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Sparse Actuator Scheduling for Discrete-Time Linear Dynamical Systems

Krishna Praveen V. S. Kondapi, Chandrasekhar Sriram, Geethu Joseph, Chandra R. Murthy

TL;DR

This work tackles sparse actuator scheduling for discrete-time linear dynamical systems by optimizing an energy-based controllability metric under a per-step sparsity cap. It introduces an $\epsilon$-regularized objective $\mathrm{Tr}\{(W_{\mathcal{S}}+\epsilon I)^{-1}\}$ and a greedy algorithm over a matroid-constrained search space, accompanied by theoretical guarantees via $\alpha$-supermodularity and submodular-optimization bounds. The approach ensures the controllability rank reaches $n$ with a finite, judiciously chosen sequence of actuators, while incurring only modest energy penalties compared to fully actuated systems, as confirmed by simulations on random graphs. Practically, the method enables energy-aware, bandwidth-efficient control in large-scale networked systems where full actuation is impractical.

Abstract

We consider the control of discrete-time linear dynamical systems using sparse inputs where we limit the number of active actuators at every time step. We develop an algorithm for determining a sparse actuator schedule that ensures the existence of a sparse control input sequence, following the schedule, that takes the system from any given initial state to any desired final state. Since such an actuator schedule is not unique, we look for a schedule that minimizes the energy of sparse inputs. For this, we optimize the trace of the inverse of the resulting controllability Gramian, which is an approximate measure of the average energy of the inputs. We present a greedy algorithm along with its theoretical guarantees. Finally, we empirically show that our greedy algorithm ensures the controllability of the linear system with a small number of active actuators per time step without a significant average energy expenditure compared to the fully actuated system.

Sparse Actuator Scheduling for Discrete-Time Linear Dynamical Systems

TL;DR

This work tackles sparse actuator scheduling for discrete-time linear dynamical systems by optimizing an energy-based controllability metric under a per-step sparsity cap. It introduces an -regularized objective and a greedy algorithm over a matroid-constrained search space, accompanied by theoretical guarantees via -supermodularity and submodular-optimization bounds. The approach ensures the controllability rank reaches with a finite, judiciously chosen sequence of actuators, while incurring only modest energy penalties compared to fully actuated systems, as confirmed by simulations on random graphs. Practically, the method enables energy-aware, bandwidth-efficient control in large-scale networked systems where full actuation is impractical.

Abstract

We consider the control of discrete-time linear dynamical systems using sparse inputs where we limit the number of active actuators at every time step. We develop an algorithm for determining a sparse actuator schedule that ensures the existence of a sparse control input sequence, following the schedule, that takes the system from any given initial state to any desired final state. Since such an actuator schedule is not unique, we look for a schedule that minimizes the energy of sparse inputs. For this, we optimize the trace of the inverse of the resulting controllability Gramian, which is an approximate measure of the average energy of the inputs. We present a greedy algorithm along with its theoretical guarantees. Finally, we empirically show that our greedy algorithm ensures the controllability of the linear system with a small number of active actuators per time step without a significant average energy expenditure compared to the fully actuated system.
Paper Structure (8 sections, 3 theorems, 32 equations, 3 figures, 1 algorithm)

This paper contains 8 sections, 3 theorems, 32 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Actuator Scheduling Problem
  3. Greedy Scheduling Algorithm
  4. Simulation Results
  5. Conclusion
  6. Proof of \ref{['prop:req_cond_to_halt']}
  7. Proof of \ref{['prop:submodular_proof']}
  8. Proof of \ref{['thm:Act_sch_guarantee']}

Key Result

Proposition 1

For a given outer loop iteration index $t$ of alg:greedy_algo_epsilon, let the $r$th (inner) iteration start with $\mathcal{T}^{(r)}$ and its search space be $\mathcal{V}^{(r)}$ be as defined in eq:search_space. Suppose that the following set, is nonempty. Then, there exists $\epsilon^*>0$ such that if $\epsilon_t<\epsilon^*$, the next iteration of the greedy algorithm satisfies Here, the schedu

Figures (3)

  • Figure 1: CDF of $10\log_{10} \left(\mathsf{Tr}\left\{\left(\boldsymbol{W}_\mathcal{S} \right)^{-1}\right\}\right)$ for varying sparsity levels with $m=n=20$.
  • Figure 2: $\frac{\mathbb{E}_{A,B} \mathsf{Tr}\left\{\boldsymbol{W}_\mathcal{S}^{-1}\right\}}{\mathbb{E}_{A,B} \mathsf{Tr}\left\{\boldsymbol{W}^{-1}\right\}}$ vs. the fraction of active actuators ($\frac{s}{m}$), averaged over 100 independent trials with $m=50$.
  • Figure 3: $\mathsf{Tr}\left\{\boldsymbol{W}_\mathcal{S}^{-1}\right\}$ as a function of sparsity ($s$), averaged over $100$ independent trials with $n=100, m=50$.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Definition 1: Supermodularity
  • Definition 2: Matroid
  • Proposition 2
  • proof
  • Theorem 1
  • proof