New math framework compares nonlinear systems with LMI efficiency
Researchers from Notre Dame define (T_e,γ,δ)-similarity to benchmark complex dynamic systems.
The paper, from authors at the University of Notre Dame, introduces a formal notion of system comparison called (T_e,γ,δ)-similarity. This metric measures how closely the outputs of two nonlinear dynamical systems match, accounting for inputs and disturbances via the L2 norm. By linking the concept to differential dissipativity, the authors show that the similarity of a nonlinear system is equivalent to the similarity of its differential dynamics. This reduces the problem to solving a Linear Matrix Inequality (LMI) feasibility problem, for which they provide necessary and sufficient conditions.
The method's utility is demonstrated in two applications: robust hierarchical control of a planar aircraft, where it guides control law design for stability, and abstract model improvement for the Moore-Greitzer jet engine model and an electronic circuit, enabling model reduction without sacrificing fidelity. This work provides a rigorous, computationally tractable framework for comparing and simplifying complex systems, with implications for control theory, robotics, and scientific modeling.
- Proposes (T_e,γ,δ)-similarity to measure output dissimilarity between nonlinear systems using L2 norms.
- Equivalence to differential dissipativity allows reformulation as a Linear Matrix Inequality (LMI) feasibility problem.
- Demonstrated on planar aircraft (hierarchical control) and Moore-Greitzer/electronic circuit models (abstraction design).
Why It Matters
Provides a rigorous, computable way to compare nonlinear systems, enabling safer and simpler control and modeling.