SISO controller design to minimize a positive combination of the l₁ and the ℋ₂ norms

We consider the problem of minimizing a given positive linear combination of the l₁ norm and the square of the ℋ₂ norm of the closed loop over all internally stabilizing controllers. The problem is analysed for the discrete-time, SISO, linear time-invariant case. It is shown that a unique optimal solution always exists, and can be obtained by solving a finite-dimensional convex optimization problem with an a priori determined dimension. It is also established that the solution is continuous with respect to changes in the coefficients of the linear combination.