Multiple Control Bandwidth Computations in Adaptive-optics

A study is made of a non{smooth matrix optimization problem arising in adaptive{ optics, which involves the optimal real{time control of a very fast{acting deformable mirror designed to compensate for atmospheric turbulence and other image degradation factors, such as wind{induced telescope vibration (windshake). The surface shape of this mirror must change rapidly to correct for time{varying optical distortions caused by these sources of image degradation. One formulation of this problem yields a functional f(U) = P n i=1 max j f(U T M j U) ii g over orthogonal matrices U, where U deenes the basis of de-formable mirror control modes to be used, and the collection of nn symmetric matrices M j characterize adaptive{optics performance over multiple control bandwidths. Selecting the orthogonal matrix U which maximizes f(U) corresponds to deening a basis of deformable mirror control modes and their associated control bandwidths. We consider rst the situation which can arise in practical applications where the matrices M j are \nearly" pairwise commutative. Besides providing useful bounds, these results lead to a simple corollary providing a closed{form solution for maximizing f in the special case where M j are simultaneously diagonalizable. The general optimization problem is approached using a heuristic Jacobi{like algorithm, and results in a previous paper on this approach by the authors are extended. Numerical tests using the algorithm indicate that in the presence of windshake, the performance of a closed{loop adaptive{optics system can be improved by the selection of distinct and independently optimized control bandwidths for separate modes of the wave{front distortion proole. These results may be relevant for the adaptive{optics system planned for the 8-meter Gemini{North telescope to be located on the mountain of Mauna