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A Stochastic Robust Adaptive Systems Level Approach to Stabilizing Large-Scale Uncertain Markovian Jump Linear Systems

SooJean Han, Minwoo M. Kim, Ieun Choo

TL;DR

This work tackles stabilizing large-scale Markovian jump linear systems under disturbances and topology changes by combining adaptive mode estimation with a stochastic multimode extension of system level synthesis. The method maintains a probabilistic belief over possible modes and uses this to weight a two-step convex optimization that yields a robust, distributed controller compatible with local information and consensus updates. Key contributions include Bayesian and norm-based mode consistency schemes, decentralized consensus strategies, and a two-phase optimization that simultaneously stabilizes all modes consistent with observed data. The approach improves robustness to mode switches, avoids abrupt destabilization, and scales to networks via separability and localized control, with demonstrated disturbance rejection on dynamic power-grid topologies.

Abstract

We propose a unified framework for robustly and adaptively stabilizing large-scale networked uncertain Markovian jump linear systems (MJLS) under external disturbances and mode switches that can change the network's topology. Adaptation is achieved by using minimal information on the disturbance to identify modes that are consistent with observable data. Robust control is achieved by extending the system level synthesis (SLS) approach, which allows us to pose the problem of simultaneously stabilizing multiple plants as a two-step convex optimization procedure. Our control pipeline computes a likelihood distribution of the system's current mode, uses them as probabilistic weights during simultaneous stabilization, then updates the likelihood via Bayesian inference. Because of this "softer" probabilistic approach to robust stabilization, our control pipeline does not suffer from abrupt destabilization issues due to changes in the system's true mode, which were observed in a previous method. Separability of SLS also lets us compute localized robust controllers for each subsystem, allowing for network scalability; we use several information consensus methods so that mode estimation can also be done locally. We apply our algorithms to disturbance-rejection on two sample dynamic power grid networks, a small-scale system with 7 nodes and a large-scale grid of 25 nodes.

A Stochastic Robust Adaptive Systems Level Approach to Stabilizing Large-Scale Uncertain Markovian Jump Linear Systems

TL;DR

This work tackles stabilizing large-scale Markovian jump linear systems under disturbances and topology changes by combining adaptive mode estimation with a stochastic multimode extension of system level synthesis. The method maintains a probabilistic belief over possible modes and uses this to weight a two-step convex optimization that yields a robust, distributed controller compatible with local information and consensus updates. Key contributions include Bayesian and norm-based mode consistency schemes, decentralized consensus strategies, and a two-phase optimization that simultaneously stabilizes all modes consistent with observed data. The approach improves robustness to mode switches, avoids abrupt destabilization, and scales to networks via separability and localized control, with demonstrated disturbance rejection on dynamic power-grid topologies.

Abstract

We propose a unified framework for robustly and adaptively stabilizing large-scale networked uncertain Markovian jump linear systems (MJLS) under external disturbances and mode switches that can change the network's topology. Adaptation is achieved by using minimal information on the disturbance to identify modes that are consistent with observable data. Robust control is achieved by extending the system level synthesis (SLS) approach, which allows us to pose the problem of simultaneously stabilizing multiple plants as a two-step convex optimization procedure. Our control pipeline computes a likelihood distribution of the system's current mode, uses them as probabilistic weights during simultaneous stabilization, then updates the likelihood via Bayesian inference. Because of this "softer" probabilistic approach to robust stabilization, our control pipeline does not suffer from abrupt destabilization issues due to changes in the system's true mode, which were observed in a previous method. Separability of SLS also lets us compute localized robust controllers for each subsystem, allowing for network scalability; we use several information consensus methods so that mode estimation can also be done locally. We apply our algorithms to disturbance-rejection on two sample dynamic power grid networks, a small-scale system with 7 nodes and a large-scale grid of 25 nodes.
Paper Structure (12 sections, 1 theorem, 20 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 12 sections, 1 theorem, 20 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Background
  4. Preliminaries on MJLS Stability
  5. System Level Synthesis
  6. Information Consensus
  7. Main Controller Framework
  8. Mode Estimation
  9. Incorporating Consensus
  10. Stochastic Multimode SLS
  11. Numerical Simulations
  12. Conclusion

Key Result

Lemma 1

For static network dynamics (eq:linear_dt_static), the following are true. First, the affine subspace described by parametrizes all possible system responses $\boldsymbol{\Phi}$ achievable by an internally stabilizing state feedback controller, where $(1/z)\mathcal{R}\mathcal{H}_{\infty}$ is the set of stable, strictly proper rational transfer matrices. Conversely, for any $\boldsymbol{\Phi}$ whi

Figures (2)

  • Figure 1: Different modes for [Top] Hexagon, and [Bottom] rectangular Grid. Some of the interconnections in Grid are asymmetric too, but we omit those details for visual cleanliness.
  • Figure 2: Sample trajectories for algorithms No. 3 and 4 in Tab. \ref{['tab:compare_architectures']} implemented on the Grid system.

Theorems & Definitions (3)

  • Lemma 1: Thm. 4.1 in anderson19
  • Remark 1: Consistency Measure Relationship
  • Remark 2: Mode Identification Tradeoff