Robust Nonlinear $\mathcal{W}_{\infty}$ Optimal Control for Input Nonaffine Systems

We propose a novel robust nonlinear
$\mathcal{W}_{\infty}$
optimal control method for dynamical systems with nonaffine control inputs. The nonlinear
$\mathcal{W}_{\infty}$
control formulation extends the classic nonlinear
$\mathcal{H}_{\infty}$
one, considering a weighted Sobolev norm of the cost variable. This approach assumes that the cost variable belongs to the weighted Sobolev space
$\mathcal{W}_{m,p,\mathbf{\Gamma}}$
, ensuring continuity and differentiability up to degree
$m$
in a certain domain
$\Omega$
. Consequently, in addition to the well-known features provided by the
$\mathcal{H}_{\infty}$
approach in terms of disturbance attenuation, the closed-loop system benefits from the enhanced transient performance. Here, the robust nonlinear
$\mathcal{W}_{\infty}$
optimal control problem is formulated via dynamic programming for increased-order systems, and a particular solution is proposed to the resulting Hamilton-Jacobi equation, along with the corresponding stability analysis. To validate the proposed method and its versatility, we provide numerical results for the control of a quadrotor. Additionally, leveraging the inherent
$\mathcal{L}_{2}$
-gain properties of our approach, we demonstrate that the resulting controller can achieve trajectory tracking with guaranteed asymptotic stability for the whole closed-loop system.