A Linear Time-Varying Inequality Approach for Prescribed Time Stability and Stabilization

Area of Science:

• Control Systems Engineering
• Nonlinear Dynamics
• Aerospace Engineering

Background:

• Traditional stability analysis methods often focus on asymptotic or exponential convergence.
• Finite-time and fixed-time stability offer faster convergence but lack unified frameworks.
• Prescribed-time stability allows for explicit control over convergence time, crucial for real-time applications.

Purpose of the Study:

• To develop a unified framework for analyzing finite-time, fixed-time, and prescribed-time stability.
• To introduce a novel linear time-varying (LTV) inequality-based approach for stability analysis.
• To solve the global prescribed-time stabilization problem for a rigid spacecraft attitude control system.

Main Methods:

• Introduction of a linear time-varying (LTV) inequality-based approach for prescribed-time stability analysis.
• Recasting existing nonlinear Lyapunov inequalities for finite- and fixed-time stability into the unified LTV framework.
• Application of the unified LTV approach to design a bounded nonlinear time-varying controller for spacecraft attitude stabilization using backstepping.

Main Results:

• Demonstration of the unified LTV inequality-based approach for analyzing various time-domain stabilities.
• Successful application of the unified framework to address the global prescribed-time stabilization of a rigid spacecraft.
• Development and validation of a bounded nonlinear time-varying controller for disturbance rejection.

Conclusions:

• The proposed LTV inequality-based approach provides a unified and effective method for finite-time, fixed-time, and prescribed-time stability analysis and stabilization.
• The developed controller ensures global prescribed-time stabilization for the spacecraft attitude control system, even under disturbance.
• Numerical simulations confirm the efficacy of the unified approach and the proposed control strategy.