A quadratic programming approach for solving the l1 multi-block problem

We present a new method to compute solutions to the general multi-block l₁ control problem. The method is based on solving a standard H₂ problem and a finite-dimensional semidefinite quadratic programming problem of appropriate dimension. The new method has most of the properties that separately characterize many existing approaches, in particular, as the dimension of the quadratic programming problem increases, this method provides converging upper and lower bounds on the optimal l₁ norm and, for well posed multi-block problems, ensures the convergence in norm of the suboptimal solutions to an optimal l₁ solution. The new method does not require the computation of the interpolation conditions, and it allows the direct computation of the suboptimal controller