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Inverse Optimal Feedback and Gain Margins for Unicycle Stabilization

Kwang Hak Kim, Velimir Todorovski, Miroslav Krstić

Abstract

The recent development of globally strict control Lyapunov functions (CLFs) for the challenging unicycle parking problem provides a foundation for pursuing optimality. We address this in the inverse optimal framework, thereby avoiding the need to solve the Hamilton$\unicode{x2013}$Jacobi$\unicode{x2013}$Bellman (HJB) equations, and establish a general result that is optimal with respect to a meaningful cost. We present several design examples that impose varying levels of penalty on the control effort, including arbitrarily bounded control. Furthermore, we show that the inverse optimal controller possesses an infinite gain margin thanks to the system being driftless, and leveraging this property, we extend the design to an adaptive controller that handles model uncertainty. Finally, we compare the performance of the non-adaptive inverse optimal controller with its adaptive counterpart.

Inverse Optimal Feedback and Gain Margins for Unicycle Stabilization

Abstract

The recent development of globally strict control Lyapunov functions (CLFs) for the challenging unicycle parking problem provides a foundation for pursuing optimality. We address this in the inverse optimal framework, thereby avoiding the need to solve the HamiltonJacobiBellman (HJB) equations, and establish a general result that is optimal with respect to a meaningful cost. We present several design examples that impose varying levels of penalty on the control effort, including arbitrarily bounded control. Furthermore, we show that the inverse optimal controller possesses an infinite gain margin thanks to the system being driftless, and leveraging this property, we extend the design to an adaptive controller that handles model uncertainty. Finally, we compare the performance of the non-adaptive inverse optimal controller with its adaptive counterpart.

Paper Structure

This paper contains 9 sections, 3 theorems, 30 equations, 5 figures.

Table of Contents

  1. Introduction
  2. $L_gV$ (gradient) controllers
  3. Inverse Optimal Stabilization
  4. Basic quadratic inverse optimality
  5. General inverse optimal designs
  6. Specific inverse optimal choices
  7. Gain margin and robustness to actuator saturation
  8. Adaptive Stabilization under Unknown Input Coefficients
  9. Conclusion

Key Result

Proposition 1

For the control system eq:unicycle_polar the positive definite, radially unbounded function is a strict CLF on $\{\rho > 0\}\times \mathbb{R}^2$.

Figures (5)

  • Figure 1: Cost-on-control functions $\eta(r)$, the respective derivative inberse $(\eta')^{-1}(r)$ and the Legendre-Fenchel transform divided by its argument $\ell\eta(r)/r$.
  • Figure 2: Inverse optimal controller trajectory with a quadratic cost on control effort \ref{['eq:eta_quad']} for several representative initial conditions (respective colors) to the target position and heading (black).
  • Figure 3: Comparison in control effort between Example \ref{['example:IOC1']} and Example \ref{['example:IOC2']} with $\varepsilon_1 = \varepsilon_2 = 1$.
  • Figure 4: Comparison in control effort between Example \ref{['example:IOC3']} and Example \ref{['example:IOC4']} with $\bar{v} = \bar{\omega} = 1$.
  • Figure 5: Comparison of state trajectories and control inputs under the quadratic cost $L_gV$ controller with both $b_1\varepsilon_1$ and $b_2\varepsilon_2$ taken as $1$ and the proposed adaptive controller with two different adaptation gains.

Theorems & Definitions (9)

  • Proposition 1: todorovski2025_CLF,restrepo2020leader
  • proof
  • Definition 1
  • Theorem 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 2