Approximated solutions to the nonlinear H2 and H∞ control approaches formulated in the Sobolev space*

Two important paradigms in control theory are the classical nonlinear $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ control approaches. Their efficiency have already been demonstrated in several applications and the background theory is well developed. Despite their many advantages, they suffer from deficiencies such as minimum settling-time and minimum overshoot. An interesting approach to solve these lacks is the formulation of both controllers in the Sobolev space. Thanks to the nature of the $\mathcal{W}_{1,2} -$ norm, the cost variable and its time derivative are taken into account in the cost functional, leading to improved transient and steady-state performance. Nevertheless, the HJB and HJBI equations that arises from the problem formulation in the Sobolev space are very hard to solve analytically. This work proposes an approach to approximate their solutions by adapting the classical Successive Galerkin Approximation Algorithms (SGAA). Numerical experiments are used to corroborate the proposed approach capacity to deal with underactuated systems when controlling the two-wheeled self-balanced vehicle.