Lyapunov function-based stability optimization for autonomous control systems operating under dynamic disturbances
Keywords:
Autonomous control, lyapunov function, stability optimization, dynamic disturbances, minimax theory, hamilton-jacobi-isaacs inequality, input-to-state stability.Abstract
Nonlinear autonomous systems functioning under bound disturbances such as wind force, vibration, and sensor noise require stability assurances that are valid under the most severe operating conditions, which cannot be established by the classical passive Lyapunov method. This paper presents a new methodology that reformulates robust stabilization as the problem of minimizing the extremal gain of the Lyapunov function in the context of the twoplayer zero-sum minimax optimization problem involving the general form of Control Lyapunov functions. The control signal minimizes the energy of the system while the adversarial disturbance minimizes the deviation of the system state. The performance index emerges through the Lyapunov derivatives that are modified to include some quadratic cost for control and disturbance. The Lyapunov parameter is solved using the method of convex sum-of-squares and results in the resolution of the Hamilton-Jacobi-Isaacs inequality, with proofs of the existence of stability of input-to-state stability and the unique optimal Lyapunov parameter vector resulting in the minimum bound energy dissipation rate. The proposed offline-online control design computes the optimal Lyapunov surface once and uses classical control algorithms for offlline optimization. The numerical implementation on an autonomous quadcopter in the canceling turbulence condition shows that the minimax controller results in 66.7% reduction of maximum lateral position deviation, 53.3% reduction of root-mean-square control effort, and delivers 2.4-times faster Lyapunov decay compared to the previously used passive Lyapunov controller while the efficiency (disturbance attenuation) increased from 0.79 to 0.31. These results confirm that recasting Lyapunov-based stabilization as a parametrized minimax optimization problem yields quantifiably superior disturbance rejection and stability margins relative to conventional passive Lyapunov design for nonlinear autonomous control systems.
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