Maximum Hands-off Control without Normality Assumption
For control theorists, this extends maximum hands-off control theory to non-normal systems, but the results are incremental as they rely on known relaxation techniques.
This paper analyzes maximum hands-off (L0-optimal) control for linear time-invariant systems without the normality assumption, using Lp-optimal control with 0<p<1 as a relaxation. It establishes existence, bang-off-bang property, relation to L1-optimal control, and continuity/convexity of the value function.
Maximum hands-off control is a control that has the minimum L0 norm among all feasible controls. It is known that the maximum hands-off (or L0-optimal) control problem is equivalent to the L1-optimal control under the assumption of normality. In this article, we analyze the maximum hands-off control for linear time-invariant systems without the normality assumption. For this purpose, we introduce the Lp-optimal control with 0<p<1, which is a natural relaxation of the L0 problem. By using this, we investigate the existence and the bang-off-bang property (i.e. the control takes values of 1, 0 and -1) of the maximum hands-off control. We then describe a general relation between the maximum hands-off control and the L1-optimal control. We also prove the continuity and convexity property of the value function, which plays an important role to prove the stability when the (finite-horizon) control is extended to model predictive control.