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Pointwise-Sparse Actuator Scheduling for Linear Systems with Controllability Guarantee

Luca Ballotta, Geethu Joseph, Irawati Rahul Thete

TL;DR

This paper tackles the problem of designing sparse actuator schedules for discrete-time linear systems under a per-step sparsity budget, with the goal of guaranteeing controllability while managing control energy. It introduces a kernel-based feasibility analysis, an improved $s$-sparse greedy algorithm that provides formal controllability guarantees when feasible, and an MCMC-based optimization layer to further reduce energy metrics. The methods operate over a horizon $h$ for state dimension $n$ and input channels $m$, ensuring the schedule yields a full-rank reachability structure and favorable Gramian-based energy metrics. Empirical results show that the proposed greedy approach reliably yields controllable schedules where naive greedy fails, and that MCMC can achieve lower energy costs at greater computational cost, offering a practical toolbox for networked control with bandwidth constraints.

Abstract

This paper considers the design of sparse actuator schedules for linear time-invariant systems. An actuator schedule selects, for each time instant, which control inputs act on the system in that instant. We address the optimal scheduling of control inputs under a hard constraint on the number of inputs that can be used at each time. For a sparsely controllable system, we characterize sparse actuator schedules that make the system controllable, and then devise a greedy selection algorithm that guarantees controllability while heuristically providing low control effort. We further show how to enhance our greedy algorithm via Markov chain Monte Carlo-based randomized optimization

Pointwise-Sparse Actuator Scheduling for Linear Systems with Controllability Guarantee

TL;DR

This paper tackles the problem of designing sparse actuator schedules for discrete-time linear systems under a per-step sparsity budget, with the goal of guaranteeing controllability while managing control energy. It introduces a kernel-based feasibility analysis, an improved -sparse greedy algorithm that provides formal controllability guarantees when feasible, and an MCMC-based optimization layer to further reduce energy metrics. The methods operate over a horizon for state dimension and input channels , ensuring the schedule yields a full-rank reachability structure and favorable Gramian-based energy metrics. Empirical results show that the proposed greedy approach reliably yields controllable schedules where naive greedy fails, and that MCMC can achieve lower energy costs at greater computational cost, offering a practical toolbox for networked control with bandwidth constraints.

Abstract

This paper considers the design of sparse actuator schedules for linear time-invariant systems. An actuator schedule selects, for each time instant, which control inputs act on the system in that instant. We address the optimal scheduling of control inputs under a hard constraint on the number of inputs that can be used at each time. For a sparsely controllable system, we characterize sparse actuator schedules that make the system controllable, and then devise a greedy selection algorithm that guarantees controllability while heuristically providing low control effort. We further show how to enhance our greedy algorithm via Markov chain Monte Carlo-based randomized optimization
Paper Structure (15 sections, 5 theorems, 30 equations, 1 figure, 2 tables, 4 algorithms)

This paper contains 15 sections, 5 theorems, 30 equations, 1 figure, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Sparse Actuator Scheduling Problem
  3. Preliminaries on Sparse Controllability
  4. Sparse Control Design Problem Formulation
  5. Design Algorithms
  6. Drawback of the Naive Greedy Algorithm
  7. Characterizing Feasible Actuator Schedules
  8. Improved $s$-Sparse Greedy Algorithm
  9. $s$-Sparse Markov Chain Monte Carlo
  10. Numerical Experiments
  11. Revisited \ref{['ex']}
  12. Experiment with Large-Scale Network
  13. Conclusion
  14. Proof of \ref{['lem:suff']}
  15. Proof of \ref{['thm:kernel-condition']}

Key Result

Proposition 1

System eq:systemmodel is $s$-sparse controllable iff it is controllable and $s \ge \max\{n-\operatorname{rk}\left\{\boldsymbol{A}\right\}, 1\}$.

Figures (1)

  • Figure 1: Cost for experiment on large-scale network. The parameters of $s$-sparse MCMC are $T_\text{in}=1$, $T_\text{min}=10^{-7}$, $\alpha=0.1$, $\texttt{it}=5000$.

Theorems & Definitions (11)

  • Proposition 1: Joseph21tac-sparseControllability
  • Definition 1
  • Proposition 2: joseph2024sparse
  • Remark 1
  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • ...and 1 more