# Finite-Time Consensus Tracking Control for Speed Sensorless Multi-Motor Systems
## 1 Introduction
To address this issue, in the absence of speed information, the finite-time consensus protocol was designed based on the estimation of speed information [25; 26]. [PERSON] et al. investigated a class of nonlinear systems with uncertain states and an observer based on impulse control to estimate external disturbances [27; 28; 29]. Through linear matrix inequalities derivation and Lyapunov functions, the theoretical results are successfully applied to multi-link robots. The concepts of the sensorless control solutions are to eliminate the hall sensors and optical encoders to obtain the position or speed information of the motors through some intelligent control algorithms [30]. In fact, the speed sensorless control schemes can reduce the encoder installed on the motor spindle and the decoder device on the control panel, and the speed signal will no longer be affected by the strong electromagnetic interference of the motor, which greatly improves the anti-interference ability and mechanical reliability of the multi-motor system [31; 32]. [PERSON] et al. proposed an adaptive observer to efficiently estimate the speed and identified the resistance with speed sensorless [33]. Considering the speed tracking performance under various uncertainties and disturbances, a sensorless speed controller for induction motors based on speed and perturbation estimation was presented by defining nonlinear dynamics and external disturbances [34]. To solve a class of high-order nonlinear systems with mismatched external disturbances and uncertainties, researchers had developed state observers to provide an on-line state estimator for control systems [35]. The unmeasured states and unmatched disturbances were estimated by an extended state observer, and a finite-time consensus protocol was designed based on the output feedback using the adding power integration method [36]. Inspired by the above mentioned works and industrial requirements, an observer-based finite-time leader-follower consensus tracking control method is proposed for the synchronous coordination control of the multi-motor system in this study. The main contributions of this paper are reflected as follows: (1) Many existing control strategies can obtain a stable synchronization system, but the obtained stable system may encounter large parameter perturbation or interference that causes loss of synchronization performance. A feedback consensus protocol is developed based on the disturbance estimation/compensation, which can actively suppress the uncertainty and chattering. Comparedto the passive anti-disturbance control strategy, the cooperative tracking performance of the multi-motor system can be improved; (2) The distributed finite-time state observers are designed to estimate the speed of each agent when only part of the motors interact locally. It means that there is no need to measure the speed of all motors and only partial agents receive the reference signal to complete the synchronization tracking process, which are easy to implement in practice. The speed sensorless control strategy avoids intensive on-line computational work and improves the mechanical reliability of the system; (3) Based on local information interaction, the proposed distributed finite-time consensus control studied the synchronization and tracking of second-order multi-agent systems with nonlinear properties and external disturbance. Compared with the existing control strategies, the proposed control method exhibits higher synchronization control accuracy through rigorous theoretical proof and numerical simulation. The remainder of this paper is organized as follows. Section 2 describes the mathematical model of the motor with parameter perturbation and necessary lemmas. The finite-time observer design and its convergence analysis are provided in Section 3. Section 4 illustrates the design of the finite-time consensus tracking protocol based on the output feedback. Moreover, the analysis of the stability of the closed-loop system is presented by using the adding a power integrator method.