Sparse quadratic regulator

We consider a control design problem aimed at balancing quadratic performance of linear systems with additional requirements on the control signal. These are introduced in order to obtain controls that are either sparse or infrequently changing in time. To achieve this objective, we augment a standard quadratic performance index with an additional term that penalizes either the ℓ₁ norm or the total variation of the control signal. We show that the minimizer of this convex optimization problem can be found by solving a two point boundary value problem (TPBVP) with non-differentiable nonlinearities. Furthermore, we employ alternating direction method of multipliers to determine the optimal controller iteratively from a sequence of linear TPBVPs. Examples are provided to illustrate the developed method.