Stabilization of Fully Actuated Nonlinear Systems: Inverse Optimal Control Design With Stability Margins
This paper formulates and solves an inverse optimal stabilization control problem for fully actuated nonlinear systems (FANSs). The control design is constructed based on the basic idea of fully actuated system (FAS) approach and Sontag’s formula, and then ensures global asymptotic stability of the closed-loop system and minimizes a cost functional that appropriately penalizes both the state and the control. The proposed scheme does not require solving Hamilton–Jacobi–Bellman (HJB) equations but the candidate control Lyapunov function (CLF) used in the control design is exactly the optimal value function. Moreover, by exploiting a connection between robustness and optimality, domination redesign of the proposed controller is shown to possess stability margins under certain assumptions and possess robustness to a class of static and dynamic input uncertainties. The effectiveness of developed result is illustrated via an application to inverse optimal attitude stabilization control of spacecraft for enhancing robustness to a type of unmodeled input dynamics. Note to Practitioners—This paper aims to establish a novel inverse optimal framework with explicit stability margins for the stabilization control of FANSs. Existing theoretical developments in nonlinear control have primarily focused on guaranteeing stability rather than optimality, since obtaining analytical solutions to the partial differential equations of the HJB equations is generally difficult or even impossible. Therefore, we propose a scheme avoiding solving the HJB equations directly, while ensuring that the candidate CLF employed in the control design coincides with the optimal value function. In contrast, the stability margins provided by optimal control of linear systems have been extensively investigated and well established, yielding significant results. However, research on nonlinear systems has largely remained confined to stability analysis, with limited attention paid to stability margins, despite their critical importance as prerequisites for robustness. In many nonlinear control designs, the absence of explicit stability margins makes the system highly vulnerable to modeling uncertainties, where even small perturbations in the control law may cause catastrophic failures, such as finite-time escape of the closed-loop dynamics. Through slight redesigns within the inverse optimality framework, explicit stability margins are established in this paper, providing rigorous theoretical guarantees that enhance robustness and practical applicability.