Comparison of Some Optimal Control Methods for the Design of Turbine Blades

This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations (2) projection operator methods in a function space which includes the frequencies of vibration and the control function (3) ε-technique penalty function transformation resulting in a highly nonlinear programming problem (4) finite difference discretization of the state equations again resulting in a nonlinear program (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.