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Maximum Hands-Off Control: A Paradigm of Control Effort Minimization

M. Nagahara, D. E. Quevedo, D. Nesic

arXiv:1408.3025v2eess.SYcs.ITmath.OC

TL;DR

The paper addresses how to achieve control objectives while minimizing the time that actuators are active. It formulates maximum hands-off control, establishes its equivalence with L1-optimal control under normality, and develops smooth and feedback extensions. The proposed feedback algorithm achieves a given sparsity rate and practical stability for linear systems with plant disturbances.

  • Problem

    The maximum hands-off objective is difficult to solve because sparsity is represented by a non-convex, discontinuous L0-based cost.

  • Method

    The paper formulates maximum hands-off control, relates it to L1-optimal control under normality, proposes L1/L2 smoothing, and develops self-triggered feedback for linear time-invariant systems.

  • Results

    Under normality, maximum hands-off control can be computed via L1-optimal control, while the feedback algorithm achieves a given sparsity rate and practical stability for linear systems.

  • Takeaways & Limitations

    The paper provides an L1-based route to sparse control and a feedback design that maintains sparsity and practical stability in the stated linear-system setting.

Abstract

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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.

I. INTRODUCTION

The paper introduces maximum hands-off control as a sparsity-based way to reduce control effort by keeping inputs exactly zero for longer intervals. It connects this objective to L1 optimization and extends it toward smooth and feedback control.

  • Hands-off control reduces effort by holding the control exactly zero over time intervals, with applications including vehicle emissions, fuel use, railway energy consumption, and network communication.
  • Maximum hands-off control seeks the sparsest admissible control by minimizing the total duration of nonzero control values.
  • The maximum hands-off problem is difficult because its L0-based objective is non-convex and discontinuous, motivating L1 convex relaxation.
  • Under normality, L1-optimal control and maximum hands-off control are equivalent, so L1 optimization can compute the sparsest solution.
  • L1/L2-optimal control provides a smooth intermediate between maximum hands-off or L1-optimal control and minimum-energy or L2-optimal control.
  • A self-triggered feedback algorithm for linear time-invariant systems achieves a specified sparsity rate and practical stability despite plant disturbances.

III. MAXIMUM HANDS-OFF CONTROL PROBLEM

The paper formulates maximum hands-off control as minimizing the support of admissible inputs while satisfying state-transfer and input constraints. The resulting problem is difficult because its objective is nonlinear and non-smooth, and removing input bounds can eliminate solutions.

  • The sparsity rate is the fraction of a finite horizon during which the control is nonzero, and smaller values indicate sparser control.
  • Admissible controls satisfy magnitude constraints and drive the nonlinear plant from a prescribed initial state to the origin at a fixed final time.
  • Maximum hands-off control minimizes the support measure of an admissible input, equivalently maximizing the time during which the control is exactly zero.
  • The maximum hands-off objective combines the sparsity rates of multiple control components using positive weights.
  • The optimization is difficult because its objective is highly nonlinear and non-smooth, motivating convex relaxation in subsequent analysis.
  • Input constraints are necessary: without them, an integrator can admit controls with arbitrarily small L0 norm, while the limiting Dirac delta is not in L1.

IV. SOLUTION TO MAXIMUM HANDS-OFF CONTROL PROBLEM

The paper replaces the difficult maximum hands-off problem with a convex L1-optimal control problem. Under normality, the resulting control is bang-off-bang, linking L1 optimization to sparse control.

  • Convex relaxation: The maximum hands-off problem is reformulated by replacing the nonconvex L0 objective with the convex L1 norm.The L1 formulation is easier to solve than the original maximum hands-off problem.
  • L1-optimal control: The L1-optimal control problem minimizes a weighted sum of the componentwise L1 norms over admissible controls.The weights λ1, ..., λm are positive and specified in advance.
  • Normality and structure: Under normality, the L1-optimal control is piecewise constant and ternary, taking values −1, 0, or 1 almost everywhere.Normality excludes nonzero singular intervals and makes the control uniquely determined by the minimum principle almost everywhere.
  • Normality and structure: For linear plants, normality can be guaranteed by a sufficient condition, although checking normality is generally difficult without solving the canonical equations.The stated sufficient condition applies to linear plants.

C. Maximum Hands-Off Control and L1 Optimality

This section proves the central equivalence between maximum hands-off and L1-optimal control. Under normality, their optimal-solution sets coincide, and linear systems additionally admit a finite switching bound.

  • Equivalence theorem: The theorem establishes that, under normality and existence of an L1 solution, the maximum hands-off and L1-optimal solution sets are identical.Thus L1 optimization can compute a sparsest admissible control under the stated assumption.
  • Equivalence theorem: The proof uses the bang-off-bang structure to show that the L1 and L0 objectives agree on the optimal control.For controls taking values −1, 0, and 1, the relevant integral expressions coincide.
  • Interpretation: The L1-to-L0 relation is presented as analogous to using L1 optimization to obtain sparsest solutions in compressed sensing.The analogy concerns the use of L1 optimality to recover an L0-optimal solution.
  • Linear-system structure: For linear systems satisfying controllability and nonsingularity assumptions, the maximum hands-off control is piecewise constant with values −1, 0, and 1.It has no direct switches between +1 and −1 or between −1 and +1.
  • Linear-system structure: The number of discontinuities is bounded by 2nm(1 + Tω/π) for the stated linear-system assumptions.Here ω is the largest imaginary part of the eigenvalues of A.

D. Discrete-time hands-off control

The discrete-time formulation defines hands-off control by minimizing the number of nonzero input samples while driving the state to the origin. Under discrete-time normality, its optimal solutions coincide with those of the corresponding L1 problem.

  • Problem formulation: The discrete-time system drives an initial state ξ to the origin over N steps using bounded control inputs.Admissible controls satisfy |ui[k]| ≤ 1 and the endpoint conditions x[0] = ξ and x[N] = 0.
  • Problem formulation: Discrete-time maximum hands-off control minimizes the number of nonzero elements in the control sequence.The associated objective uses the discrete ℓ0 count of nonzero samples.
  • L1 characterization: The discrete-time L1-optimal control is analyzed with a Hamiltonian and a dead-zone function derived from the costate.The minimum principle characterizes the control at each time step.
  • Normality: Discrete-time normality is defined through the corresponding costate condition over k = 0, 1, ..., N − 1.This assumption provides the basis for the equivalence theorem.
  • Equivalence theorem: Under discrete-time normality and existence of a solution, the maximum hands-off and ℓ1-optimal solution sets are equal.The result is stated as the discrete-time counterpart of the continuous-time equivalence.

V. L1/L2-OPTIMAL CONTROL

The paper introduces L1/L2-optimal control to balance sparsity with continuity, producing a smooth hands-off control. Its parameters trade smoothness against sparsity, with limiting cases connecting it to L1- and L2-optimal control.

  • Motivation: The L1/L2-optimal control adds regularization to the L1 cost so the control can remain continuous rather than switching abruptly.This targets applications whose actuators cannot move abruptly.
  • Problem formulation: The L1/L2-optimal control problem minimizes a weighted mixed objective over admissible controls on [0, T].The formulation uses positive weights λ_i and θ_i for each input.
  • Optimality conditions: Pontryagin’s minimum principle yields a shrinkage-based optimal-control characterization involving the costate and the ratio λ/θ.The associated Hamiltonian uses the state, control, and costate variables.
  • Continuity: The L1/L2-optimal control is continuous over the finite horizon.Continuity follows from continuity of the relevant optimal-control map and canonical-system trajectories.
  • Sparsity–smoothness trade-off: Increasing λ_i or decreasing θ_i makes input u_i(t) sparser, whereas decreasing λ_i or increasing θ_i makes it smoother.Thus the weights explicitly trade sparsity against smoothness.
  • Limiting cases and example: The L1/L2-optimal control lies between the L1-optimal (maximum hands-off) and L2-optimal controls, with the limiting cases described by Proposition 15.An example with λ_1 = θ_1 = 1 produces a continuous but sufficiently sparse control whose states approach zero by T = 10.

VI. SELF-TRIGGERED HANDS-OFF FEEDBACK CONTROL

The paper extends maximum hands-off control to feedback control for disturbed linear time-invariant systems. The proposed setting addresses the difficulty of expressing finite-horizon L1-optimal control directly as a state-feedback law.

  • Feedback extension: The paper extends maximum hands-off control from computed open-loop solutions to feedback control because model uncertainty and disturbances make state-dependent control desirable.The preceding L1-optimal solution can be computed by convex optimization after time discretization, but is difficult to express as a function of the state.
  • Implementation context: The section’s feedback setting is motivated by the gap between finite-horizon L1-optimal computation and implementing control under plant disturbances.The supplied figure captions identify comparisons involving maximum hands-off and L1/L2-optimal control, plus the resulting state trajectory.
  • Plant model: The controlled plant is modeled as a single-input linear time-invariant system with an unknown disturbance d(t).For nonlinear plants, the model can be used after linearization, with d(t) representing the linearization error.
  • Assumptions: The assumed system conditions are reachability of (A, b) and nonsingularity of A.These conditions suffice for normality of the disturbance-free L1-optimal problem for any horizon length and initial condition.

A. Sparsity Rate for Infinite Horizon Signals

The paper defines sparsity rate for infinite-horizon controls and uses self-triggered finite-horizon maximum hands-off controls to construct feedback control with a prescribed sparsity bound.

  • Definition: Sparsity rate R∞(u) measures the fraction of time an infinite-horizon control is nonzero, with values between 0 and 1.Finite-support controls have rate 0, while controls nonzero almost everywhere have rate 1.
  • Algorithm: The algorithm fixes a target rate r and applies maximum hands-off control over successive finite horizons, with self-triggered updates based on the current plant state.The next sampling time is determined by the current state and a minimum inter-sampling time prevents zero-length intervals.
  • Guarantee: The resulting infinite-horizon control satisfies R∞(u) < r.The bound follows because each finite horizon has sparsity rate at most r, and the limiting sparsity-rate lemma transfers this bound to the full control.
  • Algorithm: Each horizon uses a minimum-time control followed by zero control when needed, producing an admissible control with a longer horizon and lower support fraction.The construction compares the minimum-time control with an admissible control that extends the horizon by appending a zero segment.
  • Implementation: Minimum-time computation is required at each update and can use an efficient numerical algorithm for single-input, linear time-invariant systems.The same computation can check whether the initial state lies in the reachable set.

C. Practical Stability

Under bounded disturbances and reachability assumptions, the self-triggered feedback law provides practical stability rather than asymptotic convergence to the origin, with a bound determined by system and sparsity parameters.

  • Stability setting: Because disturbances prevent exact asymptotic stabilization, the analysis targets practical stability under bounded plant noise.Between sampling instants, the system acts open loop, while state measurements trigger feedback updates.
  • Assumptions: The stability theorem assumes bounded disturbance, an initial state in the reachable set, and a suitable invariant set contained in that reachable set.The theorem also requires conditions involving the sparsity-rate parameter and the function bounding minimum-time reachability.
  • Stability result: The analysis establishes bounded state trajectories, sampled states remaining in the set Ω, and bounded intersample behavior.These properties are obtained separately for systems with µ(A) < 0 and µ(A) > 0.
  • Tradeoff: The bound γ becomes smaller as the sparsity rate r increases, exposing a tradeoff between control sparsity and performance.The bound is deterministic and corresponds to a worst-case disturbance, though it can be reasonably tight in examples.

VII. EXAMPLE

Examples evaluate the self-triggered hands-off controller on stable and unstable nonlinear plants, with disturbances and zero-control comparisons showing sparse control alongside practical stabilization.

  • Stable plant: For the stable one-dimensional plant, the maximum hands-off control is computed through L1-optimal control under normality and bounded-disturbance assumptions.The minimum-time function and reachable set are used to verify the theorem’s conditions.
  • Stable plant: With r = 0.6 and uniform disturbance bounded by δ = 1, the feedback control has sparsity rate R∞(u) = 0.148, below the target upper bound.The corresponding figure presents the hands-off feedback control.
  • Stable plant: For the stable plant, zero control is sparsest, but the time-optimal hands-off control drives the state toward zero faster.The comparison uses trajectories for hands-off and zero control, with sampled states also shown.
  • Disturbance test: Under the worst-case disturbance d(t) = 1, zero control leaves the state at 1 while maximum hands-off control still approaches zero.The deterministic bound is described as reasonably tight for this disturbance.
  • Unstable plant: For the unstable nonlinear plant with a = 1, zero control causes divergence whereas hands-off control keeps the state close to the origin with R∞(u) = 0.1135.The simulation uses initial state x0 = 0.25 and target sparsity rate r = 0.6.

VIII. CONCLUSION

The paper introduces maximum hands-off control as sparsest admissible control, establishes its L1-optimal equivalence under normality, and extends it to feedback control with sparsity and practical-stability guarantees.

  • Conclusion: Maximum hands-off control minimizes support per unit time among admissible controls.It is also described as the sparsest or L0-optimal control.
  • Conclusion: Under normality assumptions, maximum hands-off control can be computed through L1-optimal control.This provides the paper’s finite-horizon computational connection between sparsity and L1 optimality.
  • Conclusion: For linear systems, the proposed feedback algorithm guarantees a given sparsity rate and practical stability.An example illustrates the effectiveness of the proposed control.
  • Future work: Future work includes computation without normality conditions and extension to nonlinear plants.These are identified as open cases for maximum hands-off control computation.
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