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Joint Synthesis of Trajectory and Controlled Invariant Funnel for Discrete-time Systems with Locally Lipschitz Nonlinearities

Taewan Kim, Purnanand Elango, Behcet Acikmese

Abstract

This paper presents a joint synthesis algorithm of trajectory and controlled invariant funnel (CIF) for locally Lipschitz nonlinear systems subject to bounded disturbances. The CIF synthesis refers to a procedure of computing controlled invariance sets and corresponding feedback gains. In contrast to existing CIF synthesis methods that compute the CIF with a pre-defined nominal trajectory, our work aims to optimize the nominal trajectory and the CIF jointly to satisfy feasibility conditions without the relaxation of constraints and obtain a more cost-optimal nominal trajectory. The proposed work has a recursive scheme that mainly optimize trajectory update and funnel update. The trajectory update step optimizes the nominal trajectory while ensuring the feasibility of the CIF. Then, the funnel update step computes the funnel around the nominal trajectory so that the CIF guarantees an invariance property. As a result, with the optimized trajectory and CIF, any resulting trajectory propagated from an initial set by the control law with the computed feedback gain remains within the feasible region around the nominal trajectory under the presence of bounded disturbances. We validate the proposed method via two applications from robotics.

Joint Synthesis of Trajectory and Controlled Invariant Funnel for Discrete-time Systems with Locally Lipschitz Nonlinearities

Abstract

This paper presents a joint synthesis algorithm of trajectory and controlled invariant funnel (CIF) for locally Lipschitz nonlinear systems subject to bounded disturbances. The CIF synthesis refers to a procedure of computing controlled invariance sets and corresponding feedback gains. In contrast to existing CIF synthesis methods that compute the CIF with a pre-defined nominal trajectory, our work aims to optimize the nominal trajectory and the CIF jointly to satisfy feasibility conditions without the relaxation of constraints and obtain a more cost-optimal nominal trajectory. The proposed work has a recursive scheme that mainly optimize trajectory update and funnel update. The trajectory update step optimizes the nominal trajectory while ensuring the feasibility of the CIF. Then, the funnel update step computes the funnel around the nominal trajectory so that the CIF guarantees an invariance property. As a result, with the optimized trajectory and CIF, any resulting trajectory propagated from an initial set by the control law with the computed feedback gain remains within the feasible region around the nominal trajectory under the presence of bounded disturbances. We validate the proposed method via two applications from robotics.
Paper Structure (19 sections, 2 theorems, 48 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 2 theorems, 48 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Outline
  5. Notation
  6. Problem formulation
  7. Iterative Robust Trajectory Optimization
  8. Nominal trajectory update
  9. CIF update
  10. Nonlinear dynamics
  11. Invariance of a quadratic funnel
  12. Computing the funnel via SDP
  13. Local Lipschitz constant estimation via sampling
  14. Algorithm details and summary
  15. Numerical simulation
  16. ...and 4 more sections

Key Result

theorem 1

Suppose that there exists $Q_{k}\in\mathbb{S}_{++}^{n_{x}}$, $Y_{k}\in\mathbb{R}^{n_{u}\times n_{x}}$, $\nu_{k}^{p}>0$, $\lambda_{k}^{w}>0$, and $0<\alpha<1$ such that $\lambda_k^w < \alpha$ and the following matrix inequality holds for all $k\in\mathcal{N}_{0}^{N-1}$: Then the Lyapunov condition eq:Lyapunov_condition holds for the closed loop system eq:diff_closedloop with $K_{k}=Y_{k}Q_{k}^{-1}

Figures (10)

  • Figure 1: Comparative illustration of separate synthesis and joint synthesis approahces. Left: the funnel exhibits constraint violation due to an underestimated safety margin for uncertainties. Middle: the trajectory and the funnel are feasible, but suboptimal due to an overestimated safety margin for uncertainties. Right: the proposed joint synthesis yields an optimal trajectory and funnel while satisfying the obstacle avoidance constraint.
  • Figure 2: A block diagram of the proposed method. Starting from the initial guess, the method optimizes the trajectory while considering the feasibility of the funnel. The local Lipschitz constant $\gamma_k$ of the nonlinearity around the obtained trajectory is then estimated. The next step is to optimize the funnel with the funnel constraints and the Lyapunov condition that ensures the invariance property. The entire process is repeated until both the trajectory and the funnel converge.
  • Figure 3: A block diagram of the control procedure.
  • Figure 4: Nominal trajectories and synthesized funnels (projected on position coordinates) of Model $\textrm{I}$ (top-left), Model $\textrm{II}$ (top-right), and Model $\textrm{III}$ (bottom). Each figure shows the nominal trajectory (orange line), the projection of the state ellipsoid in the funnel (blue ellipse), and the approximated funnel generated with the linear closed-loop system (brown ellipse).
  • Figure 5: Nominal trajectories and synthesized input funnels (projected on each input coordinate) of Model $\textrm{I}$ (left), Model $\textrm{II}$ (middle), and Model $\textrm{III}$ (right). The zeroth-order hold on the input is used to generate the nominal trajectory.
  • ...and 5 more figures

Theorems & Definitions (5)

  • definition 1
  • theorem 1
  • proof
  • corollary 1
  • proof