Sparsity is also important in control systems.
In particular, sparse control, called hands-off control, is proposed for saving energy and reducing CO2 emissions in control systems. For example, a hybrid vehicle, Toyota Prius (displayed above), uses this control. The internal combustion engine is stopped (control = zero; sparse control) when the vehicle is at a stop or the speed is lower than a preset threshold, and the electric motor is alternatively used. See also Hands-off Control as Green Control.
Recently, it has been proved that the sparsest (or L0 optimal) control among all admissible controls is L1 optimal (see L0/L1 Equivalence in Optimal Control). Based on this, hands-off control is extended to feedback control in
The abstract reads:M. Nagahara, D. E. Quevedo, D. Nesic, Maximum Hands-Off Control: A Paradigm of Control Effort Minimization, IEEE Transactions on Automatic Control, Vol. 61, No. 4, 2016 (to appear) (PDF is here).
In this paper, we propose a new paradigm of control, called a maximum hands-off control. A hands-off control is defined as a control that has a short support per unit time. The maximum hands-off control is the minimum support (or sparsest) per unit time among all controls that achieve control objectives. For finite horizon control, we show the equivalence between the maximum hands-off control and L1-optimal control under a uniqueness assumption called normality. This result rationalizes the use of L1 optimality in computing a maximum hands-off control. We also propose an L1/L2-optimal control to obtain a smooth hands-off control. Furthermore, we give a self-triggered feedback control algorithm for linear time-invariant systems, which achieves a given sparsity rate and practical stability in the case of plant disturbances. An example is included to illustrate the effectiveness of the proposed control.A MATLAB code for the simulation is available here.