A Critical Point-Based Variational Method for Stability Analysis of Nonlinear Control Systems

Abstract

In this paper, the novel variational pointwise critical method is proposed for the stability investigation of nonlinear control systems with constraints. The evolution of states and controls is represented in the Hilbert trajectory space, and the stability is described as the action-energy functional with the state energy, energy of the control action, energy of the terminal state, and the quadratic dynamic residual. The equilibria of systems are considered as the critical points of the action-energy functional. Local strict minima correspond to the stable equilibria, and the saddle points correspond to the critical modes of transition. The sign of the second variation gives the classification of the stability. In the case of coercivity, weak sequential lower semi-continuity, and weak closure of the admissible set, the existence of the minimizer is established by the direct method. The positive second variation guarantees the local asymptotic stability of the functional decrease along the closed-loop trajectories, and the instability is provided by the presence of the negative direction with the non zero Morse index. The notion of mountain pass minimax level is used to measure the minimal energy barrier between the stable basins. An approach based on the projected Sobolev gradient iteration is presented, for which subsequential convergence to a critical point is proven given some Lipschitz gradient and Palais–Smale conditions. The method is tested on a controlled Duffing oscillator with actuator saturation and bounded perturbations. The approach achieves a critical residual of 9.8×10⁻⁵ in 46 iterations, obtains a positive curvature threshold of 0.184 at the equilibrium, and evaluates the minimax barrier to be 0.251 energy units. Compared to the nonlinear model predictive control, it results in a 15.6% reduction in control energy and a 21.2% reduction in disturbance recovery time.