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On the Value of Linear Quadratic Zero-sum Difference Games with Multiplicative Randomness: Existence and Achievability

Songfu Cai, Xuanyu Cao

TL;DR

The paper tackles the existence and achievability of the value in an LQ zero-sum difference game for wireless networked control systems with multiplicative randomness from MIMO fading. It introduces a modified game algebraic Riccati equation (MGARE) and a PSD kernel decomposition to derive a verifiable condition for the existence of the game value, even under intermittent or almost-surely uncontrollable closed-loop dynamics. When the value exists, the minimal MGARE solution $P^{*}$ yields an explicit game value $J^{*}=\mathrm{Tr}(P^{*}(\mathbf{W}+\sum_i \mathbf{B}_i^{T}\mathbf{V}\mathbf{B}_i))$, and a closed-form, timestamp-based saddle-point policy achieves this value in a broad class of admissible strategies. The framework relaxes classical controllability requirements and provides concrete, computable criteria for the attacker weight domain $\mathbf{R}^{a}$ via the MGARE and PSD decomposition, enabling practical verification in complex WNCS settings. Numerical results illustrate tightness and applicability across diverse scenarios, highlighting how communication reliability and antenna configurations influence the existence and achievability of the game value.

Abstract

We consider a wireless networked control system (WNCS) with multiple controllers and multiple attackers. The dynamic interaction between the controllers and the attackers is modeled as a linear quadratic (LQ) zero-sum difference game with multiplicative randomness induced by the the multiple-input and multiple-output (MIMO) wireless fading. The existence of the game value is of significant importance to the LQ zero-sum game. If it exists, the value characterizes the minimum loss that the controllers can secure against the attackers' best possible strategies. However, if the value does not exist, then there will be no Nash equilibrium policies for the controllers and attackers. Therefore, we focus on analyzing the existence and achievability of the value of the LQ zero-sum game. In stark contrast to existing literature, where the existence of the game value depends heavily on the controllability of the closed-loop systems, the multiplicative randomness induced by the MIMO wireless channel fading may destroy closed-loop controllability and introduce intermittent controllability or almost sure uncontrollability. We first establish a general sufficient and necessary condition for the existence of the game value. This condition relies on the solvability of a modified game algebraic Riccati equation (MGARE) under an implicit concavity constraint, which is generally difficult to verify due to the multiplicative randomness. We next introduce a novel positive semidefinite (PSD) kernel decomposition method induced by multiplicative randomness. A verifiable tight sufficient condition is then obtained by applying the proposed PSD kernel decomposition to the constrained MGARE. Under the existence condition, we finally construct a closed-form saddle-point policy based on the minimal solution to the constrained MGARE, which is able to achieve the game value in a certain class of admissible policies.

On the Value of Linear Quadratic Zero-sum Difference Games with Multiplicative Randomness: Existence and Achievability

TL;DR

The paper tackles the existence and achievability of the value in an LQ zero-sum difference game for wireless networked control systems with multiplicative randomness from MIMO fading. It introduces a modified game algebraic Riccati equation (MGARE) and a PSD kernel decomposition to derive a verifiable condition for the existence of the game value, even under intermittent or almost-surely uncontrollable closed-loop dynamics. When the value exists, the minimal MGARE solution yields an explicit game value , and a closed-form, timestamp-based saddle-point policy achieves this value in a broad class of admissible strategies. The framework relaxes classical controllability requirements and provides concrete, computable criteria for the attacker weight domain via the MGARE and PSD decomposition, enabling practical verification in complex WNCS settings. Numerical results illustrate tightness and applicability across diverse scenarios, highlighting how communication reliability and antenna configurations influence the existence and achievability of the game value.

Abstract

We consider a wireless networked control system (WNCS) with multiple controllers and multiple attackers. The dynamic interaction between the controllers and the attackers is modeled as a linear quadratic (LQ) zero-sum difference game with multiplicative randomness induced by the the multiple-input and multiple-output (MIMO) wireless fading. The existence of the game value is of significant importance to the LQ zero-sum game. If it exists, the value characterizes the minimum loss that the controllers can secure against the attackers' best possible strategies. However, if the value does not exist, then there will be no Nash equilibrium policies for the controllers and attackers. Therefore, we focus on analyzing the existence and achievability of the value of the LQ zero-sum game. In stark contrast to existing literature, where the existence of the game value depends heavily on the controllability of the closed-loop systems, the multiplicative randomness induced by the MIMO wireless channel fading may destroy closed-loop controllability and introduce intermittent controllability or almost sure uncontrollability. We first establish a general sufficient and necessary condition for the existence of the game value. This condition relies on the solvability of a modified game algebraic Riccati equation (MGARE) under an implicit concavity constraint, which is generally difficult to verify due to the multiplicative randomness. We next introduce a novel positive semidefinite (PSD) kernel decomposition method induced by multiplicative randomness. A verifiable tight sufficient condition is then obtained by applying the proposed PSD kernel decomposition to the constrained MGARE. Under the existence condition, we finally construct a closed-form saddle-point policy based on the minimal solution to the constrained MGARE, which is able to achieve the game value in a certain class of admissible policies.
Paper Structure (25 sections, 17 theorems, 107 equations, 5 figures, 1 table)

This paper contains 25 sections, 17 theorems, 107 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. Dynamic Plant Model
  4. Wireless Communication Model
  5. Zero-sum Game Formulation
  6. Conditions for the Existence of Game Value
  7. Modified GARE
  8. Finite-Time Horizon Average Cost
  9. General Sufficient and Necessary Condition for Existence of Game Value
  10. Verifiable Sufficient Condition for Existence of Game Value
  11. Comparison and Differentiation from Existing Literature
  12. Application Examples and Tightness Analysis
  13. Achievable Schemes
  14. Numerical Results
  15. Tightness Verifications
  16. ...and 10 more sections

Key Result

Lemma 1

(Monotonicity)$\forall\mathbf{P}_{1},\mathbf{P}_{2}\in\mathcal{R}_{\mathbf{R}^{a}}$, $\mathbf{P}_{1}\preceq\mathbf{P}_{2}$ implies $f\left(\mathbf{P}_{1}\right)\preceq f\left(\mathbf{P}_{2}\right)$.

Figures (5)

  • Figure 1: Illustration of a WNCS with multiple wireless controllers and multiple wireless attackers. The remote controllers transmit the control actions to the actuators over a shared MIMO wireless communication network. The wireless attackers can intrude the wireless communication network and inject malicious attacks to the actuators.
  • Figure 2: Summary of the key contributions.
  • Figure 3: Connections between the key theorems, lemmas and corollaries.
  • Figure 4: Illustrations of the tightness of the proposed verifiable sufficient condition.
  • Figure 5: Illustrations of the bounds of $\mathrm{Tr}\left(\mathbf{P}^{*}\right)$ and $\left\Vert \mathbf{R}^{a}\right\Vert$ in Examples \ref{['Exmp: E.g.2']} and \ref{['Exmp: E.g.3']}.

Theorems & Definitions (29)

  • Remark 1
  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Remark 2
  • Remark 3
  • ...and 19 more