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

Junhui Li, Jieying Lu, Weizhou Su

TL;DR

This work develops mean-square stability and stabilizability theory for LTI MIMO systems cascaded with a linear stochastic channel modeling correlated uncertainties such as random delays and packet losses in networks. It introduces an impulse-response based model and a spectral factorization approach, culminating in a frequency-domain condition $\rho(\widehat{TW})<1$ for internal stability and a Youla-parameterization-based criterion $\rho_{\min}<1$ for stabilizability, both incorporating a coefficient of frequency variation $W(z)$. The key contributions include explicit necessary and sufficient conditions for mean-square stability and stabilizability, with specialized results for minimum-phase and nonminimum-phase plants, and a practical sufficient condition via upper triangular coprime factorization. The findings illuminate how unstable poles, NM zeros, input delays, and the stochastic channel variation jointly constrain stabilizability, offering actionable insights for networked control design in the presence of memory-bearing uncertainties. A numerical example set demonstrates how these fundamental limits manifest in parallel-channel NCSs and guides system-level tradeoffs.

Abstract

In this paper, the feedback stabilization of a linear time-invariant (LTI) multiple-input multiple-output (MIMO) system cascaded by a linear stochastic system is studied in the mean-square sense. Here, the linear stochastic system can model a class of correlated stochastic uncertainties such as channel uncertainties induced by packet loss and random transmission delays in networked systems. By proposing a key parameter called coefficient of frequency variation to characterize the correlation of the stochastic uncertainties, we present a necessary and sufficient condition of the mean-square stability for this MIMO stochastic feedback system. After then a necessary and sufficient condition for the mean-square stabilizability is provided, which reveals a fundamental limit imposed by the system's unstable poles, nonminimum-phase (NMP) zeros, relative degrees (input delays), and the coefficient of frequency variation of the stochastic uncertainties. A numerical example is presented to illustrate the fundamental constraints in the mean-square stabilizability of MIMO networked systems with parallel communication channels.

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

TL;DR

This work develops mean-square stability and stabilizability theory for LTI MIMO systems cascaded with a linear stochastic channel modeling correlated uncertainties such as random delays and packet losses in networks. It introduces an impulse-response based model and a spectral factorization approach, culminating in a frequency-domain condition for internal stability and a Youla-parameterization-based criterion for stabilizability, both incorporating a coefficient of frequency variation . The key contributions include explicit necessary and sufficient conditions for mean-square stability and stabilizability, with specialized results for minimum-phase and nonminimum-phase plants, and a practical sufficient condition via upper triangular coprime factorization. The findings illuminate how unstable poles, NM zeros, input delays, and the stochastic channel variation jointly constrain stabilizability, offering actionable insights for networked control design in the presence of memory-bearing uncertainties. A numerical example set demonstrates how these fundamental limits manifest in parallel-channel NCSs and guides system-level tradeoffs.

Abstract

In this paper, the feedback stabilization of a linear time-invariant (LTI) multiple-input multiple-output (MIMO) system cascaded by a linear stochastic system is studied in the mean-square sense. Here, the linear stochastic system can model a class of correlated stochastic uncertainties such as channel uncertainties induced by packet loss and random transmission delays in networked systems. By proposing a key parameter called coefficient of frequency variation to characterize the correlation of the stochastic uncertainties, we present a necessary and sufficient condition of the mean-square stability for this MIMO stochastic feedback system. After then a necessary and sufficient condition for the mean-square stabilizability is provided, which reveals a fundamental limit imposed by the system's unstable poles, nonminimum-phase (NMP) zeros, relative degrees (input delays), and the coefficient of frequency variation of the stochastic uncertainties. A numerical example is presented to illustrate the fundamental constraints in the mean-square stabilizability of MIMO networked systems with parallel communication channels.
Paper Structure (15 sections, 11 theorems, 71 equations, 6 figures)

This paper contains 15 sections, 11 theorems, 71 equations, 6 figures.

Table of Contents

  1. Introduction
  2. System Modeling and Problem Formulation
  3. Mean-Square Internal Stability and Stabilizability
  4. Mean-square internal stability
  5. Mean-square stabilizability
  6. Minimum phase plants with input delays
  7. Nonminimum phase plants
  8. Numerical Example
  9. Example 1
  10. Example 2
  11. Example 3
  12. Conclusion
  13. Proof of \ref{['Thm:MS_stable_condition']}
  14. Proof of \ref{['Lma:M-f']}
  15. Proof of \ref{['Cor:Stabilizability_nonminimum_triangular']}

Key Result

Lemma 1

For the plant $P$ with relative degree greater than zero in Fig. Fig:Stochastic_system, $\chi_{i,k},i=1,\cdots, m$ are independent of $u(n),n<k$.

Figures (6)

  • Figure 1: The discrete-time stochastic feedback system
  • Figure 2: LTI system with a stochastic uncertainty block
  • Figure 3: Equivalent interconnection with input-output signals
  • Figure 4: Mean-square stabilizable regions
  • Figure 5: Stabilization radius $\rho_{\min}$ vs. input delays $\tau_1$ and $\tau_2$
  • ...and 1 more figures

Theorems & Definitions (16)

  • Lemma 1: see Su2021meansquare
  • Lemma 2: see Su2021meansquare
  • Remark 1
  • Lemma 3
  • Definition 1
  • Definition 2
  • Theorem 1
  • Corollary 1
  • Lemma 4: see Horn1986Matrix
  • Lemma 5
  • ...and 6 more