Abstract
AbstractThis paper introduces the concept of an ℓ0-system gain for discrete-time LTI systems. It is shown that the ℓ0-gain is characterized by the number of non-zero entries in the impulse response of the system and hence gives a natural extension of the notion of sparsity from signals to systems. With this newly introduced system gain, we give a system theoretic explanation of the sparse closed loop response of ℓ1-optimal controlled systems by showing that the ℓ1-optimal control problem is the best convex relaxation (in the sense of Lagrangian duality) of an appropriately defined ℓ0-optimal control problem.
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