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 inThe 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.

Enjoy!