Part I. Engineering motivation: Engineering background Engineering problems; Part II. Analytic function therory: Fractional maps and Grassmannians Representing shift invariant subspaces Applications to factorization, interpolation, and approximation Further applications Matrix analogs and generalizations; Part III. Matrices: Some matrix problems in engineering Optimization, matrix inequalities, and matrix completions The spectrum of a matrix with respect to an algebra; Part IV. The general $H^\infty$ optimization problem: Nonlinear $H^\infty$ optimization Bibliography.