A new approach to the solution of the l1 control problem: the scaled-Q method

We explore an approach for solving multiple input-multiple output (MIMO) l₁ optimal control problems. This approach, which we refer to as the scaled-Q approach, is introduced to alleviate many of the difficulties facing the numerical solution of optimal l₁ control problems. In particular, the computations of multivariable zeros and their directions are no longer required. The scaled-Q method also avoids the pole-zero cancellation difficulties that existing methods based on zero-interpolation face when attempting to recover the optimal controller from an optimal closed-loop map. Because the scaled-Q approach is based on solving a regularized auxiliary problem for which the solution is always guaranteed to exist, it can be used no matter where the system zeros are (including the stability boundary). Upper and lower bounds that converge to the optimal cost are provided, and all solutions are based on finite dimensional linear programming for which efficient software exists