## 1 Introduction
[PERSON] [12] proposed an accurate and simple five-node lumped parameter thermal network (LPTN) and built the mathematical model of the LPTN, aiming at resolving the difficulty in online temperature estimation. However, the above research results are all aimed at linear feed single-drive systems, and the dynamic model of a dual linear-motor differential-drive system is not established. Many research achievements have been made on the influence of nonlinear friction on micro-feed systems [13; 14; 15; 16; 17; 18]. [PERSON] [13] introduced a delay-based nonlinear controller aimed at position control of linear ultrasonic motors, which is termed the cascade nonlinear proportional integral retarded control law, to achieve high precision and fast response in spite of the effects of inner disturbances, hysteresis, and friction phenomena. [PERSON] [14] not only incorporated various non-linear factors such as time-varying friction forces, time-varying mesh stiffness, and damping associated with internal excitations but also placed significant emphasis on analyzing the impact of non-linear external excitations, including road surface unevenness and motor torque fluctuations, on the shifting process of the drive system. The friction force at the contact interface is easily affected by factors such as surface morphology, friction coefficient, and sliding speed, especially when the linear motor is running at low speed, [PERSON] [15] proposed an orthogonal vibration decouplingmethod for low-speed regulation. [PERSON] [16] presented an asymmetric friction model and an indirect integral method (IIM), the asymmetric friction model is able to capture the nonlinear position drifting phenomenon. [PERSON] [17] proposed a novel design method for permanent magnet linear motor systems based on backstepping, in order to solve the influence of unknown factors on the system parameters of a permanent magnet linear motor including nonlinear friction, sudden load changes, thrust fluctuations, and so on. In order to avoid discontinuities, the nonlinear friction of the motor, which includes static, coulomb, and viscous terms, is considered a smooth function. [PERSON] [18] presented a state observer for an elastic joint with nonlinear friction via the information from an acceleration sensor. Many papers have established physical and mathematical models for dual servo systems [19; 20; 21; 22; 23; 24; 25]. [PERSON] [19] proposed a topological structure expanded by five improved sliding-mode observers (ISMOs) to simultaneously identify the load speed, the dual motor inertia, the load inertia, the stiffness coefficient, and the load disturbance. [PERSON] [20] proposed a novel two degrees of freedom, large range, coarse-fine parallel dual-actuation [PERSON] micropositioner (CFDFM) with low interference behavior. [PERSON] [21] investigated a difficult problem of nonlinear dynamics and motion control of a dual-flexible servo system with an underactuated hand (DFSS-UH) and designed a novel neural network sliding mode control (NNSMC) method to control the DFSS-UH. [PERSON] [22] proposed a finite-time tracking and synchronization control method for dual-motor servo systems that suffer from backlash and time-varying uncertainties, to overcome several factors that may degrade the system's performance, such as transmission backlash, parameter drift, and motor dynamic characteristic differences. [PERSON] [23] investigated the finite-time command-filtered backstepping control problem for dual-motor servo systems[PERSON] The advantages of the finite-time controller include fast convergence and high robustness, which can improve the dynamic and steady control performance of the system. [PERSON] [24] presented the design for a new differential dual-drive low-speed micro-feed mechanism and studied the difference in response of single-drive and differential dual-drive systems under the influence of friction and clearance.