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Towards optimal control of systems with backlash

Maria do Rosário de Pinho, Maria Margarida Amorim Ferreira, Georgi Smirnov

TL;DR

The paper addresses time-optimal control of mechanical systems with backlash, where inelastic impacts cause nonuniqueness in trajectories. It develops an approximation scheme that inserts large-penalty impact forces, defines backlash solutions as limits of these approximations, and applies standard necessary conditions for the approximating problems to obtain limit-based optimality relations incorporating measure-valued terms. A key contribution is a main assumption on Lebesgue density points that enables passing to the limit and deriving backlash necessary conditions, including adjoint dynamics and a Hamiltonian maximum, with a nontriviality guarantee under a stability scenario. Two explicit examples illustrate the method: a single-degree-of-freedom backlash problem and a piston–cylinder system, revealing structured control laws and phase-by-phase motion near impacts, thereby providing a principled framework for analyzing time-optimal backlash control in mechanical systems.

Abstract

In this paper we consider time-optimal control problems for systems with backlash. Such systems are described by second order differential equations coupled with restrictions modeling the inelastic shocks. A main feature of such systems is the lack of uniqueness of solution to the Cauchy problem. Here, we introduce approximation systems where the forces during the impact are taken into account. Such approximations are relevant for two reasons. Firstly, we define a set of solutions as limits of the solutions to the approximation systems. This set may be smaller than the set of of the solutions usually considered in the literature. Secondly, such approximations are adequate to derive necessary condition to the time optimal control of interest. To the best of our knowledge, this is the first attempt to derive necessary conditions of optimality for optimal control problems involving systems with backlash.

Towards optimal control of systems with backlash

TL;DR

The paper addresses time-optimal control of mechanical systems with backlash, where inelastic impacts cause nonuniqueness in trajectories. It develops an approximation scheme that inserts large-penalty impact forces, defines backlash solutions as limits of these approximations, and applies standard necessary conditions for the approximating problems to obtain limit-based optimality relations incorporating measure-valued terms. A key contribution is a main assumption on Lebesgue density points that enables passing to the limit and deriving backlash necessary conditions, including adjoint dynamics and a Hamiltonian maximum, with a nontriviality guarantee under a stability scenario. Two explicit examples illustrate the method: a single-degree-of-freedom backlash problem and a piston–cylinder system, revealing structured control laws and phase-by-phase motion near impacts, thereby providing a principled framework for analyzing time-optimal backlash control in mechanical systems.

Abstract

In this paper we consider time-optimal control problems for systems with backlash. Such systems are described by second order differential equations coupled with restrictions modeling the inelastic shocks. A main feature of such systems is the lack of uniqueness of solution to the Cauchy problem. Here, we introduce approximation systems where the forces during the impact are taken into account. Such approximations are relevant for two reasons. Firstly, we define a set of solutions as limits of the solutions to the approximation systems. This set may be smaller than the set of of the solutions usually considered in the literature. Secondly, such approximations are adequate to derive necessary condition to the time optimal control of interest. To the best of our knowledge, this is the first attempt to derive necessary conditions of optimality for optimal control problems involving systems with backlash.
Paper Structure (9 sections, 3 theorems, 119 equations, 4 figures)

This paper contains 9 sections, 3 theorems, 119 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition of solution
  3. Approximating problem
  4. Main assumption
  5. Necessary conditions
  6. The case when the target set is a stable neighborhood of an equilibrium point
  7. Examples
  8. Example 1
  9. Example 2

Key Result

Theorem 1

Let $\gamma_k\rightarrow\infty$ be a sequence of scalars and let $u_{{\gamma_k}_l}$ be a sequence of admissible controls converging in weak* topology of $L_{\infty} ([0,T],R^n)$ to some function $\hat{u}(t)\in U(t)$. For each $k_l$, let $(x_{{\gamma_k}_l}, y_{{\gamma_k}_l},v_{{\gamma_k}_l},w_{{\gamm

Figures (4)

  • Figure 1: Here, $X$ is the position of the bottom of the cylinder and $Y$ is the position of the piston. The parameters $\alpha$ and $\beta$ are the resistance coefficients while $k$ is the stiffness of the spring. Moreover, we have $V=\dot X$ and $W=\dot Y$.
  • Figure 2: Time-optimal trajectories for Example 1. The bold vertical arrow along the $w$ axis shows the jump of velocity when the trajectory reaches the boundary of the set $C=\left\{(y,w):~y\leq 0\right\}$.
  • Figure 3: Time-optimal control for Example 2.
  • Figure 4: The four main phases of the motion (with negligible friction) in Example 2. Phase 1, when the cylinder moves away from the wall while the piston moves in the opposite direction with acceleration, is on the top left figure. Phase 2, on the right top, is when the piston starts braking before collision with the bottom of the cylinder. Phase 3, on the bottom left, is when the piston collides with the bottom of the cylinder. Finally, Phase 4 depicts the situation where the piston returns to its initial position while the cylinder remains in its equilibrium position.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Definition 1
  • Example 1
  • Example 2
  • Theorem 2
  • proof
  • Theorem 3