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A Novel Parameter-Tying Theorem in Multi-Model Adaptive Systems: Systematic Approach for Efficient Model Selection

Farid Mafi, Ladan Khoshnevisan, Mohammad Pirani, Amir Khajepour

TL;DR

This work addresses the prohibitive model-count in high-dimensional multi-model adaptive systems by introducing a Parameter-Tying Theorem that enables dimension reduction through transformation to controllable canonical form. It shows how convex combinations of vertex systems remain valid under a weight transformation and provides a practical criterion for ensuring the plant lies within the convex hull of identification models. The approach is validated on coupled lateral-roll vehicle dynamics, where the number of required identification models drops from a worst-case $2^k$ to as few as two, while maintaining estimation accuracy for key states such as roll angle and roll rate. The results imply substantial computational savings and real-time feasibility for high-dimensional MMAS in applications with strong structure, such as vehicle dynamics and other coupled systems.

Abstract

This paper presents a novel theoretical framework for reducing the computational complexity of multi-model adaptive control/estimation systems through systematic transformation to controllable canonical form. While traditional multi-model approaches face exponential growth in computational demands with increasing system dimension, we introduce a parameter-tying theorem that enables significant dimension reduction through careful analysis of system characteristics in canonical form. The approach leverages monotonicity properties and coordinated parameter relationships to establish minimal sets of identification models while preserving system stability and performance. We develop rigorous criteria for verifying plant inclusion within the convex hull of identification models and derive weight transformation relationships that maintain system properties across coordinate transformations. The effectiveness of the framework is demonstrated through application to coupled lateral-roll vehicle dynamics, where the dimension reduction enables real-time implementation while maintaining estimation accuracy. The results show that the proposed transformation approach can achieve comparable performance to conventional methods while requiring substantially fewer identification models, enabling practical deployment in high-dimensional systems.

A Novel Parameter-Tying Theorem in Multi-Model Adaptive Systems: Systematic Approach for Efficient Model Selection

TL;DR

This work addresses the prohibitive model-count in high-dimensional multi-model adaptive systems by introducing a Parameter-Tying Theorem that enables dimension reduction through transformation to controllable canonical form. It shows how convex combinations of vertex systems remain valid under a weight transformation and provides a practical criterion for ensuring the plant lies within the convex hull of identification models. The approach is validated on coupled lateral-roll vehicle dynamics, where the number of required identification models drops from a worst-case to as few as two, while maintaining estimation accuracy for key states such as roll angle and roll rate. The results imply substantial computational savings and real-time feasibility for high-dimensional MMAS in applications with strong structure, such as vehicle dynamics and other coupled systems.

Abstract

This paper presents a novel theoretical framework for reducing the computational complexity of multi-model adaptive control/estimation systems through systematic transformation to controllable canonical form. While traditional multi-model approaches face exponential growth in computational demands with increasing system dimension, we introduce a parameter-tying theorem that enables significant dimension reduction through careful analysis of system characteristics in canonical form. The approach leverages monotonicity properties and coordinated parameter relationships to establish minimal sets of identification models while preserving system stability and performance. We develop rigorous criteria for verifying plant inclusion within the convex hull of identification models and derive weight transformation relationships that maintain system properties across coordinate transformations. The effectiveness of the framework is demonstrated through application to coupled lateral-roll vehicle dynamics, where the dimension reduction enables real-time implementation while maintaining estimation accuracy. The results show that the proposed transformation approach can achieve comparable performance to conventional methods while requiring substantially fewer identification models, enabling practical deployment in high-dimensional systems.
Paper Structure (31 sections, 7 theorems, 85 equations, 7 figures, 2 tables)

This paper contains 31 sections, 7 theorems, 85 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature Review
  4. Contributions
  5. Main Results
  6. Problem statement and preliminaries
  7. The Parameter-Tying Theorem
  8. Proof of the Parameter-Tying Theorem
  9. Monotonicity Analysis
  10. Characteristic Polynomial Analysis
  11. Weight Transformation Analysis
  12. Completion of the Proof
  13. Expanding the Conditions of Parameter-Tying Theorem
  14. A Criterion for Verifying Plant Inclusion in the Convex Combination of Identification Models
  15. Application to Vehicle Dynamics: State Estimation in Coupled Lateral-Roll Motion
  16. ...and 16 more sections

Key Result

Theorem 1

Let $\Sigma_p$ be the uncertain linear system defined in eq1 with parameter uncertainty vector $\mathbf{m} \in M \subset \mathbb{R}^k$. Define $\Pi_l: M \to M \setminus \{m_l\}$ as the projection operator removing the $l$-th component, and let $\hat{\mathbf{m}}_l = \Pi_l(\mathbf{m})$. Suppose that f is monotonic $\forall i,j \in \{1,\ldots,n\}$, where $a_{ij}$ are the entries of $A_p(\mathbf{m})$.

Figures (7)

  • Figure 1: Multi-model structure.
  • Figure 2: Simulated vehicle behavior in CarSim at constant speeds of: (a) 50 km/h and (b) 100 km/h.
  • Figure 3: Simulation Results for Different Target Speeds
  • Figure :
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Lemma 1: Single Parameter Extremality
  • Lemma 2: Extension to Multiple Parameters
  • proof
  • Theorem 2: Global Extremality
  • proof
  • Definition 4
  • ...and 9 more