Computationally Efficient Multi-Model Adaptive Control and Estimation for Uncertain Systems

Abstract

Multi-model adaptive technique offers a powerful framework for handling uncertainties in dynamic systems by employing multiple models that represent different operating conditions. However, despite strong theoretical foundations, practical deployment has been severely limited by the curse of dimensionality—the exponential growth in computational requirements as system complexity increases, creating a major obstacle for real-time implementation. This thesis presents a comprehensive framework to address this computational challenge through three related contributions, achieving dramatic reductions in complexity while maintaining or improving system performance.

The main insight is that traditional multi-model approaches use far more models than necessary. Instead of using every possible model combination to cover parameter uncertainties, this thesis shows that carefully chosen subsets can achieve the same coverage with much less computation. This shift from using all models to selecting the right models makes multi-model control practical for real-time systems with limited resources.

To this end, geometric methods have been developed that analyze how models cover the space of possible system behaviors. Using computational geometry principles, new Enclosed Polytope with Minimum Models (EPMM) algorithms find the smallest set of models that still covers the uncertainty space adequately. These algorithms work like placing sensors strategically—finding the fewest locations needed to monitor an entire area. An optimization framework extends this idea to continuous parameter spaces, while a transfer function method handles high-dimensional systems efficiently by focusing on input-output relationships rather than full state representations.

To overcome the fundamental scaling limitations in high-dimensional spaces, the Parameter -Tying Theorem has been developed as a theoretical innovation showing that changing to the right coordinates can considerably simplify high-dimensional uncertainty spaces. By analyzing systems in controllable canonical form and finding monotonicity properties in how parameters affect the system, the theorem proves that the number of required models can drop from exponential in the number of parameters to potentially constant. The framework extends through five conditions—including affine relationships, symmetry, and coordinated parameter variations—making it applicable beyond strictly monotonic systems.

Furthermore, a unified estimation framework has been designed to tackle the integration of physics-based and data-driven models. A consensus multi-model Kalman filter combines different model types based on how well they perform. Two methods enable proper uncertainty handling: Koopman operator-based linearization allows analytical covariance propagation for neural networks and other nonlinear data-driven models, while an ensemble-based approach provides model-independent uncertainty quantification without needing offline training. The consensus fusion automatically adjusts model weights based on prediction errors, ensuring smooth transitions between models as conditions change. Extensive experimental data from an electric all-wheel-drive vehicle under extreme conditions shows major performance improvements over traditional single-model approaches.

This thesis transforms multi-model approach from an attractive theory with limited practical use into a viable solution for real-world applications. By combining geometric insight, coordinate transformation theory, and heterogeneous model integration, it addresses the fundamental implementation barriers. The developed frameworks maintain mathematical rigor while achieving the computational efficiency needed by modern embedded systems. These advances enable robust adaptive control across diverse operating conditions, with immediate applications in autonomous vehicles, renewable energy systems, and other areas where handling uncertainty is critical. The principles established here provide a foundation for addressing high-dimensional uncertainty in complex dynamical systems across engineering fields.