Time domain fault-tolerant control method based on interconnected two-dimensional system

Abstract

The invention discloses a time domain fault-tolerant control method based on an interconnected two-dimensional system, and belongs to the technical field of time domain fault-tolerant control. The time domain fault-tolerant control method based on the interconnected two-dimensional system comprises the following steps: establishing an interconnected model system of a two-dimensional processing platform; establishing an ILC model of a two-dimensional processing platform containing an uncertain fault; designing a subsystem control law in the two-dimensional processing platform; analyzing the stability of a control algorithm of the two-dimensional interconnection system in a time domain range; designing a processing platform based on an LMI control law; designing a fault-tolerant control law of the model containing uncertainty; the fault-tolerant performance of the subsystem is verified based on a two-dimensional processing platform experiment; according to the time domain fault-tolerant control method based on the interconnected two-dimensional system, the uncertainty of the system is considered, the behavior of the system changing along with time is focused on in time domain analysis, and various uncertainty factors inside and outside the system are more comprehensively considered.

Description

Technical Field

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

[0002] Iterative Learning Control (ILC) is an effective control technology for repetitive motion systems. It summarizes the experience in the previous iteration and continuously adjusts the controller parameters to achieve precise tracking of a predetermined trajectory within a finite period. In recent years, with the wide application of networked agent systems, spatially interconnected systems have received increasing attention. Such systems are composed of multiple subsystems with similar structures, which affect each other spatially and exchange information through bidirectional communication links, and each subsystem can receive data from neighboring subsystems.

[0003] However, in actual operation, the dynamic behavior of the system often varies greatly, and the uncertainty of the system structure and parameters makes trajectory control more complex. In addition, the presence of actuator faults and external disturbances may also cause instability in the system motion, thereby affecting the control accuracy. Therefore, improving the tracking accuracy and performance of spatially interconnected systems in complex environments has become an important problem to be solved urgently. Summary of the Invention

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

[0005] To achieve the above object, the present invention provides the following technical solution: A time-domain fault-tolerant control method based on an interconnected two-dimensional system, including,

[0006] Establish an interconnected model system of a two-dimensional machining platform;

[0007] Establish an ILC model of a two-dimensional machining platform including uncertain faults;

[0008] Design the control law of the subsystems in the two-dimensional machining platform;

[0009] Analyze the stability of the control algorithm of the two-dimensional interconnected system in the time domain;

[0010] Design the control law of the two-dimensional machining platform based on LMI;

[0011] Design a fault-tolerant control law for a model including uncertainties;

[0012] Verify the fault-tolerant performance of the subsystems based on experiments on the two-dimensional machining platform.

[0013] Compared with the prior art, the beneficial effects of the present invention are as follows: The time-domain fault-tolerant control method based on an interconnected two-dimensional system takes into account the uncertainties of the system. Focusing on time-domain analysis, it pays attention to the behavior of the system changing over time, and more comprehensively considers various internal and external uncertainty factors of the system. By integrating fault-tolerant control strategies into each subsystem, global fault-tolerant control of the entire spatially interconnected system is achieved, making full use of the information interaction advantages between internal subsystems of the system, realizing refined control strategies, and improving the control accuracy and efficiency of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] 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 be obtained based on these drawings. Among them:

[0015] Figure 1 is a high-load electric linear two-dimensional displacement stage.

[0016] Figure 2 is a control trajectory diagram of the system without actuator failure.

[0017] Figure 3 is subsystem G x is a control trajectory diagram when an actuator failure occurs.

[0018] Figure 4 is a control trajectory diagram of the system without actuator failure.

[0019] Figure 5 is subsystem G y is a control trajectory diagram when an actuator failure occurs.

[0020] Figure 6 is a fault-tolerant control trajectory diagram of the two-dimensional interconnected system.

[0021] Figure 7 is a comparison diagram of the trajectory tracking error of the two-dimensional interconnected system. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0022] To make the above objects, features, and advantages of the present invention more obvious and understandable, the following will describe the specific embodiments of the present invention in detail with reference to the drawings of the specification.

[0023] Many specific details are set forth in the following description in order to fully understand the present invention, but the present invention can also be implemented in other ways different from those described herein. Those skilled in the art can make similar generalizations without departing from the connotation of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed below.

[0024] Secondly, the present invention will be described in detail with reference to the schematic diagrams. When describing the embodiments of the present invention in detail, for the convenience of explanation, the cross-sectional views showing the device structure will be enlarged locally not in accordance with the general scale, 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.

[0025] 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 manner of the present invention. The "in one embodiment" appearing in different places in this specification does not necessarily refer to the same embodiment, nor is it an independent or selectively mutually exclusive embodiment with other embodiments.

[0026] Embodiment 1

[0027] As Figures 1 to 5 shown, the present invention provides a technical solution: a time-domain fault-tolerant control method based on an interconnected two-dimensional system, including:

[0028] S1: Establish an interconnected model system of a two-dimensional processing platform;

[0029] S2: Establish an ILC model of a two-dimensional processing platform including uncertain faults;

[0030] S3: Design the control law of the subsystems in the two-dimensional processing platform;

[0031] S4: Analyze the stability of the control algorithm of the two-dimensional interconnected system in the time domain;

[0032] S5: Design the control law of the two-dimensional processing platform based on LMI;

[0033] S6: Design the fault-tolerant control law of the model including uncertainties;

[0034] S7: Experimentally verify the fault-tolerant performance of the subsystems based on the two-dimensional processing platform.

[0035] It should be noted that the interconnected model system of the two-dimensional processing platform established in the first step (S1) provides a basic framework for constructing the ILC model containing uncertain faults in the second step (S2); this framework defines the basic structure and dynamic characteristics of the system, enabling the system to maintain consistency and integrity when introducing the fault model; after constructing the ILC model containing uncertain faults, the third step (S3) designs the subsystem control law to compensate for these uncertainties and faults and ensure system performance; the design of the control law depends on the description and prediction of the system dynamics by the ILC model; after designing the subsystem control law, the fourth step (S4) analyzes the stability of the control algorithm of the two-dimensional interconnected system in the time domain to verify the effectiveness of the control law; this step is the key to ensuring that the designed control law can maintain system stability in practical applications; after analyzing the stability of the control algorithm, the fifth step (S5) designs the control law based on LMI to further optimize the control strategy; the sixth step (S6) designs the fault-tolerant control law for the model containing uncertainties, further considering the uncertainties and possible faults of the system on the basis of optimizing the control law in the previous step to improve the robustness of the system; finally, the seventh step (S7) verifies the fault-tolerant performance of the subsystem based on the two-dimensional processing platform experiment, which is a practical test of all the previous steps; this step verifies the effectiveness and practicality of the theoretical design through experiments, ensuring that the designed control law can handle uncertainties and faults in actual operation and maintain system stability and performance.

[0036] It should be noted that both the interconnected model system of the two-dimensional processing platform and the ILC model are constructed on a high-load electric linear two-dimensional displacement table. The state-space model is constructed based on the linear system and state-space description methods in modern control theory. The state-space model can describe the evolution of the internal state of the system over time and the relationship between input and output; then, based on the KYP (Kalman-Yakubovich-Popov) lemma, the asymptotic stability conditions and design parameters of the learning process are determined; finally, the control law is designed based on LMI and experiments are carried out on the experimental platform ( Figure 1 as shown). Specifically Figure 1 it is a high-load electric linear two-dimensional (XY) displacement table. Two axes of the high-load electric linear two-dimensional displacement table will be taken as the research objects. The stroke of this platform is 60 mm (+ / -30 mm), and the size is 125 mm × 125 mm × 60 mm, which is suitable for precision applications. The platform is equipped with crossed roller bearings inside, providing excellent load-bearing capacity and low friction performance.

[0037] Furthermore, establishing the interconnected model system of the two-dimensional processing platform helps to understand the operation mechanism of the experimental platform and provides an accurate theoretical basis for subsequent design and optimization. Specifically, in this step, the interconnected model system of the two-dimensional processing platform is defined as:

[0038]

[0039] In the formula, x i (p + 1) represents the state of the i-th subsystem at the next moment of p moment, x i (p) represents the state of the i-th subsystem at p moment, u i (p) represents the input of the i-th subsystem at p moment, y i (p) represents the output of the i-th subsystem at p moment, v i (p) and w i (p) represents 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. The output of each subsystem is determined by its state, so define so as to simplify formula (1).

[0040] It should be noted that when designing v i and w i The dimension of is L is the number of subsystems in the interconnected system, L = 2. R N is the vector space that distinguishes vectors v and w described below, and its vector space for distinguishing vectors v and w is:

[0041]

[0042] v i and w i are the input and output interconnection vectors further divided. v ij , v ji , w ij and w ji are the interconnection signals between subsystems i and j. 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 formula (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 distinguishes vectors v and w; S I represents a set, v represents the distinguishing vector, w represents the vector space, w ij and w ji is the interconnection signal between subsystems i and j.

[0048] The system represented by equation (1) encompasses interconnected systems in both the time and space dimensions. To eliminate the influence of spatial interconnection, the following is defined:

[0049]

[0050] In equation (4), 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, and W(p) represents the output vector. 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 measured value or output at time step p. L represents the number of sub-vectors.

[0051] Therefore, the state-space expression of system (1) can be reformulated as

[0052]

[0053] In equation (5), 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, A TT 、A TS 、B TUDenote the state transition matrix and its related terms, A ST 、A SS Denote the matrix and its related terms, C T Denote the observation matrix.

[0054] Among them

[0055]

[0056] In Equation (6), A TT 、A TS 、A ST 、A SS 、B TU 、C T respectively 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.

[0057] At this time, v i (p) and w i (p) still have an interconnection relationship, and the interconnection relationship is defined as follows:

[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] Using the connection feature of Equation (7), the interconnection 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 converting the state vector V(p) to W(p) through the permutation matrix Δ i,p , and W(p) represents another state vector or variable set 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 IRepresents a set, which is a pair of state vectors (V, W) that satisfy specific conditions

[0066] The state - space form described by Equation (5) can be reformulated as follows through Equation (8)

[0067]

[0068] where Ω = Δ i,p -A SS , l represents the number of iterations, and p represents time;

[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 T respectively 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, the steps to establish an ILC model of a two - dimensional machining platform containing an uncertain fault include:

[0071] S21: Construct an uncertainty model;

[0072] S22: Define the interconnected - system fault model;

[0073] S23: Reconstruct the fault ILC structure model of the two - dimensional machining platform.

[0074] Through S21 - S23, the system's adaptability to faults and fault - tolerance performance can be improved, and the robustness of the system can be enhanced.

[0075] Among them, to construct its uncertainty model

[0076]

[0077] In Equation (11), X l (p + 1) represents the value of the state vector at the next time step p + 1, Y l(p) represents the value of the output variable at time step p. A, B, and C represent the system matrices, corresponding to the state transition matrix, input matrix, and output matrix respectively. ΔA, ΔB, and ΔC represent the uncertain parts of the system matrices, that is, the changes in matrices A, B, and C. U l (p) represents the input signal, the value at time step p.

[0078] The expression of the uncertain structure is

[0079]

[0080] In Equation (12), E, F 1 , F 2 , F 3 are known constant matrices. H represents the uncertain perturbation, which describes the possible unknown changes or disturbances in the system and satisfies

[0081] H T H ≤ I (13)

[0082] Equation (13) is a constraint condition, indicating that the energy of the uncertain perturbation matrix H (obtained by multiplying its transpose by itself) does not exceed the identity matrix I, which is often used in control theory to limit the magnitude of uncertainty.

[0083] Furthermore, define the model of the interconnected system fault

[0084]

[0085] In Equation (14), n represents the fault variable, p and l represent the parameter variables, Γ n represents the fault coefficient, U n (p, l) represents the system output without faults;

[0086] Among them, the fault coefficient Γ n is defined as follows

[0087]

[0088] Introduction Γ n ( Γ n ≤ 1), This invention discusses the case of partial faults, that is, 0 < Γ n ≤ Γ n < 1 or This case.

[0089] When 0 < Γ n ≤ Γ n < 1, the fault coefficient is less than 1, indicating partial faults; when When it is, the fault coefficient is greater than 1, which may also indicate a specific type of fault.

[0090] In Equation (15), Γ n represents the minimum possible fault coefficient, represents the maximum possible fault coefficient.

[0091] Define

[0092]

[0093] In Equation (16), U F is a horizontally partitioned matrix composed of multiple sub - matrices, and 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.

[0094] And

[0095]

[0096]

[0097] In Equations (17) and (18), q represents the current value of the diagonal matrix, where q 1 , q 2 …q m are the elements on the diagonal, q 0 represents the initial value, and Γ n represent the upper and lower limits of the fault coefficient respectively, q n represents the average of these two values, q m0 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, Γ 0 represents the diagonal matrix, Γ 0n represents the specific element calculated through Γ n and q n |Γ 0 | is the absolute - value matrix of Γ 0

[0098] Using (16) and (17), Γ can be written as:

[0099] Γ=(I + Γ 0 )q. (19)

[0100] In Equation (19), Γ represents the fault - coefficient matrix, I represents the identity matrix, Γ 0 ​Let \( \varGamma \) represent the matrix and \( q \) represent the current value of the diagonal matrix.

[0101] where

[0102] \( |\varGamma| \) 0 \( \leq q \) 0 \( \leq I \). (20)

[0103] Therefore, the fault ILC structure model of the two - dimensional machining platform is reconstructed as follows

[0104]

[0105] In Equation (21), \( X(p) \) represents the value of the state vector at time step \( p \), \( Y(p) \) represents the value of the output vector at time step \( p \), \( A \) represents the state - transition matrix, which is used to describe how the state vector transfers from one time step to the next, \( B \) represents the input matrix, which is used to describe the influence of the input on the state, \( C \) represents the output matrix, which is used to describe how the state vector maps to the output vector, \( \varGamma \) represents the fault coefficient matrix, and \( X(p + 1) \) represents the value of the state vector at time step \( p+1 \). I (p) represents the value of the state vector at time step p, Y I (p) represents the value of the output vector at time step p, A represents the state - transition matrix, which is used to describe how the state vector transfers from one time step to the next, B represents the input matrix, which is used to describe the influence of the input on the state, C represents the output matrix, which is used to describe how the state vector maps to the output vector, Γ represents the fault coefficient matrix, X I (p + 1) represents the value of the state vector at time step p + 1.

[0106] where

[0107] A 1 \( = A \) TT \( + A \) TS (\( \Delta \) i，p \( - A \) ss ) -1 \( + \Delta A \), \( B \) 1 \( = B \) TU \( + \Delta B \), \( C \) 1 \( = C \) T \( + \Delta C \).

[0108] In the formula, \( A \) 1 represents the updated state - transition matrix, \( A \) TT , \( A \) ss and \( A \) TS represent parts of the state - transition matrix, \( B \) 1 represents the updated input matrix, \( C \) 1 represents the updated output matrix, \( \Delta \) i，p represents a specific fault at time step \( p \), \( \Delta A \) represents the uncertain part of the state - transition matrix, \( B \) TU represents a part of the input matrix, \( \Delta B \) represents the uncertain part of the input matrix, \( C \) T represents a part of the output matrix.

[0109] Furthermore, the steps for designing the control law of the subsystem in the two - dimensional machining platform include

[0110] S31: Define the ILC law, tracking error vector, and state error vector;

[0111] S32: Update the ILC law;

[0112] S33: Reconstruct the controlled system into a discrete repetitive process model.

[0113] Steps S31 - S33 help to achieve the coordinated operation between subsystems, improving the control accuracy and response speed of the entire platform.

[0114] Among them, the ILC law and tracking error vector are defined as follows

[0115] U l+1 (p) = U l (p) + ΔU l+1 (p) (22)

[0116] e l (p) = y ref (p) - y l (p) (23)

[0117] In Eqs. (22) and (23), U l+1 (p) and U l (p) represent the input signals of the (I + 1)-th and I-th iterations respectively, ΔU l+1 (p) represents the change in the input signal, e l (p) is the tracking error, representing the difference between the reference trajectory and the actual output, y ref (p) represents the reference trajectory at time step p, y l (p) represents the actual output at time step p.

[0118] Define the state error vector

[0119] η l+1 (p + 1) = X l+1 (p) - X l (p) (24)

[0120] In Eq. (24), η l+1 (p + 1) represents the state error vector at time step p + 1, X l+1 (p) represents the state vector of the (I + 1)-th iteration at time step p, X l (p) represents the state vector of the I-th iteration at time step p.

[0121] For generality, assume:

[0122] η l+1 (p + 1) = A 1 η l+1 (p) + B1 ΔU l+1 (p - 1)(25)

[0123] In Equation (25), η l+1 (p + 1) represents the state error vector at time step p + 1, A 1 represents the state transition matrix, B 1 represents the input influence matrix, ΔU l+1 (p - 1) represents the change in the input signal, η l+1 (p) represents the state error vector at time step p.

[0124] And

[0125] e l+1 (p) = -C 1 η l+1 (p + 1)+e l (p)(26)

[0126] In Equation (26), e l+1 (p) represents the tracking error of the iteration at time step p, C 1 represents the output matrix, which is used to describe how the state error is mapped to the tracking error, η l+1 (p + 1) represents the state error vector of the iteration at time step p + 1, e l (p) represents the fixed error at time step p.

[0127] The update term of Equation (22) takes the following form

[0128] ΔU l+1 (p - 1)=K 1 η l+1 (p + 1)+K 2 e l (p)+K 3 (e l (p + 1)-e l (p))(27)

[0129] In Equation (27), ΔU l+1 (p - 1) represents the change in the input signal, K 1 , K 2 and K 3 represent the control gains, η l+1 (p + 1) represents the state error vector, e l (p) and e l (p + 1) represent the tracking errors at the current and next time steps respectively.

[0130] Equation (27) is the control law, and the gain K of the control matrix in Equation (27) can be solved through the stability condition 1, K 2 , K 3 , thereby achieving the control of subsystem fault tolerance.

[0131] The control gain difference is defined as L = K 2 - K 3 , and the specific settings are as follows:

[0132]

[0133] In Equation (28), represents the state vector after update at time step p + 1, η l+1 (p + 1) represents the state error vector of the iteration at time step p + 1, e l (p) represents the tracking error at time step p of the first iteration.

[0134] The system (21) to be controlled can be reconstructed into a discrete repetitive process model

[0135]

[0136] In Equation (29), represents the state vector after update at time step p + 1, A 2 represents the state transition matrix, which is used to describe how the state vector transfers from one time step to the next, B 2 represents the input influence matrix, which is used to describe the influence of the tracking error on the state vector, e l (p) represents the tracking error at time step p of the first iteration, e l+1 (p) represents the tracking error of the iteration at time step p, C 2 represents the output matrix, which is used to describe how the state vector is mapped to the tracking error, D 2 represents the direct transfer matrix.

[0137] Among them

[0138] C 2 = [-C 1 (A 1 + B 1 ΓK 1 ) - C 1 B 1 ΓL], D 2 = I - C 1 B 1 ΓK 3 .

[0139] In Equation (15), A 2 , B 2 , C 2 and D 2is the system matrix, which is used to describe state transition, input influence, output mapping, and direct transmission respectively. A 1 , B 1 and C 1 represent the basic state transition matrix, the basic input influence matrix, and the basic output matrix respectively. Γ represents the fault coefficient matrix, K 1 and K 3 represent the control gain matrix and the control gain matrix respectively. L represents the control gain difference, and I represents the identity matrix.

[0140] Furthermore, analyzing the stability of the control algorithm of the two-dimensional interconnected system in the time domain can ensure the effectiveness of the designed control law in practical applications and avoid unstable situations during system operation. Specifically, explore and analyze the stability of the algorithm, and introduce three conditions for stability:

[0141] The necessary and sufficient condition for the linear repetitive process (1) to be stable along the channel is to satisfy

[0142] (i) ρ(D 2 ) < 1, ensuring stability in batches.

[0143] (ii) ρ(A 2 ) < 1, ensuring stability in time.

[0144] (iii) The modulus of all eigenvalues of the transfer function matrix G(z) = C 2 (zI - A 2 ) -1 B 2 + D 2 , z = e jω , is strictly less than 1.

[0145] Where ρ(D 2 ) represents the spectral radius of matrix D 2 , A 2 , B 2 , C 2 and D 2 are system matrices, which are used to describe state transition, input influence, output mapping, and direct transmission respectively. z represents a complex variable, representing a point in the frequency domain, and e jω represents a point on the unit circle, where j is the imaginary unit and ω is the angular frequency.

[0146] A sufficient condition for the discrete process in (29) to converge asymptotically along the iteration axis in the time domain is that there exists a positive definite matrix such that the following inequality holds

[0147]

[0148] In Equation (30), A 2 , B 2 , C 2 and D 2 represent the state transition matrix, the input influence matrix, the output mapping matrix, and the direct transfer matrix respectively. represents a positive definite matrix, and I represents the identity matrix.

[0149] Then the interconnected underlying subsystems satisfy the three conditions of the stability lemma, that is, the system is stable.

[0150] First, the above equation can be rewritten as

[0151]

[0152] In Equation (31), A 2 , B 2 , C 2 and D 2 represent the state transition matrix, the input influence matrix, the output mapping matrix, and the direct transfer matrix respectively. represents a positive definite matrix, and I represents the identity matrix.

[0153] Let Π = diag{I -I}, and then it can be deduced that

[0154]

[0155] G T (e jω )·G(e jω ) < I. (33)

[0156] In Equations (32) and (33), G(e jω ) is the transfer function matrix, representing the frequency response of the system, e jω is a point on the complex unit circle, where j is the imaginary unit and ω is the angular frequency, and I represents the identity matrix.

[0157] Satisfies the sufficient and necessary condition (iii) for the stability of the offline linear repetitive process.

[0158] The implicit condition for Equation (31) to hold is

[0159]

[0160] In Equation (15), A 2 , B 2 , C 2 and D 2 represent the state transition matrix, the input influence matrix, the output mapping matrix, and the direct transfer matrix respectively. represents a positive definite matrix, and I represents the identity matrix.

[0161] Known C 2 T C 2 >0, It can be deduced that Furthermore, it can be obtained that

[0162]

[0163] Equations (31) and (35) satisfy the three conditions of stability.

[0164] In equation (35), ρ(A 2 ) represents the spectral radius of matrix A 2 , that is, the modulus of the largest eigenvalue, and ρ(D 2 ) represents the spectral radius of matrix D 2 .

[0165] Furthermore, the control law design of the two-dimensional processing platform based on LMI further optimizes the control strategy, improves the dynamic performance and stability of the system, and reduces the conservatism in the design process.

[0166] Specifically, if there exist W 1 >0, W 2 >0, W 3 >0 and matrices R 1 , R 2 , R 3 , when ΔA = ΔB = ΔC = 0, the following inequality holds

[0167]

[0168] In equation (36), Φ 11 , Φ 12 and Φ 22 represent submatrices, W 1 , W 2 , W 3 represent positive definite matrices, R 1 , R 2 and R 3 represent matrices, and Ξ 1 represents a matrix composed of submatrices, which needs to satisfy the negative definite condition.

[0169] Among them

[0170]

[0171] Then the subsystem under iterative learning control is stable on both the time axis and the batch axis, and the actual output of the control system asymptotically tracks the desired trajectory. The gain matrix of the control law can be obtained as

[0172]

[0173] Step 4 can be expressed as Θ T PΘ - P < 0, where S = diag{S 1 , S 2}, and applying Schur can reconstruct Θ T PΘ - P < 0 as:

[0174]

[0175] In Equation (37), Θ represents a matrix, usually representing the state - space model of the system, P represents a positive - definite matrix, usually used to represent some kind of energy or cost function, A 2 , B 2 , C 2 , D 2 represent the state - transition matrix, input - influence matrix, output matrix, and direct - transmission matrix respectively, S represents a diagonal matrix containing positive - definite matrices S 1 and S 2 , and (*) represents some elements in the matrix, and the specific values are not given.

[0176] Left - multiply and right - multiply the above equation by to deduce

[0177]

[0178] In Equation (15), represents the inverse matrices of positive - definite matrices S 1 , S 2 , S 3 , A1, B1, C1 represent specific forms of the state - transition matrix, input - influence matrix, and output matrix respectively, Γ, K 1 , K 3 , L represent matrices, usually used to represent some specific system parameters.

[0179] Let

[0180]

[0181] The control gain can be obtained as

[0182] In the above equation, represents the inverse matrices of positive - definite matrices S 1 , S 2 , S 3 , W 1 , W 2 , W 3 represent the inverse matrices of positive - definite matrices, K 1 , K 2 , K 3denotes the control gain matrix, R 1 , R 2 , R 3 denotes system parameters or weight matrices, and L denotes the transformation matrix.

[0183] Furthermore, design a fault-tolerant control law for the model with uncertainties to improve the system's performance in the face of uncertainties and faults, and enhance the system's adaptability and reliability. The specific process is as follows:

[0184] If ΔA≠0, ΔB≠0, there exist W 1 >0, W 2 >0, W 3 >0, the matrix R 1 , R 2 , R 3 and the real numbers ρ 1 , ρ 2 satisfying the following inequalities

[0185]

[0186] In Equation (39), Ξ 1 , Ξ 2 , Ξ 3 , Ξ 4 , Ξ 5 denote submatrices, and ΔA, ΔB denote the changes in system parameters.

[0187] where the expression of Ξ 1 is Equation (37)

[0188]

[0189] At this time, the system is stable. Following the design process specified in Step Five, the specific parameters of the controller gain matrix can be determined as

[0190]

[0191] In the formula, K 1 , K 2 , K 3 denote the control gain matrix, and W 1 , W 2 , W 3 denote the inverse matrices of the positive definite matrices S1, S2, and S3.

[0192] The time-domain fault-tolerant control method based on the interconnected two-dimensional system takes into account the uncertainties of the system, focuses on the time-domain analysis and pays attention to the behavior of the system changing with time, more comprehensively considers various internal and external uncertainty factors of the system, realizes the global fault-tolerant control of the entire space interconnected system by integrating the fault-tolerant control strategy into each subsystem, makes full use of the information interaction advantage between the internal subsystems of the system, realizes a refined control strategy, and improves the control accuracy and efficiency of the system.

[0193] Embodiment 2

[0194] The following is the second embodiment of the present invention. Different from the previous embodiment: the specific process of the steps for experimentally verifying the fault-tolerant performance of the two-dimensional processing platform experimental verification subsystem is further disclosed. Verifying the fault-tolerant performance of the experimental verification subsystem can verify the practicability of the theoretical design, ensure that the system can maintain the predetermined performance in the face of actual faults, and provide reliable technical support for practical applications.

[0195] Next, the interconnectivity of the two axes of the two-dimensional mobile platform (L = 2) is discussed, and G x and G y of the two subsystems are used to illustrate the feasibility of the technology

[0196]

[0197] In the above formula, K x , K y are the gain coefficients corresponding to the x-axis and y-axis, respectively set to 0.4 and 0.5. The time constants α x , α y are set to 1.65 and 5.5. The coupling gains H xy , H yx are selected as 0.1 and 0.05. α yx , α yx represent the coupling time constants, respectively defined as 2 and 1. The system is discretized with a sampling time of T = 0.01 s.

[0198] For the two-dimensional interconnected system, fault-tolerant design needs to be carried out for each subsystem. First, the fault-tolerant control of the x-axis is discussed. The expected trajectory of the subsystem G x is set to

[0199] y d1 = sin(16πt).*exp(-2t). (41)

[0200] The time-varying fault parameter and the external disturbance signal are respectively

[0201] Γ 1 = 0.85 + 0.15sin(2πt). (42)

[0202] w = 1.5sin(2πt - 0.34π). (43)

[0203] According to Step 6, the fault-tolerant control gain matrix can be obtained as

[0204] K 1x = [-1.1733 -1.0088], K 2x = 0.5044, K 3x = 0.5044.

[0205] To further evaluate the overall tracking control performance of the system, the root mean square (RMS) of the tracking error is introduced as a performance metric:

[0206]

[0207] The smaller the RMS value, the better its convergence effect.

[0208] In the formula, T represents the total number of data time steps, and e j (p, i) represents the error at the i-th data point under the p-th parameter setting.

[0209] Next, the y-axis is discussed, and the desired trajectory is set as a segmented trajectory

[0210]

[0211] Figure 2 is the control trajectory of the system without actuator faults. As the number of iterations increases, the subsystem G x can completely track the desired trajectory at the 10th iteration. This shows that the control system can achieve accurate tracking of the target trajectory in relatively few iterations.

[0212] Figure 3 is the control trajectory of the subsystem G x when an actuator fault occurs. When the system is disturbed by a time-varying fault signal, as shown in Figure 3 (a), the tracking performance of the system will decline. However, as the number of iterations increases, the designed algorithm control system can gradually overcome these interferences and achieve accurate tracking of the desired trajectory again. This process shows that although the time-varying fault signal will affect the system performance, through the iterative learning fault-tolerant control strategy, the system can still recover and maintain the ability to track the desired trajectory.

[0213] Figure 4 is the control trajectory of the system without actuator faults. As the number of iterations increases, the subsystem G y can completely track the desired segmented trajectory at the 10th iteration. This shows that the designed control effect is good.

[0214] Figure 5 For subsystem G y The control trajectory with actuator faults, when disturbed by time-varying fault signals, such as Figure 5 As shown in (a), the tracking performance deteriorates, but as the number of iterations increases, it gradually tracks the desired trajectory again.

[0215] Figure 6 The fault-tolerant control trajectory of the two-dimensional interconnected system shows the fault-tolerant effect of the two-dimensional interconnected system. When the actuator faults between the x-axis and y-axis interfere with each other, that is, when the 31st fault occurs, the movement trajectories of its two axes start to deviate. At this time, the fault-tolerant control starts to gradually correct its movement trajectory and finally tracks the desired trajectory.

[0216] Figure 7 The trajectory tracking error of the two-dimensional interconnected system. Comparing the designed fault-tolerant control with the P-type fault-tolerant control, it is obvious that the fault-tolerant control of the present invention has a smaller convergence error and a faster speed.

[0217] The time-domain fault-tolerant control method based on the interconnected two-dimensional system takes into account the system uncertainties, focuses on the time-domain analysis which concerns the system behavior changing with time, more comprehensively considers various uncertainties inside and outside the system, realizes the global fault-tolerant control of the entire space interconnected system by integrating the fault-tolerant control strategy into each subsystem, makes full use of the information interaction advantage between subsystems inside the system, realizes a refined control strategy, and improves the control accuracy and efficiency of the system;

[0218] Taking the two-dimensional interconnected system affected by actuator faults and external disturbances as the controlled object, designing the corresponding fault-tolerant mechanism by using the information between interconnected subsystems; the present invention deeply studies the robust control method for the specification constraint uncertainties existing in the system parameters and is committed to enhancing its robustness; the present invention solves the path tracking control problem of the interconnected system under actual disturbance conditions and realizes the high-precision tracking of the desired trajectory.

[0219] Importantly, it should be noted that the construction and arrangement of the present application shown in multiple different exemplary embodiments are merely illustrative. Although only a few embodiments are described in detail in this disclosure, those who refer to this disclosure should easily understand that many modifications are possible without substantially departing from the novel teachings and advantages of the subject matter described in this application. For example, the dimensions, scales, structures, shapes and proportions of various elements, as well as parameter values such as temperature, pressure, etc., installation arrangements, use of materials, color, orientation changes, etc. For example, an element shown as integrally formed may be composed of multiple parts or elements, the position of the element may be inverted or otherwise changed, and the nature, number or position of discrete elements may be changed or altered. Therefore, all such modifications are intended to be included within the scope of the present invention. The order or sequence of any process or method steps may be changed or reordered according to alternative embodiments. In the claims, any "means-plus-function" clause is intended to cover the structures that perform the functions described herein, and not only structural equivalents but also equivalent structures. Other substitutions, modifications, changes and omissions may be made in the design, operating conditions and arrangement of the exemplary embodiments without departing from the scope of the present invention. Therefore, the present invention is not limited to a specific embodiment, but extends to various modifications that still fall within the scope of the appended claims.

[0220] In addition, in order to provide a concise description of the exemplary embodiments, all features of the actual embodiments may not 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.

[0221] It should be understood that in the development of any actual implementation, as in any engineering or design project, a large number of specific implementation decisions may 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 excessive experimentation, the development efforts will be a routine task of design, manufacturing and production.

[0222] 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 time domain fault-tolerant control method based on an interconnected two-dimensional system, characterized in that: include, Establish an interconnected model system of 2D machining platforms; Establish an ILC model of a 2D machining platform including uncertain faults; Design subsystem control laws for 2D machining platforms; Analyze the stability of control algorithms for two-dimensional interconnected systems in the time domain; Design of control law for 2D machining platform based on LMI; Design fault-tolerant control laws for models that include uncertainty; The fault-tolerant performance of the subsystem is verified based on the 2D machining platform experiment.

2. The time domain fault-tolerant control method based on interconnected two-dimensional system according to claim 1, characterized in that: The interconnected model system and ILC model of the 2D machining platform are both constructed on a high-load electric linear 2D translation stage.

3. The time domain fault-tolerant control method based on an interconnected two-dimensional system according to claim 1 or 2, characterized in that: The steps to establish an ILC model of a 2D machining platform with uncertain faults include: Construct uncertainty models; Define interconnected system fault models; Reconstruction of the 2D machining platform containing the faulty ILC structure model.

4. The time domain fault-tolerant control method based on interconnected two-dimensional system according to claim 3, characterized in that: To analyze the stability of the control algorithm of a two-dimensional interconnected system in the time domain, three conditions need to be introduced. Specifically, (i)ρ(D2)<1; ensure batch stability. (ii)ρ(A2)<1; ensuring temporal stability. (iii) The transfer function matrix is ​​G(z) = C2(zI-A2) -1 B2+D2,z=e jω , The modulus of all eigenvalues ​​of is strictly less than 1.

5. The time domain fault-tolerant control method based on interconnected two-dimensional system according to claim 4, 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.

6. The time domain fault-tolerant control method based on interconnected two-dimensional system according to claim 5, characterized in that: The uncertainty model is: Where, X l (p+1) represents the value of the state vector at the next time step p+1, Y l (p) represents the output variable value at time step p, A, B and C represent the system matrix, corresponding to the state transfer matrix, input matrix and output matrix respectively, ΔA, ΔB, ΔC represent the uncertainty part of the system matrix, that is, the change amount of matrix A, B and C, U l (p) represents the value of the input signal at time step p.

7. The time domain fault-tolerant control method based on an interconnected two-dimensional system according to any one of claims 4 to 6, Features: The fault model of the interconnected system is: Where n represents the fault variable, Γ n Indicates the failure factor.

8. The time domain fault-tolerant control method based on interconnected two-dimensional systems according to claim 7, characterized in that: The reconstruction formula of the ILC structure model of the two-dimensional machining platform containing faults is as follows: Where, X I (p) represents the value of the state vector at time step p, Y I (p) represents the value of the output vector at time step p, A represents the state transfer matrix, which is used to describe how the state vector is transferred from one time step to the next time step, B represents the input matrix, which is used to describe the impact of the input on the state, C represents the output matrix, which is used to describe how the state vector is mapped to the output vector, Γ represents the barrier coefficient matrix, X l (p+1) represents the value of the state vector at time step p+1.

9. The time domain fault-tolerant control method based on interconnected two-dimensional systems according to claim 8, characterized in that: The steps of designing the subsystem control law of the 2D machining platform include: Define the ILC law, tracking error vector and state error vector; Update ILC Law; The controlled system is reconstructed as a discrete repetitive process model.

10. The time domain fault-tolerant control method based on interconnected two-dimensional systems according to claim 9, characterized in that: The controlled system is reconstructed into a discrete repetitive process model: In the formula, represents the new state vector at time step p+1, A2 represents the state transfer matrix, which is used to describe how the state vector is transferred from one time step to the next time step, B2 represents the input influence matrix, which is used to describe the influence of the tracking error on the state vector, e l (p) represents the tracking error at time step p in the lth iteration, e l+1 (p) represents the tracking error of the iteration at time step p, C2 represents the output matrix, which is used to describe how the state vector is mapped to the tracking error, and D2 represents the direct transfer matrix.

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