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Based on the previous studies, the dynamic balance of an electrical bicycle includes two control inputs: steering and pendulum torques, and three system outputs: steering, lean and pendulum angles. Two novel reference signals are first designed so that the uncontrolled mode is simultaneously included into these two control modes. Two scaling factors for each subsystem are first employed to normalize the sliding surface and its derivative. The so-called fuzzy decentralized sliding-mode under-actuated control (FDSMUC) is first designed. Because the uncertainties of a bicycle system, caused by different ground conditions, gusts of wind, and interactions among subsystems, are often huge, an extra compensation of learning uncertainty is plunged into FDSMUC to enhance system performance. We call it as "fuzzy decentralized sliding-mode adaptive under-actuated control" (FDSMAUC). To avoid the unnecessary transient response and then destroy the balance of the bicycle, the combination of FDSMUC and FDSMAUC with a transition (i.e., fuzzy decentralized sliding-mode robust adaptive under-actuated control, FDSMRAUC) is designed. Finally, the compared simulations for an electrical bicycle among the FDSMUC, FDSMAUC and FDSMRAUC validate the efficiency of the proposed method.

Minimization of emissions of carbon dioxide and harmful pollutants and maximization of fuel economy for lean-burn spark ignition (SI) engines relies to a large extent on precise air-fuel ratio (AFR) control. However, the main challenge of AFR control is the large time-varying delay in lean-burn engines. Since the system is usually subject to external disturbances and uncertainties, a high level of robustness in AFR control design must be considered. We propose a fuzzy sliding-mode control (FSMC) to track the desired AFR in the presence of periodic disturbances. The proposed method is model free and does not need any system characteristics. Based on the fuzzy system input-output data, two scaling factors are first employed to normalize the sliding surface and its derivative. According to the concept of the if-then rule, an appropriate rule table for the logic system is designed. Then, based on Lyapunov stability criteria, the output scaling factor is determined such that the closed-loop stability of the internal dynamics with uniformly ultimately bounded (UUB) performance is guaranteed. Finally, the feasibility and effectiveness of the proposed control scheme are evaluated under various operating conditions. The baseline controllers, namely, a PI controller with Smith predictor and sliding-mode controller, are also used to compare with the proposed FSMC. It is shown that the proposed FSMC has superior regulation performance compared to the baseline controllers.

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