A flexible robot arm vibration control method based on T-S fuzzy modeling

Abstract

The application discloses a flexible mechanical arm vibration control method based on T-S fuzzy modeling, belongs to the technical field of intelligent manufacturing and distributed parameter system control, and comprises the following steps: a T-S fuzzy distributed parameter model of a flexible mechanical arm considering nonlinear disturbance and time-varying time delay is constructed; a fuzzy boundary observer is designed under the condition of limited boundary measurement, and state estimation is realized; a controller random failure modeling mechanism based on a Markov process is introduced, and a fault-tolerant boundary controller is designed; dynamic boundary sampling is realized in combination with AETM, and communication resource consumption is reduced; and LKFs and LMI methods are used to construct stability and performance criteria. The application realizes robust stability and performance guarantee of flexible mechanical arm vibration suppression under the condition of limited measurement and random controller failure.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of intelligent manufacturing and distributed parameter system control, and particularly relates to a flexible mechanical arm vibration control method based on T-S fuzzy modeling. BACKGROUND

[0002] In recent years, parabolic partial differential equation systems (PDEs) have become a common tool for characterizing spatiotemporal dynamics and are widely used in fields such as heat conduction, reaction-diffusion transport, and aerospace structure thermal regulation. However, due to the infinite-dimensional nature of distributed parameter systems (DPSs), the stability analysis and control design still face great challenges. At the same time, problems such as time delay, non-collocated sensor-actuator configuration, and nonlinear uncertainty are common in actual flexible manipulator systems, further exacerbating the complexity of controller design.

[0003] To address these challenges, researchers have proposed various analysis tools, including boundary control frameworks based on Lyapunov methods, semigroup theory, and backstepping techniques. At the same time, Takagi-Sugeno (T-S) fuzzy modeling has been gradually applied to the analysis and design of uncertain nonlinear systems due to its ability to decompose complex nonlinear systems into several local linear subsystems, which can be smoothly combined with linear control theory. This method has been extended to the boundary control and output feedback design of distributed parameter systems and has made progress in fuzzy sampled control of parabolic PDEs.

[0004] In terms of reducing communication burden, event-triggered sampled control (ETSD) has gradually gained attention by updating control or observation signals only when necessary, significantly improving resource utilization, and has been verified in ordinary differential equations and PDEs scenarios. However, most existing research assumes full-state measurement or sensor-actuator spatial collocation, and pays insufficient attention to limited measurement and non-collocated configurations. At the same time, with the popularity of networked control systems, network attack threats such as denial of service (DoS), spoofing, and replay attacks have become key factors affecting system operation. For example, some methods analyze control accuracy under DoS interference in a two-channel framework, some methods study stability problems under bandwidth constraints, and some methods propose fault-tolerant fuzzy control methods combined with event triggering, but the consideration of actuator failure is still insufficient.

[0005] In practical engineering, flexible manipulators and other actuators often experience random failures due to mechanical wear, environmental disturbances, or aging, leading to degraded system performance or even instability. To improve the robustness and safety of the system, existing research has introduced fault-tolerant control mechanisms and explored fuzzy control methods combined with random actuator failures and spoofing attacks. In addition, some research has proposed non-collocated output feedback strategies under intermittent boundary measurement conditions for reaction-diffusion systems.

[0006] With the rapid development of intelligent manufacturing, flexible manipulators have become the core equipment in automated production and precision operation, and the requirements for vibration suppression and operation stability are continuously increasing. As a typical distributed parameter system, it is difficult to obtain full state information under limited measurement conditions, and the controller is prone to cause system instability under random failure or network attack interference. The existing methods lack effective fault tolerance and vibration suppression mechanisms.

[0007] Therefore, how to realize efficient vibration control of flexible manipulators under limited measurement conditions, and maintain system stability in the presence of random controller failure and complex network environment, has become a key problem to be solved. SUMMARY

[0008] The purpose of the present application is to overcome the problems existing in the prior art, and to provide a flexible manipulator vibration suppression fault-tolerant boundary control method based on T-S fuzzy modeling under limited measurement and random controller failure conditions. It ensures that efficient vibration suppression of flexible manipulators can be achieved under the condition of limited sensor configuration and random failure of the controller, and improves the robustness and reliability of the system.

[0009] The purpose of the present application is achieved by the following technical solutions:

[0010] A flexible manipulator vibration control method based on T-S fuzzy modeling, as shown in Figure 1 , comprising:

[0011] Step 1. Establish a T-S fuzzy distributed parameter system model of the flexible manipulator;

[0012] Step 2. Based on the T-S fuzzy distributed parameter system model, use non-co-located sensors to obtain limited measurement signals at the left end boundary thereof, and construct an adaptive event-triggered sampling mechanism to output sampled signals;

[0013] Step 3. Design a fuzzy boundary observer under limited boundary measurement based on the sampled signals, for estimating the full state of the system;

[0014] Step 4. Based on the output of the fuzzy boundary observer and the sampled signals, design a non-co-located fuzzy boundary sampled controller considering Markov random failure.

[0015] As a further improvement of the present application, the step 1 is specifically as follows:

[0016] A T-S fuzzy parabolic DPSs model (including boundary actuation) of the flexible manipulator is established:

[0017] (S1)

[0018] Where, the spatial variable , For the set of all real numbers, time System status express 3D real space; For disturbances outside the boundary; Satisfies the Lipschitz condition; Here is the diffusion matrix; time delay Differentiable The initial conditions are: Right boundary From input effect, For the matching matrix, Indicates the left boundary; Indicates system state For spatial variables The first-order partial derivative, This indicates that a symmetric positive definite diffusion matrix is ​​applied to the second-order spatial derivative. , This represents the bounded external perturbation in the spatial-temporal distribution of the flexible robotic arm.

[0019] Introducing Prerequisite Variables For the Rules have:

[0020] rule :if yes as well as yes ,So:

[0021] (S2)

[0022] in This represents the corresponding fuzzy set, for the index set. Each rule ,matrix It is a known constant matrix.

[0023] The fuzzy membership function is defined as follows:

[0024]

[0025]

[0026] Fuzzy system in position Input vector at time t From each input variable Composition, the first Weight of each rule Defined as its activation strength The ratio of the activation strength of each rule to the sum of all activation strengths, while the activation strength is given by the product of the membership function values of each input variable under the rule .

[0027] Then the weighted superposition form is obtained from (S2) as

[0028] (S3)

[0029] The weight of the first rule is

[0030] (S4)

[0031] As a further improvement of the present application, step 2 is specified as follows:

[0032] The finite measurements are taken at the left end boundary and an adaptive event-triggered sampling mechanism (AETM) is adopted. First, define the boundary measurements and the sampling time sequence:

[0033] (S5)

[0034] where, is the boundary measurement at time ; is the value of the state at the left end boundary described in step 1; is the release / sampling time sequence.

[0035] In order to suppress redundant transmission and improve communication utilization, the event-triggered principle with time-varying threshold is used to determine the next trigger time:

[0036] (S6)

[0037] where, denotes the (k+1)th sampling time, k denotes the number of sampling time, h denotes the candidate time, denotes the (k+h)th candidate time with the kth sampling time as the starting point, , , is the error signal based on the latest trigger sample value and the current candidate sample value, is the trigger threshold function, denotes the boundary measurement at the sampling time , is the designed weight matrix, and the superscript T denotes the transpose of the corresponding vector or matrix.

[0038] ​The threshold function adopts a form of self-adaptive decay with error:

[0039] (S7)

[0040] wherein, are the lower / upper limits of the threshold respectively and is the error influence coefficient, and the error increase will make decrease, thereby triggering more active.

[0041] Accordingly, at the triggering moment the sampling value actually entering the control / observation loop is defined as

[0042] (S8)

[0043] wherein, is the actual sample value sent / executed at the moment ; represents the candidate sample value within the current window that makes formula (S7) true, represents the candidate sample value within the current window that makes formula (S7) not true. Through the AETM described in formulas (S5)-(S8), the redundant reporting can be adaptively inhibited while the control performance is guaranteed, the communication resource occupation and the system response speed are balanced, and the event-driven sampling input is provided for the subsequent boundary observer and non-co-located fault-tolerant boundary control law.

[0044] As a further improvement of the present application, the step 3 is specifically as follows:

[0045] At the left end boundary , an estimated type boundary measurement is acquired by using a non-co-located sensor, and a fuzzy boundary observer is constructed based on the measurement.

[0046] (S9)

[0047] wherein, is the estimation output of the observer at the moment to the left end boundary; is the system state estimated by the observer.

[0048] The fuzzy observer rule : if is and is , then:

[0049] (S10)

[0050] wherein, represents the partial derivative with respect to time , is the initial condition, denotes the spatial derivative at the left end boundary , denotes the spatial derivative at the boundary , denotes the observed state with time delay, is the time-varying time delay defined in Step 1; is the system matrix corresponding to the observer, is the system time delay matrix corresponding to the observer, , and is the observer gain matrix; is the actual sample transmitted to the observer after the event trigger in Step 2; the boundary condition remains consistent with the controlled object (left end Neumann / Robin injection, right end actuated boundary input ).

[0051] The error and boundary error injection are defined as follows:

[0052] (S11)

[0053] where, is the state estimation error; is the error injection quantity at the left end boundary for closed-loop error system modeling.

[0054] To describe the observer output under fuzzy weighting, the weighted boundary error quantity is further given:

[0055] (S12)

[0056] where, is the normalized membership weight defined in Step 1, is the premise variable vector on the observer side; is the boundary value of the error system at .

[0057] The following is the mixed attack reception model for the right end boundary (for non-co-located closed-loop docking):

[0058] (S13)

[0059] where, is the estimated output of the observer at the right end boundary ; denote the Bernoulli switch variables of spoofing attack and DoS attack respectively, and the expectation is is the fake signal mapping when attacked (there is a deviation from the real output), The signal actually received / used by the control end.

[0060] To ensure physical realizability, energy constraints are imposed on the spoofing signal:

[0061] (S14)

[0062] wherein, is a known bounded operator / matrix (such as a channel gain upper bound or a filtering operator) used to limit the energy scale of the spoofing signal to avoid unachievable abnormal inputs.

[0063] Through formulas (S9)-(S14), the present application constructs a fuzzy boundary observer framework under the conditions of limited boundary measurement and non-co-located sensing-execution, and explicitly considers the probability type hybrid network attack (spoofing attack and DoS attack) of the right end boundary, thereby providing an observation link support consistent with communication security for the subsequent "random failure fault-tolerant boundary control law" (step 4), and strengthening the robustness and implementability in industrial environment.

[0064] As a further improvement of the present application, the step 4 is specifically as follows:

[0065] For the right end boundary , a non-co-located fuzzy boundary sampling controller considering Markov random failure is designed to enhance the system stability and fault tolerance capability under network attack and actuator random degradation.

[0066] (S15)

[0067] wherein, is the boundary control input; is the control gain matrix under the fuzzy rule; = is a random degradation factor driven by a Markov chain, characterizing the degradation degree of the actuator in mode ; and is the right end boundary received signal (possibly affected by attack or failure) defined in step 3.

[0068] Suppose obeys a finite state right-continuous Markov chain with state space and transition rate matrix , then

[0069] (S16)

[0070] wherein, >0, when satisfies and

[0071] Thus, the actuator failure cases can be classified into three categories:

[0072] 1, : the actuator is normal;

[0073] 2, : the actuator is partially degraded;

[0074] : the actuator is completely failed.

[0075] Combining Step 2 and Step 3, the control input can be further described as

[0076] (S17)

[0077] where is the normalized membership function; is the state estimation of the observer at the right end boundary; , and are Bernoulli random variables for the spoofing attack and DoS attack, respectively,

[0078] Substituting the control law (S17) into the controlled PDEs model in Step 1, the closed-loop system can be obtained as

[0079] (S18)

[0080] where is the bounded disturbance input, , , are the fuzzy subsystem matrices described in Step 1.

[0081] Combining the observer and controller gains in Step 3, we can write

[0082] (S19)

[0083] where is the fuzzy membership input vector, is the weight of the th rule, is the observation error at the left end boundary , is the boundary fuzzy weight, is the scheduling dependence matrix, is the controller gain, , are the weighting parameters, is the nonlinear function term, which gives the update rule of the observation state in time and spatial directions.

[0084] Further derive error system , satisfy

[0085] (S20)

[0086] wherein, is the equivalent disturbance term introduced by the fuzzy approximation error. To reduce the disturbance influence, introduce Performance index:

[0087] (S21)

[0088] wherein, , is the performance weighting matrix, is the decay factor.

[0089] As a further improvement of the invention, the step 5 is specifically as follows:

[0090] After the completion of the non-collocated fuzzy boundary controller design considering Markov random failure described in step 4, it is also necessary to further ensure the stability of the closed-loop system under time-varying time delay, limited boundary measurement and mixed attack, and meet the expected performance index. For this purpose, the invention introduces Lyapunov-Krasovskii functional (LKFs), and derives the corresponding criterion conditions based on the linear matrix inequality (LMI) method.

[0091] First, construct LKFs to ensure the stability of the closed-loop system under time-varying time delay and random disturbance. For this purpose, the invention selects the functional in the following form:

[0092]

[0093] Wherein each component is used to describe the current energy of the system, the energy of the delay term, the boundary sampling error energy and the cumulative effect of the historical information. Specifically:

[0094] 1、 , used to represent the energy of the current state and the estimation error of the system;

[0095] 2、 used to reflect the cross-coupling relationship between the state and the error;

[0096] An integral term with exponential decay factor is introduced to capture the influence of time-varying time delay on the stability of the system;

[0097] 4、 It is used to describe the cumulative effect of historical states within the maximum time delay range, thereby improving the accuracy of stability analysis.

[0098] exist - middle, This is a combined vector of system state and observation error. The weighting function is related to the sampling interval. Let be the symmetric positive definite matrix to be found. The attenuation coefficient is... This represents the maximum time delay.

[0099] The design of this LKFs introduces time-varying delays and exponential weighting terms, which can effectively reduce the conservatism in the stability criterion and better reflect the dynamic characteristics of the system in historical intervals and spatial distribution.

[0100] Based on this, through the By deriving the derivative and combining it with the control law designed in step 4, the following performance criteria can be obtained:

[0101]

[0102] in, This indicates that the closed-loop system is in the first... The Lyapunov matrix block structure under this condition. The feasibility of this condition directly guarantees the stability of the system under different modes.

[0103] Furthermore, the above equation can be described by the following block matrix expansion:

[0104] (S23)

[0105] This shows that system stability depends not only on the principal block matrix. It is also affected by the coupling effect of time delay terms and fault-related terms. To more intuitively characterize these constraints, this invention expands them into the following form:

[0106] (S24)

[0107] Among them, each The matrix is ​​a block matrix constructed from the system model, controller gain, and observer gain. By optimizing these matrices, the impact of time delay terms, disturbance terms, and attack models on closed-loop stability can be effectively characterized.

[0108] For example, The specific form is as follows:

[0109] (S25)

[0110] where is a matrix set in LMI, , , are decision variables to be solved, and is a selection vector, is a boundary interval length, is an identity matrix, , is a known matrix, and the structure of the matrix clearly reflects the coupling characteristics of the lag term, the boundary term, and the uncertain parameters.

[0111] At the same time, when considering the hybrid attack model, energy and probability constraints need to be imposed on the deception signal and the DoS signal, so as to further ensure the physical feasibility of the controller design. For example:

[0112] (S26)

[0113] wherein, is a performance index parameter, and I is an identity matrix, and are decision variable matrices to be solved, is a weighted coefficient, is a known matrix, is a symmetric matrix, is a selection vector, is a matrix to be solved, and the overall constraint is used to ensure the feasibility of the controller design under the deception attack and the DoS attack.

[0114] Based on the above derivation, the following performance criterion can be obtained:

[0115]

[0116] wherein, , is a performance weighted matrix, is an equivalent disturbance. The inequality indicates that when the LMI condition is feasible, the closed-loop system can maintain stability and meet the performance index under the action of external disturbance.

[0117] Finally, based on the solution of the LMI, the observer gain and the controller gain can be explicitly constructed in the following form:

[0118]

[0119] wherein, is a similarity transformation matrix, , , The intermediate matrix is obtained by LMI optimization.

[0120] By constructing the LKFs and introducing the performance indicators, the stability problem of the closed-loop system under time-varying time delay, limited boundary measurement and mixed attack is converted into a set of verifiable linear matrix inequality conditions. On this basis, an extended matrix can be further introduced to equivalently transform the inequality, and linearization processing can be performed combined with the Schur complement formula, so that the originally complex cooperative design problem of the controller and the observer is converted into a standard convex optimization problem. The method not only guarantees the robust stability of the system under the dynamic event-triggered mechanism, but also meets the expected performance indicators, and achieves efficient vibration suppression and safety control objectives under the conditions of limited communication resources and random controller failure.

[0121] It should be further pointed out that the technical features corresponding to the above options can be combined or replaced with each other to form new technical solutions without conflict.

[0122] Compared with the prior art, the present application has the following advantages:

[0123] The present application proposes a fault-tolerant boundary control method based on T-S fuzzy modeling for vibration suppression of flexible manipulators under limited measurement conditions and random controller failure. The method combines AETM, fuzzy boundary observer and boundary controller considering Markov jump failure, achieving efficient vibration suppression and system stability guarantee under complex network environment and random failure conditions. Numerical simulation verification shows that the proposed method can effectively suppress the vibration of the flexible manipulator and still maintain stable operation under complex network interference and random failure conditions, with performance significantly better than the traditional periodic sampling control method, and has high engineering application value. The specific contributions include:

[0124] (1) A flexible manipulator modeling method based on T-S fuzzy modeling is proposed, which can effectively describe the dynamic characteristics of the manipulator under nonlinear, time-varying time delay and uncertain disturbance, providing a unified analysis framework for subsequent control design.

[0125] (2) A boundary sampling strategy with AETM is designed, which can dynamically adjust the trigger threshold according to the real-time error, significantly reducing the communication overhead and improving the bandwidth utilization while ensuring the control performance.

[0126] (3) A non-collocated fuzzy boundary observer is constructed, which effectively solves the observation-control mismatch problem caused by the separation of sensors and actuators in the actual system of the flexible manipulator, improving the state estimation accuracy and control reliability.

[0127] (4) In the controller design, Markov Jump Process is introduced to model the random controller failure, which can truly reflect the gain loss and function degradation of the controller in actual application, and enhance the fault tolerance of the system.

[0128] (5) Based on the new LKFs and LMI method, the sufficient conditions of system stability and performance are derived to ensure the robust stability and performance index of the closed-loop system under the conditions of mixed attack (including denial of service DoS and deception attack), limited measurement and random failure. Performance index. BRIEF DESCRIPTION OF DRAWINGS

[0129] Figure 1 A flow chart of the flexible manipulator vibration control method based on T-S fuzzy modeling according to the present application;

[0130] Figure 2 An open-loop system state 1 evolution schematic diagram of the flexible manipulator under limited boundary measurement conditions according to the present application;

[0131] Figure 3 An open-loop system state 2 evolution schematic diagram of the flexible manipulator under limited boundary measurement conditions according to the present application;

[0132] Figure 4 A flexible manipulator dynamics approximation effect schematic diagram under T-S fuzzy modeling according to the present application;

[0133] Figure 5 A dynamic characteristic schematic diagram of the random controller failure modeling mechanism according to the present application;

[0134] Figure 6 A state response curve of the closed-loop system under state 1 according to the present application;

[0135] Figure 7 A state response curve of the closed-loop system under state 2 according to the present application;

[0136] Figure 8 A boundary observer estimation error norm 1 convergence curve under mixed attack conditions according to the present application;

[0137] Figure 9 A boundary observer estimation error norm 2 convergence curve under mixed attack conditions according to the present application;

[0138] Figure 10 A curve of the boundary control input changing with time according to the present application;

[0139] Figure 11 A non-periodic DoS attack sequence schematic diagram according to the present application;

[0140] Figure 12 The communication times under the conventional trigger mechanism are shown in the figure.

[0141] Figure 13 The communication times under the trigger mechanism of the present application are shown in the figure. DETAILED DESCRIPTION

[0142] The technical solutions of the present application will be described clearly and completely below with reference to the drawings. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. The components of the embodiments of the present application described and shown in the drawings can be arranged and designed in various different configurations. Based on the embodiments in the present application, all the other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the present application.

[0143] It should be noted that the defects of the above prior art solutions are the results obtained by the inventors after practice and careful study. Therefore, the discovery process of the above problems and the solutions proposed by the embodiments of the present application to the above problems should be the contributions made by the inventors to the present application in the process of invention and creation, and should not be understood as the technical content known to those skilled in the art.

[0144] In an exemplary embodiment, a T-S fuzzy modeling based flexible manipulator vibration control method under limited measurement and random controller failure conditions is provided, which specifically includes the following parts:

[0145] 1. System structure:

[0146] (1) Distributed parameter model and boundary conditions of flexible manipulator

[0147] Consider a class of flexible manipulators described by parabolic DPSs, whose elastic vibration is depicted by formula S1, wherein, is the distributed state (such as deflection / strain, etc.) of the flexible arm, is the diffusion (stiffness / heat conduction) matrix, is the time-varying time delay and is a bounded disturbance; the left end is subjected to a natural boundary (or clamped boundary), and the right end is subjected to a boundary torque / force , is the input boundary matrix.

[0148] (2) T-S fuzzification (continuum layer)

[0149] To deal with the nonlinear term , the following premise variable vector is introduced:

[0150]

[0151] Rule If is and is , then:

[0152]

[0153] Let be the unnormalized weight, the normalized membership is as follows:

[0154]

[0155] The overall fuzzy controlled object is given by the formula .

[0156] (3) Finite boundary measurement (non-co-located) and sampling signal

[0157] Install a sensor at the left end boundary to obtain finite measurement, define

[0158]

[0159] Where is the boundary measurement matrix, is the event-triggered release time sequence (determined by the subsequent AETM), is the value of the state described in step 1 at the boundary .

[0160] Zero-order hold gets the available samples on the control side as , and the observation error is defined as

[0161] (4) Non-co-located structure of the observer side and the control side

[0162] To compensate for the mismatch caused by the non-co-located structure of "observation in , actuation in ", a fuzzy boundary observer is constructed, see step 3 for details.

[0163] 2. Boundary controller design with random failure:

[0164] During the long-term operation of the flexible manipulator, the boundary driving link (power amplifier / actuator / communication) may appear random degradation or intermittent failure, such as gain drift, packet loss / stutter, channel interruption, etc. In order to improve the robustness and fault tolerance of the system in complex environment, the invention explicitly introduces the double-channel modeling of Markov failure factor and random amplitude factor in the boundary control law.

[0165] (1) Random failure modeling

[0166] 1) Use right-continuous Markov chain to depict the switching of working / degrading / failure modes, whose infinitesimal generator is .

[0167] 2) Define degradation matrix , where represents normal, 0 represents partial degradation, and represents complete failure.

[0168] 3) Introduce random amplitude factor to describe fast small random perturbation, whose sample can come from Beta distribution or Bernoulli-Gaussian mixture (given by online diagnosis module in engineering), for reflecting the instantaneous effective gain of execution link.

[0169] (2) Boundary control law with sample-and-hold

[0170] Consider event-triggered sampling time sequence , the control side applies boundary input in interval . Combined with the right boundary estimation obtained in the last section, introduce channel delay (including calculation / transmission / execution delay), the control law takes:

[0171]

[0172] where is the proportional boundary feedback gain to be designed, when mixed attack such as DoS / spoofing exists, it can be replaced by measured at the receiving end (see signal constraint in step 3), the form of the control law remains unchanged.

[0173] (3) Parameters and implementation points

[0174] 1) Refreshed by online fault diagnosis / health assessment module;

[0175] 2)Obtained offline once by solving LMI described later, directly calculated according to (S28) when deployed ;

[0176] 3, function stability analysis:

[0177] To ensure that the above controller can still guarantee closed-loop stability and performance under the joint action of time-varying time delay, limited boundary measurement, mixed attack and random failure, the present application constructs LKFs and gives LMI criterion.

[0178] (1) LKFs construction and meaning​

[0179] Construct LKFs as follows:

[0180]

[0181] where each component accounts for: current distribution state energy , estimation error energy , weighted integral term of historical time delay interval, event-triggered hold error term, and boundary energy coupling term. This structure can capture the influence of spatial distribution and time delay simultaneously, and reduce conservatism through exponential weight.

[0182] (2) Performance index and differential inequality

[0183] Under external disturbance , introduce index, and derive along closed-loop trajectory:

[0184]

[0185] where is selected performance output (including boundary displacement / velocity or combination), > 0 is desired decay level. Equation (S27) guarantees that when disturbance energy is bounded, closed-loop response energy is subject to constraint, i.e. satisfies robust performance.

[0186] (3) LMI and solvability conditions

[0187] The time delay, sample-and-hold, and boundary coupling terms appearing in equations (S18) and (S19) are unified upper bounded by extended matrix inequality, integral inequality, and boundary partition identity, and affine relaxation and Schur complement processing are used for product terms containing , to obtain standard LMI set:

[0188]

[0189] where , are block matrices constructed by , observer gain, LKFs parameters, trigger weight, time delay upper bound, etc.

[0190] Engineering implementation: the standard LMI set formula is solved as a convex optimization problem in MATLAB, directly obtaining controller gain and observer gain , The formula is guaranteed to be valid along the closed-loop trajectory, thereby obtaining asymptotic stability and performance.

[0191] 4. Example verification

[0192] In order to further verify the effectiveness of the flexible manipulator vibration suppression fault-tolerant boundary control method based on T-S fuzzy modeling under the condition of limited measurement and random controller failure proposed in the present application, the present embodiment is verified by numerical examples.

[0193] (1) System model setting:

[0194] The dynamics of the flexible manipulator under boundary constraints can be represented as a distributed parameter system with nonlinear terms, time delay effects and external disturbances. The dynamics of the system in the space-time domain is described as follows:

[0195]

[0196] where, , denotes the partial derivative with respect to time , , denotes the second-order partial derivative with respect to space , denotes the time delay function, , is the system parameter, denotes the control input, denotes the external disturbance input, is the boundary condition, denotes the spatial derivative condition at the boundary , , is the initial condition.

[0197] (2) Fuzzy modeling and parameter selection:

[0198] Based on the T-S fuzzy modeling method, the above distributed parameter system is converted into a fuzzy sub-model weighted form. The premise variable , is a constant parameter, and the fuzzy membership function is constructed as follows:

[0199]

[0200] From this, the fuzzy weighted model of the flexible manipulator and the corresponding parameter matrix , etc. can effectively represent the dynamic evolution characteristics under different time delays and external disturbances.

[0201] (3) Controller and observer design:

[0202] In the case of considering random controller failure, factors such as actuator gain attenuation, local signal loss and boundary measurement limitation are introduced. By solving the LMI constraint condition, the boundary fault-tolerant controller gain and fuzzy observer gain are obtained as follows:

[0203]

[0204]

[0205] The above results are obtained by solving the MATLAB tool, which guarantees the robust stability under the condition of time-varying time delay and random controller failure.

[0206] Results and effect description

[0207] After completing the design and stability analysis of the control method, the effectiveness and superiority of the proposed method are further verified by numerical examples. Based on the distributed parameter model of the nonlinear flexible manipulator, different comparative experimental scenes are constructed under the conditions of time-varying time delay, limited boundary measurement and random controller failure, and the relevant result images are drawn.

[0208] Firstly, Figures 2-3 The spatio-temporal evolution of state estimation error without observer and controller intervention is shown. It can be seen that the error trajectory shows obvious periodic oscillation, and its amplitude gradually increases with time, indicating that the system is highly sensitive to external disturbance and attack, and it is difficult to obtain accurate state estimation, and the system stability is difficult to guarantee. Figure 4 The dynamics approximation effect of the flexible manipulator under the state of the two error systems of T-S fuzzy modeling is given.

[0209] Secondly, Figure 5 The dynamic characteristic diagram of the random controller failure modeling mechanism is given. It can be observed that the attack signal presents obvious periodic disturbance characteristics in time domain, which further aggravates the uncertainty and instability of the system, and highlights the importance of resisting deception attacks in actual operation.

[0210] After introducing the designed fuzzy observer and non-co-located boundary controller, Figures 6-7 The state response curves under the state of the two closed-loop systems are given, which show the state evolution results of the closed-loop system. The results show that the state trajectory of the flexible manipulator can quickly converge and remain in the stable interval, which shows that the proposed control strategy can effectively suppress system oscillation, achieve ideal spatio-temporal regulation performance, and still maintain robustness under the action of time-varying time delay, external disturbance and deception attack.

[0211] Further, Figures 8-9The diagram illustrates the norm curves of two state estimation errors under the combined effects of a DoS attack (blocking) and a spoofing attack. The pink shaded area represents the interval of the DoS attack, and the red square dots represent the moments of the spoofing attack. It can be seen that although the system error fluctuates significantly in the early stages of the attack, it gradually stabilizes and remains within a finite range over time. This demonstrates that the method proposed in this invention can still guarantee acceptable estimation accuracy and stability under mixed attack conditions.

[0212] Figure 10 The evolution curves of the boundary control input under mixed attack conditions are presented, where different colored segments represent Markov random switching modes. It can be seen that the control input can be adjusted promptly under both deception and DoS attacks, and maintains an effective response under the Markov random switching mechanism, demonstrating that the method of this invention possesses good attack adaptability and switching robustness.

[0213] Figure 11 The transmission availability curves for aperiodic DoS attacks are presented. As can be seen from the figure, the duration and intervals of the attacks are irregular, realistically reflecting the communication congestion characteristics under complex operating environments. The method of this invention can still maintain stable operation under such complex attack conditions, verifying the rationality and practicality of the proposed attack modeling and defense mechanism.

[0214] at last, Figure 12 and Figure 13 The differences in transmission behavior between the traditional fixed-threshold event triggering mechanism and the proposed adaptive event triggering mechanism were compared. The traditional fixed-threshold mechanism triggers frequently under interference, resulting in dense and irregular transmission intervals; while the adaptive mechanism proposed in this invention can dynamically adjust the threshold according to the error, making the triggering sequence more orderly and significantly reducing the transmission frequency, thereby reducing the communication burden while ensuring control performance.

[0215] The vibration suppression and fault-tolerant boundary control method for flexible robotic arms based on TS fuzzy modeling proposed in this invention can effectively achieve robust stability and performance index constraints of the system under complex conditions such as limited measurement, time-varying time delay, external disturbances and mixed attacks, demonstrating high engineering application value and promotion prospects.

[0216] The above detailed embodiments are a description of the present invention. It should not be considered that the specific embodiments of the present invention are limited to these descriptions. For those skilled in the art, several simple deductions and substitutions can be made without departing from the concept of the present invention, and all of these should be considered to fall within the protection scope of the present invention.

Claims

1. A T-S fuzzy modeling based vibration control method for a flexible manipulator, characterized in that, The method comprises the following steps: Step 1. A T-S fuzzy distributed parameter system model of a flexible manipulator is established; the T-S fuzzy distributed parameter system model is composed of weighted superposition of a plurality of fuzzy rules, each rule corresponds to a local linear subsystem; the state of the T-S fuzzy distributed parameter system model satisfies boundary conditions: The left end boundary is a natural boundary or a clamping boundary, and the right end boundary is affected by a boundary control input; Step 2. Based on the T-S fuzzy distributed parameter system model, a non-co-located sensor is used to obtain a limited measurement signal at the left end boundary of the T-S fuzzy distributed parameter system model, and an adaptive event-triggered sampling mechanism is constructed to output a sampled signal; the trigger condition of the adaptive event-triggered sampling mechanism is: wherein, denotes the (k+1)th sampling instant, k denotes the number of the sampling instant, h denotes a candidate instant, denotes the (k+1)th candidate instant starting from the kth sampling instant denotes the (k+1)th candidate instant starting from the kth sampling instant denotes the (k+1)th candidate instant starting from the kth sampling instant denotes the error signal, denotes a trigger threshold function, denotes the boundary measurement at the sampling instant denotes the boundary measurement at the sampling instant denotes the weighting matrix, the superscript T denotes the transpose of the corresponding vector; Step 3. A fuzzy boundary observer under limited boundary measurement is designed based on the sampled signal, which is used to estimate the full state of the system; the output of the fuzzy boundary observer is used to estimate the state of the right end boundary, and the mixed attack receiving model of the right end boundary is: where, denotes the actual received signal at the control end, denotes the Bernoulli switching variable for the spoofing attack, denotes the Bernoulli switching variable for the denial-of-service attack, is the estimated output of the fuzzy boundary observer at the right end boundary, is the mapping of the fake signal when under attack; Step 4. Based on the output of the fuzzy boundary observer and the sampled signal, a non-co-located fuzzy boundary sampled controller considering Markov random failure is designed; the non-co-located fuzzy boundary sampled controller is as follows: wherein, is a boundary control input, is a stochastic degradation factor driven by a Markov chain, is a control gain matrix under fuzzy rules, denotes the mode of the actuator.

2. The vibration control method for a flexible robot arm based on T-S fuzzy modeling according to claim 1, wherein The method further comprises the following steps: Step 5. A stability criterion of the system is derived by using Lyapunov-Krasovskii functional and linear matrix inequality method.

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