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A probabilistic scheduling algorithm for networked control systems

Meghna Singh, Atreyee Kundu

TL;DR

This work addresses scheduling in networked control systems with limited bandwidth by introducing a probabilistic scheduling logic that assigns the shared network to disjoint plant subsets with specified probabilities, ensuring stochastic stability of all plants when modeled as Markovian jump linear systems. It provides necessary and sufficient LMIs to certify stability for each plant under a given schedule and offers an offline procedure to design both the scheduling policy and static state-feedback gains $K_i$ via LMI feasibility. The approach is validated with numerical experiments showing stabilization and scalability, highlighting the trade-off between controller design and offline search complexity. The results enable stable operation of multiple unstable plants over a shared network and set the stage for future work incorporating communication uncertainties.

Abstract

This paper deals with the design of scheduling logics for Networked Control Systems (NCSs) whose communication networks have limited capacity. We assume that only a subset of the plants can communicate with their controllers at any time instant. Our contributions are twofold. First, we present a probabilistic algorithm to design scheduling logics that, under certain conditions on the plant and controller dynamics and the capacity of the network, ensure stochastic stability of each plant in an NCS. Second, given the plant dynamics and the capacity of the network, we design static state-feedback controllers such that the conditions for stability under our scheduling logics are satisfied. The main apparatus for our analysis is a Markovian jump linear system representation of the individual plants in an NCS. Our stability conditions involve sets of matrix inequalities. We present numerical experiments to demonstrate our results.

A probabilistic scheduling algorithm for networked control systems

TL;DR

This work addresses scheduling in networked control systems with limited bandwidth by introducing a probabilistic scheduling logic that assigns the shared network to disjoint plant subsets with specified probabilities, ensuring stochastic stability of all plants when modeled as Markovian jump linear systems. It provides necessary and sufficient LMIs to certify stability for each plant under a given schedule and offers an offline procedure to design both the scheduling policy and static state-feedback gains via LMI feasibility. The approach is validated with numerical experiments showing stabilization and scalability, highlighting the trade-off between controller design and offline search complexity. The results enable stable operation of multiple unstable plants over a shared network and set the stage for future work incorporating communication uncertainties.

Abstract

This paper deals with the design of scheduling logics for Networked Control Systems (NCSs) whose communication networks have limited capacity. We assume that only a subset of the plants can communicate with their controllers at any time instant. Our contributions are twofold. First, we present a probabilistic algorithm to design scheduling logics that, under certain conditions on the plant and controller dynamics and the capacity of the network, ensure stochastic stability of each plant in an NCS. Second, given the plant dynamics and the capacity of the network, we design static state-feedback controllers such that the conditions for stability under our scheduling logics are satisfied. The main apparatus for our analysis is a Markovian jump linear system representation of the individual plants in an NCS. Our stability conditions involve sets of matrix inequalities. We present numerical experiments to demonstrate our results.
Paper Structure (6 sections, 3 theorems, 17 equations, 3 figures, 1 table, 4 algorithms)

This paper contains 6 sections, 3 theorems, 17 equations, 3 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Main results
  4. Stabilizing scheduling logics
  5. Numerical experiments
  6. Conclusion

Key Result

Theorem 1

Consider an NCS described in §s:prob_stat. Suppose that Assumption a:divisibility holds. Let $\gamma$ be a scheduling logic obtained from Algorithm algo:sched_design. Each plant $i$ in e:plants is stochastically stable under $\gamma$ if and only if the following conditions hold: for each $i\in c_j$, where $\mathcal{P}^i = p_{c_j}P_{i_s} + (1-p_{c_{j}})P_{i_u}$.

Figures (3)

  • Figure 1: Block diagram of NCS
  • Figure 2: $\left\lVert x_1(t)\right\rVert^{2}$ versus $t$
  • Figure 3: $\left\lVert x_2(t)\right\rVert^{2}$ versus $t$

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 7 more