Inverse variational identification methods for hidden parameter estimation in nonlinear engineering dynamic models

Abstract

Hidden parameter estimation in nonlinear engineering dynamic systems is seen as an ill-name inverse variational problem instead of just being a direct least-squares fit. In other words, a cost functional that penalizes trajectory mismatch and then minimizes this cost function under a system of nonlinear dynamic constraints and inequality bounds defined by physical admissibility conditions, such as stresses, displacements and damping limits ensuing from physical admissibility constraints. Then, using a Lagrangian duality argument, the study reduces constrained program to a standard minimax problem formulation and introduces an appropriate characterization of stationarity by means of generalized Karush-Kuhn-Tucker conditions based on Clarke subdifferential calculus allowing one to deal with non-smooth boundary conditions. The study shown that through the use of generalized Mangasarian-Fromovitz conditions along with metric subregularity of the constraint map, one can guarantee both boundedness and local uniqueness of the set of Lagrange multipliers which prevents the pathological Lagrange multiplier blow-up as a consequence of failure of classical Linear Independence Constraint Qualification. The paper proposes a projected gradient minimax algorithm and proves a descent lemma under bounded measurement noise establishing linear convergence of the algorithm to the hidden parameter vector. Then apply our approach to a nonlinear engineering example of hidden stiffness coefficients and compare the proposed framework with classical formulation in terms of numerical stability of multiplication norm as well as the convergence of the parameters and error of the trajectory.