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Towards a constructive framework for control theory

Pavel Osinenko

TL;DR

A fairly general framework for proof techniques in analysis and synthesis of control systems based on constructive analysis which explicitly states that every computation be doable only up to a finite precision thus accounting for computational uncertainty is presented.

Abstract

This work presents a framework for control theory based on constructive analysis to account for discrepancy between mathematical results and their implementation in a computer, also referred to as computational uncertainty. In control engineering, the latter is usually either neglected or considered submerged into some other type of uncertainty, such as system noise, and addressed within robust control. However, even robust control methods may be compromised when the mathematical objects involved in the respective algorithms fail to exist in exact form and subsequently fail to satisfy the required properties. For instance, in general stabilization using a control Lyapunov function, computational uncertainty may distort stability certificates or even destabilize the system despite robustness of the stabilization routine with regards to system, actuator and measurement noise. In fact, battling numerical problems in practical implementation of controllers is common among control engineers. Such observations indicate that computational uncertainty should indeed be addressed explicitly in controller synthesis and system analysis. The major contribution here is a fairly general framework for proof techniques in analysis and synthesis of control systems based on constructive analysis which explicitly states that every computation be doable only up to a finite precision thus accounting for computational uncertainty. A series of previous works is overviewed, including constructive system stability and stabilization, approximate optimal controls, eigenvalue problems, Caratheodory trajectories, measurable selectors. Additionally, a new constructive version of the Danskin's theorem, which is crucial in adversarial defense, is presented.

Towards a constructive framework for control theory

TL;DR

A fairly general framework for proof techniques in analysis and synthesis of control systems based on constructive analysis which explicitly states that every computation be doable only up to a finite precision thus accounting for computational uncertainty is presented.

Abstract

This work presents a framework for control theory based on constructive analysis to account for discrepancy between mathematical results and their implementation in a computer, also referred to as computational uncertainty. In control engineering, the latter is usually either neglected or considered submerged into some other type of uncertainty, such as system noise, and addressed within robust control. However, even robust control methods may be compromised when the mathematical objects involved in the respective algorithms fail to exist in exact form and subsequently fail to satisfy the required properties. For instance, in general stabilization using a control Lyapunov function, computational uncertainty may distort stability certificates or even destabilize the system despite robustness of the stabilization routine with regards to system, actuator and measurement noise. In fact, battling numerical problems in practical implementation of controllers is common among control engineers. Such observations indicate that computational uncertainty should indeed be addressed explicitly in controller synthesis and system analysis. The major contribution here is a fairly general framework for proof techniques in analysis and synthesis of control systems based on constructive analysis which explicitly states that every computation be doable only up to a finite precision thus accounting for computational uncertainty. A series of previous works is overviewed, including constructive system stability and stabilization, approximate optimal controls, eigenvalue problems, Caratheodory trajectories, measurable selectors. Additionally, a new constructive version of the Danskin's theorem, which is crucial in adversarial defense, is presented.
Paper Structure (8 sections, 8 theorems, 14 equations)

This paper contains 8 sections, 8 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. The framework
  3. Optimal control and stabilization
  4. Danskin's theorem
  5. Selector theorems
  6. System analysis
  7. Overview of case studies
  8. Conclusion

Key Result

Theorem 1

Consider $\mathcal{U}$, the space of all equi-Lipschitz and equi-bounded functions from a compact set $\mathbb{X} \subset \mathbb{R}^n$ to $\mathbb{R}^m$, and $J$, a uniformly continuous (cost) functional on $\mathcal{U}$ ("equi" here to mean having a common Lipschitz constant and a common bound, re

Theorems & Definitions (17)

  • Theorem 1: Constructive functional EVT Osinenko2018-constr-aEVTfnc-Euclid
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Theorem 2: Constructive Danskin's theorem
  • proof
  • Definition 1: Representable inverse
  • Definition 2: Simple set-valued function
  • Definition 3: Regular set-valued function
  • ...and 7 more