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

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.