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Minimax Optimal Adaptive Control for Systems on Cones

Anders Rantzer

TL;DR

The paper develops minimax adaptive control for systems constrained to positive cones, framing the problem as a dual-control dynamic game to handle worst-case disturbances and uncertain parameters. By proving a cone-based Bellman equivalence, it provides exact solutions to the general problem and its linear-quadratic specialization, enabling data-compressed, history-dependent control laws that balance exploration and exploitation. An explicit construction is given for a family of linear systems, demonstrating a concrete controller with guaranteed performance and an explicit cone \(\mathcal{Q}\) leading to an exact solution. The work advances the theory of adaptive control on positive cones with a rigorous minimax/dynamic-programming foundation and practical LQ instances.

Abstract

The theory of optimal control on positive cones has recently identified several new problem classes where the Bellman equation can be solved explicitly, in analogy with classical linear quadratic control. In this paper, the idea is extended to minimax adaptive control, yielding exact solutions to instances of the Bellman equation for dual control. In particular, this allows for optimization of the fundamental tradeoff between exploration and exploitation.

Minimax Optimal Adaptive Control for Systems on Cones

TL;DR

The paper develops minimax adaptive control for systems constrained to positive cones, framing the problem as a dual-control dynamic game to handle worst-case disturbances and uncertain parameters. By proving a cone-based Bellman equivalence, it provides exact solutions to the general problem and its linear-quadratic specialization, enabling data-compressed, history-dependent control laws that balance exploration and exploitation. An explicit construction is given for a family of linear systems, demonstrating a concrete controller with guaranteed performance and an explicit cone leading to an exact solution. The work advances the theory of adaptive control on positive cones with a rigorous minimax/dynamic-programming foundation and practical LQ instances.

Abstract

The theory of optimal control on positive cones has recently identified several new problem classes where the Bellman equation can be solved explicitly, in analogy with classical linear quadratic control. In this paper, the idea is extended to minimax adaptive control, yielding exact solutions to instances of the Bellman equation for dual control. In particular, this allows for optimization of the fundamental tradeoff between exploration and exploitation.

Paper Structure

This paper contains 7 sections, 2 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. Solution to the General Problem.
  5. The Linear Quadratic Special Case
  6. Example with Explicit Solution
  7. Acknowledgements

Key Result

Theorem 1

Consider proper cones $\mathcal{X}\subseteq\mathbb{R}^{N_y}$, $\mathcal{U}\subseteq\mathbb{R}^{N_y+N_v}$ and $\mathcal{V}, \mathcal{S}\subseteq\mathbb{R}^{2N_y+N_v}$. Suppose that for every $y\in\mathcal{X}$ there exists $v$ with $(y,v)\in\mathcal{U}$. Assume also that the cones $\mathcal{U}$ and $\ is a linear function of $(z,y,v)$ whenever finite. Then the following conditions are equivalent: M

Figures (1)

  • Figure 1: We want a feedback controller that works for all system parameters within the given bounds. Nonlinear adaptive controllers can do much better by estimating $(A,B)$ and use the estimate for control. The objective of this paper is to derive such controllers using a dynamic game formulation.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2