Riemannian Optimal Control and Model Matching of Linear Port-Hamiltonian Systems

This paper presents a unified controller design method for $H^2$ optimal control and model matching problems of linear port-Hamiltonian systems. The controller design problems are formulated as optimization problems on the product manifold of the set of skew symmetric matrices, the manifold of the symmetric positive definite matrices, and Euclidean space. A Riemannian metric is chosen for the manifold in such a manner that the manifold is geodesically complete, i.e., the domain of the exponential map is the whole tangent space for every point on the manifold. In order to solve these problems, the Riemannian gradients of the objective functions are derived, and these gradients are used to develop a Riemannian steepest descent method on the product manifold. The geodesic completeness of the manifold guarantees that all points generated by the steepest descent method are on the manifold. Numerical experiments illustrate that our method is able to solve the two specified problems.