.
  • Figure 5: (Top) The results of trajectories propagated from randomly selected samples within the funnel entry. (Bottom) Time history of Lyapunov function $V$, as defined in \ref{['eq:Lyapunov_function']}, for each trajectory sample.
  • ...and 5 more figures

Theorems & Definitions (16)

  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 6 more