Synchronous tracking control method for multi-motor driving system based on adjacent cross coupling

Abstract

The invention discloses a synchronous tracking control method for a multi-motor driving system based on adjacent cross coupling, which comprises the following steps of: firstly, establishing a multi-motor driving system dynamic model containing backlash nonlinearity and external disturbance, and designing a finite time disturbance observer to estimate the total disturbance of the system; a tracking controller is designed by combining a first-order command filter based on the thought of a backstepping method, the problem of differential explosion is eliminated, tracking errors are constrained by using a finite time preset performance function, transient performance and finite time convergence of the tracking errors are ensured, and an adjacent cross-coupling control strategy is adopted, so that the tracking precision is improved. And a hyperbolic function is introduced to adjust synchronous control input, and a self-adaptive adjacent cross-coupling synchronous controller is designed, so that the actual control input of each motor is the sum of the mean value of tracking control input and the synchronous control input of the motor. According to the method, the coupling influence between synchronization and tracking is reduced, rapid synchronization among multiple motors is realized, and finally, the stability of the system is proved by using the Lyapunov theorem.

Description

Technical Field

[0001] The present invention belongs to the technical field of motor synchronization and load tracking control for multi-motor drive systems, and particularly relates to a synchronous tracking control method for multi-motor drive systems based on adjacent cross-coupling. Background Art

[0002] In modern engineering, single-motor drive systems are widely used in industrial, medical, and military fields. With the development of science and technology, due to low driving force and high cost, single-motor drive systems can no longer meet the requirements of modern industrial development.

[0003] With the improvement of industrial production technology, multi-motor drive systems have been widely used, especially in the control of large inertia and high-power systems, and remarkable results have been achieved. The multi-motor drive system uses multiple motors to jointly drive the load system, with greater driving power and load capacity. At the same time, the multi-motor drive system can better cope with load changes and fluctuations in output power, improving the system control performance.

[0004] As a strongly coupled and highly nonlinear system, the multi-motor drive system will inevitably be adversely affected by various factors such as backlash nonlinearity, external disturbances, and unmodeled dynamics during its operation, thereby reducing the system control performance and load tracking performance, and even causing the drive system to be unstable. In addition, due to factors such as load changes and external disturbances in the multi-motor drive system, the positions and speeds of the motors will be out of sync, resulting in collisions between the motors, affecting the system performance, and even causing damage to the drive system. Therefore, how to design a controller to solve problems such as load tracking, motor synchronization, nonlinear compensation, and disturbance observation of multi-motor drive systems with unknown disturbances is of great significance for achieving precise control of multi-motor drive systems. Summary of the Invention

[0005] Aiming at the multi-motor drive system with strong coupling, nonlinearity, and external disturbances, the present invention provides a synchronous tracking control method for multi-motor drive systems based on adjacent cross-coupling, which can ensure the transient performance and finite-time convergence of the motor tracking error, reduce the coupling effect between synchronization and tracking, achieve fast synchronization between multi-motors, and thus improve the response speed, accuracy, and stability of the system.

[0006] To achieve the above object, the technical solution adopted by the present invention is: a synchronous tracking control method for multi-motor drive systems based on adjacent cross-coupling, including the following steps: Step 1: Establish a dynamic model of the multi-motor drive system with backlash nonlinearity and external disturbances, describe the transmission torque between the motor and the load based on a continuously differentiable backlash model, and then construct the state-space equation of the multi-motor drive system and decompose it into a load subsystem and a multi-motor subsystem; Step 2: Design finite-time disturbance observers for the load side and the multi-motor subsystem, and estimate the total disturbance of the load side and the total disturbance of each motor subsystem respectively; Step 3: Design a finite-time preset performance tracking controller based on the backstepping method and a first-order command filter. The tracking controller uses the finite-time preset performance function to constrain the tracking error; Based on the adjacent cross-coupling synchronization control structure, introduce a hyperbolic function to adjust the synchronization control input, and design an adaptive adjacent cross-coupling synchronization controller; Output the total disturbance estimation value of each motor subsystem to its respective synchronization controller for feedforward compensation, and output the total disturbance estimation value of the load side to the tracking controller for feedforward compensation, so that the actual control input of each motor is the sum of the mean value of the tracking control input and the synchronization control input of this motor.

[0007] Furthermore, the dynamic model of the multi-motor drive system in Step 1 is: where, J mi and J l are the moment of inertia of the motor and the moment of inertia of the load respectively, b mi and b l are the viscous friction coefficient of the motor and the viscous friction coefficient of the load respectively, θ mi and θ l are the angular displacement of the motor and the angular displacement of the load respectively, T dmi and T dl are the disturbing torque on the motor side and the disturbing torque on the load side respectively, n is the total number of motors, i is the serial number of the current motor, u i is the input torque of the i-th motor, T i is the transmission torque between the i-th motor and the load.

[0008] Furthermore, the backlash model in Step 1 adopts the following approximate dead zone function: where, k>0 is the torque torsion coefficient, α is the backlash width, z i =θ mi -θ l represents the angular displacement difference between the motor and the load, and r = 2 / α is a positive constant.

[0009] Furthermore, the state space equation of the multi-motor drive system in Step 1 is: where, x1, x2, x 3i , x 4iThey respectively represent the load position, load speed, the position of the i-th motor, and the speed of the i-th motor. y = x1 indicates that the output value y of the multi-motor drive system is the load position x1.

[0010] Since the multi-motor drive system is composed of multiple motors jointly driving the load through a transmission mechanism, it can be regarded as a cascade system composed of a multi-motor subsystem and a load subsystem. Therefore, the multi-motor drive system is written as the following load subsystem and multi-motor subsystem: The load subsystem is: The multi-motor subsystem is: Where X1 = [x1 x2] T , X 2i = [x 3i x 4i T , C1 = C2 = [1 0], and y1 and y 2i are the load output and the output of each motor respectively.

[0011] The expressions for the total disturbance at the load end and the total disturbance of each motor subsystem are: In the formula, F1 is the total disturbance at the load end, and f 2i is the total disturbance of the i-th motor subsystem.

[0012] Furthermore, the finite-time disturbance observer designed for the load end in step 2 is: In the formula, m1 > 0 and m2 > 0 represent the observer gains, c1 = p1 / q1, where p1 and q1 are positive odd integers and satisfy p1 < q1 < 2p1, and respectively represent the estimated values of x2 and F1; The finite-time disturbance observer of the multi-motor subsystem is designed as: In the formula, m3 > 0 and m4 > 0 represent the observer gains, c2 = p2 / q2, where p2 and q2 are positive odd integers and satisfy p2 < q2 < 2p2, and respectively represent the estimated values of x 4i and f 2i respectively.

[0013] Furthermore, the finite-time preset performance function in step 3 is designed as: In the formula, a1, a2, b1, b2, b3, ρ ∞ and t p are preset normal constants, and ρ ∞ is the steady-state value, representing the maximum convergence range of the error, and t p is the convergence time to converge to the steady-state value.

[0014] Furthermore, the first-order command filter in the third step is designed as: In the formula, ω>0 is the design parameter, represents the state of the command filter, represents the input of the filter, represents the output of the filter.

[0015] Furthermore, the control law of the tracking controller in the third step is: where, k4 is the tracking controller parameter, e3 and e4 are the tracking errors, and η3 is the virtual control law; The control law of the synchronization controller is: u si =-J mi k s ψ i , where, the adaptive gain e si is the synchronization error between the motors.

[0016] Furthermore, the design of the finite-time preset performance tracking controller in the third step includes: first, defining the load tracking error, converting the tracking error into an unconstrained error through an error transformation function, then keeping the tracking error within the boundaries specified by the finite-time preset performance function, and then using the backstepping method to design the virtual control law step by step, and combining the first-order command filter to eliminate the differential explosion problem in the backstepping process, and further generating the tracking control input.

[0017] The beneficial effects of the present invention are:

[0018] (1) The present invention first establishes the dynamic model of the multi-motor drive system with backlash nonlinearity and external disturbances. By using the finite-time disturbance observer, it can accurately estimate the total disturbances at the load end and the motor end within a predetermined time, and inject the estimated values of the total disturbances at the load end and the motor end into the control inputs of the tracking controller and the synchronization controller respectively for feedforward compensation to offset the influence of the disturbances, reduce the periodic error caused by the gear clearance, and enhance the anti-interference ability of the system.

[0019] (2) The present invention designs a finite-time preset performance function to transform the tracking error, shortening the convergence time of the load tracking error and improving the steady-state accuracy. Based on the idea of the backstepping method and combined with a first-order command filter, a tracking controller is designed. The backstepping hierarchical control is adopted to gradually stabilize the load position, speed, and motor state. Combining with the command filter to avoid differential explosion, it ensures the transient performance and finite-time convergence of the tracking error.

[0020] (3) The present invention adopts an adjacent cross-coupling control strategy. Each motor exchanges information with its left and right neighbors, reducing the dependence on a single node and decreasing the multi-motor synchronization error. A hyperbolic function is introduced to adjust the synchronization control input, reducing the coupling effect between synchronization and tracking, achieving fast synchronization among multi-motors, and reducing the collision probability between motors.

[0021] (4) The present invention can automatically adjust the adaptive gain k s with the change of the synchronization error. According to the properties of the hyperbolic tangent function, when the synchronization error is too large, the adaptive gain k s will tend to a constant a and will not continuously increase the control input as the error increases, thus avoiding damage to the motor drive equipment due to excessive synchronization control input and reducing the waste of energy. When the synchronization error decreases to tend to 0, the adaptive gain k s will also tend to 0, thereby reducing the coupling effect of synchronization control on tracking control.

[0022] (5) Finally, the actual control rate of each motor includes the tracking control of the load and the synchronization control of the motor. Therefore, the system can simultaneously ensure the load tracking performance and the synchronization performance among multi-motors. The stability of the system is proved by the Lyapunov theorem. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0024] Figure 1 is a schematic diagram of gear meshing of the multi-motor drive system of the present invention;

[0025] Figure 2 is a comparison diagram of the dead zone and the approximation function of the present invention;

[0026] Figure 3 is a comparison diagram of the finite-time preset performance function and the traditional preset performance function of the present invention;

[0027] Figure 4This is the structural diagram of the controller for the multi-motor drive system of the present invention. Detailed implementation manners

[0028] The following specific embodiments are given to further clearly, completely, and detailedly illustrate the technical solutions of the present invention. This embodiment is the best embodiment based on the technical solutions of the present invention, but the protection scope of the present invention is not limited to the following embodiments.

[0029] It should be noted that unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which this application belongs.

[0030] The present invention proposes a synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling. The multi-motor drive system is composed of multiple motors that jointly drive a load through a gear link, so that the load runs along a specified trajectory. Taking a four-motor drive system as an example, its transmission gear structure diagram is as Figure 1 shown, where O1, O2, O3, and O4 are the drive systems, and O0 is the driven system. The synchronous tracking control method for the multi-motor drive system based on adjacent cross-coupling specifically includes the following contents: Step 1: Establish a dynamic model of the multi-motor drive system with backlash nonlinearity and external disturbances. Based on a continuously differentiable backlash model, describe the transmission torque between the motor and the load, and then construct the state space equation of the multi-motor drive system, and decompose it into a load subsystem and a multi-motor subsystem; Step 2: Design finite-time disturbance observers for the load end and the multi-motor subsystem, and estimate the total disturbance at the load end and the total disturbance of each motor subsystem respectively; Step 3: Design a finite-time preset performance tracking controller based on the backstepping method and a first-order command filter. The tracking controller uses a finite-time preset performance function to constrain the tracking error; based on the adjacent cross-coupling synchronization control structure, introduce a hyperbolic function to adjust the synchronization control input, and design an adaptive adjacent cross-coupling synchronization controller; output the total disturbance estimation value of each motor subsystem to its respective synchronization controller for feedforward compensation, and output the total disturbance estimation value of the load end to the tracking controller for feedforward compensation, so that the actual control input of each motor is the sum of the mean of the tracking control input and the synchronization control input of this motor.

[0031] The following will detail each step of the present invention:

[0032] The dynamic model of the multi-motor drive system with backlash nonlinearity and external disturbances established in Step 1 is: In the formula, J mi and J l are the rotational inertia of the motor and the rotational inertia of the load respectively, bmi and b l are the viscous friction coefficients of the motor and the load respectively, and θ mi and θ l are the angular displacements of the motor and the load respectively, and T dmi and T dl are the disturbance torques on the motor side and the load side respectively. n is the total number of motors, i is the serial number of the current motor, and u i is the input torque of the i-th motor, and T i is the transmission torque between the i-th motor and the load.

[0033] The transmission torque between the motor and the load can be described by the following dead zone function: In the above formula, k is the torque torsion coefficient, α is the backlash width, and z i is the angular displacement error between the motor and the load. Since the dead zone function has non-differentiable characteristics, a smooth, continuous and differentiable approximate dead zone function is introduced to replace the original dead zone function.

[0034] The backlash model in Step 1 adopts the following approximate dead zone function: In the formula, k>0 is the torque torsion coefficient, α is the backlash width, and z i = θ mi - θ l and r = 2 / α is a positive constant. The curves of the dead zone function and the approximate dead zone function (i.e., the approximation function) are as Figure 2 shown.

[0035] Let That is, the load position, load speed, motor position, and motor speed are selected as state variables. According to Equations (1) and (3), the state space equation of the multi-motor drive system is: In the formula, x1, x2, x 3i , x 4i represent the load position, load speed, the position of the i-th motor, and the speed of the i-th motor respectively. y = x1 represents the output value y of the multi-motor drive system, which is the load position x1.

[0036] Assume that the parameters of all motors are the same. Since the multi-motor drive system is composed of multiple motors that jointly drive the load through a transmission mechanism and is regarded as a cascade system composed of a multi-motor subsystem and a load subsystem, the multi-motor drive system is written as the following load subsystem and multi-motor subsystem:

[0037] The load subsystem is: The multi-motor subsystem is as follows: Wherein, X1 = [x1 x2] T , X 2i = [x 3i x 4i T , C1 = C2 =

[10] , y1 and y 2i are the load output and the output of each motor respectively.

[0038] The expressions for the total disturbance at the load end and the total disturbance of each motor subsystem are: Wherein, F1 is the total disturbance at the load end, and f 2i is the total disturbance of the i-th motor subsystem.

[0039] In the second step, in order to handle the situation where some system states are unmeasurable and there are unknown disturbances, a finite-time disturbance observer is designed. The finite-time disturbance observer designed for the load end is: Wherein, m1 > 0 and m2 > 0 represent the observer gains, c1 = p1 / q1, p1 and q1 are positive odd integers, and satisfy p1 < q1 < 2p1, and represent the estimated values of x2 and F1 respectively; The finite-time disturbance observer of the multi-motor subsystem is designed as: Wherein, m3 > 0 and m4 > 0 represent the observer gains, c2 = p2 / q2, p2 and q2 are positive odd integers, and satisfy p2 < q2 < 2p2, and represent the estimated values of x 4i and f 2i respectively.

[0040] The traditional Prescribed Performance Function (PPF) is usually defined as: ρ(t) = (ρ0 - ρ ∞ )e -kt + ρ ∞ , where the parameters ρ0 and ρ ∞ are the initial value and the final value of the performance function respectively, k > 0 is the convergence rate, and ρ0 > ρ ∞ > 0.

[0041] It can be noted that in the traditional PPF, the initial error e(0) needs to satisfy the initial condition Moreover, the performance function ρ(t) decreases exponentially. When the overshoot value at the initial moment of the system is too large, the error e(t) may exceed the constraint boundary of the performance function. At the same time, the traditional PPF converges to the steady-state value within infinite time and cannot guarantee the finite-time convergence of the error.

[0042] In order to ensure the tracking error of the system, improve the convergence speed of the performance function, and achieve the convergence of the error within a finite time within the preset performance boundary, the present invention designs a finite-time prescribed performance function (FTPPF) based on hyperbolic cotangent and hyperbolic tangent to transform the tracking error, ensuring that both the transient performance and the steady-state performance of the system can meet the preset requirements.

[0043] The finite-time prescribed performance function is as follows: In the formula, a1, a2, b1, b2, b3, ρ ∞ and t p are pre-designed positive constants, ρ ∞ is the steady-state value, representing the maximum convergence range of the error, and t p is the convergence time to the steady-state value.

[0044] The comparison diagram between the finite-time prescribed performance function and the traditional prescribed performance function is as Figure 3 shown.

[0045] In order to solve the "differential explosion" problem existing in the design process of the traditional backstepping method, a first-order command filter is designed to estimate the differential of the virtual control signal in the backstepping process. The design of the first-order command filter is as follows: In the formula, ω>0 is the design parameter, represents the state of the command filter, represents the input of the filter, represents the output of the filter.

[0046] The structure diagram of the multi-motor drive system controller in the third step is as Figure 4As shown below. First, the finite-time disturbance observer is used to estimate the total disturbances at the load side and the motor side respectively. Subsequently, the finite-time preset performance function is used to constrain the system tracking error. To eliminate the "differential explosion" problem of the traditional backstepping method, the first-order command filter is combined with the backstepping method to achieve the tracking of the load. Then, based on the adjacent cross-coupled synchronization control structure, a hyperbolic function is introduced to design an adaptive adjacent cross-coupled synchronization controller to achieve the synchronous operation of multiple motors. Finally, the actual control rate of each motor is the sum of the average value of the total tracking control rate and the synchronous control rate of each motor. The design steps are as follows:

[0047] The design steps of the finite-time preset performance tracking controller are as follows:

[0048] Step 1: Define the tracking error \(e_1 = x_1 - y\) d , \(e_2 = x_2 - \eta_1\), where \(\eta\) i (\(i = 1, 2, 3\)) is the virtual control input, and \(y\) d is the desired load position. Define the state variable The control input To achieve the preset performance constraint, the tracking error should satisfy the following inequality: where \(\delta\) and are positive constants, and \(- δρ (t)\) and represent the lower and upper bounds of the maximum overshoot respectively.

[0049] To design the preset performance control using the finite-time preset performance function, a smooth and strictly monotonically increasing function \(S(\xi)\) is designed to transform the constrained tracking error into an equivalent unconstrained error. Define the error transformation function as: where \(\xi\) is the transformed error. According to (13), \(S(\xi)\) has the following properties: According to (13) and (15), (12) can be equivalent to: According to (13) and (17), the transformed error \(\xi(t)\) can be obtained by inverse transformation as: where \(\lambda(t) = e(t) / \rho(t)\). The derivative of \(\xi(t)\) is: where is a bounded variable. Through the error transformation (18), the original preset performance control problem is transformed into the stability problem of the transformed error ξ. According to (19), the error transformation of the tracking error e1 is performed to obtain its transformed error ξ1 as follows:[[]] where λ1 = e1 / ρ. The derivative of ξ1 is:[[]] where Design the Lyapunov function V1 as:[[]] Its derivative is:[[]] Let y d The derivative of y can be obtained through a first-order command filter d as:[[]] Design the virtual control law η1 as:[[]] Then

[0050] Step 2: The derivative of e2 is:[[]] Let η1 pass through a first-order command filter, and the derivative of η1 can be obtained as:[[]] Design the Lyapunov function V2 as:[[]] Its derivative is:[[]] Design the virtual control law η2 as:[[]] Then

[0051] Step 3: The derivative of e3 is:[[]] Let η2 pass through a first-order command filter, and the derivative of η2 can be obtained as Design the Lyapunov function \(V_3\) as: Its derivative is: Design the virtual control law \(\eta_3\) as: Then

[0052] Step 4: The derivative of \(e_4\) is: Let \(\eta_3\) pass through a first-order command filter to obtain the derivative of \(\eta_3\) as: Design the Lyapunov function \(V_4\) as: Its derivative is: Design the tracking control input \(u\) as: where \(k\) i > 0 (i = 1, 2, 3, 4) are design parameters.

[0053] The design steps of the adaptive adjacent cross-coupling synchronization controller are as follows: The multi-motor synchronization error \(e\) under adjacent cross-coupling si is: The synchronization control input \(u\) si can be expressed as: According to the multi-motor subsystem (6), the derivative of the synchronization error can be obtained as: where Design the adaptive synchronization controller \(u\) according to the synchronization error (44) si as: u si = -J mi k s \(\psi\) i (47) where The adaptive gain \(k\) s can be expressed as:

[0054] Adaptive gain k s will change continuously with the change of the synchronization error. According to the properties of the hyperbolic tangent function, when the synchronization error is too large, the adaptive gain k s will tend to the constant a and will not continuously increase the control input as the error increases, thus avoiding the damage of the motor drive device due to too large synchronization control input and reducing the waste of energy; when the synchronization error decreases to tend to 0, the adaptive gain k s will also tend to 0, thereby reducing the coupling effect of the synchronization control on the tracking control.

[0055] The present invention designs a finite-time preset performance tracking controller based on the backstepping method and an adaptive synchronization controller based on adjacent cross-coupling, and proves the stability of the system by using the Lyapunov theorem. The specific steps are as follows: Lemma 1: For the load subsystem (5) and the finite-time disturbance observer (8), it can be ensured that the disturbance estimation error converges to the neighborhood with the origin as zero within the finite time t f and the convergence time t f satisfies: where, Λ1 is a symmetric positive definite matrix, and are positive definite matrices, Lemma 2: For the first-order command filter (11), if the input signal and its derivative are continuous and bounded, then there exist constants ζ>0 and satisfying l is a very small positive constant.

[0056] Substituting Equation (43) into (42) gives: From Lemma 2, it can be obtained that: Using the Young's inequality, we have: Substituting Equation (52) into Equation (51) gives: It can be further written as: where: where $\xi_1$ is the converted tracking error, and its conversion principle is to use a finite-time preset performance function to constrain the original error. Therefore, the converted error $\xi_1$ is a bounded function with a very small boundary. According to Theorem 1, the disturbance observation error is finite-time bounded; according to Lemma 2, the filtering error $l$ i is bounded. Therefore, both $\varLambda$ and $\varXi$ are bounded.

[0057] Therefore, all signals of the closed-loop system are semi-globally ultimately uniformly bounded, and the tracking error converges to a very small neighborhood restricted by the preset performance boundary within a finite time.

[0058] Design the Lyapunov function $V$ s as follows: Taking the derivative of it and substituting it into Equation (46), we can get: where $k$ s $- 2\mu>0$, and $\mu$ is a positive number. The solution of Equation (57) is:

[0059] From this, it can be obtained that the synchronization error signal is bounded and convergent, and the synchronization error is semi-globally ultimately uniformly bounded. Therefore, under the action of the synchronization control law $u$ si the multi-motor system can achieve synchronous operation.

[0060] Through fast disturbance suppression, strong error constraint, intelligent synchronization strategy and low-complexity design, the present invention realizes high-precision, high-reliability and high-energy efficiency control of the multi-motor drive system, and is particularly suitable for fields with strict requirements for synchronization and anti-interference such as industrial automation and intelligent equipment.

[0061] It should be noted that the parts not described in detail in the above embodiments are all prior arts. The above are only the preferred embodiments of the present invention, and do not impose any form of limitation on the present invention. Although the present invention has been disclosed above with the preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some changes or modifications to equivalent embodiments by using the disclosed technical content within the scope of the technical solution of the present invention. However, any simple modification, equivalent change and modification made to the above embodiments based on the technical essence of the present invention should be covered by the protection scope of the present invention.

Claims

1. A synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling, characterized in that, It includes the following steps: Step 1: Establish a dynamic model of a multi-motor drive system with backlash non-linearity and external disturbances. Based on a continuously differentiable backlash model, describe the transmission torque between the motors and the load, and then construct the state-space equation of the multi-motor drive system, and decompose it into a load subsystem and a multi-motor subsystem; Step 2: Design finite-time disturbance observers for the load end and the multi-motor subsystem respectively to estimate the total disturbance at the load end and the total disturbance of each motor subsystem; Step 3: Design a finite-time prescribed performance tracking controller based on the backstepping method and a first-order command filter. The tracking controller uses a finite-time prescribed performance function to constrain the tracking error; Based on the adjacent cross-coupling synchronization control structure, introduce a hyperbolic function to adjust the synchronization control input, and design an adaptive adjacent cross-coupling synchronization controller; Output the estimated value of the total disturbance of each motor subsystem to its respective synchronization controller for feedforward compensation, and output the estimated value of the total disturbance at the load end to the tracking controller for feedforward compensation, so that the actual control input of each motor is the sum of the mean of the tracking control input and the synchronization control input of this motor.

2. The synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling according to claim 1, wherein The dynamic model of the multi-motor drive system in Step 1 is: Where, J mi and J l are the moment of inertia of the motor and the moment of inertia of the load respectively, b mi and b l are the viscous friction coefficient of the motor and the viscous friction coefficient of the load respectively, θ mi and θ l are the angular displacement of the motor and the angular displacement of the load respectively, T dmi and T dl are the disturbing torque on the motor side and the disturbing torque on the load side respectively, n is the total number of motors, i is the serial number of the current motor, u i is the input torque of the i-th motor, T i is the transmission torque between the i-th motor and the load.

3. The synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling according to claim 2, wherein, The backlash model in Step 1 adopts the following approximate dead zone function: where \(k > 0\) is the torque torsion coefficient, \(\alpha\) is the backlash width, \(z\) i =\(\theta\) mi -\(\theta\) l represents the angular displacement difference between the motor and the load, and \(r = 2 / \alpha\) is a positive constant.

4. The synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling according to claim 3, characterized in that The state-space equation of the multi-motor drive system in Step 1 is: Wherein, x1, x2, x 3i , x 4i respectively represent the load position, load speed, the position of the i-th motor, and the speed of the i-th motor. y = x1 indicates that the output value y of the multi-motor drive system is the load position x1; The load subsystem is: The multi-motor subsystem is: wherein, X1 = [x1 x2] T , X 2i = [x 3i x 4i T , C1 = C2 = [10], y1 and y 2i are the load output and each motor output respectively;​ The expressions of the total disturbance at the load end and the total disturbance of each motor subsystem are: where, F1 is the total disturbance at the load end, and f 2i is the total disturbance of the i-th motor subsystem.

5. The synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling according to claim 4, wherein The finite-time disturbance observer designed for the load end in Step 2 is: where, m1>0 and m2>0 represent the observer gains, c1 = p1 / q1, p1 and q1 are positive odd integers, and satisfy p1 < q1 < 2p1, and represent the estimated values of x2 and F1, respectively; The finite-time disturbance observer design of the multi-motor subsystem is: where, m3>0 and m4>0 represent observer gains, c2 = p2 / q2, p2 and q2 are positive odd integers, and satisfy p2 < q2 < 2p2, and respectively represent the estimated values of x 4i and f 2i .

6. The synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling according to claim 5, characterized in that The finite-time prescribed performance function in Step 3 is designed as: where a1, a2, b1, b2, b3, ρ ∞ and t p are preset normal constants, and ρ ∞ is the steady-state value, representing the maximum convergence range of the error, and t p is the convergence time to converge to the steady-state value.

7. The synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling according to claim 5 or 6, characterized in that The first-order command filter in Step 3 is designed as: where ω > 0 is the filter parameter, represents the state of the command filter, represents the input of the filter, represents the output of the filter.

8. The synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling according to claim 7, characterized in that, The control rate of the tracking controller in the third step is as follows: where k4 is the parameter of the tracking controller, e3 and e4 are the tracking errors, and η3 is the virtual control rate; The control rate of the synchronization controller is: u si =-J mi k s ψ i , where Adaptive gain e si is the synchronization error between motors.

9. The synchronous tracking control method for a multi-motor drive system based on adjacent cross-coupling according to claim 1, characterized in that The design of the finite-time prescribed performance tracking controller in Step 3 includes: First, define the load tracking error, and convert the tracking error into an unconstrained error through an error transformation function, then keep the tracking error within the boundary specified by the finite-time prescribed performance function, and then use the backstepping method to design the virtual control law step by step, and combine the first-order command filter to eliminate the differential explosion problem in the backstepping process, and then generate the tracking control input.

Images