Optimal Active Control of a Deformable Mirror

Abstract

A new control design method for active shape control of a dynamic flexible structures with piezoelectric inclusions is developed. It is applied mainly for applications such as deformable mirrors of an adaptive optics system, space-based imaging systems and other applications where deformable mirrors constitute part of the system. In this method, the dynamics of the flexible structure — the mirror — is included in the controller design in the form of algebraic equations of motion. The algebraic optimal control laws are derived in an explicit form for a general time-varying distributed parameter system with piezoelectric inclusions. The essence of the approach is based on using space-time assumed mode expansions of the generalized coordinates and inputs, or in other words the Rayleigh—Ritz method extended to both space and time dimensions in conjunction with variational work—energy principles that govern the physical system. An optimal tracking controller was designed for shaping the surface of the deformable mirror used in an adaptive optics system. The desired time-changing surface trajectory can be supplied analytically or by point data that can be curve fitted using Zernike polynomials for wavefront distortions for direct use. One feature of this method is that it accommodates the dynamic behavior of the structure instead of considering only the static shape control. It is shown through illustrated examples that the controller can operate at much higher control frequencies than what is currently reported in the literature.