This paper introduces the Motor-Transmission Drive System as a benchmark example for the safety analysis of hybrid systems. In particular, we illustrate the problem of checking the gear meshing duration and the impact impulse (both of which we refer to as safety) of the Motor-Transmission Drive System. We aim to provide a complete problem description to which different verification tools or approaches for safety analysis can be applied and compared. For this reason, we first elaborate on a hybrid automaton (HA) model of the Motor-Transmission Drive System to describe the gear meshing process with uncertain initial states, and then we specify the safety property of interest. Next, we clarify the characteristic phenomena exhibited by the benchmark which make the verification problem hard to solve. Finally, we show some simulation results to illustrate the influences of the initial states on the safety property. This benchmark example can help the researchers and engineers to find appropriate methods for safety verification of this kind of hybrid system.

Motor-transmission coupled drive system is attractive for battery and hybrid electric vehicles. In such a system, the motor rotor is directly connected to the transmission input shaft and the active-synchronization technique is implemented to assist the speed synchronization; therefore, the gear-shifting characteristics are different from those of traditional manual and automated mechanical transmissions. In this work, we present a methodology for modeling the gear-shifting process and analyzing its characteristics in a motor-transmission coupled drive system. We treat the engaging of sleeve and desired clutch gear as a two-phase process—sleeve first interacting with synchro ring and then with clutch gear, respectively, and investigate all possible interaction ways in each phase. The movement of each part is governed by multibody dynamics, and the speed jumps caused by shifting impacts are described using the Poisson coefficient of restitution. We then develop a hybrid automaton (HA) model to couple the continuous-time evolutions and the discrete transitions of state variables, which cover all interaction ways of sleeve, synchro ring, and clutch gear. Based on this model, we carry out simulations in matlab to analyze the effects of two control parameters—the relative rotational speed of sleeve and desired clutch gear, and the shifting force—on shifting performance. Simulation and bench test results show that the optimal control parameters are located in the domain where the relative rotational speed is negative with small absolute value, which means the sleeve will not be locked out by synchro ring and can engage with the desired clutch gear smoothly.

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Motor transmission-based drive systems are attractive for electric vehicles but, as the motor is directly connected to the transmission shaft which meshes with the gears, controlling gear shifts is challenging. In this paper, we present a methodology for synthesis and verification of open-loop optimal control of the electric motor in a motor-transmission drive system. The key steps in this methodology are (a) developing a continuous-time model of the trajectory of the sleeve during the meshing process based on appropriate coefficients of restitution, (b) discrete-time controller synthesis for finitely many initial states using model predictive control (MPC) and (c) verification of the synthesized controller for a higherfidelity continuous time hybrid automaton model. First, we develop a model of the motor-transmission drive system as a continuous-time hybrid automaton (CHA) with uncertain initial states. Next, this model is transformed to a piece-wise affine (PWA) form for solving an optimal control problem using the multi-parametric toolbox (MPT). Finally, the delay bound for the synthesized controller is verified by computing a bounded time over-approximation of the reach set using an existing algorithm for deterministic linear hybrid automata. Our results show that on the average our synthesized controller can shorten the meshing duration by 71.05% and reduce impacts impulse by 85.72% compared to an existing controller and the sleeve can mesh with the gear within a desired time from every initial state.

Non‐synchronizer electric‐driven mechanical transmission (EMT) has advantages of low cost and simple mechanical structure, which is a key assembly in electric vehicles. In such an system, as the synchronizer is removed from the mechanical transmission, big impacts between the sleeve and clutch gear arise during a gear shift, which cause a long power‐off interruption. In this article, our contribution is to propose a novel control strategy to avoid impacts by achieving both zero relative rotational speed and angle differences between the sleeve and clutch gear. To obtain this strategy, we first derive the dynamic model of the gear‐shifting process as a hybrid automaton (HA) model. Based on the HA model, we can see that impacts can be eliminated when the rotational speed and angle differences become zeros. Then, to obtain zero rotational speed and angle differences and reduce the gear‐shifting time, a time‐optimal control law to synchronize the rotational speed and angle is first solved based on Pontryagin's minimum principle. Meanwhile, to resolve the solution difficulty brought by the period‐varied rotational angle difference, we introduce an optimal initial angle difference, and then use an incremental angle difference to replace the period‐varied one. Based on that, a model tracking strategy is proposed to enhance the system's antidisturbance capacity. For the gear‐shifting actuator, a dual bang‐bang control law is solved to save the gear‐shifting time. Finally, we carry out simulations and bench tests to validate the control strategy. Results show that the control strategy can indeed reduce impacts and power‐off time.

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