[Paper Review] On Stability of the Linearized Spacecraft Attitude Control System

Bin Zhou|arXiv (Cornell University)|Apr 1, 2015

Adaptive Control of Nonlinear Systems11 references4 citations

TL;DR

This paper provides necessary and sufficient conditions for polynomial and Lyapunov stability of the linearized spacecraft attitude control system, enabling semi-global and global stabilization via saturated linear state feedback. The key contribution is a parameter-dependent stability analysis based on inertia ratios and orbital dynamics, ensuring stability under actuator saturation.

ABSTRACT

This note is concerned with the stability and stabilization of the linearized spacecraft attitude control system. Necessary and sufficient conditions are respectively provided to guarantee that the considered systems are polynomially stable and stable in the Lyapunov sense. These two classes of conditions guarantee that the linearized spacecraft attitude control system can be respectively stabilized semi-globally and globally by saturated linear state feedback.

Motivation & Objective

• To analyze the stability properties of the linearized spacecraft attitude control system under realistic actuator constraints.
• To identify necessary and sufficient conditions for polynomial and Lyapunov stability in terms of spacecraft inertia parameters.
• To enable the design of saturated linear state feedback controllers that ensure semi-global and global stabilization.
• To bridge the gap between linear control design and practical implementation by accounting for actuator saturation effects.
• To provide a parameterized framework for assessing stability based on inertia ratios and orbital rate.

Proposed method

• Derives the linearized spacecraft dynamics from the nonlinear attitude model using perturbation around a reference orbit.
• Transforms the system into a block-diagonal form via similarity transformations to decouple rotational dynamics along principal axes.
• Applies Lyapunov stability theory to analyze the eigenvalue distribution of the system matrix A, focusing on non-positive real parts.
• Uses structured Lyapunov equations with block-structured positive definite solutions to verify stability conditions.
• Introduces parameterized stability conditions involving inertia ratios σ₁, σ₂, σ₃ and their combinations φ₁, φ₂, and φ₂²−16φ₁.
• Establishes that global stabilization is possible under Lyapunov stability, and semi-global stabilization under polynomial stability.

Experimental results

Research questions

• RQ1 Under what conditions on the spacecraft inertia parameters is the linearized attitude control system polynomially stable?
• RQ2 When is the linearized system Lyapunov stable, ensuring neutral stability in the absence of external disturbances?
• RQ3 How do the inertia ratios σ₁, σ₂, σ₃ affect the stability properties of the linearized system?
• RQ4 Can saturated linear feedback achieve global or semi-global stabilization under these stability conditions?
• RQ5 What is the role of the orbital rate ω₀ and the inertia matrix in determining the system's eigenvalue structure?

Key findings

• The linearized system is polynomially stable if and only if σ₂ ≥ 0, φ₁ = σ₁σ₃ ≥ 0, φ₂ = 3σ₁ + σ₃σ₁ + 1 ≥ 0, and φ₂² − 16φ₁ ≥ 0.
• The eigenvalues of the system matrix A are purely imaginary and given by s₁,₂ = ±√(3σ₂)ω₀i, s₃,₄ = ±√[(φ₂ + √(φ₂²−16φ₁))/2]ω₀i, and s₅,₆ = ±√[(φ₂ − √(φ₂²−16φ₁))/2]ω₀i.
• Lyapunov stability is achieved when the same conditions hold, and the system admits a positive definite solution to the Lyapunov equation.
• Under Lyapunov stability, the system can be globally stabilized using saturated linear state feedback, even with time-varying periodic control matrices.
• Under polynomial stability, the system can be semi-globally stabilized via saturated linear feedback, as per established control theory.
• The inertia ratios σ₁, σ₂, σ₃ are not independent and can be fully parameterized by two ratios β₁ = Jₓ/Jᵧ and β₂ = Jᵧ/J_z.