A sliding mode control method for ball screw feeding system based on disturbance observer

Abstract

A sliding mode control strategy based on disturbance observer. Firstly, a disturbance observer is designed to observe the disturbance and system state of ball screw feeding system in the case that the disturbance of the system can be described by an external system. Then, according to the reference trajectory of the given numerical control machine tool worktable and the observation value of the non-matching disturbance obtained by the disturbance observer, the rotation trajectory of the motor is obtained by analyzing the system state equation. Finally, a sliding mode control method is designed to achieve the tracking control of the numerical control machine tool based on the observation value of the disturbance and system state and the rotation trajectory of the motor, which effectively overcomes the influence of non-matching disturbance on the tracking performance of the system, thereby improving the tracking accuracy of the ball screw feeding system.

Description

Technical Field

[0001] This invention relates to the field of CNC machine tool control, specifically to a sliding mode control method for a ball screw feed system based on a disturbance observer. Background Technology

[0002] The ball screw feed system, as a key component of CNC machine tools, boasts advantages such as simple structure, high dynamic and static rigidity, and high transmission efficiency. Its working principle involves converting the rotary motion of a servo motor into the linear motion of the worktable, which then drives the worktable to cut the workpiece according to a given reference trajectory. Therefore, the controller's primary task is to enable the worktable to achieve high-precision trajectory tracking. Its tracking accuracy and speed performance directly determine the working accuracy and efficiency of the CNC machine tool. However, during the operation of a CNC machine tool, it is susceptible to disturbances, including both matched and mismatched disturbances, making high-precision trajectory tracking control difficult.

[0003] For the trajectory tracking problem of ball screw feed systems, traditional control methods, such as PID and P-PI cascaded control, can achieve good tracking performance. However, with the increasing performance requirements of high-end CNC machine tools in the manufacturing industry, machine tool feed systems with ball screws as the main drive component are developing towards high speed and high precision, and traditional control methods can no longer fully meet current control requirements. Therefore, research on high-precision trajectory tracking control of ball screw feed systems is of great significance for practical engineering applications. Sliding mode control, as a type of variable structure control, mainly consists of approaching motion and sliding mode motion. Correspondingly, the design of the controller is mainly divided into the design of the sliding surface and the design of the control law. For the sliding surface, we can pre-set it. When the system enters the sliding mode motion, the system performance is determined by the characteristics of the sliding surface, which makes the system highly robust to parameter disturbances and external disturbances. Due to its strong robustness and simple structure, sliding mode control is particularly suitable for trajectory tracking control of ball screw feed systems. However, the robustness of sliding mode control is usually only for matched disturbances. When the disturbance does not meet the matching conditions, the robustness of the system will no longer exist, and thus it will be unable to accurately track the given trajectory.

[0004] To address this, the present invention, based on a disturbance observer, designs a motor rotation trajectory generation system by observing the system state and disturbances. Then, through the ingenious design of the sliding surface parameters, it realizes the exponential tracking control of the worktable and the motor, thereby improving the tracking performance of the ball screw feed system in the presence of mismatched disturbances.

[0005] The existing technologies are as follows:

[0006] Comparison with the technology of patent CN105305913A "A novel anti-disturbance following controller for ball screw feed systems";

[0007] I. Patent CN105305913A achieves motor position and load position control based on active disturbance rejection technology and proportional-integral (PI) control. It utilizes an extended state observer to observe the total disturbance received by the system and designs an active disturbance rejection controller based on the total disturbance observation, position command, and state observation. This patent can only observe matched total disturbances, and the designed controller can only solve the impact of matched disturbances on the system's tracking performance. However, in the actual operation of the ball screw feed system, it is also affected by unmatched disturbances. Our patent considers both matched and unmatched disturbances received by the system. By introducing the observation value of the unmatched disturbance into the sliding surface, a sliding mode controller is cleverly designed to overcome the impact of unmatched disturbances on the system's tracking performance.

[0008] II. Patent CN105305913A designed a motor position command generator based on a system model. This design is acceptable because it does not consider the mismatched disturbances existing between the motor and load channels. Our patent considers the mismatched disturbances within these channels and designs a trajectory generation system to generate the trajectory that the motor needs to track. The effectiveness of this trajectory generation system is demonstrated through analysis of the sliding mode and simulation experiments.

[0009] Technical comparison with patent CN103837339A "A Dynamic Servo Force Drive System";

[0010] I. Patent CN103837339A designs an adaptive sliding mode control algorithm with a disturbance observer. This algorithm, based on a disturbance observer, is designed to address system uncertainties and disturbances that satisfy matching conditions. It effectively counteracts the impact of matched disturbances on system performance, and introducing the observed values ​​of the matched disturbances into the controller also reduces switching gain. However, this patent does not consider system uncertainties and disturbances that do not satisfy matching conditions. Our patent, on the other hand, considers unmatched disturbances and designs a sliding mode controller, effectively solving the impact of unmatched disturbances on system tracking performance and improving the system's tracking accuracy. Summary of the Invention

[0011] To address the impact of mismatched disturbances on the tracking performance of ball screw feed systems in CNC machine tools, this invention proposes a sliding mode control method for ball screw feed systems based on a disturbance observer, which aims to improve the robustness of the ball screw feed system to mismatched disturbances and its tracking accuracy.

[0012] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0013] A sliding mode control method for a ball screw feed system based on a disturbance observer includes the following steps:

[0014] Step 1: When the system disturbance is described by an external system, design a disturbance observer to observe the disturbance and unmeasurable states experienced by the system. The external system is described as follows:

[0015]

[0016] Step 2: Based on the observed values ​​of the mismatched disturbance obtained by the disturbance observer and the tracking trajectory that the CNC machine tool table needs to track, design the rotational displacement trajectory that the motor needs to track.

[0017] Step 3: To address the unmatched disturbances experienced by the system, a sliding mode controller is designed by introducing the observed values ​​of the unmatched disturbances into the sliding surface, and the effectiveness of the proposed sliding mode controller is verified based on the simulation model.

[0018] As a further improvement of the present invention, the rotational displacement trajectory generation system for the motor to be tracked described in step 2 is designed to enable the worktable of the CNC machine tool ball screw feed system to track a given reference trajectory x. 1d The tracking, based on the analysis of the system's state-space equations, is based on x. 1d And the observations of the unmatched perturbation obtained in step 1. The rotational displacement trajectory x that the motor needs to track was designed. 2d The trajectory generation system is as follows:

[0019]

[0020] Where, x 1d It is the reference trajectory that the worktable needs to follow, obtained by the CNC system through analyzing the CAD model of the part to be machined. x represents the observed values ​​of the mismatched disturbance in the ball screw feed system obtained from the disturbance observer. 2d This refers to the rotational trajectory that the motor needs to track when the designed system overcomes the effects of mismatched disturbances to complete the tracking target of the worktable.

[0021] As a further improvement to the present invention, the design of the sliding mode controller described in step 3 is as follows;

[0022] First, the rotation trajectory x obtained from step 2 is... 2d By introducing a sliding surface, and then using the design of the sliding surface, the table and motor are jointly controlled. That is, the table and motor of the ball screw feed system are controlled to exponentially convergently track the reference trajectory x. 1d and the rotational displacement trajectory x obtained from step 2 2d Thus, even in the presence of mismatched disturbances, the ball screw feed system achieves high-precision target tracking. The designed sliding surface is described as follows:

[0023]

[0024] in,

[0025]

[0026] The observed values ​​of x are represented by parameters c1 and c3, which are chosen such that the characteristic equation... The eigenvalues ​​are all less than zero in real parts.

[0027] Compared with the prior art, the present invention, employing the above technical solution, has the following technical effects:

[0028] (1) This invention takes the tracking control of a ball screw feed system as its background and proposes a trajectory generation system under the condition of non-matching disturbance, that is, in order to realize the tracking control of the worktable to a given reference trajectory x 1d During tracking, the rotational displacement trajectory of the motor was designed based on the system equations. This trajectory generation is not limited to disturbances, but also applies to uncertainties in the system model. We can use an observer to simultaneously observe the uncertainties and disturbances in the channel and generate the rotational displacement trajectory of the motor under these conditions. Of course, we need to consider the solvability and boundedness of the trajectory. Under normal circumstances, the disturbances and uncertainties we are studying are satisfied.

[0029] (2) The present invention designs a sliding mode controller based on a disturbance observer. By designing the sliding mode surface, the ball screw feed system table and motor can achieve exponential tracking of the reference trajectory, effectively solving the coupling between the two subsystems, enabling the table and motor to be controlled independently, and ensuring the tracking speed and tracking accuracy. Attached Figure Description

[0030] Figure 1 This is a flowchart of the invention;

[0031] Figure 2 This is a control model diagram of the present invention;

[0032] Figure 3 This is a state and disturbance observation diagram of the present invention;

[0033] Figure 4 This is a diagram showing the tracking effect of the workbench in this invention;

[0034] Figure 5 This is a diagram illustrating the motor tracking effect of the present invention. Detailed Implementation

[0035] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0036] The sliding mode control method based on a disturbance observer according to the present invention will be further described below with reference to the accompanying drawings and examples. The present invention provides a sliding mode control scheme based on a disturbance observer for a ball screw feed system affected by unmatched disturbances. The overall flowchart is shown below. Figure 1 As shown, a disturbance observer is first designed to observe the disturbances and uncertainties experienced by the system. Then, by transmitting the observed system state and disturbance values, along with the given reference trajectory that the table needs to track, to the trajectory generation system, the rotational displacement trajectory that the motor needs to track can be obtained. Finally, the design of a sliding mode tracking controller effectively overcomes the impact of mismatched disturbances on the system's tracking performance, achieving tracking control of the ball screw feed system. The main steps include:

[0037] Step 1: When the system disturbance can be described by the external system, design a disturbance observer to observe the disturbance and unmeasurable state of the system.

[0038] This invention is introduced by selecting a flexible body model of a ball screw feed system considering the first-order shaft-torsional coupled vibration mode as the research object. The model is described as follows:

[0039]

[0040] in,

[0041]

[0042] k represents the overall axial stiffness of the system, m1 and m2 represent the equivalent mass of the rolling element and the equivalent mass of the moving element, respectively, b1 represents the viscous damping of the motor and bearings, b2 represents the viscous damping of the linear guide, and x1 and x2 represent the linear displacement of the ball screw table and the rotational displacement of the motor, respectively. and Let d1 and d2 represent the linear speed of the ball screw table and the rotational speed of the motor, respectively. u represents the motor torque control command, and d1 and d2 represent the disturbances experienced by the system. From the disturbance matrix D, it can be intuitively seen that disturbance d1 is a matched disturbance, and d2 is a non-matched disturbance. For the matched disturbance d1, since it and the control input u are in the same channel, we can add a compensation term to the control input u to cancel it out. For the non-matched disturbance d2, we choose to observe it using a disturbance observer and introduce a sliding surface for cancellation.

[0043] Without loss of generality, we discuss here the perturbations that can be described by the following additional system:

[0044]

[0045] In practice, besides disturbances, the system state is sometimes inconvenient to measure or very costly to measure. Therefore, we need to design an observer to simultaneously observe the state and disturbances, and use the estimates to design the sliding mode control law. This can significantly reduce the switching gain of sliding mode control and improve tracking accuracy. This invention designs a disturbance observer, as follows:

[0046]

[0047] in, Let L represent the observed values ​​of x, y, ξ, and d respectively. x and L d The observation gain matrix that needs to be designed can be described as:

[0048] By subtracting the system model (1) and the observation model (3), we can obtain the observation error model as follows:

[0049]

[0050] in, Our goal is to design a suitable observation gain matrix L. x and L d This ensures the exponential stability of the observation error system. The observation error matrix is ​​defined as follows:

[0051]

[0052] From the observation error system (4), we can obtain:

[0053]

[0054] in,

[0055] Choose the Lyapunov function V1 as:

[0056]

[0057] Where P is a positive definite matrix.

[0058] Taking the derivative of function V1, we get:

[0059]

[0060] in,

[0061] To ensure the stability of the observation error system, the observation gain matrix L needs to satisfy the following Lyapunov inequality:

[0062]

[0063] At this point we can obtain:

[0064]

[0065] From the above equation, it can be found that when the observation gain matrix L satisfies Lyapunov's inequality (8), the observation error system is exponentially stable. At this time, the observed value of the observation system (3) converges exponentially to the true value of the original system (1).

[0066] Step 2: Based on the observed values ​​of the mismatched disturbance obtained by the disturbance observer and the tracking trajectory that the CNC machine tool table needs to track, derive and generate the rotational displacement trajectory that the motor needs to track.

[0067] Here we generate the rotational displacement trajectory that the motor needs to track when the workbench tracks a given reference trajectory. According to the system model (1), we can obtain:

[0068]

[0069] Right now

[0070] At this point, when the workbench tracks the given reference trajectory, i.e., x1 = x 1d At this time, the rotational displacement trajectory of the motor is taken as x. 2d Then x 2d The following equation should be satisfied:

[0071]

[0072] Since the disturbance d2 cannot be directly measured, we need to introduce the observed value of the disturbance d2 obtained in step 1. Therefore, the trajectory x is generated by the following equation. 2d :

[0073]

[0074] in, This represents the observation error of the observation system (3) with respect to the disturbance d2, as can be seen from the introduction in step 1. Since the exponential convergence is zero, we can use it here. To replace d2 for generating the rotational displacement trajectory x of the motor 2d Based on the trajectory generation system (12), we can determine the trajectory based on the input reference trajectory x. 1d Design a motor to track the rotational displacement trajectory x. 2d This information is then input to the sliding mode control system in step 3 to enable the CNC machine tool's ball screw feed system table to respond to a given reference trajectory x. 1d Tracking.

[0075] Step 3: To address the unmatched disturbances experienced by the system, a sliding mode controller is designed by introducing the observed values ​​of the unmatched disturbances into the sliding mode surface, and the tracking control effect is verified based on the simulation model.

[0076] The sliding surface is selected as follows:

[0077]

[0078] in, The parameters c1 and c3 are chosen only if the characteristic equation The eigenvalues ​​are all less than zero in real parts.

[0079] Control law design process:

[0080] Differentiating the sliding surface (13) yields:

[0081]

[0082] in,

[0083] To achieve exponential convergence of the sliding surface, we can choose the following exponential convergence law:

[0084] u=-m1[c1f1+c2f2+c3f3+f4+Ks+εsgn(s)] (15)

[0085] Here, K and ε are positive numbers of appropriate size.

[0086] It is easy to select a Lyapunov function The stability of the control rate (15) will be demonstrated, but will not be elaborated here.

[0087] Stability analysis of sliding mode motion:

[0088] When the system enters the sliding surface, i.e., s = 0, by combining s = 0 and the observation system (3), we can obtain:

[0089]

[0090] Will And by substituting the trajectory generation system (12) into (16), we can obtain:

[0091]

[0092] In step 1, we have already proven the exponential stability of the observation system (3), and based on the selection of c1 and c3, at this point... It also converges exponentially to zero, therefore the workbench can quickly track a given reference trajectory x. 1dAt this point, based on the sliding surface s = 0, we can obtain:

[0093]

[0094] In formula (18), and The exponent tends to zero, therefore It also has an exponential tendency toward zero, meaning the motor can quickly track the rotational displacement trajectory x. 2d Thus, the stability of the sliding surface has been proven.

[0095] Simulation experiments verify:

[0096] Based on the proposed control method, a simulation model of sliding mode tracking control for a CNC machine tool ball screw feed system based on a disturbance observer was built on the MATLAB / Simulink platform. The model framework is as follows: Figure 2 As shown, it mainly consists of a trajectory generation system, a disturbance system, a disturbance observer, a ball screw feed system, and a sliding mode controller. The simulation parameters are as follows:

[0097] m1 = 1.3016 (V·s) 2 ·m -1 m2 = 0.1484 (V·s) 2 ·m -1 k = 4.1814 × 10 4 (V·m -1 c = 5.3550 (V·s·m) -1 b1 = 8.0954 × 10 -4 (V·s·m -1 b2 = 1.6103 (V·s·m) -1 ), x 1d =5sint(m), α=5, K=80, ε=5, c1=50, c3=50. The system state and the observed effects of the disturbance are as follows: Figure 3 As shown, it can be observed that the observed values ​​can quickly approximate the true values ​​of the system, and the ball screw feed system table for a given reference trajectory x 1d The tracking effect is as follows Figure 4 As shown, the motor is related to the rotational displacement trajectory x output by the trajectory generation system. 2d The tracking effect is as follows Figure 5 As shown, it can be observed that the proposed sliding mode tracking control based on the disturbance observer has a good control effect on the ball screw feed system affected by unmatched disturbances, and can achieve control over the trajectory x. 1d and x 2d Index tracking.

[0098] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any modifications or equivalent changes made based on the technical essence of the present invention shall still fall within the scope of protection claimed by the present invention.

Claims

1. A sliding mode control method for a ball screw feed system based on a disturbance observer, comprising the following steps, characterized in that: Step 1: When the system disturbance is described by an external system, design a disturbance observer to observe the disturbance and unmeasurable states experienced by the system. The external system is described as follows: ； Step 2: Based on the observed values ​​of the mismatched disturbance obtained by the disturbance observer and the tracking trajectory that the CNC machine tool table needs to track, design the rotational displacement trajectory that the motor needs to track. Step 3: To address the unmatched disturbances experienced by the system, a sliding mode controller is designed by introducing the observed values ​​of the unmatched disturbances into the sliding surface, and the effectiveness of the proposed sliding mode controller is verified based on the simulation model. The design of the sliding mode controller described in step 3 is as follows; First, the rotation trajectory obtained in step 2 is... By introducing a sliding surface, and then through the design of the sliding surface, the table and motor are jointly controlled. That is, the table and motor of the ball screw feed system are controlled to exponentially convergently track the reference trajectory. and the rotational displacement trajectory obtained from step 2 Thus, even in the presence of mismatched disturbances, the ball screw feed system achieves high-precision target tracking. The designed sliding surface is described as follows: ； in, ； express Observed values, parameters The selection only needs to make the characteristic equation The eigenvalues ​​are all less than zero in real parts.

2. The tracking control scheme for a ball screw feed system subjected to unmatched disturbances based on a disturbance observer according to claim 1, characterized in that: The rotational displacement trajectory generation system described in step 2, which requires the motor to track, is used to enable the worktable of the CNC machine tool's ball screw feed system to track a given reference trajectory. The tracking, based on the analysis of the system's state-space equations, is based on... And the observations of the unmatched perturbation obtained in step 1. The rotational displacement trajectory that the motor needs to track was designed. The trajectory generation system is as follows: ； in, It is the reference trajectory that the worktable needs to follow, obtained by the CNC system through analyzing the CAD model of the part to be machined. These are the observations of the mismatched disturbance in the ball screw feed system obtained from the disturbance observer. This refers to the rotational trajectory that the motor needs to track when the designed system overcomes the effects of mismatched disturbances to complete the tracking target of the worktable.

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