# Finite-Time Consensus Tracking Control for Speed Sensorless Multi-Motor Systems
## 1 Introduction  
In the era of Industry 4.0, the growing scale and complexity of industrial production make it necessary to coordinate multiple motors to complete complex production tasks in many areas, such as printing, textile and high-precision machining [1]. The synchronous control accuracy of the multi-motor system directly affects the production efficiency and product quality. Improving the precision of the synchronous control can achieve significant benefits, including low cost, high adaptivity, and easy maintenance [2, 3]. The existing synchronous control schemes of the multi-motor system mainly include uncoupled synchronous control and coupling synchronous control [4, 5, 6, 7]. The uncoupled synchronous control strategy assumes that each team member can track the same command signal. Thus, the control method is simple and easy to implement. However, because members cannot share information, the control precision and robustness are not satisfactory when the motor system is subjected to external disturbances and parameter perturbation [8]. To our best knowledge, [PERSON] first proposed the cross-coupling control scheme [9]. The distinguishing feature of this method is that the difference of position or velocity is used as the feedback signal of every team member to compensate for the deviation with the reference value. A novel method based on cross-coupling angle compensation was presented, which can realize the angle compensation of permanent magnet synchronous motors in offline or online conditions [10]. A cross-coupling control strategy based on second-order terminal sliding mode was applied to aim to improve the synchronizationaccuracy of dual-motor [11]. [PERSON] et al. proposed a cross-coupled control strategy for the consensus tracking of multi-motor [12]. Moreover, researchers conducted considerable works on coupling synchronous control with modern control theory [13; 14]. However, these control schemes inevitably produced a large number of intensive online calculations due to parameter coupling. To this end, it is necessary and meaningful to find a satisfactory synchronous control method with high synchronization performance. The synchronization tracking problem of the multi-agent system can be regarded as the individual state variables gradually converge to the given desired state of the system, namely the goal is to asymptotically synchronize [15; 16]. Compared with infinite settling time, the finite-time consensus control of multi-agent systems has been widely investigated in recent years because of its superior performance in improving the precision of synchronous control, and faster convergence rate [17]. In practice, the many required control systems need fast dynamic response such as multi-motor, robot operation, and spacecraft systems and the system error converges to zero in a finite time [18]. Unfortunately, most existing finite-time consensus control methods only focus on simple integrator models, such as first-order, second-order, and high-order integrators [19; 20; 21]. [PERSON] et al. proposed an observer-based variable structure consensus control to achieve the high-precision synchronous control [22]. Given the inevitable existence of parameter drift and unmodeled dynamics in an actual motor system, the desired performance is impossible to achieve by the application of a simplified-model-based consensus control in an actual multi-motor system such as a shaftless-driven printing press [23]. Moreover, when the speed of the leader changes, the followers should obtain the acceleration information of the leader, which is difficult or even impossible to quickly achieve in practical engineering [23; 24]. Nevertheless, the angular position or velocity is assumed to be measurable in the above-cited research works. In practice, precise measurements may not be accessible and may be subject to the failure of onboard sensors or environmental perturbations.