DC motors are easy to track speed and position and have a wide speed range, making them widely used in industrial production, such as in cold rolling mills. During the rolling process in a cold rolling mill, the load on the DC motor varies due to fluctuations in the thickness of the incoming steel sheet. Ensuring that the system speed remains essentially constant under external load disturbances and changes in internal parameters—that is, improving the system's robustness—is of significant engineering importance to guarantee production quality and output. DC motor speed control systems are often treated as linear systems, but in reality, they contain various nonlinearities, such as armature reaction nonlinearity, excitation nonlinearity, and saturation nonlinearity. This paper primarily considers armature reaction nonlinearity. Using a design method for nonlinear system state feedback linearization, and through appropriate nonlinear state and input transformations, the comprehensive problem of the speed nonlinear control system is transformed into a comprehensive problem of a linear system. A sliding mode variable structure control method, which is highly robust to system parameter changes and external disturbances, is then employed to achieve DC motor speed tracking control. Mathematical Model of Controlled Object in DC Motor Speed ​​Loop Considering the actual situation of armature reaction or armature reaction that cannot be completely canceled by the linear compensation winding, the motor excitation will have the following basic relationship[2][3]: cmφ=a+bi=kfif+bi (1) Where: cmφ is the torque current coefficient corresponding to the motor flux, if is the excitation current, kf is the conversion coefficient between the excitation current and cmφ, which is usually a constant; a=kfif, which is a constant when the DC motor is in conventional voltage regulation and speed regulation without field weakening speed regulation; b is a very small negative number, representing the reaction of the armature current i on the excitation flux. Therefore, the controlled object model of the DC motor speed control system under this case is nonlinear. The dynamic structure diagram of the controlled object in the DC motor speed control system is shown in Figure 1: [align=center] Figure 1 Dynamic structure diagram of the controlled object in the DC motor speed control system[/align] In Figure 1, in addition to the current loop regulator, there is a compensation circuit cmφe/cmφ. Its purpose is to ensure that the open-loop gain of the entire DC motor speed control system remains unchanged during conventional voltage regulation and field weakening speed regulation. Here, cmφe is the rated torque constant, and cmφ is the actual torque constant of the motor. lt is the current regulator, ks is the thyristor amplification factor, β is the current feedback factor, r is the total armature circuit resistance, is the electromagnetic time constant of the motor, and l is the total armature circuit inductance. When designing the current regulator of a DC motor, since the electromagnetic time constant tl in the actual system is generally much smaller than the electromechanical time constant, the back EMF can be ignored when designing the current loop regulator. Using the method given in reference [4], the current regulator lt is designed as: (2) and the current loop is simplified as: (3) where ts is the time constant of the rectifier and toi is the time constant of the current filter, which are 0.00167 seconds and 0.001 seconds respectively. Thus, the block diagram of the DC motor can be simplified as shown in Figure 2. [align=center] Figure 2 Equivalent structure diagram of the controlled object of DC motor speed[/align] Selecting the armature current i and the motor speed ω of the DC motor as state variables x1 and x2, the state space model is: (4) where ω—rotor angular velocity, j—moment of inertia, f—viscous friction coefficient, t1—electromagnetic torque, and t1—load torque. Design of speed loop nonlinear controller State feedback linearization For a single-input single-output nonlinear control system, the state or input/output of a class of nonlinear systems can be linearized through nonlinear state transformation and input transformation. Consider the following single-input single-output nonlinear control system [6] (5) In order to study the dynamic response relationship between input and output, we differentiate y with respect to t until the first occurrence of the control input u in the derivative term of y: (6) The following is an input-output feedback linearization of the DC motor speed control system considering armature reaction nonlinearity. Substituting equation (1) into equation (4), we obtain the following state equation set: (10) Thus, the input-output linearization of the DC motor speed control system with armature reaction nonlinearity can be realized. Speed ​​loop variable structure controller design Under certain conditions, the sliding mode of the variable structure control method is invariant to the changes in disturbance and parameters. This is the problem that robust control needs to solve. For the linear system after state feedback precise linearization, the variable structure control method can be applied. The switching function is defined as follows: Simulation Study and Results The parameters of the DC motor driving the upper roller of a cold rolling mill in a certain factory are as follows: rated power of the motor is 1500 kW, rated current ie = 1720 A, total armature circuit resistance r = 0.0314 Ω; total armature circuit inductance l = 0.0003 H; moment of inertia of the motor j = 1542 kg·m²; b = -0.00109, thyristor amplification factor ks = 152, kf = 0.27, rated excitation current ie = 113.29 A, rated motor torque constant cmφe = kfife + bie = 29.1 nm/A. Based on the above data, the designed system is simulated using MATLAB software. [align=center]Figure 3 Step signal tracking curve Figure 4 Switching function curve[/align] Figure 3 shows the case where the DC motor tracks the step signal nd=9.55ωd=600rpm. The load disturbance is set to t1=(29100+2910sin(3.14t)) (equivalent to the rolling load disturbance caused by fluctuations in the incoming material thickness during rolling, fluctuating within a range of about 60% of the rated load). Figure 3 shows that there is no overshoot during speed tracking, and the dynamic process is good. Figure 4 shows the switching function curve against time. Figure 3 shows that the trajectory fluctuates around 0, indicating that the designed variable structure controller has good robustness to changes in system parameters. Simulation results show that the speed variable structure control system based on input-output feedback linearization designed in this paper has good tracking performance, and the system also has good tracking performance when the external load and internal system parameters change.