[Paper Review] Mean-Square Input-Output Stability and Stabilizability of a Networked Control System with Random Channel Induced Delays

Weizhou Su, Junhui Li+1 more|arXiv (Cornell University)|Aug 29, 2021

Stability and Control of Uncertain Systems39 references4 citations

TL;DR

This paper proposes a frequency-domain framework for analyzing mean-square input-output stability and stabilizability in SISO networked control systems with random channel-induced delays and packet dropout. By modeling the unreliable channel via an i.i.d. stochastic process and introducing the 'frequency response of variation,' the authors derive a necessary and sufficient condition for mean-square stabilizability that depends on the interaction between the channel's uncertainty and the plant’s unstable poles, generalizing the small-gain theorem to stochastic multiplicative uncertainties.

ABSTRACT

This work mainly investigates the mean-square stability and stabilizability for a single-input single-output networked linear feedback system. The control signal in the networked system is transmitted over an unreliable channel. In this unreliable channel, the data transmission times, referred to as channel induced delays, are random values and the transmitted data could also be dropout with certain probability. The channel induced delays and packet dropout are modeled by an independent and identically distributed stochastic process with a fixed probability mass function. At the channel terminal, a linear combination of data received at one sampling time is applied to the plant of the networked feedback system as a new control signal. To describe the uncertainty in the channel, a concept so called frequency response of variation is introduced for the unreliable channel. With the given linear receiving strategy, a mean-square stability criterion is established in terms of the frequency response of variation of the unreliable channel for the networked feedback system. It is shown by this criterion that the mean-square stability is determined by the interaction between the frequency response of variation and the nominal feedback system. The role played by the random channel induced delays is the same as that played by a colored additive noise in an additive noise channel with a signal-to-noise ratio constraint. Moreover, the mean-square input-output stabilizability via output feedback is studied for the networked system. When the plant in the networked feedback system is minimum phase, an analytic necessary and sufficient condition is presented for its mean-square input-output stabilizability. It turns out that the stabilizability is only determined by the interaction between the frequency response of variation of the channel and unstable poles of the plant.

Motivation & Objective

• To address the challenge of maintaining stability in networked control systems where data transmission is impaired by random delays and packet dropout.
• To model the unreliable communication channel as an i.i.d. stochastic process with a fixed probability mass function, capturing both delay and dropout effects.
• To develop a frequency-domain characterization of channel uncertainty using the 'frequency response of variation' to describe the relative deviation of the channel from its nominal behavior.
• To establish a mean-square input-output stability criterion that depends on the interaction between the channel’s frequency response of variation and the nominal feedback system.
• To derive a necessary and sufficient condition for mean-square input-output stabilizability via output feedback when the plant is minimum phase, linking stabilizability to unstable poles and channel uncertainty.

Proposed method

• Model the networked control system as a discrete-time linear time-invariant (LTI) system with stochastic multiplicative uncertainty due to random channel-induced delays and packet dropout.
• Introduce the concept of 'frequency response of variation' to describe the relative deviation of the channel’s transfer function from its nominal value, capturing the stochastic behavior of delays and dropouts.
• Apply a linear receiving strategy at the actuator that combines data received at the same sampling time, enabling a tractable input-output analysis.
• Derive a mean-square input-output stability criterion by analyzing the power gain of the closed-loop system, showing that stability depends on the interaction between the frequency response of variation and the nominal system’s transfer function.
• Establish a necessary and sufficient condition for mean-square stabilizability by solving a convex optimization problem involving the stabilizability index, defined as the minimum achievable power gain of the control signal.
• Use the Youla parameterization framework to parameterize all stabilizing controllers and analyze the stability boundary in terms of the norm of the Youla parameter.

Experimental results

Research questions

• RQ1 How does random channel-induced delay and packet dropout affect the mean-square input-output stability of a networked control system?
• RQ2 Can a frequency-domain representation of channel uncertainty—specifically, the 'frequency response of variation'—be used to characterize the stability of networked systems with stochastic delays?
• RQ3 What is the necessary and sufficient condition for mean-square input-output stabilizability of a minimum-phase SISO networked system with random delays and dropouts?
• RQ4 How does the relative degree of the plant influence the mean-square stabilizability index, and what is the critical relative degree beyond which stabilizability is lost?
• RQ5 To what extent does improper controller design, as measured by the Youla parameter norm, lead to instability in the mean-square sense?

Key findings

• The mean-square input-output stability of the networked system is determined by the interaction between the frequency response of variation of the unreliable channel and the nominal feedback system, generalizing the small-gain theorem to stochastic multiplicative uncertainties.
• The role of random channel-induced delays is mathematically equivalent to that of a colored additive noise in an additive noise channel with a signal-to-noise ratio constraint.
• For a minimum-phase plant, the necessary and sufficient condition for mean-square stabilizability is that the stabilizability index—defined as the minimum achievable power gain of the control signal—must be less than one.
• In the numerical example with relative degree τ = 1, the stabilizability index is 0.1728, indicating that the system is mean-square stabilizable.
• When the relative degree increases to τ = 5, the stabilizability index exceeds one, implying that the system becomes mean-square unconditionally unstable under the given controller class.
• Monte Carlo simulations confirm that as the norm of the Youla parameter increases toward 1, the average power of the control signal diverges, validating the theoretical stability boundary.