Polynomial approximation-based solution estimation for robust control optimization in dynamic engineering systems

Abstract

Robust optimal control of dynamic engineering systems under an extremely high level of parametric uncertainty and bounded exogenous disturbances still has high computational loads when applied through classical dynamic programming and Hamilton-Jacobi-Bellman partial differential equations, both of which are fraught with the difficulty of the curse of dimensionality while the state dimension grows. This work presents the robust synthesis problem as a minimax optimization problem over admissible sets of control actions and disturbances while proposing a polynomial approximation framework such that the state trajectories along with feedback control laws are expressed via the truncated orthogonal bases including both Legendre polynomials and Chebyshev polynomials and polynomial chaos expansion. The weighted-residual Galerkin projection changes the differential constraints into algebraic equations with the finite dimension, while the minimax optimization task is transformed into a semidefinite relaxation problem for the polynomial coefficients. The spectral convergence of the approximation error is achieved for the analytic vector fields along with the sub-optimality bound that links finite order solutions with the substance of the saddle-point control. The complexity analysis shows that the scaling by coefficients allows replacing exponential growth of the grid with polynomial growth with respect to the dimensionality of the state variables. The numerical experiments on the trajectory controlling of hypersonic vehicles, vibration suppression of flexible manipulators, and pitch control of wind turbines have demonstrated the worst-case savings in terms of costs of up to 41.6 percent in comparison to the baseline methods of H-infinity control and model predictive control demonstrating the accuracy and feasibility of the proposed estimation approach.