[Paper Review] Proofs of "LQG Control For MIMO System Over Multiple TCP-like Erasure Channels"

Emanuele Garone, Bruno Sinopoli+2 more|ArXiv.org|Sep 11, 2009

Cybersecurity and Information Systems3 citations

TL;DR

This paper provides rigorous proofs for the infinite-horizon LQG control of MIMO systems over multiple TCP-like erasure channels, establishing the existence and convergence of the control Riccati equation. It derives a closed-form control law via a novel matrix algebraic approach and proves stability using a Lyapunov-like argument based on a linear operator acting on covariance matrices.

ABSTRACT

Here we will provide the proofs of the results stated in the Infinite Horizon LQG Control section of [1] by focusing on the control law and the related MARE. The analysis of the observation case can be achieved in a dual way and it is partially covered by [2].

Motivation & Objective

• To rigorously prove the existence and convergence of the control Riccati equation for MIMO systems under multiple TCP-like erasure channels.
• To establish the optimality of the derived control law through minimization of a quadratic cost function over channel state realizations.
• To demonstrate the monotonicity and convergence of the Riccati iteration using a linear operator on covariance matrices.
• To derive an equivalent LMI condition for stability and performance, enabling numerical verification of the control policy.
• To extend the analysis to the dual observation case, partially covered via duality principles.

Proposed method

• Derives the control Riccati equation (MARE) using expectation over all channel state subsets, weighted by erasure probabilities.
• Introduces an auxiliary function φ(K,X) that aggregates the expected cost over all channel state realizations, parameterized by the feedback gain K.
• Proves that the optimal gain K̄_X minimizes φ(K,X), leading to the closed-form solution K̄_X = -AᵀXB̄N [Σ_I η_I²(N_I(U+BᵀXB)N_I)]⁻¹.
• Uses a linear operator L(Y) = Σ_I η_I²(F_I Y F_Iᵀ) to model the evolution of the covariance matrix, enabling convergence analysis.
• Applies Schur complement and congruence transformations to convert the Riccati inequality into a linear matrix inequality (LMI).
• Establishes equivalence between the existence of a stabilizing solution and the feasibility of the derived LMI.

Experimental results

Research questions

• RQ1 Does the infinite-horizon LQG control problem for MIMO systems over multiple erasure channels admit a unique stabilizing solution?
• RQ2 Can the optimal control law be expressed in closed form, and is it globally optimal over all channel state realizations?
• RQ3 Under what conditions does the Riccati iteration converge to a unique fixed point?
• RQ4 Is there an equivalent LMI condition that characterizes the existence of a stabilizing solution?
• RQ5 How does the system's covariance evolve under the control law, and what guarantees stability?

Key findings

• The optimal control gain K̄_X is derived as the minimizer of the auxiliary function φ(K,X), ensuring the Riccati equation is satisfied.
• The Riccati iteration converges to a unique fixed point S̄ for any initial covariance S₀ ≥ 0, under the condition that a positive-definite solution exists.
• The convergence is proven via a Lyapunov-like argument using the linear operator L(Y), showing that Lᵏ(W) → 0 as k → ∞ under spectral radius < 1.
• The operator L(Y) is shown to be linear, monotonic, and contractive under the condition that Y > L(Y) for some positive-definite Y.
• An equivalent LMI condition is derived, proving that the existence of a stabilizing solution is equivalent to the feasibility of a specific LMI with variables Y and Z.
• The LMI formulation enables numerical verification of the stabilizing solution and provides a computational framework for controller design.