Flexible mechanical arm vibration control method based on T-S fuzzy modeling

Abstract

The invention discloses a flexible mechanical arm vibration control method based on T-S fuzzy modeling, and belongs to the technical field of intelligent manufacturing and distributed parameter system control. The flexible mechanical arm vibration control method comprises the steps that a flexible mechanical arm T-S fuzzy distributed parameter model considering nonlinear disturbance and time-varying and time-lag is constructed; designing a fuzzy boundary observer under a limited boundary measurement condition to realize state estimation; introducing a controller random failure modeling mechanism based on a Markov process, and designing a fault-tolerant boundary controller; dynamic boundary sampling is realized in combination with AETM, and the consumption of communication resources is reduced; and stability and performance criteria are constructed by using LKFs and LMI methods. According to the method, the robust stability and performance guarantee of vibration suppression of the flexible mechanical arm are realized under the conditions of limited measurement and failure of the random controller.

Description

Technical Field

[0001] This invention relates to the field of intelligent manufacturing and distributed parameter system control technology, and in particular to a vibration control method for a flexible robotic arm based on TS fuzzy modeling. Background Technology

[0002] In recent years, parabolic partial differential equation systems (PDEs) have become a common tool for characterizing spatiotemporal dynamics and have been widely applied in fields such as heat conduction, reaction-diffusion transport, and thermal regulation of aerospace structures. However, due to the infinite-dimensional nature of distributed parameter systems (DPSs), their stability analysis and control design still face significant challenges. Furthermore, practical flexible robotic arm systems commonly suffer from time delays, non-coordinated sensor-actuator configurations, and nonlinear uncertainties, further complicating controller design.

[0003] To address these challenges, researchers have proposed various analytical tools, including boundary control frameworks based on Lyapunov methods, semigroup theory, and backpropagation techniques. Meanwhile, Takagi-Sugeno (TS) fuzzy modeling, due to its ability to decompose complex nonlinear systems into several locally linear subsystems and thus smoothly integrate with linear control theory, has been increasingly applied to the analysis and design of uncertain nonlinear systems. This method has been extended to boundary control and output feedback design for distributed parameter systems and has made progress in the research of fuzzy sampling control of parabolic PDEs.

[0004] Event-triggered sampling control (ETSD) has gained increasing attention for reducing communication overhead. It significantly improves resource utilization by updating control or observation signals only when necessary and has been validated in ordinary differential equations and PDE scenarios. However, most existing studies assume full state measurability or spatial co-location of sensors and actuators, neglecting finite measurements and non-co-location configurations. Meanwhile, with the proliferation of networked control systems, network attack threats (such as denial-of-service (DoS), spoofing, and replay attacks) have become critical factors affecting system operation. For example, some methods have analyzed control accuracy against DoS interference in dual-channel frameworks, some have studied stability issues under bandwidth constraints, and some have proposed fault-tolerant fuzzy control methods combining event triggering; however, consideration of actuator failure remains insufficient.

[0005] In practical engineering, actuators such as flexible robotic arms often experience random failures due to mechanical wear, environmental interference, or aging, leading to decreased system performance or even instability. To improve the robustness and safety of these systems, existing research has introduced fault-tolerant control mechanisms and explored fuzzy control methods that combine random actuator failures with deception attacks. Furthermore, some studies have proposed non-co-located output feedback strategies for reaction-diffusion systems under intermittent boundary measurement conditions.

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

[0007] Therefore, how to achieve efficient vibration control of flexible robotic arms under limited measurement conditions and maintain system stability in random controller failure and complex network environments has become a critical problem that urgently needs to be solved. Summary of the Invention

[0008] The purpose of this invention is to overcome the problems existing in the prior art and provide a fault-tolerant boundary control method for vibration suppression of flexible robotic arms based on TS fuzzy modeling under conditions of limited measurement and random controller failure. This ensures efficient vibration suppression of the flexible robotic arm even under limited sensor configuration and random controller failure, and improves the robustness and reliability of the system.

[0009] The objective of this invention is achieved through the following technical solution: A vibration control method for a flexible robotic arm based on TS fuzzy modeling, such as Figure 1 As shown, it includes: Step 1. Establish the TS fuzzy distributed parameter system model of the flexible robotic arm; Step 2. Based on the TS fuzzy distribution parameter system model, a non-co-position sensor is used at its left boundary to acquire a limited measurement signal, and an adaptive event-triggered sampling mechanism is constructed to output the sampled signal; Step 3. Based on the sampled signal, design a fuzzy boundary observer under finite boundary measurement to estimate the full state of the system; Step 4. Based on the output of the fuzzy boundary observer and the sampling signal, design a non-co-located fuzzy boundary sampling controller that considers Markov random failure.

[0010] As a further improvement to the present invention, step 1 is specifically as follows: Establish a TS fuzzy parabolic DPSs model (including boundary actuation) for the flexible robotic arm: (S1) Among them, spatial variables , 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.

[0011] Introducing prerequisite variables For the Rules have: rule :if yes as well as yes ,So: (S2) in This represents the corresponding fuzzy set, for the index set. Each rule ,matrix It is a known constant matrix.

[0012] The fuzzy membership function is defined as follows: 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 sum of the activation intensities of all rules, where the activation intensity is... Then the membership function values ​​of each input variable under this rule are used. The product of the products is given.

[0013] Then, from (S2), we obtain the weighted superposition form: (S3) Among them, the first The weights of the rules are: (S4) As a further improvement of the present invention, step 2 is specifically as follows: At the left boundary Limited measurements are acquired using an adaptive event-triggered sampling mechanism (AETM). First, the boundary measurement and sampling time sequence is defined: (S5) in, For at any time Boundary measurement quantity; The state described in step 1 is at the left boundary. The value at; This is the release / sampling time sequence.

[0014] To suppress redundant transmissions and improve communication utilization, an event-triggered principle with a time-varying threshold is used to determine the next triggering time: (S6) in, This represents the (k+1)th sampling time, where k represents the sampling time number and h represents the candidate time. Indicates the k-th sampling time Starting from the first One candidate moment, , This is based on the error signal between the latest triggered sample and the current candidate sample. To trigger the threshold function, Indicates at the sampling time Boundary measurement quantity, The weighted matrix obtained from the design is given by the superscript T, which indicates the transpose of the corresponding vector or matrix.

[0015] The threshold function adopts an adaptive decay form as the error decreases: (S7) in, They are respectively below the threshold / upper limit and This is the error influence coefficient; an increase in error will cause... Reduce, thereby triggering a more positive response.

[0016] Therefore, at the triggering time The sampled value that actually enters the control / observation loop is defined as (S8) in, For at any time The actual sample value sent / executed; This represents the candidate sample values ​​that make equation (S7) true within the current window. This represents the candidate sample value that makes equation (S7) invalid within the current window. Through the AETM described in equations (S5)-(S8), redundant reporting can be adaptively suppressed while ensuring control performance, balancing communication resource consumption and system response speed, and providing event-driven sampling input for subsequent boundary observers and non-co-located fault-tolerant boundary control laws.

[0017] As a further improvement to the present invention, step 3 is specifically as follows: At the left boundary A non-co-located sensor is used to obtain estimated boundary measurements, and a fuzzy boundary observer is constructed based on these measurements.

[0018] (S9) in, For the observer at time The estimated output for the left boundary; The system state estimated by the observer.

[0019] Fuzzy Observer Rules :if yes as well as yes ,So: (S10) in, Indicates time The partial derivatives, As initial conditions, Indicates the left boundary Spatial derivative at point, Indicates at the boundary Spatial derivative at point, This indicates the observation state with time delay. The time-varying delay defined in step 1; The system matrix corresponding to the observer. The system time delay matrix corresponding to the observer is... , and The observer gain matrix; The actual sample values ​​transmitted to the observer after the event in step 2 is triggered; the boundary conditions remain consistent with the controlled object (Neumann / Robin injection at the left end, and actuation boundary input at the right end). effect).

[0020] Error and boundary error injection are defined as follows: (S11) in, This represents the state estimation error; This represents the error injection amount at the left boundary, used for modeling the closed-loop error system.

[0021] To describe the observer output under fuzzy weighting, the weighted boundary error is further given: (S12) in, The normalized membership weights defined in step 1, The observer-side premise variable vector; For the error system in Boundary values ​​at that point.

[0022] The following is a hybrid attack reception model for the right boundary (used for non-co-located closed-loop docking): (S13) in, For the observer at the right boundary The estimated output; Let Bernoulli switch variables represent spoofing attacks and DoS attacks respectively, with expected values ​​of This is a spoofed signal mapping for when attacked (which deviates from the actual output). The signals actually received / used by the control unit.

[0023] To ensure physical realizability, an energy constraint is imposed on the deception signal: (S14) in, These are known bounded operators / matrices (such as upper bounds on channel gain or filtering operators) used to limit the energy scale of spoofed signals and avoid unrealizable anomalous inputs.

[0024] Using formulas (S9)-(S14), this invention constructs a fuzzy boundary observer framework under the conditions of finite boundary measurement and non-co-location sensing-execution, and explicitly considers probabilistic hybrid network attacks (spoofing attacks and DoS attacks) on the right boundary. This provides observation link support consistent with communication security for the subsequent "random failure tolerance boundary control law" (step 4), and enhances robustness and feasibility in industrial environments.

[0025] As a further improvement to the present invention, step 4 is specifically as follows: Regarding the right boundary A non-co-located fuzzy boundary sampling controller is designed to consider Markov random failures in order to enhance the system stability and fault tolerance under network attacks and actuator random degradation.

[0026] (S15) in, For boundary control input; The control gain matrix under fuzzy rules; = A stochastic degradation factor driven by a Markov chain. Characterizing the actuator in mode The degree of degradation below; Receive the signal for the right boundary defined in step 3 (which may be affected by attacks or failures).

[0027] Assumption Follows a finite-state right-continuous Markov chain, with state space The transfer rate matrix is Then it satisfies (S16) in, >0, when Timely satisfaction and Therefore, actuator failures can be classified into three categories: 1. The actuator is working normally; 2. Actuator part degradation; The actuator has completely failed.

[0028] Combining steps 2 and 3, the control input can be further described as follows: (S17) in, Normalized membership function; State estimation of the observer at the right boundary; , These are Bernoulli random variables for spoofing attacks and DoS attacks, respectively. Mapping of fake signals under attack.

[0029] Substituting the control law (S17) into the controlled PDEs model from step 1, the closed-loop system form is obtained as follows: (S18) in, For a bounded disturbance input, , , The fuzzy subsystem matrix described in step 1.

[0030] Combining the observer and controller gains from step 3, it can be written as: (S19) in The input vector is the fuzzy membership degree. For the first The weight of each rule, For the left boundary Observational error at that location For boundary fuzzy weights, For the scheduling dependency matrix, For controller gain, , For weighted parameters, The term is a nonlinear function, which gives the overall update law of the observed state in both time and space.

[0031] Further derivation yields the error system ,satisfy (S20) in, This is the equivalent perturbation term introduced by the fuzzy approximation error. To reduce the impact of the perturbation, we introduce... Performance metrics: (S21) in, , For performance weighting matrix, This is the attenuation factor.

[0032] As a further improvement to the present invention, step 5 is specifically as follows: After completing the design of the non-co-located fuzzy boundary controller considering Markov random failures as described in step 4, it is still necessary to further ensure the stability of the closed-loop system under time-varying delays, finite boundary measurements, and mixed attacks, and to meet the expected requirements. Performance metrics. To this end, this invention introduces Lyapunov-Krasovsky functionals (LKFs) and derives the corresponding criterion conditions based on the linear matrix inequality (LMI) method.

[0033] First, LKFs are constructed to ensure the stability of the closed-loop system under time-varying delays and random disturbances. To this end, this invention selects functionals of the following form: Each component is used to describe the system's current energy, delay term energy, boundary sampling error energy, and the cumulative effect of historical information, respectively. Specifically: 1. , Energy used to characterize the current state of the system and the estimation error; 2. Used to reflect the cross-coupling relationship between state and error; An integral term with an exponential decay factor is introduced to capture the impact of time-varying delays on system stability; 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.

[0034] 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.

[0035] 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.

[0036] 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: 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.

[0037] Furthermore, the above equation can be described by the following block matrix expansion: (S23) 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: (S24) 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.

[0038] For example, The specific form is as follows: (S25) in For a matrix set in LMI, , , Let be the decision variables to be determined, and Indicates the selection of a vector. Indicates the length of the boundary interval. Represents the identity matrix. , The matrix is ​​known, and its structure clearly reflects the coupling characteristics of lag terms, boundary terms, and uncertain parameters.

[0039] At the same time, when considering hybrid attack models, energy and probability constraints need to be imposed on spoofing signals and DoS signals to further ensure the physical feasibility of the controller design. For example: (S26) in, Representing performance index parameters, I is the identity matrix. and Let be the matrix of decision variables to be determined. Indicates the weighting coefficient. Given a matrix, It is a symmetric matrix. Indicates the selection of a vector. Let be the matrix to be determined. The overall constraints are used to ensure the feasibility of the controller design under deception attacks and DoS attacks.

[0040] Based on the above derivation, the following performance criteria can be obtained: in, , For performance weighting matrix, This is an equivalent disturbance. The inequality shows that, when the LMI condition is feasible, the closed-loop system can remain stable under external disturbances and satisfy the following conditions. Performance metrics.

[0041] Finally, based on the solution of the LMI, the observer gain and controller gain can be explicitly constructed in the following form: in, Let be the similarity transformation matrix. , , This is the intermediate matrix obtained through LMI optimization. This result provides a workable design method for practical implementation.

[0042] By constructing LKFs and introducing performance metrics, this invention transforms the stability problem of closed-loop systems under time-varying delays, finite boundary measurements, and mixed attacks into a set of verifiable linear matrix inequalities. Based on this, extended matrices can be introduced to perform equivalent transformations on the inequalities, and Schur's complement formula can be used for linearization, thus transforming the originally complex controller-observer co-design problem into a standard convex optimization problem. This method not only guarantees the robust stability of the system under dynamic event-triggered mechanisms but also meets the expected... The performance indicators achieve the goal of efficient vibration suppression and safe control under conditions of limited communication resources and random controller failure.

[0043] It should be further noted that the technical features corresponding to the above options can be combined or substituted to form new technical solutions if there is no conflict.

[0044] Compared with the prior art, the beneficial effects of the present invention are: This invention addresses the vibration suppression problem of flexible robotic arms under finite measurement conditions and random controller failures, proposing a fault-tolerant boundary control method based on TS fuzzy modeling. This method combines AETM, a non-coordinated fuzzy boundary observer, and a boundary controller considering Markov jump failures, achieving efficient vibration suppression and system stability assurance under complex network environments and random fault conditions. Numerical simulations verify that the proposed method effectively suppresses the vibration of the flexible robotic arm and maintains stable operation under complex network interference and random fault conditions, significantly outperforming traditional periodic sampling control methods and possessing high engineering application value. Specific contributions include: (1) A flexible robotic arm modeling method based on TS fuzzy modeling is proposed, which can effectively characterize the dynamic characteristics of the robotic arm under nonlinear, time-varying time delay and uncertain disturbances, and provides a unified analysis framework for subsequent control design.

[0045] (2) A boundary sampling strategy with AETM was designed. This mechanism can dynamically adjust the trigger threshold according to the real-time error, which can significantly reduce communication overhead and improve bandwidth utilization while ensuring control performance.

[0046] (3) A non-co-position fuzzy boundary observer was constructed, which effectively solved the observation-control mismatch problem caused by the separation of sensor and actuator positions in the actual system of flexible robotic arm, and improved the state estimation accuracy and control reliability.

[0047] (4) Introducing the Markov Jump Process into the controller design to model the failure of the stochastic controller can truly reflect the loss of gain and functional degradation that may occur in the actual application of the controller, thereby enhancing the fault tolerance of the system.

[0048] (5) Based on novel LKFs and LMI methods, sufficient conditions for system stability and performance are derived, ensuring the robust stability of the closed-loop system under mixed attacks (including denial-of-service DoS and spoofing attacks), finite measurements, and random failure conditions. Performance metrics. Attached Figure Description

[0049] Figure 1 This is a flowchart of a vibration control method for a flexible robotic arm based on TS fuzzy modeling according to the present invention. Figure 2 This is a schematic diagram of the evolution of state 1 of the open-loop system of the flexible robotic arm under finite boundary measurement conditions in this invention; Figure 3 This is a schematic diagram of the evolution of state 2 of the open-loop system of the flexible robotic arm under finite boundary measurement conditions in this invention; Figure 4 This is a schematic diagram of the dynamic approximation effect of the flexible robotic arm under TS fuzzy modeling in this invention; Figure 5 This is a schematic diagram illustrating the dynamic characteristics of the random controller failure modeling mechanism in this invention; Figure 6 This is the state response curve of the closed-loop system under state 1 of the present invention; Figure 7 This is the state response curve of the closed-loop system under state 2 of the present invention; Figure 8 The convergence curve of the boundary observer estimation error norm 1 under mixed attack conditions in this invention; Figure 9 The convergence curve of the boundary observer estimation error norm 2 under mixed attack conditions in this invention; Figure 10 This is a curve showing how the boundary control input of this invention changes over time. Figure 11 This is a schematic diagram of the non-periodic DoS attack sequence of the present invention; Figure 12 This diagram illustrates the number of communications under a traditional triggering mechanism. Figure 13 This is a schematic diagram illustrating the number of communications under the triggering mechanism of this invention. Detailed Implementation

[0050] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. The components of the embodiments of this application described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0051] It should be noted that the defects in the solutions in the prior art are all the results of the inventors' practice and careful research. Therefore, the discovery process of the above problems and the solutions proposed by the embodiments of this application in the following text should be the inventors' contributions to this application in the process of invention and creation, and should not be understood as technical content known to those skilled in the art.

[0052] In an exemplary embodiment, a vibration control method for a flexible robotic arm based on TS fuzzy modeling is provided under conditions of finite measurement and random controller failure, specifically including the following parts: 1. System Structure: (1) Distributed parameter model and boundary conditions of flexible robotic arm Consider a class of flexible robotic arms described by parabolic DPSs, whose elastic vibrations are characterized by equation S1, where, This refers to the distributed states of the flexible arm (such as deflection / strain). For diffusion (stiffness / thermal conductivity) matrix, For time-varying and time-delayed For bounded perturbations; left end Using natural boundaries (or clamped boundaries), right end Subject to boundary moment / force effect, This is the input boundary matrix.

[0053] (2) TS blurring (continuum layer) To handle nonlinear terms The following prerequisite variable vector is introduced: rule :like yes as well as yes ,but: make The following are the unnormalized weights, and the normalized membership degrees are as follows: The overall fuzzy controlled object is determined by the formula Provided.

[0054] (3) Finite boundary measurement (non-co-location) and sampling signal A sensor is installed at the left boundary to obtain finite measurements, defined as follows: in For the boundary measurement matrix, The event-triggered release time sequence (determined by subsequent AETM). The state described in step 1 is at the boundary The value at that location.

[0055] Zero-order hold yields control-side available samples. The observation error is defined as (4) Non-co-located structure between the observer side and the control side To compensate for "observation in , to drive To address the mismatch caused by the non-co-location of "", a fuzzy boundary observer is constructed, as detailed in step 3.

[0056] 2. Boundary controller design with random failures: During the long-term operation of a flexible robotic arm, the boundary drive link (power amplifier / actuator / communication) may experience random degradation or intermittent failures, such as gain drift, packet loss / stuttering, and channel interruption. To improve the robustness and fault tolerance of the system in complex environments, this invention explicitly introduces dual-channel modeling of Markov failure factors and random amplitude factors into the boundary control law.

[0057] (1) Random failure modeling 1) Using a right-continuous Markov chain The infinitesimal generator is used to characterize the switching between working / degrading / failure modes. .

[0058] 2) Define the degeneracy matrix ,in This indicates normal operation; 0 < For partial degradation, It is completely ineffective.

[0059] 3) To describe fast, small-amplitude random disturbances, a random amplitude factor is introduced. Its samples can come from a Beta distribution or a Bernoulli-Gaussian mixture (provided in engineering by an online diagnostic module) to reflect the instantaneous effective gain of the execution link.

[0060] (2) Boundary control law with sample-and-hold Consider the event-triggered sampling time sequence The control side uses zero-order maintenance within the range Apply boundary input. Combine this with the right-hand boundary estimate obtained in the previous section. Introducing channel delay (Including computation / transmission / execution delays), control law is taken as follows: in, For the proportional boundary feedback gain to be designed, in the presence of mixed attacks such as DoS / spoofing, it can be... Replaced with the measurement at the receiving end (See signal constraints in step 3), the form of the control law remains unchanged.

[0061] (3) Parameters and key points of implementation 1) Refreshed by the online fault diagnosis / health assessment module; 2) The solution is obtained offline in one step through LMI (described later), and during deployment, it is calculated directly according to (S28). ; 3. Function stability analysis: To ensure that the above controller can maintain closed-loop stability and performance under the combined effects of time-varying delays, finite boundary measurements, mixed attacks and random failures, this invention constructs LKFs and provides LMI criteria.

[0062] (1) Construction and meaning of LKFs Construct the following LKFs: Each component is included in: the energy of the current distribution state. Estimation error energy The structure includes a weighted integral term for the historical time delay interval, an event-triggered hold-up error term, and a boundary energy coupling term. This structure can simultaneously capture the effects of spatial distribution and temporal delay, and reduces conservatism through exponential weighting.

[0063] (2) Performance index and differential inequality External disturbances Under the influence of, The index is derived to show the trajectory along the closed loop: in For the selected performance output (including boundary displacement / velocity or combination). >0 represents the desired attenuation level. Equation (S27) guarantees that when the disturbance energy is bounded, the closed-loop response energy is affected by... Constraints, i.e., satisfying Robust performance.

[0064] (3) LMI transformation and solvability conditions The time delay, sample-and-hold, and boundary coupling terms appearing in equations (S18) and (S19) are unified and conservatively bounded by extended matrix inequalities, integral inequalities, and boundary partition identities. Furthermore, for terms containing... , The product terms are treated with affine relaxation and Schur complement to obtain the standard LMI group: in , For the reason A block matrix constructed with observer gain, LKFs parameters, trigger weights, and upper delay bounds.

[0065] Engineering Implementation: The standard LMI set of formulas is solved as a convex optimization problem in MATLAB, directly yielding the controller gain that satisfies robustness constraints. and observer gain , Solving for the post-hoc control law and the LKFs formula ensures that the formula holds along the closed-loop trajectory, thus yielding asymptotic stability and... performance.

[0066] 4. Example Verification To further verify the effectiveness of the flexible robotic arm vibration suppression fault-tolerant boundary control method based on TS fuzzy modeling under finite measurement and random controller failure conditions proposed in this invention, this embodiment is verified in conjunction with numerical examples.

[0067] (1) System model setting: Considering the dynamic behavior of a flexible robotic arm under boundary constraints, it can be represented as a distributed parameter system with nonlinear terms, time delay effects, and external disturbances. The dynamic form of this system in the space-time domain is described as follows: in, , Indicates time The partial derivatives, , Represents space The second-order partial derivative, Represents the time delay function. , For system parameters, Indicates control input, This indicates an external disturbance input. For boundary conditions, Indicates at the boundary Spatial derivative condition at , , These are the initial conditions.

[0068] (2) Fuzzy modeling and parameter selection: Based on the TS fuzzy modeling method, the above distributed parameter system is transformed into a weighted form of fuzzy sub-models. Prerequisite variables are selected. , Assuming constant parameters, the fuzzy membership function is constructed as follows: Therefore, the fuzzy weighted model and corresponding parameter matrix of the flexible robotic arm can be obtained. These methods can effectively characterize the dynamic evolution characteristics under different time delays and external disturbances.

[0069] (3) Controller and observer design: Considering the failure of the stochastic controller, factors such as actuator gain attenuation, local signal loss, and boundary measurement limitations are introduced. By solving the LMI constraints, the boundary fault-tolerant controller gain and the fuzzy observer gain are obtained, respectively: The above results were obtained using MATLAB tools, ensuring robust stability under time-varying delay and controller random failure conditions.

[0070] Results and Effects After completing the design and stability analysis of the control method, this invention further verifies the effectiveness and superiority of the proposed method through numerical examples. Based on the distributed parameter model of the nonlinear flexible robotic arm, different comparative experimental scenarios were constructed under time-varying time delay, finite boundary measurement, and stochastic controller failure conditions, and relevant result images were plotted.

[0071] first, Figures 2-3 The spatiotemporal evolution of the state estimation error is shown without observer and controller intervention. It can be seen that the error trajectory exhibits obvious periodic oscillations, and its amplitude gradually increases over time, indicating that the system is highly sensitive to external disturbances and attacks, making it difficult to obtain accurate state estimates and ensuring system stability. Figure 4 The dynamic approximation effect of the flexible manipulator under two error system states in TS fuzzy modeling is presented.

[0072] Secondly Figure 5A schematic diagram of the dynamic characteristics of the stochastic controller failure modeling mechanism is presented. It can be observed that the attack signal exhibits obvious periodic disturbance characteristics in the time domain, further exacerbating the uncertainty and instability of the system, highlighting the importance of combating deception attacks in actual operation.

[0073] After introducing the designed fuzzy observer and non-co-located boundary controller Figures 6-7 State response curves for two closed-loop system states are presented, demonstrating the state evolution of the closed-loop system. The results show that the state trajectory of the flexible robotic arm can converge quickly and remain within the stable range, indicating that the proposed control strategy can effectively suppress system oscillations, achieve ideal spatiotemporal regulation performance, and maintain robustness under time-varying delays, external disturbances, and deception attacks.

[0074] Furthermore, Figures 8-9 The 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.

[0075] 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.

[0076] 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.

[0077] 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.

[0078] 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.

[0079] 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 vibration control method for a flexible robotic arm based on TS fuzzy modeling, characterized in that, include: Step 1. Establish the TS fuzzy distributed parameter system model of the flexible robotic arm; Step 2. Based on the TS fuzzy distribution parameter system model, a non-co-position sensor is used at its left boundary to acquire a limited measurement signal, and an adaptive event-triggered sampling mechanism is constructed to output the sampled signal; Step 3. Based on the sampled signal, design a fuzzy boundary observer under finite boundary measurement to estimate the full state of the system; Step 4. Based on the output of the fuzzy boundary observer and the sampling signal, design a non-co-located fuzzy boundary sampling controller that considers Markov random failure.

2. The vibration control method for a flexible robotic arm based on TS fuzzy modeling according to claim 1, characterized in that, It also includes the following steps: Step 5. Derive the stability criterion of the system using Lyapunov-Krasovsky functionals and linear matrix inequalities.

3. The vibration control method for a flexible robotic arm based on TS fuzzy modeling according to claim 1, characterized in that, The TS fuzzy distribution parameter system model is composed of a weighted superposition of multiple fuzzy rules, with each rule corresponding to a local linear subsystem; the state of the TS fuzzy distribution parameter system model satisfies the following boundary conditions: The left boundary is a natural boundary or a clamping boundary, while the right boundary is controlled by the boundary input.

4. The vibration control method for a flexible robotic arm based on TS fuzzy modeling according to claim 1, characterized in that, The triggering condition for the adaptive event-triggered sampling mechanism is: ,in, This represents the (k+1)th sampling time, where k represents the sampling time number and h represents the candidate time. Indicates the k-th sampling time Starting from the first One candidate moment, Indicates the error signal. This indicates the trigger threshold function. Indicates at the sampling time Boundary measurement quantity, This represents a weighted matrix, and the superscript T indicates the transpose of the corresponding vector.

5. The vibration control method for a flexible robotic arm based on TS fuzzy modeling according to claim 4, characterized in that, In step 3, the output of the fuzzy boundary observer is used to estimate the right-hand boundary state. The hybrid attack receiver model for the right-hand boundary is as follows: ,in, This indicates the signal actually received by the control terminal. The Bernoulli switch variable represents the deception attack. The Bernoulli switch variable represents a denial-of-service attack. This is the estimated output of the fuzzy boundary observer at the right boundary. This is a mapping of fake signals when attacked.

6. The vibration control method for a flexible robotic arm based on TS fuzzy modeling according to claim 5, characterized in that, The non-co-location fuzzy boundary sampling controller is as follows: ,in, For boundary control input, A stochastic degradation factor driven by a Markov chain. The control gain matrix under fuzzy rules. Indicates the mode of the actuator.

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