Approximation of low rank solutions for linear quadratic control of partial differential equations

Abstract

Algebraic Riccati equations (ARE) of large dimension arise when using approximations to design controllers for systems modeled by partial differential equations. We use a modified Newton method to solve the ARE that takes advantage of several special features of these problems. The modified Newton method leads to a right-hand side of rank equal to the number of inputs regardless of the weights. Thus, the resulting Lyapunov equation can be more efficiently solved. The Cholesky-ADI algorithm is used to solve the Lyapunov equation resulting at each step. The algorithm is straightforward to code. Performance is illustrated with a number of standard examples. An example on controlling the deflection of the Euler-Bernoulli beam indicates that for weakly damped problems a low rank solution to the ARE may not exist. Further analysis supports this point.