Iterative Solution of Algebraic Riccati Equations for Damped Systems

Algebraic Riccati equations (ARE) of large dimension arise when using approximations to design controllers for systems modelled by partial differential equations. We use a modified Newton method to solve the ARE. Since the modified Newton method leads to a right-hand side of rank equal to the number of inputs, regardless of the weights, the resulting Lyapunov equation can be more efficiently solved. A low-rank Cholesky-ADI algorithm is used to solve the Lyapunov equation resulting at each step. The algorithm is straightforward to code. Performance is illustrated with an example of a beam, with different levels of damping. Results indicate that for weakly damped problems a low rank solution to the ARE may not exist. Further analysis supports this point