The Algebraic Theory of Operator Matrix Polynomials with Applications to Aeroelasticity in Flight Dynamics and Control

This paper develops an algebraic framework for operator matrix polynomials and demonstrates its application to control-design problems in aeroservoelastic systems. We present constructive spectral-factorization and linearization tools (block spectral divisors, companion forms and realization algorithms) that enable systematic block-pole assignment for large-scale MIMO models. Building on this theory, an adaptive block-pole placement strategy is proposed and cast in a practical implementation that augments a nominal state-feedback law with a compact neural-network compensator (single hidden layer) to handle un-modeled nonlinearities and uncertainty. The method requires state feedback and the system’s nominal model and admits Laplace-domain analysis and straightforward implementation for a two-degree-of-freedom aeroelastic wing with cubic stiffness nonlinearity and Roger aerodynamic lag is validated in MATLAB R2023a. Comprehensive simulations (Runge–Kutta 4) for different excitations and step disturbances demonstrate the approach’s advantages: compared with Eigenstructure assignment, LQR and H₂-control, the proposed method achieves markedly better robustness and transient performance (e.g., closed-loop ‖H(iω)‖₂ ≈ 4.64, condition number χ ≈ 11.19, and reduced control efforts μ ≈ 0.41, while delivering faster transients and tighter regulation (rise time ≈ 0.35 s, settling time ≈ 1.10 s, overshoot ≈ 6.2%, steady-state error ≈ 0.9%, disturbance-rejection ≈ 92%). These results confirm that algebraic operator-polynomial techniques, combined with a compact adaptive NN augmentation, provide a well-conditioned, low-effort solution for robust control of aeroelastic systems.