Stabilization of Infinite-Dimensional Systems with Input/Output Quantization and Packet Loss

Core Concepts

The author proposes quantization schemes for stabilizing infinite-dimensional discrete-time systems subject to packet loss in the sensor-to-controller channel. The closed-loop system achieves exponential convergence under suitable conditions on the quantization error bounds and the duration bound of packet loss.

Abstract

The key highlights and insights of the content are: The author considers abstract infinite-dimensional linear systems, especially regular linear systems, with finite-dimensional input and output spaces. The continuous-time plant is connected to a zero-order hold and a generalized sampler. The author proposes quantization schemes for the discretized system with packet loss in the sensor-to-controller channel. The quantizers dynamically adjust the zoom parameters to ensure that the plant input and output are contained in the quantization regions and the quantization errors converge to zero. The author shows that the closed-loop state starting in a given region exponentially converges to zero if the bounds of quantization errors and packet-loss duration satisfy suitable conditions. This is achieved by a discretization-based approach. The author develops methods for approximately computing the norms of the operators representing the system dynamics, which are used in the quantizer design. The author focuses on the case where the plant is diagonalizable with finite-dimensional input and output spaces. For sampled-data regular linear systems, the author shows that the exponential convergence at sampling times can be extended to the continuous-time interval [0, ∞).

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Key Insights Distilled From

Stabilization of infinite-dimensional systems under quantization and packet loss

by Masashi Waka... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.00911.pdf

Deeper Inquiries

How can the proposed quantization schemes be extended to handle more general classes of infinite-dimensional systems, such as those with unbounded input/output operators or non-diagonalizable generators

The proposed quantization schemes can be extended to handle more general classes of infinite-dimensional systems by considering various modifications and adaptations. For systems with unbounded input/output operators, one approach could involve incorporating saturation functions or constraints to ensure that the quantized values remain within a bounded range. This can help address the challenges posed by unbounded operators and prevent instability due to unbounded quantization errors. In the case of non-diagonalizable generators, alternative methods such as model reduction techniques or spectral factorization approaches can be employed to approximate the system dynamics with a diagonalizable operator. By transforming the system into a more manageable form, the quantization schemes designed for diagonalizable operators can be applied effectively. Additionally, utilizing generalized eigenvectors or Jordan canonical forms can help in handling non-diagonalizable operators and extending the quantization schemes to such systems. Overall, the key idea is to adapt the quantization schemes to the specific characteristics of the system, incorporating suitable modifications to accommodate unbounded operators or non-diagonalizable generators while maintaining stability and performance.

What are the potential trade-offs between the conservativeness of the conditions on quantization errors and packet-loss duration, and the achievable convergence rate of the closed-loop system

The potential trade-offs between the conservativeness of the conditions on quantization errors and packet-loss duration and the achievable convergence rate of the closed-loop system are crucial considerations in designing robust control strategies for infinite-dimensional systems. A more conservative approach, with stricter conditions on quantization errors and packet-loss duration, may lead to a more robust closed-loop system that can tolerate higher levels of disturbances and uncertainties. However, this conservatism could come at the cost of a slower convergence rate or reduced performance in terms of tracking accuracy or disturbance rejection. On the other hand, relaxing the conditions on quantization errors and packet-loss duration may improve the convergence rate and overall performance of the closed-loop system. Still, it could make the system more susceptible to disturbances, quantization errors, or packet losses, potentially compromising stability and robustness. Finding the right balance between these trade-offs is essential in designing effective control strategies for infinite-dimensional systems. It often involves a careful analysis of the system dynamics, performance requirements, and the specific constraints imposed by quantization and packet loss, aiming to achieve an optimal trade-off between robustness and performance.

Can the techniques developed in this work be applied to the stabilization of infinite-dimensional systems under other types of data-rate constraints, such as event-triggered control or intermittent observations

The techniques developed in this work can be applied to the stabilization of infinite-dimensional systems under other types of data-rate constraints, such as event-triggered control or intermittent observations, with appropriate modifications and adaptations. For event-triggered control, where control actions are triggered based on certain events or conditions rather than at fixed time intervals, the quantization schemes can be adjusted to account for the event-based nature of control updates. By incorporating event-triggering mechanisms into the design of dynamic quantizers and updating rules, the closed-loop system can achieve stability and performance objectives while minimizing communication and computation resources. Similarly, for systems with intermittent observations, where sensor measurements are only available at certain time instances, the quantization schemes can be tailored to handle the intermittent nature of data transmission. By designing quantization strategies that account for irregular observation intervals and packet losses during these intervals, the closed-loop system can adapt to the intermittent data-rate constraints while ensuring stability and performance requirements are met. Overall, by customizing the quantization schemes and control strategies to suit the specific characteristics of event-triggered control or intermittent observations, the techniques developed in this work can be effectively applied to stabilize infinite-dimensional systems under different types of data-rate constraints.