Fault-tolerant control method for spatial interconnection system based on frequency domain

Abstract

The invention discloses a frequency domain-based fault-tolerant control method for a spatial interconnection system, and belongs to the technical field of system fault-tolerant control. The fault-tolerant control method for the spatial interconnection system based on the frequency domain is characterized by comprising the following steps: establishing an interconnection model system of a two-dimensional processing platform based on a high-load electric linear two-dimensional displacement platform; based on the interconnection model system, constructing a fault-containing ILC model of the two-dimensional processing platform; setting a control law of a subsystem in the two-dimensional processing platform according to the ILC model containing the fault; verifying whether a control algorithm of the two-dimensional interconnection system is in a limited frequency domain range or not; according to the method, an ILC iterative learning control module is used for repeating control of an operation process, so that system errors can be quickly converged in finite time; the fault-tolerant capability is designed from the perspective of the whole system by utilizing the learning capability of ILC.

Description

Technical Field

[0001] The present invention relates to the technical field of system fault-tolerant control, and particularly to a fault-tolerant control method for a spatially interconnected system based on the frequency domain. Background Art

[0002] The industrial Internet of Things remote communication infinite vision nanoscale machining platform (two-dimensional machining platform) is a device for precise positioning and movement, and its main function is to perform positioning on a plane. This platform can be used to execute some tasks that require high precision and high load capacity, such as precision machining and optical testing. The device itself usually does not have a moving function, but realizes precise linear or two-dimensional displacement on the plane through an electric drive system. Therefore, the two-dimensional machining platform has important practical value in the fields of industry, medicine, scientific research, etc.

[0003] Due to its high precision, high load capacity and stability, this kind of device usually requires complex structural design and control algorithms. The two-dimensional machining system includes two moving axes (X-axis and Y-axis). When performing biaxial movement, these two axes form a non-linear interconnected system, and the position and movement state of each other affect each other, thereby affecting the position and attitude of the entire machining platform. Although the interaction between the X-axis and the Y-axis may be relatively simple, overall, the entire system may exhibit complex and variable behaviors. With the improvement of the safety performance requirements for process control systems, fault-tolerant control can further ensure the stability and reliability of the system. Therefore, it is particularly important to conduct fault-tolerant research on such mechanical systems. Summary of the Invention

[0004] In view of the problems existing in the above-mentioned existing technologies, the present invention is proposed.

[0005] To achieve the above object, the present invention provides the following technical solution: A fault-tolerant control method for a spatially interconnected system based on the frequency domain, including,

[0006] Based on a high-load electric linear two-dimensional displacement stage, establish an interconnected model system of the two-dimensional machining platform;

[0007] Based on the interconnected model system, construct an ILC model of the two-dimensional machining platform with faults;

[0008] According to the ILC model with faults, set the control law of the subsystems in the two-dimensional machining platform;

[0009] Verify whether the control algorithm of the two-dimensional interconnected system is within the finite frequency domain;

[0010] According to the verification result and the control law based on LMI, optimize the control law of the two-dimensional machining platform;

[0011] Based on the two-dimensional machining platform, experimentally test the subsystems.

[0012] Compared with the prior art, the beneficial effects of the present invention are as follows: The fault-tolerant control method for the spatial interconnection system based on the frequency domain utilizes the control of the ILC iterative learning control module for the repeated operation process, so as to quickly converge the system error within a limited time; by using the learning ability of ILC, the fault-tolerant ability is designed from the perspective of the entire system, and this fault-tolerant strategy is improved through ILC to make it more intelligent and adaptive. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. 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 also be obtained based on these drawings. Among them:

[0014] Figure 1 is a high-load electric linear two-dimensional (XY) displacement table.

[0015] Figure 2 is the spectrum of the desired trajectory of subsystem G 1 in the frequency domain range.

[0016] Figure 3 is the output trajectory of subsystem G 1 at different iteration times.

[0017] Figure 4 is the sudden actuator failure diagram of subsystem G 1 at the 31st time.

[0018] Figure 5 is subsystem G 1 when affected by actuator failure.

[0019] Figure 6 is subsystem G 2 spectrum diagram of the desired trajectory.

[0020] Figure 7 is the tracking curve diagram under noise interference.

[0021] Figure 8 is the desired trajectory diagram of the subsystem.

[0022] Figure 9 is the fault working trajectory diagram of the two-dimensional processing platform.

[0023] Figure 10 is the comparison test diagram of frequency domain and finite frequency domain fault tolerance. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0024] To make the above objects, features, and advantages of the present invention more apparent and understandable, the following detailed description of the specific embodiments of the present invention will be given in conjunction with the accompanying drawings of the specification.

[0025] In the following description, many specific details are set forth in order to provide a thorough understanding of the present invention. However, the present invention may be practiced in other ways different from those described herein. Those skilled in the art can make similar extensions without departing from the spirit of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed below.

[0026] Secondly, the present invention will be described in detail with reference to the schematic diagrams. When detailing the embodiments of the present invention, for ease of explanation, the cross-sectional views showing the device structure will be enlarged locally out of proportion, and the schematic diagrams are only examples and should not limit the scope of protection of the present invention herein. In addition, in actual production, three-dimensional spatial dimensions including length, width, and depth should be included.

[0027] Furthermore, the so-called "one embodiment" or "embodiment" herein refers to a specific feature, structure, or characteristic that may be included in at least one implementation of the present invention. The phrase "in one embodiment" appearing in different places in this specification does not necessarily refer to the same embodiment, nor is it an embodiment that is separate or selectively mutually exclusive with other embodiments.

[0028] Embodiment 1

[0029] The present invention provides a technical solution: a fault-tolerant control method for a space-interconnected system based on the frequency domain, including,

[0030] S1: Based on a high-load electric linear two-dimensional displacement stage, establish an interconnected model system of the two-dimensional processing platform;

[0031] S2: Based on the interconnected model system, construct an ILC model of the two-dimensional processing platform with faults;

[0032] S3: According to the ILC model with faults, set the control law of the subsystems in the two-dimensional processing platform;

[0033] S4: Verify whether the control algorithm of the two-dimensional interconnected system is within the finite frequency domain;

[0034] S5: According to the verification results and the control law based on LMI, optimize the control law of the two-dimensional processing platform;

[0035] S6: Based on the two-dimensional processing platform, experimentally test the subsystems.

[0036] The interconnected model system of the two-dimensional processing platform established in the first step provides the basis for constructing the ILC model with faults in the second step; the interconnected model system is a framework for understanding and simulating the operation of the entire processing platform, while the ILC model, on this basis, takes into account the possible fault situations in actual operation to achieve more precise control and optimization; after constructing the ILC model with faults, the third step of designing the subsystem control law has a basis. The control law needs to be designed according to the characteristics of the ILC model to ensure that each subsystem can still work in coordination and maintain the stability and performance of the overall system in the presence of faults; after designing the subsystem control law, the fourth step of analyzing the stability of the control algorithm in the finite frequency domain is to verify whether the control law designed in the third step is effective; this step is a key link to ensure that the designed control law can maintain the system stability under specific operating conditions; after analyzing the stability of the control algorithm, the fifth step of designing the control law based on LMI can further optimize the designed control law; finally, the sixth step of experimentally verifying the fault tolerance performance of the subsystem based on the two-dimensional processing platform is a practical test of the previous five steps. This step verifies the effectiveness and practicality of the theoretical design through experiments to ensure that the designed control law can handle faults and maintain the stability and performance of the system in actual operation.

[0037] It should be noted that the formula of the interconnected model system of the two-dimensional processing platform is

[0038]

[0039] where x i (p + 1) represents the state of the i-th subsystem at the next moment of p, and x i (p) represents the state of the i-th subsystem at p, u i (p) represents the input of the i-th subsystem at p, y i (p) represents the output of the i-th subsystem at p, v i (p) and w i (p) represent the spatial interconnection effect of the subsystem, represents the transfer matrix of the system, represents the observation matrix of the subsystem, represents the direct transfer matrix.

[0040] Among them, the dimension of each v i and w i is L is the number of subsystems in the interconnected system. In the two-dimensional processing platform, L = 2. R N is the vector space that distinguishes vectors v and w described as follows;

[0041]

[0042] Furthermore, vi and w i are further partitioned input and output interconnection vectors, v ij , v ji , w ij and w ji are the interconnection signals between subsystems i and j, p represents the moment (time step). For such interconnection signals, let

[0043]

[0044] or

[0045] (w(p), v(p)) ∈ S I , for all p ≥ 0

[0046] where S I = {(w, v) ∈ R N × R N : w ji = w ij , for i, j = 1, …, L}

[0047] In Equation (3), v ij (p) represents a certain value between the i-th and j-th elements at time step p, w ij (p) represents another value between the i-th and j-th elements at time step p, v ji (p) represents a certain value between the j-th and i-th elements at time step p, w ji (p) represents another value between the j-th and i-th elements at time step p, w(p) represents a vector of dimension, representing the state or variable at time step p, v(p) represents a vector of dimension, representing the state or variable at time step p, R N represents the vector space that differentiates vectors v and w; S I represents a set, v represents the differentiating vector, w represents the vector space, w ij and w ji are the interconnection signals between subsystems i and j.

[0048] The system described by Equation (1) is an interconnected system containing two variables of time and space. It is necessary to transform the underlying subsystems in the interconnected system into a one-dimensional system with respect to time by removing the interconnection variables. Therefore, it is necessary to re-formulate System (1) and make the following definitions:

[0049]

[0050] In the formula, X(p) represents the state vector, a vector composed of multiple sub-vectors, and each sub-vector x i(p) is a row vector representing the state vector at time step p, V(p) represents the disturbance or noise vector, W(p) represents the output vector, and each sub-vector w i (p) is a row vector representing another variable at time step p, U(p) represents the control input vector, and each sub-vector u i (p) is a row vector representing the control input at time step p, Y(p) represents the output vector, and each sub-vector y i (p) is a row vector representing the measurement or output at time step p, and L represents the number of sub-vectors.

[0051] According to Equation (4), the state-space model (1) of the system can be rewritten as:

[0052]

[0053] where X(p + 1) represents the value of the state vector at the next time step p + 1, W(p) represents the value of the output vector at the current time step p, Y(p) represents the measurement or observation vector at the current moment, X(p) represents the current state, V(p) represents the disturbance or noise vector, U(p) represents the control input vector at the current moment, and A TT 、A TS 、B TU represent the state transition matrix and its related terms, A ST 、A SS represent the output matrix and its related terms, and C T represents the observation matrix.

[0054] where

[0055]

[0056] where A TT 、A TS 、A ST 、A SS 、B TU 、C T represent the diagonal matrix of the state transition matrix, the diagonal matrix describing the relationship between the state vector and the disturbance or noise vector, the diagonal matrix describing the influence of the state vector on the output vector, the diagonal matrix describing the influence of the disturbance or noise vector on the output vector, the diagonal part of the matrix of the influence of the control input on the state vector, and the diagonal part of the observation matrix, respectively.

[0057] At this time, Equation (5) is a one-dimensional model containing only the time variable. However, the periodic interconnection relationship of v i (p) and w i (p) of the spatial interconnection effect of the subsystem still exists, so the following periodic interconnection is defined:

[0058]

[0059] In Equation (7), represents the value of the (i + 1)-th variable at time step p, represents the value of the i-th variable at time step p, represents the value of the (i - 1)-th variable at time step p.

[0060] By connecting features using Equation (7), the interconnected variable relationship can be obtained as:

[0061] Δ i,p V(p) = W(p) (8)

[0062] In Equation (8), Δ i，p V(p) represents the conversion of the state vector V(p) to W(p) through the permutation matrix Δ i,p and W(p) represents another state vector or set of variables at time step p.

[0063] Among them, Δ i,p is the permutation matrix of V and W, and it can be obtained that

[0064] (V, W) ∈ S I (9)

[0065] In the formula, S I represents a set, which represents the pair of state vectors (V, W) that satisfy specific conditions.

[0066] The present invention hopes to design an iterative learning fault-tolerant controller so that the interconnected system can be controlled. For this purpose, through Equation (8), the state-space form described by Equation (5) can be reformulated as a model with an ILC structure as

[0067]

[0068] Among them Ω = Δ i，p -A SS , l represents the number of iterations, and p represents the time step.

[0069] In Equation (10), X l (p + 1) represents the value of the state vector at the next time step p + 1, X l (p) represents the value of the state vector at the current time step p, U l (p) represents the value of the control input vector at the current time step p, Y l (p) represents the value of the output vector at the current time step p, A TT 、A TS 、B TU 、C Trespectively represent the main part of the state transition matrix, the matrix describing the relationship between the state vector and the disturbance or noise vector, the main part of the matrix of the influence of the control input on the state vector, and the main part of the observation matrix. A, B, and C respectively represent the state transition matrix, the matrix of the influence of the control input on the state vector, and the observation matrix. Ω represents the permutation matrix, and A SS is the matrix of the influence of the disturbance or noise vector on the output vector.

[0070] Furthermore, based on the interconnected model system, the steps for constructing the ILC model of the two-dimensional machining platform with faults include

[0071] S21: Construct the ILC model based on the state-space model of the interconnected model system;

[0072] S22: Input the control input of the machining system as the fault signal into the interconnected model system to obtain the fault coefficient;

[0073] S23: Reconstruct the ILC model of the two-dimensional machining platform with faults according to the fault coefficient;

[0074] It should be noted that iterative learning control (ILC) is a control method suitable for repetitive operation processes. It corrects the current control signal by using the previous state and output information of the system so as to quickly converge the system error within a limited time. By utilizing the learning ability of ILC, the fault tolerance ability can be designed from the perspective of the entire system, especially when the subsystem faces actuator faults. The key to fault-tolerant control lies in ensuring that the system can still operate effectively under partial faults. Through ILC, this fault-tolerant strategy can be improved to make it more intelligent and adaptive.

[0075] Specifically, when the two-dimensional machining platform is moving, its subsystems face many disturbances and uncertainties. Designing an appropriate fault-tolerant control system can detect and correct errors during the transmission process. For the control input U n (p, l), which represents an input signal of the interconnected system fault, the following fault model is defined:

[0076]

[0077] n represents a variable in which a fault occurs. Among them, the fault coefficient Γ n satisfies the following conditions

[0078]

[0079] Let the parameters in this fault model Γ n ( Γ n ≤1), It is known that an unknown scalar Γ is set n varies within a known range. When Γ n = 1, it corresponds to no system fault, that is Γ n = 0 corresponds to a complete fault; 0 < Γ n ≤ Γ n <1 or corresponds to a partial fault of the subsystem.

[0080] Define

[0081]

[0082] In Equation (13), U F is a horizontally partitioned matrix composed of multiple sub - matrices. Each sub - matrix U m F is a part of the matrix, where i ranges from 1 to m, and Γ n represent the upper and lower limits of the fault coefficient respectively.

[0083] And

[0084]

[0085] In Equations (14) and (15), q represents a diagonal matrix, q 0 represents the initial condition, and Γ n represent the upper and lower limits of the fault coefficient respectively. q n represents the average value of these two values, and q n0 represents the ratio of the difference to the sum of these two values, which is usually used to represent a certain relative change or ratio.

[0086] Introduce the symbols

[0087]

[0088] Using (13) - (16), the fault coefficient Γ can be written as

[0089] Γ=(I + Γ 0 )q(17)

[0090] In Equation (17), Γ represents the fault coefficient, Γ 0 represents an unknown vector, representing the fault coefficient or uncertainty factor under the initial condition, and q represents a vector;

[0091] where

[0092] |Γ 0 |≤q 0 ≤I (18)

[0093] In the present invention, it is assumed that the upper and lower bounds of the failure range of each actuator are known, i.e., Γ n and Therefore, an additional parameter q n As an entry in the vector q, can be used to scale the original range of the unknown scalar Γ n so that Γ in (13) can be regarded as a structured uncertainty of the form (17), whose known and unknown vectors are q 0 and Γ 0 .

[0094] Therefore, the two-dimensional machining platform contains a fault ILC structure model reconstruction as follows

[0095]

[0096] In Equation (19), Γ represents the fault coefficient, X l (p + 1) represents the value of the state vector at the next time step p + 1, X l (p) represents the value of the state vector at the current time step p, U l (p) represents the value of the control input vector at the current time step p, Y l (p) represents the value of the output vector at the current time step p, and A, B, and C represent the state transition matrix, the influence matrix of the control input on the state vector, and the observation matrix, respectively.

[0097] Furthermore, according to the ILC model, the steps of setting the control law of the subsystem in the two-dimensional machining platform include

[0098] S31: Determine the state error vector and the tracking error vector according to the ILC model with faults;

[0099] S32: Set the control law of the subsystem in the two-dimensional machining platform based on the state error vector and the tracking error vector;

[0100] Among them, the formulas for the state error vector and the tracking error vector are as follows:

[0101] η l+1 (p + 1) = X l+1 (p) - X l (p) (20)

[0102] e l (p) = y ref (p) - y l (p) (21)

[0103] In Equations (19) and (20), η l+1 (p + 1) represents the state error vector, el (p) represents the tracking error vector, X l+1 (p) represents the value of the state vector at iteration i + 1 and time step p, specifically representing the state of the system at iteration i + 1 and time step p, X l (p) represents the value of the state vector at iteration i and time step p, specifically representing the state of the system at iteration i and time step p, y ref (p) represents the value of the reference output at time step p, y l (p) represents the value of the actual output at iteration i and time step p;

[0104] Define

[0105] ΔU l+1 (p) = U l+1 (p) - U l (p) (22)

[0106] In equation (22), ΔU l+1 (p) represents the state change at iteration i + 1 and time step p, U l+1 (p) represents the state of the system at iteration i + 1 and time step p, U l (p) represents the value of the state vector at iteration i and time step p.

[0107] Design the following control law:

[0108] ΔU l+1 (p) = K 1 μ l+1 (p + 1) + K 2 μ l+1 (p) + K 3 e l (p) (23)

[0109] In equation (23), ΔU l+1 (p) represents the value of the change in control input at time step p, μ l+1 (p + 1) represents the predicted value of the state vector at iteration l + 1 and time step p + 1, μ l+1 (p) represents the value of the state vector at iteration 1 + 1 and time step p, e l (p) represents the value of the tracking error vector at iteration 1 and time step p, K 1 , K 2 , K 3 represents the control gain to be designed, K 1 μ l+1 (p + 1) considers the predicted value of the future state and is used for feedforward control, K 2 μ l+1 (p) considers the value of the current state and is used for feedback control, K 3 el (p) takes into account the current tracking error for error correction.

[0110] Where

[0111] μ l (p) = y l (p - 1) - y l-1 (p - 1) = Cη l (p) (24)

[0112] In Equation (24), μ l (p) represents the state error at iteration i and time step p, y l (p - 1) represents the actual output of the system at iteration i and time step p - 1, y l-1 (p - 1) represents the actual output of the system at iteration i - 1 and time step p - 1, p - 1 represents what, C represents the observation matrix, describing how the state vector affects the output vector, η l (p) represents the state change at iteration i and time step p.

[0113] To rewrite the system into a discrete - type system, set

[0114]

[0115] In Equation (25), represents the value of the state vector at iteration i + 1 and time step p + 1, η l+1 (p + 1) represents the value of the state change at iteration 1i + 1 and time step p + 1, η l+1 (p) represents the value of the state change at iteration i + 1 and time step p, η l+1 (p - 1). represents the value of the state change at iteration i + 1 and time step p - 1.

[0116] The role of the control law is used to describe the update and representation method of the state vector, especially constructing the state vector through the state change. Specifically: is a vector containing three elements, respectively representing the state changes at different time steps; this representation method helps to better understand and process state information in the analysis and design of control systems; these formulas are usually used in the analysis and design of control systems, especially in iterative learning control (ILC), for evaluating and improving the performance of the system.

[0117] Thus, the controlled system can be reconstructed into a discrete repetitive process model

[0118]

[0119] In Equation (26), Denotes the state of the system at iteration i+1 and time step p+1, Denotes the state transition matrix, Denotes the influence matrix of the control input on the state, Denotes the state of the system at iteration i+1 and time step p, e l (p) denotes the difference between the actual output and the reference output at iteration i and time step p, e l+1 (p) denotes the difference between the actual output and the reference output at iteration i+1 and time step p, Denotes the observation matrix, Denotes the direct transfer matrix.

[0120] Where

[0121]

[0122] In the formula, Denotes the updated state transition matrix, A denotes the original state transition matrix, B denotes the influence matrix of the control input on the state, Denotes the updated control input matrix, Γ denotes the gain matrix, K 1 , K 2 , K 3 Denotes the control gain coefficient, Denotes the updated observation matrix, C denotes the observation matrix, I denotes the identity matrix.

[0123] It should be noted that, considering that an additional term has been added in the control method as a necessary compensation means to make up for the influence of not assuming that the state vector is available for the control method, the control law (23) can be rewritten by analysis as

[0124] ΔU l+1 (p-1) = K 1 Cη l+1 (p) + K 2 Cη l+1 (p-1) + K 3 e l (p) (27)

[0125] In formula (27), K 1 , K 2 , K 3 Are the control gains to be designed.

[0126] The purpose of this step is to verify whether the control algorithm of the subsystem in the two-dimensional processing platform is within the finite frequency domain. The specific verification process: To verify whether the parameter matrix in formula (26) satisfies Lemma 3 of stability, it is necessary to ensure that this matrix satisfies the three conditions listed in the lemma. The designed linear matrix inequalities (35) and (36) are to prove that these conditions are satisfied. First, use Lemma 5 for transformation to ensure that condition i) is satisfied. Then, combine Lemma 4 for further transformation to ensure that condition ii) is satisfied. Finally, the third condition of the stability lemma is satisfied by formula (45). Therefore, it can be concluded that the system is stable.

[0127] It should be noted that the control algorithm refers to the design process, and the control law refers to formula (27).

[0128] Furthermore, the steps to verify whether the control algorithm of the two-dimensional interconnected system is within the finite frequency domain include

[0129] S41: Determine that the matrix needs to satisfy the three conditions of the lemma;

[0130] S42: Construct linear matrix inequalities to prove that the three conditions are satisfied;

[0131] S43: Verify whether the two-dimensional interconnected system satisfies the three conditions of the stability lemma;

[0132] Verifying whether the control algorithm of the two-dimensional interconnected system is within the finite frequency domain is used to analyze the stability of the control algorithm of the two-dimensional interconnected system within the finite frequency domain.

[0133] Regarding the stability of the two-dimensional interconnected system model described by formula (26) along the number of test times, before designing the ILC algorithm that monotonically converges on the two-dimensional processing platform within a specific frequency range, it is necessary to introduce the lemmas that appear in the following text:

[0134] Lemma 1 Given matrices Θ = Θ T , X, Y of appropriate dimensions, then for any Δ that satisfies Δ T Δ ≤ I, the following inequality

[0135] Θ + XΔY + Y T Δ T X T <0 (28)

[0136] In formula (28), Θ represents a constant matrix, usually representing some fixed parameters or matrices in the system, X and Y represent variable matrices, which can represent the state feedback gain of the system or other control parameters, Δ represents an unknown perturbation matrix, usually representing the uncertainty in the system or external interference, and T represents the transpose of the matrix.

[0137] The necessary and sufficient condition for it to hold is that there exists ε > 0 such that

[0138] Θ + ε 2 XX T + ε -2 Y T Y < 0 (29)

[0139] In equation (29), Θ represents a constant matrix, usually representing some fixed parameters or matrices in the system, X and Y represent variable matrices, which can represent the state feedback gain or other control parameters of the system, Δ represents an unknown perturbation matrix, usually representing the uncertainty or external disturbance in the system, T represents the transpose of the matrix, and ε represents a positive small parameter.

[0140] Lemma 2 Ω = P - A in equation (10) ss is invertible if and only if S I ∩ S B = {0},

[0141]

[0142] In equation (30), S B represents the set containing all vector pairs (W, V) that satisfy specific conditions, W and V represent N - dimensional real vectors, S′ B represents the set, I M represents the M - order identity matrix, represents the vector, and I represents the unit vector.

[0143] Lemma 3 The necessary and sufficient condition for the linear repetitive process (1) to be stable along the channel is that it satisfies

[0144] (i) ρ(D) < 1

[0145] (ii) ρ(A) < 1

[0146] (iii) The modulus of all eigenvalues of the transfer function matrix G(z) = C(zI - A) -1 B + D, z = e jω , is strictly less than 1.

[0147] Lemma 4 For the given matrix For matrices Λ, Σ with the column dimension of M, there exists a matrix E such that the inequality (31) holds

[0148] Ψ + sym{Λ T WΣ} < 0 (31)

[0149] In equation (31), Ψ represents a constant matrix, sym(X) represents the symmetric part of the matrix X, Λ represents a parameter matrix, W represents a variable matrix, and ∑ represents a parameter matrix.

[0150] The necessary and sufficient condition for its establishment is the following two projection inequalities

[0151]

[0152] In formula (32), Λ ⊥ : represents the transformation matrix, ∑ ⊥ represents the transformation matrix, and represent the results of the action of the gradient operator.

[0153] Lemma 5 Given any and the symmetric matrix Π, assume that For all hold, where -π ≤ ω 1 ≤ ω 2 ≤ π, the following two inequalities can be obtained as equivalent

[0154] i)

[0155]

[0156] In formula (33), G(e jω represents the value of the transfer function on the unit circle, I represents the identity matrix, and Π represents the weight or transformation matrix.

[0157] ii) There exist a positive definite matrix Q > 0 and a symmetric matrix P such that

[0158]

[0159] where,[[]] is selected as shown in the following table,

[0160] In formula (34), represents the transformed matrix, I represents the identity matrix, ξ represents the weight, represents, and Π represents the transformation matrix.

[0161] Table 1 provides the w and values for low, medium, and high frequencies.

[0162] Table 1

[0163]

[0164] Next, explore and analyze the stability of the algorithm:

[0165] If there exist S > 0 and E = diag{E 1 E 2} with appropriate dimensions in the linear matrix inequalities (35) and (36), satisfying the negative definite relationship, such as

[0166]

[0167] In equations (35) and (36), Ξ 1 , Ξ 2 , Ξ 3 represents a specific parameter, and (*) represents a symmetric term. represents a certain transformation value of the state transition matrix, and S represents the weight.

[0168] Among them

[0169]

[0170] then the interconnected subsystem in the two-dimensional processing platform can converge stably when facing actuator faults.

[0171] The main reasons for stability are as follows:

[0172] By performing an equivalent transformation on equation (34) through Lemma 5, the following relationship can be obtained

[0173]

[0174] In equation (37), represents the transformed state transition matrix and input matrix, ζ 1 , ζ 2 , ζ 3 represents a specific parameter, I represents the identity matrix, represents the transformed output matrix and direct transfer matrix, (*) represents a symmetric term.

[0175] Simplify it and re-express it as

[0176]

[0177] Among them

[0178] In equation (38), represents the transformed state transition matrix and input matrix, ζ 1 , ζ 2 , ζ 3 represents a specific parameter, I represents the identity matrix, represents the transformed output matrix and direct transfer matrix, (*) represents a symmetric term, -I represents the negative of the identity matrix, and Λ ⊥ represents a matrix.

[0179] According to Lemma 4, if

[0180]

[0181] In equation (39), denotes a transformation, sym{·} denotes the symmetric part, and Λ ⊥ denotes a matrix, ∑ denotes a specific parameter, and E usually denotes the uncertainty matrix.

[0182] If it holds, then there must exist

[0183]

[0184] In equation (40), Λ ⊥ denotes the transformation matrix, and ∑ ⊥ denotes the specific parameter matrix.

[0185] One of the conditions for satisfying equation (40) has been obtained from equation (38). Therefore, there exists Σ ⊥ such that

[0186]

[0187] where Σ = [0 I 0].

[0188] The following inequality can be deduced

[0189]

[0190] Equation (42) is equivalent to

[0191]

[0192] It can be known from Lemma 5 that From the implicit conditions for Lemma 5 to hold, we can obtain So equation (43) holds. Thus, equation (41) holds, and the stability condition i) is satisfied simultaneously.

[0193] Next, it is deduced that is equivalent to the following equation

[0194]

[0195] Through the schur lemma, equation (44) is transformed into

[0196]

[0197] where E 1 = diag{E 11 E 12}, and substituting it in, equation (35) can be obtained.

[0198] When Π = diag{I -I} in equation (34), (45) satisfies Lemma 5, and G(e jω ) T G(ejω ) < I, satisfying stability condition iii).

[0199] Given S > 0, thus equation (36) can deduce Satisfying stability condition ii). Thus, it can be known that equations (35) and (36) satisfy the three stability conditions of Lemma 3, so the interconnected subsystem is stable.

[0200] Furthermore, step 5: According to the verification result and the LMI-based control law, optimize the control law of the two-dimensional machining platform, specifically:

[0201] The control law of LMI refers to the linear matrix inequality, which is a matrix inequality similar to (35)(36)(52)(53). The reason for optimizing the control law of the subsystem in the two-dimensional machining platform by the control law of LMI is that the (35)(36) inequalities contain the coupling terms of the product of the controller gain matrices and the product terms of the unknown matrix E and the controller gain matrix, and cannot be directly solved by the LMI toolbox. It needs to be transformed into a linear matrix inequality through transformation, and after the transformation is completed, the control matrix of the system can be solved.

[0202] Next, from step four and Lemma 4, it can be obtained that the following equations must hold

[0203]

[0204] where Σ 2 = [I 0], Substituting it in, we can get

[0205]

[0206] Thus, it is proved that (46) holds, and at the same time, it is deduced that As can be seen from step four, where the parameter E = diag{E 1 E 2}

[0207]

[0208] In equation (48), T represents the transformation matrix, -I represents the negative of the identity matrix, E represents the perturbation or uncertainty matrix, represents the transpose of the state transition matrix, and (*) represents the symmetric term.

[0209] This type of inequality is usually used to design robust controllers to ensure the stability of the system in the presence of uncertainties; it is used to design controllers that can handle model uncertainties. It is also used to solve for optimal control parameters to ensure that the system meets specific performance metrics; these formulas are commonly used in control system analysis and design, especially in robust control and optimization problems, to improve the performance of the system.

[0210] Premultiply Equation (48) by and then postmultiply by the transpose of, we can obtain

[0211]

[0212] In Equation (49), represents the error matrix, E represents the perturbation or uncertainty matrix, A represents the state transition matrix, and T represents the transformation matrix.

[0213] Set We can get

[0214]

[0215] Given where

[0216] As can be seen from Step 4, the following inequality holds

[0217]

[0218] In Equation (51), ζ 1 , ζ 2 , ζ 3 represents specific parameters or transformation matrices, E represents the perturbation or uncertainty matrix, represents the transformed state transition matrix, input matrix, output matrix, and direct transfer matrix.

[0219] Premultiply and postmultiply the above equation by and its transpose, and let where We can derive Equation (53).

[0220] Therefore, the sufficient condition for the discrete repetitive process of the two-dimensional machining platform to asymptotically converge along the iteration axis in each frequency domain interval is the existence of and X 1 , X 2 , X 3 such that the following inequality holds

[0221]

[0222] In Equation (52), Ω11 , Ω 12 , Ω 22 represents a specific parameter or transformation matrix, γ 11 , γ 12 , γ 22 : represents another set of specific parameters or transformation matrices, and (*) represents symmetric terms.

[0223] wherein

[0224]

[0225] From the above LMIs, it can be obtained that X 3 = K 3 , then the control law gain matrix is K 3 = X 3 .

[0226] After the above transformation by equations (46)-(51), the new linear matrix inequalities (52)(53) are steps for further optimizing the control law design of the subsystem in the two-dimensional machining platform based on LMI.

[0227] The present invention converts a two-dimensional space interconnected system into a one-dimensional equivalent model by enhancing the discrete space model variables coupled to multiple subsystems. Based on the conditions of the equivalent model, combined with finite frequency domain analysis and ILC technology, the stability and convergence of the interconnected subsystems are deeply analyzed, and the anti-interference and fault tolerance capabilities of the system are improved. Tests are carried out on a single-chip XY platform to verify the effectiveness of the proposed algorithm. Fault tolerance control is of great significance in dealing with actuator faults and external disturbances.

[0228] Furthermore, in step 6, based on the two-dimensional machining platform, the subsystem is experimentally tested, which is mainly used to experimentally verify the fault tolerance performance of the subsystem based on the two-dimensional machining platform.

[0229] Specifically, the steps of experimentally testing the subsystem based on the two-dimensional machining platform include

[0230] S61: Construct the reference trajectory of the subsystem;

[0231] S62: Set the time-varying fault parameters and external disturbance signals;

[0232] S63: According to the gain matrix for optimizing the control law of the two-dimensional machining platform;

[0233] S64: Determine the root mean square error RMS as an index for measuring the fault estimation error.

[0234] To verify the effectiveness of the fault-tolerant control method designed in the present invention, considering the non-linear interconnection of two axes of a two-dimensional machining platform (as shown in Figure 1 ), the feasibility of the following two subsystems is verified

[0235]

[0236] In Equation (54), G 1 represents the first subsystem, G 2 represents the second subsystem, and s represents the conversion from the time domain to the frequency domain.

[0237] First, a fault-tolerant control test is carried out on the subsystem G 1 . The reference trajectory y 1 (x-axis) required by the subsystem G d1 is defined as

[0238] y d1 = 0.8(sin(3t) + sin(15t)) 2 (55)

[0239] The time-varying fault parameter is set to

[0240] Γ 1 = 0.759 + sin(2πt) (56)

[0241] The external disturbance signal w is set to

[0242] w = 1.5sin(2πt - 0.34π) (57)

[0243] According to Step 5, the gain matrix of the subsystem fault-tolerant control is K 11 = [-1.3735 -0.8083], K 21 = 0.3256, K 31 = 0.0248.

[0244] The present invention introduces the root mean square error RMS as an index for measuring the fault estimation error. It is defined as follows:

[0245]

[0246] In Equation (58), t represents time, k represents a parameter, and T represents the total number of samples.

[0247] The present invention designs an iterative fault-tolerant algorithm, transforms the controlled system model into a repetitive process model, and combines the generalized KYP primitive to study the convergence conditions of the system in different frequency ranges. The interconnected subsystems affected by actuator faults and external disturbances are regarded as the controlled objects, and the stability of their models is discussed in the finite frequency domain. The present invention solves the path tracking problem of the interconnected system under actual disturbances and faults, and realizes high-precision tracking of the desired trajectory.

[0248] Embodiment 2

[0249] To illustrate the beneficial effects of the present solution, this embodiment provides a comparative experiment. Specifically, the existing technical solutions mainly involve the research on iterative learning fault-tolerant control based on 2D system theory. This includes the research on fault-tolerant control in the full frequency domain, such as finite frequency domain fault detection and fault-tolerant control in networked control systems. In addition, there is also the robust dissipative iterative learning fault-tolerant control for the actuator fault multi-rate sampling intermittent process. These methods use the state space model sampled at a slow rate to describe the multi-sampling rate process through improved techniques, and design control strategies based on 2D system theory.

[0250] As Figure 1 shown, a high-load electric linear two-dimensional (XY) displacement stage is provided, which can have precise motion control and durability. The stroke of this stage is 60 mm (+ / - 30 mm), and the size is 125 mm * 125 mm * 60 mm, suitable for various precision applications. It combines a direct drive iron core linear motor with a non-contact optical linear encoder to ensure high precision and stable feedback. The stage form is built-in with crossed roller bearings, providing excellent load-bearing capacity and low friction performance. Through experiments, it is not difficult to find that as shown in Figure 3, when the subsystem G 1 is not disturbed by the fault signal (56) and only affected by the external disturbance (57), the tracking curve of the output trajectory of the subsystem G 1 at different iteration times. It can be clearly seen that the proposed algorithm can track the desired trajectory at the 30th iteration, and its tracking effect is good. Figure 4 What is shown is that when the actuator of the subsystem G 1 suddenly fails at the 31st time, and its fault parameter is (65), at this time, the output trajectory of the subsystem G 1 changes from Figure 2 b to Figure 3 a. The tracking accuracy significantly decreases due to the fault, but as the number of iterations increases, the tracking performance of the subsystem G 1 gradually improves, and finally accurately tracks the desired trajectory as shown in Figure 3 a, which reflects the good fault-tolerant ability of the control algorithm. Figure 5 Show the subsystem G 1Subject to the error effect during actuator failure. This curve clearly demonstrates the ability of the designed fault-tolerant technology to accurately and effectively repair faults in the presence of fault variations. Figure 6 For subsystem G 2 Spectrum diagram of the desired trajectory. Figure 7 Illustrated are the tracking curves when the system is not interfered by the fault signal (60) and only by the noise interference (61). The output trajectories at the 10th and 30th times are shown respectively, and it can be seen that the output trajectories converge.

[0251] Figure 8 Shown is that when the actuator of the subsystem fails suddenly at the 31st time, with its fault parameter being (60), the tracking accuracy of the subsystem decreases significantly due to the fault. However, as the number of iterations increases, the subsystem finally accurately tracks the desired trajectory as Figure 8 shown in b. Figure 9 Shows that when the two-dimensional processing platform has faults simultaneously in two axes (as shown by the blue line), the platform deviates from the original working trajectory (as shown by the red line). In this case, the fault-tolerant control gradually takes effect, enabling the platform to finally re-track the original working trajectory (as shown by the black dashed line).

[0252] After comparison, as Figure 10 Shows the comparison test results under the full-frequency domain and finite-frequency domain fault-tolerant control strategies. Through comparative analysis, it can be clearly observed that under the action of the finite-frequency domain iterative learning control strategy, the output tracking error of the system shows a faster convergence speed and achieves a better tracking effect. This result intuitively proves the effectiveness of the finite-frequency domain iterative learning control strategy in improving the system performance.

[0253] When conducting the comparative experiment, the same conditions include the consistency of the system model, external interference, and the type of actuator failure. In the experiment, the fault-tolerant designs within the finite-frequency domain range and the full-frequency domain designed by the present invention are compared to ensure that when both face the same external interference and actuator failure, the comparison of the error convergence speed and accuracy is carried out.

[0254] This frequency-domain-based fault-tolerant control method for spatially interconnected systems uses the ILC iterative learning control module to control the repeated operation process, so as to quickly converge the system error within a limited time; utilizes the learning ability of ILC to design the fault-tolerant ability from the perspective of the entire system, and improves this fault-tolerant strategy through ILC to make it more intelligent and adaptive.

[0255] In addition, to provide a concise description of the exemplary embodiments, not all features of the actual embodiments may be described, that is, those features that are not relevant to the currently considered best mode of implementing the present invention, or those features that are not relevant to the implementation of the present invention.

[0256] It should be understood that, during the development of any actual implementation, such as in any engineering or design project, numerous specific implementation decisions can be made. Such development efforts may be complex and time-consuming, but for those of ordinary skill in the art who benefit from this disclosure, without undue experimentation, the development efforts will be routine work in design, manufacturing, and production.

[0257] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the technical solutions of the present invention, and they should all be covered within the scope of the claims of the present invention.

Claims

1. A frequency domain-based spatial interconnection system fault-tolerant control method, characterized in that: include, Based on the high-load electric linear 2D translation stage, an interconnected model system of 2D machining platform is established; Based on the interconnected model system, an ILC model with faults on a 2D machining platform is constructed; According to the ILC model with faults, the control law of the subsystem in the 2D machining platform is set; Verify that the control algorithm of the two-dimensional interconnected system is within the limited frequency domain; According to the verification results and the control law based on LMI, the control law of the 2D machining platform is optimized; Based on the two-dimensional processing platform, the subsystem is experimentally tested.

2. The frequency domain-based spatial interconnection system fault-tolerant control method according to claim 1, characterized in that: The interconnected model system formula of the two-dimensional processing platform is: Among them, x i (p+1) represents the state of the ith subsystem at the next moment after time p, x i (p) represents the state of the i-th subsystem at time p, u i (p) represents the input of the ith subsystem at time p, y i (p) represents the output of the ith subsystem at time p, v i (p) and w i (p) represents the spatial interconnection of subsystems, represents the transfer matrix of the system, represents the observation matrix of the subsystem, represents a direct transfer matrix.

3. The frequency domain-based spatial interconnection system fault-tolerant control method according to claim 1 or 2, characterized in that: Based on the interconnected model system, the steps of constructing the ILC model of the 2D machining platform with faults include: Construct an ILC model based on the state space model of the interconnected model system; The control input of the machining system is input into the interconnected model system as a fault signal to obtain the fault coefficient; According to the fault coefficient, the ILC model of the 2D machining platform with faults is reconstructed; Among them, the state space model formula of the interconnected model system is: Where X(p+1) represents the value of the state vector at the next time step p+1, W(p) represents the value of the output vector at the current time step p, Y(p) represents the measurement or observation vector at the current moment, X(p) represents the current state, V(p) represents the interference or noise vector, U(p) represents the control input vector at the current moment, and A TT , A TS , B TU represents the state transfer matrix and its related items, A ST , A SS Represents the matrix and its related terms, C T represents the observation matrix; Among them, the ILC model formula is: Where, X l (p+1) represents the value of the state vector at the next time step p+1, X l (p) represents the value of the state vector at the current time step p, U l (p) represents the value of the control input vector at the current time step p, Y l (p) represents the value of the output vector at the current time step p, A, B, and C represent the state transfer matrix, the influence matrix of the control input on the state vector, and the observation matrix, respectively; Among them, the failure coefficient is, Γ=(I+Γ0)q In the formula, Γ represents the fault coefficient, Γ0 represents the unknown vector, and q represents the vector; Among them, the ILC model formula for reconstructing the faulty 2D machining platform is: Where Γ represents the failure coefficient, X l (p+1) represents the value of the state vector at the next time step p+1, X l (p) represents the value of the state vector at the current time step p, U l (p) represents the value of the control input vector at the current time step p, Y l (p) represents the value of the output vector at the current time step p, A, B, and C represent the state transfer matrix, the influence matrix of the control input on the state vector, and the observation matrix, respectively.

4. The frequency domain-based spatial interconnection system fault-tolerant control method according to claim 3, characterized in that: According to the ILC model, the steps to set the subsystem control law in the 2D machining platform include: According to the ILC model with faults, a state error vector and a tracking error vector are determined; Based on the state error vector and the tracking error vector, the subsystem control law of the two-dimensional machining platform is set; Among them, the formulas of state error vector and tracking error vector are as follows: η l+1 (p+1)X l+1 (p)-X l (p) e l (p)=y ref (p)-y l (p) Where η l+1 (p+1) represents the state error vector, e l (p) represents the tracking error vector,; Among them, the subsystem control law formula is: D.U. l+1 (p-1)=K1Cη l+1 (p)+K2Cη l+1 (p-1)+K3e l (p) In the formula, K1, K2, K3 are the control gains to be designed, ΔU l+1 (p) represents the value of the change in the control input at time step p, μ l+1 (p+1) represents the predicted value of the state vector at iteration l+1 and time step p+1, μ l+1 (p) represents the value of the state vector at iteration l+1 and time step p, e l (p) represents the value of the tracking error vector at iteration l and time step p.

5. The frequency domain-based spatial interconnection system fault-tolerant control method according to claim 4, characterized in that: The steps to verify whether the control algorithm of a two-dimensional interconnected system is within the finite frequency domain include: Determine the three conditions of the lemma that the matrix must satisfy; Construct linear matrix inequalities and prove that the three conditions are satisfied; Verify whether the two-dimensional interconnected system satisfies the three conditions of the stability lemma; Among them, the matrix is ​​the parameter matrix in the discrete system.

6. The frequency-domain-based spatial interconnection system fault-tolerant control method according to claim 5, characterized in that: The three conditions are, (i)ρ(D)<1 (ii)ρ(A)<1 (iii) The transfer function matrix is ​​G(z) = C(zI-A) -1 B+D,z=e jω , The modulus of all eigenvalues ​​of is strictly less than 1.

7. The frequency domain-based spatial interconnection system fault-tolerant control method according to claim 6, characterized in that: The linear matrix inequality is, Wherein, Ξ1, Ξ2, Ξ3 represent specific parameters, (*) represents symmetric terms, Represents a certain transformation value of the state transfer matrix, and S represents the weight.

8. The frequency domain-based spatial interconnection system fault-tolerant control method according to claim 7, characterized in that: The control law of LMI refers to the linear matrix inequality.

9. The frequency domain-based spatial interconnection system fault-tolerant control method according to claim 8, characterized in that: Based on the 2D processing platform, the steps of experimental verification subsystem include: Construct reference trajectories for subsystems; Setting time-varying fault parameters and external interference signals; According to the gain matrix of the control law of the optimized two-dimensional machining platform; The root mean square error (RMS) is determined as an indicator for measuring the fault estimation error.

10. The frequency domain-based spatial interconnection system fault-tolerant control method according to claim 9, characterized in that: The reference trajectory of the subsystem is, the d1 =0.8(sin(3t)+sin(15t)) 2 。

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