Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning and Control from Similarity Between Heterogeneous Systems: A Behavioral Approach

Chenchao Wang, Deyuan Meng

TL;DR

This paper defines similarity between heterogeneous linear time-varying system behaviors through principal angles between their admissible-behavior subspaces and introduces similarity indexes as $SI(\\mathcal{B}_{1,x_1},\\mathcal{B}_{2,x_2})=\cos\\Theta(\\mathcal{W}_1,\\mathcal{W}_2)$. It then develops a projection-based learning control framework that leverages guest-system experience to guide the host system, with efficient computation via the SVD of $H_1^{\rm T}H_2$ and the affine-decomposition $\\mathcal{B}_{i,x_i}=\\mathcal{W}_i+w_{i,\text{off}}$. The key result shows that the optimal host trajectory $w_h$ is the orthogonal projection of the guest trajectory $w_g$ onto the host’s admissible set, given by $w_h=H_1UD\overline{g}+P_{\\mathcal{W}_1}(w_{2,\text{off}}-w_{1,\text{off}})+w_{1,\text{off}}$ with $w_g=H_2V\overline{g}+w_{2,\text{off}}$, enabling task transfer without repeated trial-and-error when similarity is high. Simulations validate the approach on multiple tasks and illustrate the influence of similarity indexes on transfer efficacy, highlighting practical benefits for fast, data-efficient control of heterogeneous systems.

Abstract

This paper proposes basic definitions of similarity and similarity indexes between heterogeneous linear systems and presents a similarity-based learning control strategy. By exploring geometric properties of admissible behaviors of linear systems, the similarity indexes between two admissible behaviors of heterogeneous systems are defined as the principal angles between their subspace components, and an efficient strategy for calculating the similarity indexes is developed. By leveraging the similarity indexes, a similarity-based learning control strategy is proposed via projection techniques. With the application of the similarity-based learning control strategy, host system can efficiently accomplish the same tasks by leveraging the successful experience of guest system, without the necessity to repeat the trial-and-error process experienced by the guest system.

Learning and Control from Similarity Between Heterogeneous Systems: A Behavioral Approach

TL;DR

This paper defines similarity between heterogeneous linear time-varying system behaviors through principal angles between their admissible-behavior subspaces and introduces similarity indexes as . It then develops a projection-based learning control framework that leverages guest-system experience to guide the host system, with efficient computation via the SVD of and the affine-decomposition . The key result shows that the optimal host trajectory is the orthogonal projection of the guest trajectory onto the host’s admissible set, given by with , enabling task transfer without repeated trial-and-error when similarity is high. Simulations validate the approach on multiple tasks and illustrate the influence of similarity indexes on transfer efficacy, highlighting practical benefits for fast, data-efficient control of heterogeneous systems.

Abstract

This paper proposes basic definitions of similarity and similarity indexes between heterogeneous linear systems and presents a similarity-based learning control strategy. By exploring geometric properties of admissible behaviors of linear systems, the similarity indexes between two admissible behaviors of heterogeneous systems are defined as the principal angles between their subspace components, and an efficient strategy for calculating the similarity indexes is developed. By leveraging the similarity indexes, a similarity-based learning control strategy is proposed via projection techniques. With the application of the similarity-based learning control strategy, host system can efficiently accomplish the same tasks by leveraging the successful experience of guest system, without the necessity to repeat the trial-and-error process experienced by the guest system.
Paper Structure (8 sections, 5 theorems, 43 equations, 7 figures)

This paper contains 8 sections, 5 theorems, 43 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Similarity and Similarity Indexes
  4. Similarity-Based Learning Via Projection
  5. Simulation Examples
  6. Example 1: Effectiveness in Multiple Tasks
  7. Example 2: Effectiveness with Different Similarity Indexes
  8. Conclusions and Future Works

Key Result

Lemma 1

For the discrete LTV system $\Sigma_{i,\mathbb{T}}$ with the initial state $x_i(0)=x_i$, the admissible behavior $\mathcal{B}_{i,x_i}$ is an affine set.

Figures (7)

  • Figure 1: The mechanism of the similarity-based learning control.
  • Figure 2: Less similar affine hyperplanes in $\mathbb{R}^3$.
  • Figure 3: More similar affine hyperplanes in $\mathbb{R}^3$.
  • Figure 4: An example for illustrating the machenism of similarity-based learning control
  • Figure 5: Inputs and outputs of $\Sigma_{1,\mathbb{T}}$ and $\Sigma_{2,\mathbb{T}}$ for reference $r^1(t)$.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Definition 1
  • Lemma 2
  • Remark 3
  • Definition 2
  • Remark 4
  • Remark 5
  • ...and 11 more