Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Linear Programming Framework for Optimal Event-Triggered LQG Control

Zahra Hashemi, Dipankar Maity

TL;DR

The paper addresses the challenge of optimally scheduling sensor-to-controller communications in stochastic LQG control with a transmission cost. It reformulates the inherently nonlinear, bilinear scheduling problem into an exact MILP by introducing auxiliary binary monomials and applying McCormick relaxation, and embeds this MILP within an MPC framework for online adaptation. The main contributions include an exact MINP-to-MILP transformation, one-step transmission certificates, and proofs that the MPC scheduler outperforms any deterministic policy, with substantial computational speedups demonstrated in simulations. The results offer a scalable, structure-exploiting approach for resource-aware control in networked systems and lay the groundwork for extensions to multi-agent settings.

Abstract

This letter explores intelligent scheduling of sensor-to-controller communication in networked control systems, particularly when data transmission incurs a cost. While the optimal controller in a standard linear quadratic Gaussian (LQG) setup can be computed analytically, determining the optimal times to transmit sensor data remains computationally and analytically challenging. We show that, through reformulation and the introduction of auxiliary binary variables, the scheduling problem can be cast as a computationally efficient mixed-integer linear program (MILP). This formulation not only simplifies the analysis but also reveals structural insights and provides clear decision criteria at each step. Embedding the approach within a model predictive control (MPC) framework enables dynamic adaptation, and we prove that the resulting scheduler performs at least as well as any deterministic strategy (e.g., periodic strategy). Simulation results further demonstrate that our method consistently outperforms traditional periodic scheduling.

A Linear Programming Framework for Optimal Event-Triggered LQG Control

TL;DR

The paper addresses the challenge of optimally scheduling sensor-to-controller communications in stochastic LQG control with a transmission cost. It reformulates the inherently nonlinear, bilinear scheduling problem into an exact MILP by introducing auxiliary binary monomials and applying McCormick relaxation, and embeds this MILP within an MPC framework for online adaptation. The main contributions include an exact MINP-to-MILP transformation, one-step transmission certificates, and proofs that the MPC scheduler outperforms any deterministic policy, with substantial computational speedups demonstrated in simulations. The results offer a scalable, structure-exploiting approach for resource-aware control in networked systems and lay the groundwork for extensions to multi-agent settings.

Abstract

This letter explores intelligent scheduling of sensor-to-controller communication in networked control systems, particularly when data transmission incurs a cost. While the optimal controller in a standard linear quadratic Gaussian (LQG) setup can be computed analytically, determining the optimal times to transmit sensor data remains computationally and analytically challenging. We show that, through reformulation and the introduction of auxiliary binary variables, the scheduling problem can be cast as a computationally efficient mixed-integer linear program (MILP). This formulation not only simplifies the analysis but also reveals structural insights and provides clear decision criteria at each step. Embedding the approach within a model predictive control (MPC) framework enables dynamic adaptation, and we prove that the resulting scheduler performs at least as well as any deterministic strategy (e.g., periodic strategy). Simulation results further demonstrate that our method consistently outperforms traditional periodic scheduling.

Paper Structure

This paper contains 8 sections, 2 theorems, 48 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Optimal Controller
  4. Optimal Communication Protocol
  5. Mixed Integer Linear Program Reformulation
  6. Efficiency and Optimality of MPC
  7. Case Study: Double Integrator
  8. Conclusion

Key Result

Theorem 1

Consider eq:discrete-system, the error dynamics eq:error-dynamics, and the cost eq:cost-function2 with $\Gamma_k:=L_k^{\tiny \intercal} S_k L_k\succeq0$. For $k\in\{0,\ldots,T-1\}$ define the tail Gramian Let $\mathrm{Ben}_k(e_k^s)$ be the expected cost reduction obtained by sending at time $k$ (i.e., skip minus send). Then Consequently, if $e_k^{s{\tiny \intercal}}\Gamma_k e_k^s\ge\lambda$ then

Figures (7)

  • Figure 1: System architecture illustrating the interaction among the plant, scheduler, network, and controller. At each time step $k$, the scheduler selects a binary transmission decision $\theta_k$. Based on this decision, the corresponding measurement $z_k \in \{x_k, \varnothing\}$ is transmitted to the controller which computes the input $u_k$
  • Figure 2: Average optimization time (seconds) vs. problem size $n$ on a log scale for a horizon $T=10$; shaded regions indicate ± one standard deviation across trials (some of them are not visible due to the log scale and small values).
  • Figure 3: Transmission schedules under offline vs. MPC strategies for a representative disturbance realization.
  • Figure 4: Estimation error trajectories under offline vs. MPC scheduling. The lighter shade vertical lines represent the scheduling time which are the same as the ones in Fig. \ref{['fig:schedule-comparison']}.
  • Figure 5: Average communication count under offline vs. MPC scheduling across different values of $\lambda$ and noise covariance.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 1: One--Step Optimality Certificates
  • proof
  • Theorem 2: MPC dominates open-loop deterministic schedules
  • proof