A Fixed-Time Control Method for Multi-Motor Servo Systems Based on Multi-Faceted Sliding Mode

Abstract

The present invention discloses a fixed-time control method for a multi-motor servo system based on multi-surface sliding mode, including the modeling of a multi-motor servo system affected by friction and unknown time-varying time delay, the design of a fixed-time tracking controller, the design of a fixed-time synchronization controller, and the design of a fixed-time convergence controller. The present invention discloses a synchronization and tracking control method for a multi-motor servo system based on multi-surface sliding mode. Its technology includes the modeling of a multi-motor servo system, the design of a fixed-time tracking controller, the design of a fixed-time synchronization controller, and the design of a fixed-time convergence controller. In view of the load tracking control problem in a multi-motor servo system, the present invention designs a fixed-time tracking controller based on multi-surface sliding mode; in order to achieve speed synchronization control among multiple motors, a fixed-time synchronization controller based on multi-surface sliding mode is designed. The present invention can effectively solve the synchronization and tracking control problems of a multi-motor servo system affected by friction and unknown time-varying time delay.

Description

Technical Field

[0001] The present invention relates to the modeling of a multi-motor servo system with friction and unknown time-varying delays, the design of a fixed-time tracking controller, the design of a fixed-time synchronization controller, and the design of a fixed-time convergence controller. Background Art

[0002] The problems of synchronization and load tracking control of multi-motors have always been widespread problems faced by motor drive systems. In recent years, many advanced control strategies have been used for the problems of multi-motor synchronization and tracking control. For example, ["Synchronization and tracking control for multi-motor driving servo systems with backlash and friction" (Wei Zhao, Xuemei Ren; Xuehui Gao, Int. J. Robust Nonlinear Control 2016; 26: 2745–2766.)] considered the existence of backlash and friction in a multi-motor servo system and proposed a switching controller combining backlash and friction compensation. However, due to factors such as multi-motor linkage and transmission links, there may be a delay effect in the multi-motor servo system. The existence of the time-delay effect brings new challenges to the control of the motor servo system. So far, the synchronization and tracking control problems of multi-motor drive systems with friction and unknown time-varying delays have not been fully studied. Therefore, it will be more challenging to cope with the influence of unknown time-varying delays while compensating for friction, and it also increases the difficulty of the design of control algorithms. Summary of the Invention

[0003] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a fixed-time control method for a multi-motor servo system based on multi-surface sliding mode to solve the modeling of a multi-motor servo system with friction and unknown time-varying delays, the design of a fixed-time tracking controller, the design of a fixed-time synchronization controller, and the design of a fixed-time convergence controller.

[0004] The algorithm adopted by the present invention to solve the above problems is that for the load tracking control problem in a multi-motor servo system, a fixed-time tracking controller based on multi-surface sliding mode is designed; in order to achieve speed synchronization control between multi-motors, a fixed-time synchronization controller based on multi-surface sliding mode is designed. The present invention can effectively solve the synchronization and tracking control problems of a multi-motor servo system under the influence of friction and unknown time-varying delays.

[0005] For the modeling of the multi-motor servo system under the influence of the friction and unknown time-varying delays, for the i-th motor, considering external disturbances and unknown time-varying delays, the vehicle dynamic model becomes:

[0006]

[0007]

[0008] For the design of the fixed-time tracking controller, for the \(i\)-th motor, a multi-surface sliding mode method based on RBF network friction compensation is adopted to design a tracking controller in the following form:

[0009]

[0010] where the parameter adaptation law is:

[0011]

[0012] where \(s_1\) and \(s_2\) are two designed sliding surfaces; \(x\) 4i represents the speed of the \(i\)-th motor, and \(y\) d is the specified desired trajectory; is the estimated value of the friction nonlinear function \(f\) m , \(h = [h\) j T is the output of the Gaussian basis function; \(\eta_0\), \(\omega\), \(\gamma\), and \(\kappa\) are all positive tuning parameters; \(sgn(\cdot)\) is the sign function, is a known positive number;

[0013] \(a_1 = 1 / J\) m , \(a\) 2i \(= b\) i / J, \(a_3 = 1 / J\); \(\mu := [\mu_1, \mu_2]\), \(\beta = diag\{=\beta_1, \beta_2\}\), \(\mu\) i , \(\beta\) i (\(i = 1, 2, 3, 4\)) are all positive tuning parameters;

[0014] For the design of the fixed-time synchronization controller, for the \(i\)-th motor, a multi-surface sliding mode method is adopted to design a synchronization controller as follows:

[0015]

[0016] where \(s\) s1i , \(s\) s2i are two designed sliding surfaces; \(\eta\) m , \(\mu\) s1i , \(\mu\) s2i , \(\mu\) s3i , \(\mu\) s4i , are all positive tuning parameters; \(m = i, i + 1\) represents two driving motors, \(i = 1, 3, \ldots, 2n - 1\).

[0017] ​For the design of the fixed-time convergence controller, a controller is designed for the i-th motor by comprehensively considering the tracking and synchronization problems as follows: u i = u ti + ιu si ,

[0018] where ι is the synchronization parameter and satisfies:

[0019]

[0020] where χ is a positive parameter, and s s is the position error between two motors.

[0021] The object of the present invention is achieved as follows.

[0022] The present invention relates to the modeling of a multi-motor servo system under the influence of friction and unknown time-varying delays, the design of a fixed-time tracking controller, the design of a fixed-time synchronization controller, and the design of a fixed-time convergence controller. The present invention discloses a synchronization and tracking control method for a multi-motor servo system based on multi-surface sliding mode. Its technology includes the modeling of a multi-motor servo system, the design of a fixed-time tracking controller, the design of a fixed-time synchronization controller, and the design of a fixed-time convergence controller. The present invention designs a fixed-time tracking controller based on multi-surface sliding mode for the load tracking control problem in a multi-motor servo system; in order to achieve the speed synchronization control between multiple motors, a fixed-time synchronization controller based on multi-surface sliding mode is designed. The present invention can effectively solve the synchronization and tracking control problems of a multi-motor servo system under the influence of friction and unknown time-varying delays. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 is a schematic structural diagram of the control system of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0024] The following describes the specific embodiments of the present invention with reference to the drawings, so that those skilled in the art can better understand the present invention. It should be particularly noted that in the following description, when the detailed description of known functions and designs may dilute the main content of the present invention, these descriptions will be omitted here.

[0025] The following takes the connected vehicle system as an example and describes the technical solution of the present invention in detail with reference to the drawings.

[0026] Figure 1 is a schematic structural diagram of the control system of the present invention

[0027] The following takes the multi-motor servo system as an example and describes the technical solution of the present invention in detail with reference to the drawings.

[0028] Such as Figure 1As shown in the figure, the present invention relates to the modeling of a multi-motor servo system with friction and unknown time-varying time delay, the design of a fixed-time tracking controller, the design of a fixed-time synchronization controller, and the design of a fixed-time convergence controller.

[0029] System modeling: Consider a group of multi-motor servo systems consisting of 1 load and 2n motors. The dynamic equation of the i-th motor is as follows:

[0030]

[0031]

[0032] And

[0033]

[0034] Among them, J i , θ i , b i and u i respectively represent the moment of inertia, angular position, speed, viscous friction coefficient, and input torque of the i-th th (i = 0, 1, 2,..., 2n) motor, and it is assumed that J1 = J2 =... = J 2n = J; J m , θ m and respectively represent the moment of inertia, angular position, and speed of the load; f m represents the friction torque acting on the load; T i represents the transmission torque between the i-th th (i = 0, 1, 2,..., 2n) motor and the load, k i and c i are the torsional coefficient and damping coefficient respectively, and i represents the i-th th (i = 0, 1, 2,..., 2n) motor. Here, a continuous, smooth, and differentiable function is used to replace the dead zone function to represent T i ; i represents the i-th th (i = 0, 1, 2,..., 2n) motor.

[0035] Considering the existence of external disturbances and unknown time-varying time delays, the dynamic model of the i-th th motor becomes:

[0036]

[0037] Among them, d i (t) and respectively represent the external disturbance and unknown time-varying time delay term existing in the i-th th motor; Denote the time-delay state variable, where τ i (t) is an unknown state time-varying delay; h i (·) is an unknown continuous function, ζ i is an unknown positive constant, while is a known positive continuous function; d i (t) is a bounded disturbance, satisfying |d i (t)| ≤ D, D > 0; i represents the i th (i = 0, 1, 2, …, 2n)th motor.

[0038] For the i (i = 0, 1, 2, …, 2n)th motor, select the position and speed of the load and the motor as the system states, and define the state variables:

[0039]

[0040] Then the dynamic equation of the i-th motor can be written as the following state-space model:

[0041]

[0042] where, a1 = 1 / J m , a 2i = b i / J, a3 = 1 / J, and i = 1, 2, …, 2n.

[0043] Fixed-time tracking controller design: Design the first sliding surface

[0044] s1 = y - y d (6)

[0045] Design the corresponding reaching law

[0046]

[0047] Design the second sliding surface

[0048]

[0049] where, μ := [μ1, μ2], = βdiag{β1, β2},

[0050] u i , β i (i = 1, 2) are both positive tuning parameters, and sgn(·) is the sign function.

[0051] Select an appropriate Lyapunov-Krasovskii functional to compensate for the unknown time-varying delay uncertainties, and the L-K functional is designed as:

[0052]

[0053] wherein, is a known normal constant.

[0054] Use the RBF neural network to compensate for the friction nonlinearity, The parameter adaptation law is:

[0055]

[0056] wherein, γ and κ are positive tuning parameters; h = [h j T is the output of the Gaussian basis function, is the estimated weight of the neural network

[0057] Furthermore, the tracking controller is designed as:

[0058]

[0059] wherein, y d is the specified desired trajectory, is a known positive number; μ i , β i (i = 3, 4) are all positive tuning parameters;

[0060] Fixed-time synchronization controller design: Design the first sliding surface

[0061] s s1 = [s s11 , s s12 , …, s s1n T = [θ1 - θ2, θ3 - θ4, …, θ 2n-1 - θ 2n T (12)

[0062] Design the corresponding reaching law

[0063]

[0064] wherein, are all positive definite matrices, are all positive tuning parameters.

[0065] Design the second sliding surface

[0066]

[0067] ​​​Select an appropriate Lyapunov-Krasovskii functional to compensate for the unknown time-varying delay uncertainties. The L-K functional is designed as follows:

[0068]

[0069] Furthermore, the synchronization controller is designed as:

[0070]

[0071] where μ m , μ s1i , μ s2i , μ s3i , μ s4i , are all positive tuning parameters; m = i, i + 1 represents two driving motors, and i = 1, 3, …, 2n - 1.

[0072] Fixed-time convergence controller design: The overall controller design considering both tracking and synchronization problems is as follows:

[0073] u i = u ti + ιu si (17)

[0074] where ι is the synchronization parameter and satisfies:

[0075]

[0076] where χ is a positive parameter, and s s is the position error between the two motors, such as the position error between O1 and O2.

[0077] Although the above describes the illustrative specific embodiments of the present invention for the convenience of those skilled in the art to understand the present invention, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions made using the concept of the present invention are within the scope of protection.

Claims

1. A fixed-time control method for a multi-motor servo system based on multi-surface sliding mode, characterized in that, It includes the modeling of a multi-motor servo system under the influence of friction and unknown time-varying delays, the design of a fixed-time tracking controller, the design of a fixed-time synchronization controller, and the design of a fixed-time convergence controller; The modeling of the multi-motor servo system under the influence of friction and unknown time-varying delays is as follows: Consider a group of multi-motor servo systems consisting of 1 load and 2n motors. The dynamic equation of the i-th motor is as follows: And Among them, J i , θ i , b i and u i respectively represent the moment of inertia, angular position, speed, viscous friction coefficient and input torque of the i th (i = 0, 1, 2, …, 2n)th motor, and it is assumed that J1 = J2 = … = J 2n = J; J m , θ m and respectively represent the moment of inertia, angular position and speed of the load; f m represents the frictional torque acting on the load; T i represents the transmission torque between the i th (i = 0, 1, 2, …, 2n)th motor and the load, k i and c i are the torsional coefficient and damping coefficient respectively, and i represents the i th (i = 0, 1, 2, …, 2n)th motor. Here, a continuous, smooth and differentiable function is used to replace the dead zone function to represent T i ; i represents the i th (i = 0, 1, 2, …, 2n)th motor; Considering the existence of external disturbances and unknown time-varying delays, the dynamic model of the i-th th motor becomes: where d i (t) and respectively represent the external disturbance and the unknown time-varying delay term existing in the i-th th motor; represents the delay state variable, where τ i (t) is the unknown state time-varying delay; h i (·) is an unknown continuous function, ζ i is an unknown positive constant, and is a known positive continuous function; d i (t) is a bounded disturbance, satisfying |d i (t)| ≤ D, D > 0; i represents the i-th th (i = 0, 1, 2, …, 2n) motor; The design of the fixed-time tracking controller is as follows: For the i-th motor, a multi-surface sliding mode method based on radial basis network friction compensation is adopted to design the following tracking controller: Among them, the parameter adaptation law is: Among them, s1 and s2 are two designed sliding surfaces; x 4i represents the speed of the i-th motor, and y d is the specified desired trajectory; is the estimated value of the friction nonlinear function f m , and h = [h j T is the output of the Gaussian basis function;​ η0, ω, γ, and κ are all positive tuning parameters; sgn(·) is the sign function is a known positive number, a1 = 1 / J m , a 2i = b i / J, a3 = 1 / J; μ := [μ1, μ2], β = diag{β1, β2}, μ j , β j (j = 1, 2, 3, 4) are all positive tuning parameters; 2. The fixed-time control method for a multi-motor servo system based on multi-surface sliding mode according to claim 1, wherein The design of the fixed-time synchronization controller is as follows: For the i-th motor, a multi-surface sliding mode method is adopted to design the synchronization controller in the following form: Among them, s s1i , s s2i are two designed sliding mode surfaces, η m , μ s1i , μ s2i , μ s3i , μ s4i , are all positive tuning parameters, m = i, i + 1 represents two driving motors, and i = 1, 3, …, 2n - 1.

3. The fixed-time control method for a multi-motor servo system based on multi-surface sliding mode according to claim 1, wherein The design of the fixed-time convergence controller is as follows: For the i-th motor, a controller is designed by comprehensively considering the tracking and synchronization problems as follows: u i = u ti + ιu si Among them, ι is the synchronization parameter and satisfies: where X is a positive parameter, and s s is the position error between the two motors.

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