Optimal Scheduling for Feedback Control in Resource-Constrained Networked Systems

Core Concepts

The core message of this article is to design an optimal stationary scheduling law that minimizes the long-term tradeoff between regulation cost and communication cost in resource-constrained networked control systems.

Abstract

The article investigates the infinite-horizon optimal scheduling for resource-aware networked control systems by addressing the rate-regulation tradeoff. It considers a scenario where the sensor and the controller communicate via a networked channel, and the transmission scheduling problem is formulated as a Markov decision process on an unbounded continuous state space controlled by scheduling decisions.
The key highlights and insights are:

The authors leverage the separation principle to fix the control law as a certainty equivalence controller and focus on the optimal scheduling design.
They show that the optimal solution of the average-cost minimization problem exists and is a canonical triplet that includes the optimal stationary scheduling law, the optimal average cost, and the differential cost function.
The authors use value iteration to obtain the differential cost and show that the iteration sequence is convergent. They also show that the solution on a truncated state space is identical to the solution of the original optimization problem.
The authors design the optimal stationary scheduling law based on the value of information (VoI) metric, which quantifies the uncertainty of decision makers with respect to control task achievement. They show that the error dynamics is bounded under the VoI-based scheduling law, guaranteeing the stochastic stability of the closed-loop system.
The authors prove that the differential cost, its expectation, and the VoI metric are symmetric functions when the system matrix is diagonalizable. Therefore, the VoI-based scheduling law is also symmetric.
By analyzing the dynamic behavior of the iterative algorithm, the authors show that the differential cost, its expectation, and the VoI metric are monotone and quasi-convex when the system matrix is diagonalizable. They then prove that the VoI-based optimal scheduling law is of threshold type and quadratic form, which is easy to compute and implement yet preserves optimality.

Stats

The system matrices A, B, C are assumed to be of appropriate dimensions, where the pair (A, B) is controllable and the pair (A, C) is observable.
The process noise wk and the measurement noise vk are independent Gaussian processes with zero mean and positive semidefinite variances W and V, respectively.
The initial state x0 is a random vector with mean ¯x0 and positive semidefinite covariance R0.

Quotes

"The optimal scheduling law based on VoI is shown to be deterministic and stationary, of which expression is obtained using value iteration."
"The closed-loop system under the designed scheduling law is stochastically stable."
"When the system matrix is diagonalizable, the VoI function is monotone and quasi-convex. Based on this, the optimal scheduling law is shown to be of threshold type and quadratic form."

Key Insights Distilled From

Infinite-horizon optimal scheduling for feedback control

by Siyi Wang,Sa... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2402.08819.pdf

Deeper Inquiries

How can the proposed VoI-based scheduling framework be extended to handle more complex system dynamics, such as nonlinear or time-varying systems

The proposed VoI-based scheduling framework can be extended to handle more complex system dynamics, such as nonlinear or time-varying systems, by incorporating advanced control techniques. For nonlinear systems, the scheduling law can be adapted to account for the nonlinearity in the system dynamics. This can involve using nonlinear control strategies, such as feedback linearization or adaptive control, to ensure stability and performance in the presence of nonlinearities. Additionally, techniques like model predictive control (MPC) can be employed to handle time-varying dynamics by optimizing control actions over a finite time horizon based on a dynamic model of the system.

What are the potential challenges and limitations in implementing the optimal scheduling law in practical networked control applications, and how can they be addressed

Implementing the optimal scheduling law in practical networked control applications may face challenges and limitations related to computational complexity, communication delays, and system uncertainties. To address these challenges:

Computational Complexity: Efficient algorithms and optimization techniques can be utilized to reduce the computational burden of solving the optimization problem. Approximation methods, such as reinforcement learning or online optimization, can be employed to find near-optimal solutions in real-time.
Communication Delays: Strategies like event-triggered control or predictive control can be used to mitigate the effects of communication delays. These approaches allow for control decisions to be made based on available information, reducing the reliance on real-time communication.
System Uncertainties: Robust control techniques, such as H-infinity control or robust MPC, can be implemented to handle uncertainties in the system. By designing controllers that are robust to variations in system parameters, the performance of the control system can be improved under uncertain conditions.

Given the insights on the symmetric and monotonic properties of the VoI function, are there any connections or implications to other areas of control theory, such as robust control or optimal control

The symmetric and monotonic properties of the VoI function have implications for other areas of control theory, such as robust control and optimal control:

Robust Control: The symmetry of the VoI function can be leveraged in robust control design to ensure stability and performance under uncertain conditions. By considering the symmetry of the system dynamics, robust controllers can be designed to maintain stability in the presence of uncertainties.
Optimal Control: The monotonicity of the VoI function can be utilized in optimal control problems to guide the selection of control actions that lead to improved system performance. By incorporating the monotonic properties into the optimization process, optimal control laws can be designed to achieve desired control objectives efficiently.