A Networked Contour Tracking Controller Based on Finite-Time Disturbance Estimation

Abstract

The present invention discloses a networked contour tracking controller based on finite-time disturbance estimation for multi-axis contour tracking control of multi-axis automation equipment, including: an input signal internal model, a state feedback controller, an equivalent input disturbance estimator, and a terminal sliding mode observer corresponding to each servo axis, as well as a cross-coupling controller between the axes. The input signal internal model is used to process the input signal to improve the single-axis trajectory tracking accuracy; the state feedback controller calculates the state feedback gain by using the pole placement method; the equivalent input disturbance estimator is used to estimate the total disturbance composed of the system network disturbance and the external disturbance; the terminal sliding mode observer is used to achieve fast estimation of the system state; the cross-coupling controller is used to improve the accuracy of multi-axis contour control; finally, the control input of the contour tracking controller with contour error compensation and total disturbance compensation is obtained. The present invention realizes high-precision contour tracking control of the networked motion system.

Description

Technical Field

[0001] This application belongs to the technical field of networked multi-axis motion control, and particularly relates to a networked contour tracking controller based on finite-time disturbance estimation. Background Art

[0002] In various multi-axis automation equipment such as numerical control machine tools, industrial robots, cutting machines, and engraving machines, the most common processing task requirement is to achieve multi-axis contour tracking control, so as to complete the precise processing of products on a two-dimensional plane and in three-dimensional space. The performance of multi-axis contour tracking control will directly affect the processing quality of products. It is not only affected by the single-axis tracking performance, but also related to the synchronization performance of multiple axes. At the same time, external disturbances such as unmodeled dynamics, load inertia changes, and tool wear will affect the accuracy of multi-axis contour tracking control. In severe cases, it will cause poor processing quality or even workpiece scrapping and other adverse consequences. Therefore, how to improve the accuracy of contour tracking control has always been a hot issue of concern in the industrial and academic circles.

[0003] On the other hand, with the rapid development of network communication technology, networked control systems based on real-time industrial Ethernet have received more and more attention. Introducing network technology into the motion control system overcomes the disadvantages of traditional motion control systems such as complex wiring, difficult maintenance, and poor scalability. However, the introduction of the network has also brought some new problems that need to be solved urgently. One of them is the uncertainty-induced time delay caused by the network, which may seriously affect the contour tracking and synchronization control performance of multi-axis control. Therefore, how to solve the combined action of external disturbances and network-induced time delay in multi-axis systems is the key to achieving high-precision multi-axis contour tracking control in a network environment, and it has important practical significance for improving the performance of networked multi-axis motion control systems and realizing the domestic substitution of similar controller products.

[0004] An effective solution is to regard the time delay as a network disturbance and use the disturbance estimation technology to achieve the real-time estimation and compensation of the combined action of network disturbances and external disturbances. In the past few decades, several disturbance estimation technologies have been proposed, such as disturbance observers, unknown input observers, extended state observers, and equivalent input disturbance methods. Among them, the method based on the extended state observer has the least requirement for the system mathematical model, but is only applicable to integral series systems and has limited suppression performance for mismatched disturbances. Although a generalized extended state observer has been proposed to overcome the above disadvantages, it only considers a class of constant disturbances. The idea of equivalent input disturbance is to estimate an equivalent disturbance on the control input channel, whose influence on the system output is the same as that of the entire disturbance, which means it can suppress the influence of both matched and mismatched disturbances. In addition, the method based on equivalent input disturbance does not require the model of the disturbance or the inverse dynamics of the controlled object, and the design of the control system is very simple. Therefore, the equivalent input disturbance method and its improved methods have been successfully applied to various systems to suppress the influence of disturbances and nonlinearities. However, the existing methods based on equivalent input disturbance are mostly linear and the state estimation error converges asymptotically, which cannot further meet the requirements of nonlinear convergence and finite-time control of the state estimation error for high-precision contour tracking control tasks in the network environment. Summary of the Invention

[0005] In order to suppress the interference of network-induced time delay and external disturbances on the multi-axis contour tracking control performance of a networked multi-axis system, and at the same time solve the deficiency of the existing disturbance suppression method based on equivalent input disturbance in the convergence speed of the state estimation error, this application proposes a networked contour tracking controller based on finite-time disturbance estimation. Considering the combined influence of time-varying network-induced time delay and external disturbances on the system, an improved equivalent input disturbance estimator and a terminal sliding mode observer are designed. It not only gives the real-time estimation of the equivalent disturbance on the control input channel of the system, but also ensures the fast and finite-time convergence of the state estimation error. Furthermore, a state feedback control law with disturbance compensation function is designed, and combined with a cross-coupling controller to further improve the control accuracy of the contour tracking error, and finally achieve high-precision contour tracking control of the networked multi-axis motion control system.

[0006] The technical solution adopted by this application to solve its technical problems is as follows:

[0007] A networked contour tracking controller based on finite-time disturbance estimation, which is used for the multi-axis contour tracking control of multi-axis automation equipment, includes: an input signal internal model corresponding to each servo axis, a state feedback controller, an equivalent input disturbance estimator, a terminal sliding mode observer, and a cross-coupling controller between the axes, where:

[0008] The input signal internal model is used to process the input signal to improve the single-axis trajectory tracking accuracy. Its processing process is represented by the following formula:

[0009]

[0010] In the formula, x Ri (t) is the state of the internal model, represents the first derivative of x Ri (t), r i (t) is the input signal, y i (t) is the system output, A Ri and B Ri are the internal model matrices;

[0011] The state feedback controller calculates the state feedback gain using the pole placement method, and then calculates the state feedback control output u fi (t), which is represented by the following formula:

[0012]

[0013] In the formula, represents the observed value of the system state x i (t), K Ri and K Pi are the state feedback gains;

[0014] The equivalent input disturbance estimator is used to estimate the total disturbance composed of the system network disturbance and the external disturbance, and obtain the total disturbance estimate value which is represented by the following formula:

[0015]

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] In the formula, L a [] and respectively represent the Laplace transform and the inverse transform, s is the Laplace operator, is the estimate value of the equivalent input disturbance, is the total disturbance estimate value after filtering processing, and respectively represent and The Laplace transform of H i (s) is and The transfer function between K Hi is the gain coefficient, denotes the Moore-Penrose generalized inverse matrix, i.e., B i and C i are system matrices, u fi (t) is the state feedback control output, u i (t) is the control input, L i is the terminal sliding mode observer gain, is the observation error, v i (Δx i (t), t) is the nonlinear switching term of the terminal sliding mode observer;

[0022] The terminal sliding mode observer is used to achieve fast estimation of system state, and is expressed by the formula as follows:

[0023]

[0024] In the formula, denotes the observed value of x i (t), denotes The first derivative of u fi (t) is the state feedback control output, L i is the terminal sliding mode observer gain, is the observation error, y i ((t) is the system output, is the observed value of the system output, and the nonlinear switching term of the terminal sliding mode observer is v i (Δx i (t), t); The sliding mode surface is selected as Then the nonlinear switching term v i is further expressed as v i =[v 1i v 2i T :

[0025]

[0026] In the formula, sgn(·) is the sign function, α i and β i are positive integers, p and q are odd numbers, and satisfy p>q;

[0027] ​The cross-coupling controller is used to improve the accuracy of multi-axis contour control. By calculating the input contour error amount, the contour error compensation amount for each single axis is obtained, and it is expressed by the following formula:

[0028] u′ fi (t) = u fi (t) + u ci (t) (11)

[0029]

[0030]

[0031] e i (t) = r i (t) - y i (t) (14)

[0032] In the formula, u fi (t) is the state feedback control output, u ci (t) represents the compensation amount of single-axis contour tracking error control, u′ fi (t) represents the state feedback control output after adding the contour error compensation, e i (t) is the single-axis trajectory tracking error, ε(t) is the contour tracking error, c i is the cross-coupling control gain, k P , k I , k D are the cross-coupling PID control parameters;

[0033] In summary, the system control input u i (t) of the contour tracking controller with contour error compensation and total disturbance compensation can be obtained as follows:

[0034]

[0035] A networked contour tracking controller based on finite-time disturbance estimation proposed in this application has the following beneficial effects: 1) It can effectively suppress the interference of network-induced time delay and bounded external disturbances on the contour tracking control performance, and at the same time improve the deficiency of the existing interference suppression method based on equivalent input disturbance in the convergence speed of state estimation error. 2) It improves the structure of the traditional equivalent input disturbance estimator and gives the constraint conditions that the estimator parameters need to satisfy under the new structure, which not only solves the causality problem in the equivalent input disturbance estimation method, but also realizes the decoupled design of the equivalent input disturbance estimator and the terminal sliding mode observer. 3) For the single-axis trajectory tracking control task and multi-axis contour control task of the networked motion system, a state feedback control law with disturbance compensation and a cross-coupling control strategy are designed respectively, and finally the high-precision contour tracking control of the networked motion system is realized. Description of the Drawings

[0036] Figure 1 is a framework diagram of a finite-time equivalent input disturbance estimator;

[0037] Figure 2 is a framework diagram of a contour tracking controller based on a finite-time equivalent input disturbance estimator and a cross-coupling control strategy;

[0038] Figure 3 is a comparison diagram of the control effect of single-axis tracking error;

[0039] Figure 4 is a comparison diagram of the control effect of contour tracking error. Detailed Implementation Manner

[0040] In order to make the objectives, technical solutions, and advantages of the present application clearer and more understandable, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0041] In one embodiment, as Figure 1 and Figure 2 shown, a networked contour tracking controller based on finite-time disturbance estimation is proposed for multi-axis contour tracking control of multi-axis automation equipment, including: an input signal internal model, a state feedback controller, an equivalent input disturbance estimator, and a terminal sliding mode observer corresponding to each servo axis, as well as a cross-coupling controller between the axes.

[0042] The input signal internal model is used to process the input signal to improve the single-axis trajectory tracking accuracy, and its processing process is represented by the following formula:

[0043]

[0044] In the formula, x Ri (t) is the state of the internal model, represents the first-order differential of x Ri (t), r i (t) is the input signal, y i (t) is the system output, A Ri and B Ri are internal model matrices;

[0045] The state feedback controller calculates the state feedback gain using the pole placement method, and then calculates the state feedback control output u fi (t), which is represented by the following formula:

[0046]

[0047] In the formula, Represents the system state x i (t) observation value, K Ri and K Pi are state feedback gains;

[0048] The equivalent input disturbance estimator is used to estimate the total disturbance composed of system network disturbances and external disturbances to obtain the total disturbance estimate value It is expressed by the formula as follows:

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055] In the formula, L a [] and respectively represent the Laplace transform and the inverse transform, s is the Laplace operator, is the estimated value of the equivalent input disturbance, is the total disturbance estimate value after filtering processing, and respectively represent and Laplace transforms of, H i (s) is and transfer function between, K Hi is the gain coefficient, represents the Moore-Penrose generalized inverse matrix, that is B i and C i are system matrices, u fi (t) is the state feedback control output, u i (t) is the control input, L i is the terminal sliding mode observer gain, is the observation error, v i (Δx i (t), t) is the nonlinear switching term of the terminal sliding mode observer;

[0056] The terminal sliding mode observer is used to achieve fast estimation of system state estimation, and is expressed by the formula as follows:

[0057]

[0058] In the formula, represents the observed value of x i (t), represents the first-order differential of, u fi (t) is the feedback control input, L i is the terminal sliding mode observer gain, is the observation error, y i ((t) is the system output, is the observed value of the system output, and the nonlinear switching term of the terminal sliding mode observer is v i (Δx i (t), t); The sliding mode surface is selected as Then the nonlinear switching term v i is further expressed as v i =[v 1i v 2i T :

[0059]

[0060] In the formula, sgn(·) is the sign function, α i and β i are positive integers, p and q are odd numbers, and p > q is satisfied;

[0061] The cross-coupling controller is used to improve the accuracy of multi-axis contour control. The contour error compensation amount for each single axis is calculated by inputting the contour error amount, and is expressed by the formula as follows:

[0062] u′ fi (t)=u fi (t)+u ci (t) (11)

[0063]

[0064]

[0065] e i (t)=r i (t)-y i (t) (14)

[0066] In the formula, u fi (t) is the state feedback control output, u ci (t) represents the compensation amount of single-axis contour tracking error control, u′ fi (t) represents the state feedback control output after adding the contour error compensation, e i ​(t) is the uniaxial trajectory tracking error, ε(t) is the contour tracking error, and c i is the cross-coupling control gain, and k P , k I , k D are the cross-coupling PID control parameters;

[0067] In summary, the system control input u i (t) of the contour tracking controller with contour error compensation and total disturbance compensation can be obtained as follows:

[0068]

[0069] The design process of the above-mentioned networked contour tracking controller based on finite-time disturbance estimation is as follows:

[0070] Step S1: Establish an equivalent input disturbance state space model of the n-axis networked motion control system.

[0071] The state space model of the n-axis networked motion control system is expressed as:

[0072]

[0073] In the formula, i = 1,..., n, and x i (t) = [x 1i x 2i T represents the system state of the i-th axis, x 1i and x 2i are the system position and velocity respectively, is the first derivative of x i (t), u i is the system control input, y i is the system output, A i , B i , C i are system matrices with appropriate dimensions, τ i represents the time delay caused by network induction, d τi (t) represents the network disturbance, and d i (t) represents the bounded external disturbance caused by unmodeled dynamics and load changes, etc., and B di represents the gain matrix corresponding to the external disturbance. At the same time, the system (16) satisfies the following assumption conditions: First, the system composed of (A i , B i , C i ) is observable and controllable. Second, the time delay caused by the network does not destroy the stability of the system;

[0074] At the same time, the concept of equivalent input disturbance is introduced, and d ei ​(t) is the equivalent input disturbance of the system (16), i.e., d ei (t) has an impact on the system output equivalent to the network disturbance d τi and the external disturbance d i (t) on the system output y i The total impact. Then, the state - space model of the system (16) can be rewritten as:

[0075]

[0076] Thus, the final state - space model is obtained.

[0077] Step S2: Design a stabilizing linear state observer to achieve the stable - state estimation of the nominal system.

[0078] Design the linear state observer of the system (17) as:

[0079]

[0080] Wherein, represents the observed value of the system state x i (t), represents the first - order differential of, y i (t) is the system output, is the observed value of the system output, u fi (t) is the feedback control input, and L i is the observer gain.

[0081] Subsequently, use pole placement to design the observer gain L i . Considering that the system composed of (A i , C i ) is observable, the poles of the observer can be placed at any position in the left - hand half of the complex plane. For the convenience of observer design and analysis, all poles are placed at - p0 here. The characteristic polynomial of the linear part of the observer can be expressed as:

[0082] λ i (s) = |sI-(A i -L i C i )|=(s + p0) 2 (19)

[0083] Then, the observer gain L i =[2p0 p0 2 T .

[0084] ​Step S3: Introduce the terminal sliding mode (TSM) technique and design a terminal sliding mode observer.

[0085] Define the terminal sliding mode function as v i (σ), and select the sliding mode surface as where Δx i (t) is the observed value of the system state and the error from the system state x i (t), and Δx i (t) = [Δx 1i Δx 2i T ; then the sliding mode function can also be expressed as v i (σ) = v i (Δx i (t), t), and v i = [v 1i v 2i T . Further, design v i as follows:

[0086]

[0087] where sgn(·) is the sign function, α i and β i are positive integers, p and q are odd numbers, and p > q. Then, the TSM observer applicable to system (17) can be obtained as:

[0088]

[0089] Therefore, by combining the terminal sliding mode strategy and introducing the nonlinear switching term v i (Δx i (t), t), when the system observation error Δx i (t) is small, the observation system still has a large convergence approaching law, making the system state observation error Δx i (t) converge to zero within a finite time; compared with the exponential stability of the linear observer, the finite-time stability based on the terminal sliding mode can give the system a clear upper bound on the convergence time, thereby realizing fast system state and disturbance signal estimation.

[0090] Step S4: Design an improved equivalent input disturbance (IEID) estimator to achieve accurate estimation of the total system disturbance.

[0091] Define the estimated value of the equivalent input disturbance d ei (t) as ​​The estimation error is Then there is:

[0092]

[0093] In the formula, represents the first-order differential of Δx i (t).

[0094] Meanwhile, Equation (17) can be further expressed as:

[0095]

[0096] According to Equation (21) and Equation (23), the least-squares solution of the equivalent input disturbance estimation value can be obtained as:

[0097]

[0098] In the formula, represents the Moore-Penrose generalized inverse matrix, that is,

[0099] Furthermore, to solve the causality brought about by using the current control input in Formula (24), a decoupling transfer function H (s) is designed such that: i (s) makes:

[0100]

[0101] In the formula, H i (s) is the transfer function between and . and respectively represent and 's Laplace transform, s is the Laplace operator, is the disturbance estimation value after the output of the transfer function H i (s). It can be seen from Formula (10) that when |H i (jω i )| >> 1 holds (ω i is the angular frequency of the disturbance estimation ), then there is:

[0102]

[0103] Then, it can make

[0104]

[0105] Thus, an accurate estimation of the system total disturbance signal is achieved.

[0106] Through the above steps of design, a finite-time system state observer and disturbance estimation based on the state-feedback control output \(u fi (t)\) and the system output \(y i (t)\) are constructed. And the disturbance estimation

[0107] Step S5: Design a stabilizing state-feedback controller to achieve stable tracking control of the system.

[0108] Given the reference input \(r i (t)\), first, the input signal internal model is adopted to improve the single-axis trajectory tracking accuracy. The internal model system is designed as follows:

[0109]

[0110] where \(x Ri (t)\) is the state of the internal model system, denotes the first derivative of \(x Ri (t)\), \(y i (t)\) is the system output, \(A Ri and \(B Ri are system matrices with appropriate dimensions.

[0111] By combining Equation (17) and Equation (128), the single-axis closed-loop system can be obtained as follows:

[0112]

[0113] The pole placement method is used to design the state-feedback controller gains \(K Ri and \(K Pi where \(K Ri is the state-feedback gain of the internal model system, and \(K Pi is the state-feedback gain of the observer system. Then, the single-axis state-feedback control output can be expressed as:

[0114]

[0115] Step S6: Design a control law with disturbance compensation to achieve a robust control strategy based on finite-time disturbance estimation.

[0116] Based on the disturbance estimation a negative compensation amount for it is added to the state-feedback control output \(u fi (t)\) to obtain the control input \(u i (t)\) of the single-axis system with disturbance compensation as:

[0117]

[0118] Then, it can be designed as Figure 1The networked single-axis tracking control method based on finite-time disturbance estimation as shown.

[0119] Step S7: Design a cross-coupled controller, construct a complete scheme of the networked contour tracking controller based on finite-time disturbance estimation, and achieve high-precision multi-axis contour tracking control.

[0120] According to Figure 2 , the high-precision contour tracking controller designed in this application includes an internal model system, a state feedback controller, an improved equivalent input disturbance (IEID) estimator, a terminal sliding mode (TSM) observer, and a cross-coupled control (CCC).

[0121] Combined with the single-axis tracking controller designed in the above steps ( Figure 1 ), design the cross-coupled controller as follows:

[0122] u′ fi (t)=u fi (t)+u ci (t) (32)

[0123]

[0124]

[0125] e i (t)=r i (t)-y i (t) (35)

[0126] In the formula, (i = 1,..., n, u fi (t) is the output of the state feedback control, u ci (t) represents the compensation amount of the single-axis contour tracking error control, u′ fi (t) represents the output of the state feedback control after adding the contour error compensation, r i (t) is the given reference input, y i (t) is the system output, e i (t) is the single-axis trajectory tracking error, ε(t) is the contour tracking error, c i is the cross-coupling control gain, k P , k I , k D are the cross-coupling PID control parameters;

[0127] Furthermore, in the output u′ of the state feedback control after adding the contour error compensation fiAdd the negative compensation amount of the disturbance estimation to (t) to obtain the final system control input u as follows: i (t) is:

[0128]

[0129] Therefore, based on the proposed networked contour tracking controller, high-precision contour tracking control is achieved through the effective compensation of the coupling error and external disturbance.

[0130] The effectiveness and superiority of the method are verified through the following case:

[0131] Set the two-axis motion control system to track the following desired trajectory

[0132]

[0133] where the desired trajectory is a circular trajectory with a radius R = 1 and the center coordinates o1 = 0, o2 = 0, and its approximate contour tracking error ε0(t) is expressed as

[0134]

[0135] where

[0136] Considering that the single-axis transfer functions of the two-axis motion control system are respectively

[0137]

[0138]

[0139] After further converting to the state-space equation form, we have B1 = [0 4.41] T , B2 = [0 4.10] T , C1 = C2 = [1 0]. According to Equation (28), the internal model of the two-axis motion control system is set as B R1 = B R2 = [0 1] T . In addition, configure the terminal sliding mode observer parameters as L1 = [3.01 17.04] T , L2 = [3.01 15.77] T , α1 = α2 = 0.57, β1 = 5.37, β2 = 5.19, p = 7, q = 1; configure the estimator gain K H1 = K H2 = 100; given the state feedback gain K through pole placement R1 = K R2 = [-386.43 -323.35], KP1 = K P2 = [73.13 -4.89]; The PID parameters of the configured cross-coupling controller are k P = 100, k I = 0, k D = 100. The bounded external disturbances given in the case are d1(t) = d2(t) = sin(5πt) + cos(3πt) + sin(0.5πt), 10 ≤ t ≤ 20, and the time delays caused by network induction are τ1 = τ2 = 0.001. The effectiveness and superiority of the three strategies of "improved equivalent input disturbance + linear observer", "improved equivalent input disturbance + terminal sliding mode observer", and "improved equivalent input disturbance + terminal sliding mode observer + cross-coupling controller" are compared and verified, and the single-axis tracking error of the system is as Figure 3 shown, and the contour tracking error is as Figure 4 shown. From Figure 3 and Figure 4 it can be seen that the two-axis motion control system affected by external bounded disturbances and network-induced time delays can effectively suppress the disturbances under the above three control strategies, and has high single-axis tracking accuracy and contour control accuracy. Through comparison, it shows that the control strategy of the present invention based on "improved equivalent input disturbance + terminal sliding mode observer + cross-coupling controller" can effectively improve the contour control performance of the networked two-axis motion control system compared with the other two strategies, and achieve high-precision multi-axis contour tracking control.

[0140] The above embodiments only represent several implementation manners of the present application, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention patent. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several deformations and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the patent of the present application should be subject to the appended claims.

Claims

1. A networked contour tracking controller based on finite-time disturbance estimation for multi-axis contour tracking control of multi-axis automation equipment, characterized in that, The networked contour tracking controller based on finite-time disturbance estimation includes: an input signal internal model corresponding to each servo axis, a state feedback controller, an equivalent input disturbance estimator, a terminal sliding mode observer, and a cross-coupling controller between axes, where: The input signal internal model is used to process the input signal to improve the single-axis trajectory tracking accuracy, and its processing process is represented by the following formula: where x Ri (t) is the state of the internal model, denotes the first-order differential of x Ri (t), r i (t) is the input signal, y i (t) is the system output, A Ri and B Ri are the internal model matrices; The state feedback controller calculates the state feedback gain using the pole placement method, and then calculates the state feedback control output u fi (t), which is expressed by the formula as follows: wherein, represents the observed value of the system state x i (t), and K Ri and K Pi are state feedback gains; The equivalent input disturbance estimator is used to estimate the total disturbance composed of the system network disturbance and the external disturbance, and obtain the total disturbance estimation value It is expressed by the formula as follows: where L a [] and denote the Laplace transform and the inverse Laplace transform respectively, s is the Laplace operator, is the estimated value of the equivalent input disturbance, is the estimated value of the total disturbance after filtering, and denote the Laplace transforms of and respectively, H i (s) is and the transfer function between, K Hi is the gain coefficient, denotes the Moore - Penrose generalized inverse matrix, i.e., B i and C i are the system matrices, u fi (t) is the state - feedback control output, u i (t) is the control input, L i is the terminal sliding - mode observer gain, is the observation error, v i (Δx i (t), t) is the nonlinear switching term of the terminal sliding - mode observer; The terminal sliding mode observer is used to achieve fast estimation of system state, and is represented by the following formula: wherein, represents the observed value of x i (t), represents the first-order differential of, u fi (t) is the state feedback control output, L i is the terminal sliding mode observer gain, is the observation error, y i (t) is the system output, is the observed value of the system output, and the nonlinear switching term of the terminal sliding mode observer is v i (Δx i (t), t); the sliding mode surface is selected as then the nonlinear switching term v i is further expressed as v i =[v 1i v 2i T :​ where sgn(·) is the sign function, α i and β i are positive integers, p and q are odd numbers, and p > q is satisfied, and Δx 1i is the first component of the observation error; The cross-coupling controller is used to improve the accuracy of multi-axis contour control, and calculates the contour error compensation amount for each single axis through the input contour error amount, and is represented by the following formula: u′ fi (t) = u fi (t) + u ci (t) (11) e i r(t) i y(t) i (14) where, u fi (t) is the state feedback control output, u ci (t) represents the compensation amount of the single-axis contour tracking error control, u′ fi (t) represents the state feedback control output after adding the contour error compensation, e i (t) is the single-axis trajectory tracking error, ε(t) is the contour tracking error, c i is the cross-coupling control gain, k P , k I , k D are the cross-coupling PID control parameters; In summary, the system control input u of the contour tracking controller with contour error compensation and total disturbance compensation can be obtained as follows: i (t) is expressed as follows:

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