Nash game‐based adaptive robust control design optimization for the underactuated mechanical system with fuzzy evidence theory

Abstract

This article proposes a novel Nash game‐theoretical optimal adaptive robust control design approach to address the constraint‐following control problem for the uncertain underactuated mechanical systems with fuzzy evidence theory. First, the uncertainty is considered bounded and the bound is unknown but lies in a specified fuzzy evidence number. Second, a deterministic adaptive robust control scheme is proposed based on the servo constraint following control method, which renders the uncertain underactuated mechanical system to follow the specified constraints accurately with deterministic performance (guarantee uniform boundedness and uniform ultimate boundedness). It is shown that the designed self‐adjusting leakage‐type adaptive law can compensate for the uncertainty and avoid overcompensation. Third, based on the performance analysis and the fuzzy evidence description of uncertainty, the Nash game theory is introduced into the multi‐parameter optimization design for the two tunable control gains selected as two players. The cost functions for two players are relevant to the system constraint‐following performance and control cost. Then we can obtain the optimal control gains by seeking the Nash equilibrium which is always proved to exist. Ultimately, the simulation results on the two‐wheeled self‐balancing robot demonstrate the availability of the proposed control scheme and the optimal design approach for the underactuated mechanical systems with uncertainties.

Fixed‐time linear quadratic adaptive sliding mode control for a class of cart‐pendulum robots

“…In recent years, many kinds of advanced control methods have been developed to compensate for the effects of the imperfections. In terms of a large number of relevant stability and tracking results for various systems, numerous efficient control methods are developed based on the techniques of sliding-mode control, [1][2][3][4] adaptive control, [5][6][7] robust control, [8][9][10] observer-based control, 11,12 intelligent control, [13][14][15] model predictive control 16,17 and so forth.…”

Iterative feedback tuning for optimal repetitive constraint‐following control of uncertain mechanical systems using Udwadia–Kalaba theory

“…Related work to this article is References 24‐27. In Reference 24, the authors presented an adaptive robust control strategy for constraint following of underactuated MSs.…”

“…The two tunable control parameters are optimized using a nash‐game approach. Reference 25 proposed an ARC for the underactuated MSs. Modeling the uncertainty of the underactuated MSs in a fuzzy method, the authors then optimize the tunable control gains using the nash‐game theory.…”

“…While they show low efficiency when addressing 0 order holonomic constraint‐following errors. Second, extant control optimization strategies like the fuzzy optimization, 17 the cooperative game theory, 21 the nash game theory, 25 and the Stackelberg game theory 26 are all model‐based. When dealing with periodic constraints, the repetitive tasks provide valuable input–output data for the self‐learning of the controller, and thus a more practical perspective to optimize controllers.…”

Data‐based learning control for optimization of nonlinear systems

“…11 A novel optimal constraint-following controller is proposed for uncertain mechanical systems. 12 The third group of papers [13][14][15][16][17][18][19] focuses on robustness on data-based optimal learning control. A novel Nash game-theoretical optimal adaptive robust control design approach is proposed to address the constraint-following control problem for the uncertain underactuated mechanical systems with fuzzy evidence theory.…”