Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Control of Hybrid Systems with Submersive Resets

William Clark, Maria Oprea, Aden Shaw

Abstract

Hybrid dynamical systems are systems which posses both continuous and discrete transitions. Assuming that the discrete transitions (resets) occur a finite number of times, the optimal control problem can be solved by gluing together the optimal arcs from the underlying continuous problem via the "Hamilton jump conditions." In most cases, it is assumed that the reset is a diffeomorphism (onto its image) and the corresponding Hamilton jump condition admits a unique solution. However, in many applications, the reset results in a drop in dimension and the corresponding Hamilton jump condition admits zero/infinitely many solutions. A geometric interpretation of this issue is explored in the case where the reset is a submersion (onto its image). Necessary conditions are presented for this case along with an accompanying numerical example.

Optimal Control of Hybrid Systems with Submersive Resets

Abstract

Hybrid dynamical systems are systems which posses both continuous and discrete transitions. Assuming that the discrete transitions (resets) occur a finite number of times, the optimal control problem can be solved by gluing together the optimal arcs from the underlying continuous problem via the "Hamilton jump conditions." In most cases, it is assumed that the reset is a diffeomorphism (onto its image) and the corresponding Hamilton jump condition admits a unique solution. However, in many applications, the reset results in a drop in dimension and the corresponding Hamilton jump condition admits zero/infinitely many solutions. A geometric interpretation of this issue is explored in the case where the reset is a submersion (onto its image). Necessary conditions are presented for this case along with an accompanying numerical example.
Paper Structure (7 sections, 4 theorems, 48 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 7 sections, 4 theorems, 48 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Optimal Control of Hybrid Systems
  3. Systems with Submersive Resets
  4. Uniqueness of the Co-state Jump
  5. Bouncing Ball with an "Internal Variable"
  6. Computing the Admissible Co-States
  7. Conclusion

Key Result

Theorem 1

Let $(\bar{x},\bar{u})$ be a controlled hybrid trajectory over the interval $[t_0,T]$ such that there exists finitely many resets and the reset times are uniformly separated. If this trajectory is optimal, then there exists a lifted hybrid curve $(\bar{x},\bar{p}):[t_0,T]\to T^*M$ such that away fro While at resets, the curve jumps according to Finally, the curve is subject to the terminal condit

Figures (6)

  • Figure 1: Schematic of the dynamics described by \ref{['eq:sHDS']}, where $M$ is given by the ambient space, $S$ is a $2$ dimensional surface, and its image under $\Delta$ is a curve. Two different controlled trajectories emanating from the same initial point may be mapped to the same state at the reset.
  • Figure 2: The foliation, $\mathcal{F}$, of the guard by the reset along with $T_x\mathcal{F}\subset T_x\mathcal{S}$.
  • Figure 3: Depiction of $\Xi_x$ for $x \in M$, where $\varphi_{\tau(x,v_1)}(x,v_1)$ is perpendicular to $T_y\mathcal{S}$ for some $y$ whereas $\varphi_{\tau(x,v_2)}(x,v_2)$ is not.
  • Figure 4: A schematic for the optimal control problem with two resets. The initial co-state must be admissible, $p_0\in \Xi_{x_0}$. Following the first reset, the new co-state must be contained in $\Xi_{x_1^+}\cap \tilde{\Delta}(p_1^-)$. After the second (and final) reset, the co-state no longer needs to be admissible. Rather, it needs to be chosen such that the terminal condition is satisfied.
  • Figure 5: A plot of $\Xi_{(0,1,1)}$ (the yellow surface) along with $\tilde{\Delta}(0,-1,0.1,1,1,0)$ (the blue curve). In this example, there exists a single intersection point.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1: Hybrid maximum principle
  • Proposition 1
  • Theorem 2
  • Proposition 2