CLOT Norm Minimization for Continuous Hands-off Control

TL;DR

This paper introduces a novel control method using the CLOT norm to achieve continuous, sparse, and practical control signals for systems, improving upon traditional bang-off-bang controls by balancing sparsity and continuity.

Contribution

The paper proposes the CLOT norm for continuous sparse control, demonstrating its advantages over existing methods and analyzing its continuity properties and extensions with state constraints.

Findings

CLOT control is continuous and sparser than elastic net control.

Numerical simulations confirm the effectiveness of CLOT in producing sparse, continuous controls.

The difference between successive CLOT control values is O(√h), indicating a form of continuity.

Abstract

In this paper, we consider hands-off control via minimization of the CLOT (Combined $L$-One and Two) norm. The maximum hands-off control is the $L^0$-optimal (or the sparsest) control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration. In general, the maximum hands-off control is a bang-off-bang control taking values of $\pm 1$ and $0$. For many real applications, such discontinuity in the control is not desirable. To obtain a continuous but still relatively sparse control, we propose to use the CLOT norm, a convex combination of $L^1$ and $L^2$ norms. We show by numerical simulations that the CLOT control is continuous and much sparser (i.e. has longer time duration on which the control takes 0) than the conventional EN (elastic net) control, which is a convex combination of $L^1$ and squared $L^2$ norms. We also prove that the CLOT control is continuous in the sense that, if $O(h)$ denotes the sampling period, then the difference between successive values of the CLOT-optimal control is $O(\sqrt{h})$, which is a form of continuity. Also, the CLOT formulation is extended to encompass constraints on the state variable.