Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sparsity-Constrained Linear Quadratic Regulation Problem: Greedy Approach with Performance Guarantee

Shumpei Nishida, Kunihisa Okano

TL;DR

This work introduces a greedy method to find a suboptimal solution to a linear quadratic regulation problem, and establishes bounds on the submodularity ratio and curvature, which enable it to offer a practical performance guarantee for the greedy algorithm.

Abstract

We study a linear quadratic regulation problem with a constraint where the control input can be nonzero only at a limited number of times. Given that this constraint leads to a combinational optimization problem, we adopt a greedy method to find a suboptimal solution. To quantify the performance of the greedy algorithm, we employ two metrics that reflect the submodularity level of the objective function: The submodularity ratio and curvature. We first present an explicit form of the optimal control input that is amenable to evaluating these metrics. Subsequently, we establish bounds on the submodularity ratio and curvature, which enable us to offer a practical performance guarantee for the greedy algorithm. The effectiveness of our guarantee is further demonstrated through numerical simulations.

Sparsity-Constrained Linear Quadratic Regulation Problem: Greedy Approach with Performance Guarantee

TL;DR

This work introduces a greedy method to find a suboptimal solution to a linear quadratic regulation problem, and establishes bounds on the submodularity ratio and curvature, which enable it to offer a practical performance guarantee for the greedy algorithm.

Abstract

We study a linear quadratic regulation problem with a constraint where the control input can be nonzero only at a limited number of times. Given that this constraint leads to a combinational optimization problem, we adopt a greedy method to find a suboptimal solution. To quantify the performance of the greedy algorithm, we employ two metrics that reflect the submodularity level of the objective function: The submodularity ratio and curvature. We first present an explicit form of the optimal control input that is amenable to evaluating these metrics. Subsequently, we establish bounds on the submodularity ratio and curvature, which enable us to offer a practical performance guarantee for the greedy algorithm. The effectiveness of our guarantee is further demonstrated through numerical simulations.
Paper Structure (10 sections, 13 theorems, 48 equations, 4 figures, 1 algorithm)

This paper contains 10 sections, 13 theorems, 48 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Greedy algorithm and (non-)submodular function maximization
  4. Explicit form of the optimal control input and the optimal cost
  5. Performance guarantee of the greedy algorithm
  6. Numerical examples and comparison with the existing results
  7. Control performance of the greedy solution
  8. Performance guarantee of the greedy algorithm
  9. Conservativeness of the established guarantee
  10. Conclusion

Key Result

Proposition 1

Let $f$ be a monotone nondecreasing set function with submodularity $\gamma \in [0,1]$ and curvature $\alpha \in [0,1]$. Then, Algorithm algo:greedy algorithm enjoys the following approximation guarantee for solving Problem prob:set function optimization problem:

Figures (4)

  • Figure 1: Feedback system
  • Figure 2: LQR cost defined by \ref{['eq:obj function']} versus the number of times $d$ that the control inputs are applied.
  • Figure 3: Lower bounds on the approximation ratio versus the spectral norm of $A$. The blue solid line represents the mean for 1000 simulations, and the shadowed region around the line visualizes standard deviations from the mean.
  • Figure : The greedy algorithm for Problem \ref{['prob:set function optimization problem']}.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2: Das2011
  • Definition 3: Bian2017
  • Proposition 1: Bian2017
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 2
  • Remark 2
  • Lemma 2
  • ...and 11 more