Stackelberg-Game-Oriented Optimal Control for Bounded Constrained Mechanical Systems: A Fuzzy Evidence-Theoretic Approach

Abstract

This article proposes a novel Stackelberg-game-oriented optimal control approach to address the bounded constraint-following control problem for uncertain mechanical systems. First, the uncertainties (possibly fast time-varying) in the system are assumed to be bounded with an unknown boundary, which lies in a specified fuzzy evidence number. In practical engineering, bounded system performance is always demanded, such as the inequality constraint. A diffeomorphism transformation approach is proposed to transform the constrained system into a restructured one satisfying the bounded constraint. Second, we propose an adaptive robust control oriented by the constraint-following control to render the restructured system to follow the specified constraints accurately with deterministic performance (guaranteeing uniform boundedness and uniform ultimate boundedness). The self-adjusting adaptive law (leakage-type) can compensate for the uncertainties and avoid overcompensation. Third, a Stackelberg-game-oriented optimization approach is proposed to obtain the optimal control parameters based on the fuzzy evidence theory. In the optimization approach, the two control parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> are considered as two players with respective cost functions related to system performance and control cost. Furthermore, the optimization problem is solved by obtaining the Stackelberg strategy, which is proved to exist in analytic form. Ultimately, the permanent magnet synchronous linear motor system simulation is presented to show the design process and the excellent performance of the proposed optimal control scheme.