Characterization of maximum hands-off control

Abstract Maximum hands-off control aims to maximize the length of time over which zero actuator values are applied to a system when executing specified control tasks. To tackle such problems, recent literature has investigated optimal control problems which penalize the size of the support of the control function and thereby lead to desired sparsity properties. This article gives the exact set of necessary conditions for a maximum hands-off optimal control problem using an L 0 -norm, and also provides sufficient conditions for the optimality of such controls. Numerical example illustrates that adopting an L 0 cost leads to a sparse control, whereas an L 1 -relaxation in singular problems leads to a non-sparse solution.

[1]  John L. Casti.  Introduction to the Mathematical Theory of Control Processes, Volume I: Linear Equations and Quadratic Criteria, Volume II: Nonlinear Processes , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[2]  Fredi Tröltzsch,et al.  Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls , 2014, SIAM J. Control. Optim..

[3]  W. Brockett,et al.  Minimum attention control , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[4]  Daniel E. Quevedo,et al.  Maximum Hands-Off Control: A Paradigm of Control Effort Minimization , 2014, IEEE Transactions on Automatic Control.

[5]  John T. Betts,et al.  Practical Methods for Optimal Control and Estimation Using Nonlinear Programming , 2009 .

[6]  Frank Allgöwer,et al.  ℓ0-System Gain and ℓ1-Optimal Control , 2011 .

[7]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[8]  H. Maurer,et al.  On L1‐minimization in optimal control and applications to robotics , 2006 .

[9]  Roland Herzog,et al.  Approximation of sparse controls in semilinear equations by piecewise linear functions , 2012, Numerische Mathematik.

[10]  Massimo Fornasier,et al.  Sparse Stabilization and Control of Alignment Models , 2012, 1210.5739.

[11]  K. Kunisch,et al.  A convex analysis approach to optimal controls with switching structure for partial differential equations , 2016, 1702.07540.

[12]  S. Kahne,et al.  Optimal control: An introduction to the theory and ITs applications , 1967, IEEE Transactions on Automatic Control.

[13]  Daniel E. Quevedo,et al.  Maximum hands-off control and L1 optimality , 2013, 52nd IEEE Conference on Decision and Control.

[14]  F. Clarke.  Functional Analysis, Calculus of Variations and Optimal Control , 2013 .

[15]  M. Fornasier,et al.  Sparse stabilization and optimal control of the Cucker-Smale model , 2013 .

[16]  Tamer Basar,et al.  Sparsity based feedback design: A new paradigm in opportunistic sensing , 2011, Proceedings of the 2011 American Control Conference.

[17]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[18]  Edward N. Hartley,et al.  Terminal spacecraft rendezvous and capture with LASSO model predictive control , 2013, Int. J. Control.

[19]  Victor M. Becerra,et al.  Optimal control , 2008, Scholarpedia.

[20]  Daniel E. Quevedo,et al.  Sparse Packetized Predictive Control for Networked Control Over Erasure Channels , 2013, IEEE Transactions on Automatic Control.

[21]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[22]  Richard Bellman,et al.  Introduction to the mathematical theory of control processes , 1967 .

[23]  Georg Stadler,et al.  Elliptic optimal control problems with L1-control cost and applications for the placement of control devices , 2009, Comput. Optim. Appl..

[24]  Fu Lin,et al.  Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers , 2011, IEEE Transactions on Automatic Control.