Fault-tolerant control method based on polyhedral uncertain interconnection system

Abstract

The invention discloses a fault-tolerant control method based on a polyhedral uncertain interconnection system, and the method comprises the steps: building a spatial interconnection system state space model with polyhedral uncertainty and external interference, and constructing a dynamic model through representing an actuator fault as a structured uncertainty parameter with a known boundary; constructing an iterative learning control increment model containing an actuator fault based on the model, and defining a system operation batch, a finite time period, an expected trajectory and a tracking error; designing an I LC updating law, and further exporting a linear discrete repetition process model represented by an extended state vector; for repetitive and non-repetitive disturbances, calculating a controller gain matrix which ensures that the system has robust stability and H-infinity performance indexes by solving a linear matrix inequality; the high-precision tracking performance and the strong fault-tolerant capability of the system under the conditions of multi-source uncertainty, interference and actuator faults are verified through online experimental operation and root mean square error calculation.

Description

Technical Field

[0001] This invention belongs to the field of automatic control technology, specifically relating to a fault-tolerant control method based on a polyhedral uncertain interconnected system. Background Technology

[0002] In the field of precision machining and manufacturing, such as the motion axis units of multi-axis gantry milling machines and the multi-printhead array system of 3D printers, multiple units exhibit strong mechanical coupling and dynamic interaction during coordinated motion. These systems are typically modeled as spatially interconnected systems, characterized by interrelationships between subsystems through boundary variables, and model parameters are susceptible to external factors such as workload and temperature changes, exhibiting significant polyhedral uncertainty. Furthermore, actuators may experience performance degradation or even complete failure due to mechanical wear, aging, or sudden malfunctions during long-term operation, further increasing the complexity of system control.

[0003] Existing control methods face the following challenges when dealing with such problems:

[0004] Insufficient uncertainty modeling: Traditional control strategies often fail to fully consider the changes of parameters in the polyhedral uncertainty set, making it difficult to guarantee the robustness of the system under all operating conditions;

[0005] Interconnection effects are difficult to handle: the strong coupling between subsystems complicates centralized control design, while decentralized control often ignores interconnection effects, leading to performance degradation;

[0006] Limited fault tolerance: Most methods do not incorporate actuator failures into the controller design phase, but instead take passive compensation measures after a failure occurs, resulting in a large response delay;

[0007] Weak disturbance suppression performance: When both repetitive disturbances (such as periodic processing forces) and non-repetitive disturbances (such as random external disturbances) exist, existing iterative learning control (ILC) methods lack a unified framework for disturbance suppression and stability analysis.

[0008] Therefore, there is an urgent need for a fault-tolerant control method that can uniformly handle polyhedral uncertainties, dynamic interconnections, actuator failures, and mixed-type disturbances, in order to improve the tracking accuracy and operational reliability of complex interconnected systems under fault conditions. Summary of the Invention

[0009] In view of this, the main objective of the present invention is to provide a fault-tolerant control method based on a polyhedral uncertain interconnection system.

[0010] To achieve the above objectives, the technical solution of the present invention is implemented as follows:

[0011] A fault-tolerant control method based on a polyhedral uncertain interconnected system, the method comprising:

[0012] S1. System modeling steps: Establish a state-space model of a spatially interconnected system with polyhedral uncertainties and external disturbances; based on the actuator fault model, characterize the fault as a structured uncertainty parameter with known boundaries to obtain a unified dynamic system model that includes uncertainties, disturbances, interconnection effects and actuator faults;

[0013] S2. ILC Model Construction Steps: Based on the unified dynamic system model obtained in S1, establish an iterative learning control (ILC) model that includes actuator faults; define the system running batch (k) and finite working cycle (p), and construct an ILC system incremental model that includes fault parameters; define the desired output trajectory and tracking error to obtain a system incremental model suitable for ILC analysis and design;

[0014] S3. Control Law Design Steps: Based on the system incremental model obtained in S2, design an ILC update law; and substitute the control law into the system incremental model to obtain a linear discrete repetitive process model represented by an extended state vector;

[0015] S4. Stability analysis steps for repetitive disturbances: Based on the linear discrete repetitive process model obtained in S3, solve the linear matrix inequality (LMI) constraints for repetitive disturbances, and calculate the controller gain matrix that can guarantee the robust stability and H∞ performance index of the system under repetitive disturbances. This gain matrix will be used for system control.

[0016] S5. Non-repetitive disturbance stability analysis steps: Based on the same linear discrete repetitive process model obtained in S3 and the solution method in S4, solve the LMI constraint conditions for non-repetitive disturbances, and calculate the controller gain matrix that can guarantee the robust stability and H∞ performance index of the system under non-repetitive disturbances. The gain matrix will be used for the control of the space interconnection system.

[0017] S6. Experimental verification steps: Based on the controller gain matrix calculated in S5, online operation is performed. By calculating the root mean square error (RMSE) performance index, the tracking performance and fault tolerance performance of the spatial interconnection system under the conditions of polyhedral uncertainty, external interference and actuator failure are verified.

[0018] Preferably, the state-space model of the spatial interconnection system is as follows:

[0019]

[0020] Where p represents time, i∈{1,2,...n} represents the index of the subsystem in space, and x i (p) represents the state of the i-th subsystem at time p, where n is the number of subsystems, u i(p) represents the input of the i-th subsystem at time p, y i (p) represents the output of the i-th subsystem at time p, d i (p) represents external disturbances to the subsystem, v i (p) and w i (p) Characterizes the spatial interconnection of subsystems.

[0021] Preferably, in step S1, the spatial interconnection system is converted into a standard form by defining a lifting vector, and its equivalent state-space model is expressed as:

[0022] The coefficient matrices have the following structure:

[0023]

[0024] Preferably, in step S2, the actuator fault model is defined as:

[0025] α = 1, 2, ..., m, where the failure coefficient Γ α satisfy Their ranges are respectively Γ α ≤1, It is known, that is, let the unknown variable Γ be. α Variation within a known range, when Γ α =1 corresponds to no system faults. Γ α =0 corresponds to a complete failure; 0 < Γ α ≤Γ α <1 or For the corresponding partial faults, the fault model is integrated into the system model obtained in S1, and then reconstructed into the ILC system incremental model described in S2.

[0026] Preferably, in step S3, the ILC update law is designed as follows:

[0027] U k+1 (p)=U k (p)+Δ k+1 (p),

[0028] Where Δ k+1 (p) represents the correction amount designed based on historical error data;

[0029] By defining the extended state vector Substituting the system incremental model obtained from S2, the linear discrete repetitive process model required for subsequent stability analysis is derived:

[0030]

[0031] Preferably, in step S4, the stability of the nominal discrete repetitive process under repetitive perturbations is addressed by solving the linear matrix inequality (LMI): in

[0032]

[0033] Υ6 = -sym(F4).

[0034] The controller gain matrix that guarantees the robust stability and H∞ disturbance attenuation performance of the system is calculated. The controller gain matrix is ​​the direct input of the S6 experimental verification step. For uncertain discrete repetitive processes, the LMI is required to hold for all vertices i = 1, ..., l of the convex polyhedron.

[0035] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0036] This invention first models actuator failure as a structured uncertainty and integrates it into the system's iterative learning control framework. Then, by transforming the system model into a linear discrete repetitive process, the controller design problem is transformed into a problem of solving LMIs. By solving these LMIs offline, the controller gain can be obtained, which can simultaneously guarantee the stability of the system under repetitive and non-repetitive disturbances and has specified fault tolerance performance. Finally, to verify the effectiveness of the algorithm, experiments are conducted on an interconnected system. Attached Figure Description

[0037] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and, together with their descriptions, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0038] Figure 1 The present invention provides output response curves of subsystem 1 under repetitive disturbance in different operating batches.

[0039] Figure 2 The present invention provides output response curves of subsystem 2 under repetitive disturbance in different operating batches.

[0040] Figure 3 For designing curves;

[0041] Figure 4 The curve showing the change of the RMS of an uncertain system with the increase of test batches under repeated perturbations;

[0042] Figure 5The present invention provides output response curves of subsystem 1 under non-repetitive perturbation in different operating batches;

[0043] Figure 6 The present invention provides the output response curves of subsystem 2 under non-repetitive disturbance in different operating batches;

[0044] Figure 7 The present invention provides the output response curves of subsystem 3 under non-repetitive disturbance in different operating batches;

[0045] Figure 8 The processing curve is for non-repetitive interference. Detailed Implementation

[0046] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0047] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, article, or apparatus that includes that element.

[0048] The core of this invention lies in: firstly, modeling actuator failure as a structured uncertainty and integrating it into the system's iterative learning control framework; then, transforming the system model into a linear discrete repetitive process, and converting the controller design problem into a problem of solving LMIs; by solving these LMIs offline, the controller gain can be obtained, which can simultaneously guarantee the stability of the system under repetitive and non-repetitive disturbances and possess specified fault-tolerant performance; finally, to verify the effectiveness of the algorithm, experimental verification is conducted on an interconnected system.

[0049] This invention provides a fault-tolerant control method based on a polyhedral uncertain interconnected system, the method comprising:

[0050] S1. Establish a state-space model of a spatial interconnected system with polyhedral uncertainties and external disturbances; based on the actuator fault model, characterize the fault as a structured uncertainty parameter with known boundaries.

[0051] S2. Establish an ILC model that includes actuator faults: Define system operation batches and finite working cycles, construct an incremental ILC system model that includes fault parameters; define the expected output trajectory and tracking error.

[0052] S3. Design the control law for the space interconnection system: Design an update law and substitute it into the system incremental model to obtain a linear discrete repetitive process model represented by an extended state vector. The output of this model provides the basis for subsequent stability analysis and controller design.

[0053] S4. Analyze the stability of the control algorithm under repetitive disturbances: Based on the linear discrete repetitive process model, solve the linear matrix inequality (LM I) constraints for repetitive disturbances, and calculate the controller gain matrix that guarantees the robust stability and H∞ performance index of the system. This step is completed before the system is put into operation, and the obtained gain will be used for control.

[0054] S5. Analysis of the stability of the control algorithm under non-repetitive disturbances: Based on S4, for the LMI constraint condition of non-repetitive disturbances, the controller gain matrix that guarantees the robust stability and H∞ performance index of the system is calculated. The obtained gain will be used for the control of the space interconnection system.

[0055] S6. Experimental verification of the fault-tolerant performance of the space interconnection system: Based on the online operation described in S5, the tracking performance and fault-tolerant performance of the space interconnection system under conditions of polyhedral uncertainty, external interference, and actuator failure are verified by calculating the root mean square error (RMSE) performance index.

[0056] Step 1: Establish a state-space model of a spatial interconnected system with polyhedral uncertainties and external disturbances.

[0057] This invention considers a class of spatial interconnection systems with polyhedral uncertainty and external disturbances.

[0058]

[0059] in Represents the state matrix, Indicates interconnected input v i The influence matrix on the state, Indicates the disturbance input d i The influence matrix on the state, Indicates control input u i The influence matrix on the state, Indicates that w i The matrix, Let represent the output matrix and the direct transfer matrix, θ represent the time-varying parameters of the system, p represent time, i∈{1,2,...n} represent the index of the subsystem in space, and x represent the time-varying parameters of the system.i (p) represents the state of the i-th subsystem at time p, where n is the number of subsystems, u i (p) represents the input of the i-th subsystem at time p, y i (p) represents the output of the i-th subsystem at time p, d i (p) represents external disturbances to the subsystem, v i (p) and w i (p) Characterizes the spatial interconnection of subsystems.

[0060] In practical applications, Typically, it is a zero matrix. Furthermore, the uncertain model matrix in system (1) belongs to a convex bounded uncertain domain, which can be represented as...

[0061]

[0062] in

[0063]

[0064] Among them G j j = 1, ..., l represents the vertices of the convex polyhedron, l is the number of vertices, and θ j It is a weighting coefficient, with a value ranging from 0 to 1.

[0065] To facilitate system analysis, the space interconnection system is converted into a standard form by defining the following lift vector.

[0066]

[0067] Where X(p) represents the overall state vector, V(p) represents the overall interconnect input vector, D(p) represents the overall external disturbance vector, U(p) represents the overall control input vector, W(p) represents the overall interconnect output vector, and Y(p) represents the overall measurement output vector.

[0068] Based on the above vector definition, the equivalent state-space model of this uncertain spatial interconnection system can be expressed as follows:

[0069]

[0070] in

[0071]

[0072] Where A 11 (θ) represents the system state matrix, A 12 (θ) represents the system interconnection input matrix, B 11 (θ) represents the system disturbance input matrix, B 12 (θ) System control input matrix. A 21A represents the system interconnection output matrix. 22 C1 represents the system interconnection output matrix, and C2 represents the system measurement output matrix. This represents the state matrix of the i-th subsystem. This represents the interconnection input matrix of the i-th subsystem. Let represent the perturbation input matrix of the i-th subsystem. Let represent the control input matrix of the i-th subsystem.

[0073] It should be noted that equation (4) needs to be simplified due to the inclusion of interconnected variables and polyhedral uncertainties.

[0074] Utilizing the interconnection property (3) and the corresponding boundary condition v + (1) = w - (1)=0,v - (n)=w + (n) = 0, which allows establishing relationships between interconnected variables.

[0075]

[0076] Where η is the permutation matrix representing the spatial interconnection structure. Combining equations (5) and (4), the state-space model of the system can be derived.

[0077]

[0078] The coefficient matrices have the following structure:

[0079]

[0080] The core output of this step is a unified system dynamic model that incorporates uncertainties, disturbances, interconnection effects, and actuator failures. This model forms the basis and prerequisite for all work in the second step.

[0081] The second step is to establish an ILC model that includes actuator failures.

[0082] Space interconnection systems face numerous disturbances and uncertainties. Fault-tolerant control systems can detect and correct errors during transmission, and this applies to the system's control input U. α (p,k), An input signal representing a fault in an interconnected system is defined by the following fault model:

[0083]

[0084] Where α represents a variable indicating a failure, k represents the current batch of the system in operation, and the failure coefficient Γ α The following conditions must be met

[0085] Their ranges are respectively Γ α ≤1, It is known, that is, let the unknown variable Γ be. α Variation within a known range, when Γ α =1 corresponds to no system faults. Γ α =0 corresponds to a complete failure; 0 < Γ α ≤Γ α <1 or This corresponds to a partial fault. For example, when the output of the faulty brake is greater than the output of the normal controller, it indicates partial degradation or abnormality of the brake.

[0086] Define the following matrix

[0087]

[0088] U F Represents the uncertain input matrix, The upper bound matrix representing uncertainty. Γ Γ represents the lower bound matrix of uncertainty, and Γ represents the actual uncertainty matrix.

[0089] and

[0090]

[0091] Where q represents the scaling matrix, q α q represents the scaling factor for uncertainty under fault α. α0 This represents the normalized bias of the uncertainty under fault α.

[0092] Introduction symbols

[0093]

[0094] Where Γ 0α This represents the normalized uncertainty under fault α, which indicates the actual uncertainty relative to the nominal value q. α The relative deviation, |Γ0| represents the absolute value matrix of the normalized uncertainty.

[0095] Using the above formula, Γ can be written as

[0096] Γ=(I+Γ0)q (13)

[0097] in

[0098] |Γ0|≤q0≤I (14)

[0099] In this invention, it is assumed that the upper and lower bounds of the fault range of each actuator are known, i.e. Γα and Therefore, an additional parameter q α As an entry in vector q, it is used to adjust the initial range of the unknown scalar Γ0. Thus, in (9), the fault model Γ can be regarded as a structured uncertainty with known q0 and unknown vector Γ0, as shown in equation (13).

[0100] The subsystem contains faults. The ILC structural model is reconstructed as follows:

[0101]

[0102] Where k represents the current batch of the system, and p∈[0,t] represents the finite working cycle of each batch of the system.

[0103] Therefore, all the work in the second step is to reconstruct the general system model from the first step into a form that is very suitable for iterative learning control analysis and design, thereby providing tools and clarifying the objectives for designing specific control laws in the third step.

[0104] The third step is to design the control laws for the spatial interconnection system.

[0105] For the iterative learning control strategy, the following update law is designed.

[0106] U k+1 (p)=U k (p)+Δ k+1 (p) (16)

[0107] Where Δ k+1 (p) represents the correction amount designed based on historical error data. The desired output trajectory Y is set. r (p), then the tracking error of the system during the (k+1)th run can be expressed as

[0108] e k+1 (p)=Y r (p)-Y k+1 (p) (17)

[0109] To simplify the analysis process, the following incremental variables are defined, including the perturbation error vector.

[0110]

[0111] β k+1 (p+1)=D k+1 (p)-D k (p) (19)

[0112] The analysis makes the following reasonable assumption: the initial state of the system is consistent in each run, i.e., Y r (0)=Y k(0)=CX k (0) and initial state deviation Based on this, the system state increment satisfies:

[0113] and

[0114]

[0115] The update mechanism of the ILC algorithm is designed as follows.

[0116]

[0117] in, K1, K2, and K3 are the gain matrices to be designed. The update term consists of feedback information and PD-type previous tracking error information.

[0118] Define extended state vector Let K = K2 - K3, and substitute equation (22) into system equations (20) and (21), we can derive the following linear discrete repetitive process model.

[0119]

[0120] The coefficient matrices are defined as follows:

[0121]

[0122] Based on this discrete repetitive process model, subsequent work will focus on system stability analysis.

[0123] The fourth step is to analyze the stability of the control algorithm under repetitive disturbances.

[0124] If the nominal discrete repetitive process (23) is subjected to repetitive perturbation, i.e., β k+1 A sufficient condition for batch stability at the lower edge of (p) = 0 is that there exist matrices S = diag{S1,S2} > 0, S3 > 0, F = diag{F1,F2,F3,F4}, W1, W2,W3, such that the following LMI holds.

[0125]

[0126] in

[0127]

[0128] Υ6 = -sym(F4).

[0129] Then the gain of the ILC learning law is

[0130]

[0131] If the discrete repetitive process (23) is uncertain in the repetitive perturbation, i.e., β k+1 The necessary and sufficient condition for batch stability at the lower edge of the action (t) = 0 is that there exist matrices S = diag{S1,S2} > 0, S3 > 0, F = diag{F1,F2,F3,F4}, W1,W2,W3, such that the following LMIs hold for any j = 1,…,l.

[0132]

[0133] in

[0134]

[0135] Then the gain of the ILC learning law is

[0136]

[0137] Although this step has addressed the stability issue of discrete repetitive processes caused by repetitive perturbations, the impact of non-repetitive perturbations has not yet been discussed. Therefore, step five proposes LMI conditions to ensure batch robust stability of the system under non-repetitive perturbations, for both nominal and uncertain cases.

[0138] Step 5: Analyze the stability of the control algorithm under non-repetitive disturbances.

[0139] This step focuses on discrete repetitive processes (23) with external disturbances, and systematically analyzes their behavior under non-repetitive disturbances (i.e., β). k+1 Dynamic performance under the action of (p)≠0). First, for the nominal system, robust stability and H∞ disturbance attenuation performance criteria based on LMI are established. Furthermore, for the case of parameter uncertainty, the necessary and sufficient conditions for ensuring that the system simultaneously satisfies robust stability and a specified H∞ performance level are derived, and the gain matrix of the ILC learning law is explicitly constructed accordingly. Next, rigorous conclusions and controller synthesis methods for the nominal system and the uncertain system are given respectively.

[0140] If, for the nominal discrete repetitive process (23), there exist positive definite diagonal matrices S = diag{S1,S2}>0, S3>0, and W1,W2,W3 and block diagonal matrices F = diag{F1,F2,F3,F4} such that the following linear matrix inequality holds, then the system under non-repetitive disturbances, i.e., β... k+1 (p)≠0 can achieve batch robust stability and possess H ∞ Disturbance attenuation performance.

[0141]

[0142] in

[0143]

[0144] Θ7=-γ 2 I,Θ8=-sym(F4)

[0145] If a feasible solution exists for LMI, then the gain matrix of the learning law (22) of ILC can be determined by equation (26).

[0146] If we consider the existence of a discrete, repetitive process with uncertain parameters (23), when the system is subjected to a non-repetitive disturbance (β) k+1 When (p)≠0) is applied, it simultaneously satisfies robust stability and H along the batch direction. ∞ The necessary and sufficient condition for disturbance attenuation performance is that there exist positive definite diagonal matrices S = diag{S1,S2}>0, S3>0, and block diagonal matrices F = diag{F1,F2,F3,F4} and gain matrices W1,W2,W3, such that for all i = 1,...,l, the following linear inequality holds.

[0147]

[0148] in

[0149]

[0150] If the above LMIs have feasible solutions, then the gain matrix of the ILC algorithm can be expressed as:

[0151]

[0152] Step 6: Experimentally verify the fault-tolerant performance of the space interconnection system.

[0153] To verify the effectiveness of the fault-tolerant control method proposed in this invention, an experiment was conducted on a space interconnection system consisting of three subsystems.

[0154] The experiment is divided into two parts: repeated disturbance test and non-repeated disturbance test. The core purpose of this experiment is to verify the following two technical effects: (1) whether the designed controller can stabilize the system and accurately track the desired trajectory under disturbance and fault; (2) whether the method has effective fault tolerance capability.

[0155] The first step is the repetitive perturbation test, where the repetitive perturbation is set to...

[0156]

[0157] According to step four, its control matrix can be obtained as follows:

[0158] K1=[-1.65100.0837], K2=1.47873, K3=0.0272.

[0159] RMS is introduced as an indicator to evaluate the tracking performance of space interconnection systems.

[0160]

[0161] Set the time-varying actuator fault to

[0162] Γ = 0.85 + 0.35sin(πp).

[0163] Next, the fault-tolerant control effects of the two subsystems will be tested separately:

[0164] Depend on Figure 1 The output response curves of subsystem 1 under repetitive perturbation in different batches are shown. Since the repetitive perturbation is canceled out during the batch process, the control input is continuously optimized with the increase of the number of iterations, the output completely tracks the desired trajectory, and the root mean square error converges to zero along the batch, verifying the effectiveness of the method. Then, fault tolerance testing is performed on subsystem 2, such as... Figure 2 As shown, the same conclusion can be obtained.

[0165] Figure 3 To design the machining curve, data shows that in the absence of actuator failure, the system can perfectly track its reference trajectory by the 30th iteration. However, when the actuator fails, the operating curve deviates significantly from the machining target. Assume the system suffers an actuator failure in the 31st iteration. Figure 4 The fault-tolerant control chart shows that the fault-tolerant control begins to take effect. In the 32nd and 33rd iterations, its running curve gradually approaches the processing curve and finally fully tracks it, proving the effectiveness of the algorithm.

[0166] Figure 4 This study demonstrates the variation of the RMS of an uncertain system with increasing number of test batches under repetitive perturbations. The error curve exhibits a gradual decay and convergence characteristic, asymptotically approaching zero. In the initial experimental phase, the system is significantly affected by repetitive perturbations, resulting in a large tracking error. As the number of iteration batches increases, the fault-tolerant control mechanism continuously suppresses the perturbation effect and compensates for system uncertainty, leading to a continuous reduction in error. Even in the event of actuator failure, the system can still continuously correct deviations through iterative learning, demonstrating good robustness and fault tolerance.

[0167] Next, we tested the fault tolerance effect of non-repetitive perturbations.

[0168] Set the non-repetitive perturbation as

[0169]

[0170] The time-varying actuator fault is

[0171] Γ = 0.65 + 0.15sin(πp)

[0172] According to step five, its control matrix can be obtained as follows:

[0173] K1=[-1.89090.0920], K2=0.9314, K3=0.332

[0174] This invention conducted operational observations on three subsystems. When the system is subjected to non-repetitive perturbations, the tracking performance is significantly affected in the initial stage. For example... Figure 5 , Figure 6 and Figure 7 As shown in neutron diagram a, the output trajectory exhibits strong jitter and irregular motion, deviating significantly from the set value. With iterative learning, the control algorithm gradually suppresses disturbances and optimizes the input, and the outputs of each subsystem gradually approach the desired trajectory, ultimately achieving accurate tracking (see...). Figure 5 , Figure 6 , Figure 7 (See subgraph b in the diagram). To further verify the system's fault tolerance, an actuator fault was artificially introduced in the 31st iteration. At this point, the system output deviated significantly, and the tracking performance temporarily decreased (see...). Figure 5 , Figure 6 , Figure 7 (See subgraph c in the original text). However, based on the invented fault-tolerant control mechanism, the system can adjust the control strategy online, effectively compensate for the impact of faults, and ultimately make the output trajectory reconverge to the reference trajectory (see...). Figure 5 , Figure 6 , Figure 7 (See subgraph d in the diagram). The above results consistently demonstrate that the control method proposed in this invention still possesses good learning recovery capability and robust performance when facing non-repetitive disturbances and sudden actuator failures.

[0175] The goal is to process into Figure 8 The red curve in the graph shows that when an actuator malfunctions in the system, the output machining trajectory deviates significantly from the reference trajectory and fluctuates greatly, exhibiting irregular movement. However, after adding fault-tolerant control to the system, its output trajectory gradually tracks the reference trajectory, demonstrating good fault-tolerance performance.

[0176] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention.

Claims

1. A fault-tolerant control method based on a polyhedral uncertain interconnection system, characterized by, The method comprises: S1. System modeling step: establishing a state space model of a spatial interconnected system with polytopic uncertainty and external disturbance; based on an actuator fault model, characterizing the fault as a structured uncertainty parameter with known boundaries, obtaining a unified dynamic system model containing uncertainty, disturbance, interconnected effects and actuator faults; S2. ILC model construction step: based on the unified dynamic system model obtained in S1, establishing an iterative learning control (ILC) model containing actuator faults; defining system running batch (k) and limited working period (p), constructing an ILC system incremental model containing fault parameters; defining expected output trajectory and tracking error, obtaining a system incremental model suitable for ILC analysis and design; S3. Control law design step: based on the system incremental model obtained in S2, designing an ILC update law; and substituting the control law into the system incremental model to obtain a linear discrete repetitive process model represented by an extended state vector; S4. Repetitive disturbance stability analysis step: based on the linear discrete repetitive process model obtained in S3, solving the linear matrix inequality (LMI) constraint condition for repetitive disturbance respectively, calculating the controller gain matrix that can guarantee the robust stability and H performance index of the system under repetitive disturbance, which will be used for system control; S5. Non-repetitive disturbance stability analysis step: based on the same linear discrete repetitive process model obtained in S3 and the solving method of S4, solving the LMI constraint condition for non-repetitive disturbance, calculating the controller gain matrix that can guarantee the robust stability and H performance index of the system under non-repetitive disturbance, which will be used for control of the spatial interconnected system; S6. Experimental verification step: based on the controller gain matrix calculated in S5, performing online operation, verifying the tracking performance and fault tolerance performance of the spatial interconnected system under the condition of polytopic uncertainty, external disturbance and actuator fault by calculating the root mean square error (RMSE) performance index.

2. The fault-tolerant control method based on polyhedral uncertain interconnection system according to claim 1, wherein, The state space model of the spatial interconnected system is: where p denotes the time, i∈{1,2,...n} denotes the serial number of the subsystem in space, x i (p) denotes the state of the i-th subsystem at time p, n is the number of subsystems, u i (p) denotes the input of the i-th subsystem at time p, y i (p) denotes the output of the i-th subsystem at time p, d i (p) denotes the external disturbance of the subsystem, v i (p) and w i (p) represents the spatial interconnection of the subsystem.

3. The fault-tolerant control method based on polyhedral uncertain interconnection system according to claim 1 or 2, characterized in that, In the step S1, the spatial interconnected system is converted into a standard form by defining a lifting vector, and its equivalent state space model is represented as: Wherein each coefficient matrix has the following structure:

4. The fault-tolerant control method based on polyhedral uncertain interconnection system according to claim 3, wherein, In the step S2, the actuator fault model is defined as: where the fault coefficient Γ α satisfies whose ranges are Γ α ≤1, is known, i.e. let the unknown variable Γ α vary within a known range, when Γ α = 1 corresponds to no fault of the system, Γ α = 0 corresponds to a complete fault; 0 < Γ Γ α ≤ Γ α < 1 or corresponds to a partial fault, which is incorporated into the system model obtained by S1, and then reconstructed into the ILC system incremental model described by S2.

5. The fault-tolerant control method based on polyhedral uncertain interconnection system according to claim 4, wherein, In the step S3, the ILC update law is designed as: U k+1 (p) = U k (p) + Δ k+1 (p), where Δ k+1 (p) represents a correction designed based on historical error data; By defining an extended state vector And substituting the obtained system increment model of S2, the linear discrete repetitive process model required for subsequent stability analysis is derived:

6. The fault-tolerant control method based on polyhedral uncertain interconnection system according to claim 5, wherein, In the step S4, the stability of the nominal discrete repetitive process under the repetitive disturbance is solved by solving a linear matrix inequality (LMI): wherein The controller gain matrix that guarantees the robust stability and H∞ disturbance attenuation performance of the system is calculated, which is the direct input of the S6 experimental verification step; for uncertain discrete repetitive processes, the LMI must be true for all vertices i = 1, …, l of the convex polyhedron.

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