Necessary and Sufficient Conditions for Mean-Square Stability and Stabilizability of Linear Time-Invariant and Stochastic Systems in Feedback

Основные понятия

The paper presents necessary and sufficient conditions for the mean-square stability and stabilizability of a linear time-invariant (LTI) multiple-input multiple-output (MIMO) system cascaded by a linear stochastic system, which can model correlated stochastic uncertainties such as random transmission delays and packet dropouts in networked control systems.

Аннотация

The paper studies the feedback stabilization of an LTI MIMO system connected to a linear stochastic system in the mean-square sense. The linear stochastic system is used to model a class of correlated stochastic uncertainties, such as those induced by packet loss and random transmission delays in networked systems.
The key contributions are:

A general impulse response-based model is established to describe the MIMO networked system with correlated stochastic uncertainties.

A necessary and sufficient condition for the mean-square stability of the system is derived using the proposed model, which generalizes the mean-square small-gain theorem.

Necessary and sufficient conditions for the mean-square stabilizability of the system are provided, revealing the fundamental limits imposed by the plant's unstable poles, non-minimum-phase zeros, relative degrees (input delays), and the coefficient of frequency variation of the stochastic uncertainties.

For minimum-phase plants with input delays and non-minimum-phase plants, computationally efficient solutions to the mean-square stabilizability problem are presented.

The results provide insights into the design and analysis of networked control systems with correlated stochastic uncertainties.

Статистика

The paper does not contain any explicit numerical data or statistics. It focuses on the theoretical analysis and derivation of stability and stabilizability conditions.

Цитаты

"A general impulse response based model is established for MIMO networked systems with correlated stochastic uncertainties such as random transmission delays and packet dropouts."
"Mean-square stabilizability conditions, both necessary and sufficient, for a class of MP and NMP plants with correlated stochastic uncertainties are derived, which explicitly reveals the intrinsic effect of the plant's characteristic (i.e., unstable poles, input delay, and NMP zeros) and the coefficient of frequency variation of the correlated stochastic uncertainties on the closed-loop stabilizability."

Ключевые выводы из

Mean-Square Stability and Stabilizability for LTI and Stochastic Systems Connected in Feedback

by Junhui Li,Ji... в arxiv.org 05-06-2024

https://arxiv.org/pdf/2402.04074.pdf

Дополнительные вопросы

What are some practical applications of the proposed stability and stabilizability conditions in the design of networked control systems

The proposed stability and stabilizability conditions have practical applications in the design of networked control systems, especially in scenarios where communication networks introduce stochastic phenomena like random transmission delays and packet disorder. These conditions can be used to ensure the mean-square stability and stabilizability of the system, which is crucial for maintaining system performance and stability in the presence of uncertainties. For example, in networked control systems for industrial automation or autonomous vehicles, where communication delays and uncertainties can impact the control performance, these conditions can help in designing robust and reliable control strategies.

How can the results be extended to handle more complex network topologies or communication protocols

The results can be extended to handle more complex network topologies or communication protocols by incorporating additional factors into the analysis. For instance, in networked control systems with multiple communication channels, each with different characteristics such as varying delay distributions or packet loss rates, the framework can be adapted to account for these variations. By considering the individual properties of each channel and their impact on the overall system dynamics, the stability and stabilizability conditions can be customized to suit the specific network topology. Additionally, the framework can be extended to include feedback loops, multiple controllers, or decentralized control architectures commonly found in large-scale networked systems.

Can the framework be adapted to consider other types of stochastic uncertainties beyond random transmission delays, such as Markovian switching or time-varying parameters

The framework can be adapted to consider other types of stochastic uncertainties beyond random transmission delays, such as Markovian switching or time-varying parameters, by modifying the modeling assumptions and system equations accordingly. For example, in the case of Markovian switching, where the system parameters or dynamics switch between different modes based on a Markov process, the stability and stabilizability conditions can be formulated to account for the probabilistic nature of the switching behavior. Similarly, for systems with time-varying parameters, the framework can be extended to include parameter variations over time and analyze their impact on system stability. By incorporating these additional uncertainties, the framework can provide a more comprehensive analysis of the system's behavior under varying conditions.