Fault-tolerant asynchronous control method for single-connecting-rod mechanical arm system based on double-variable decision

Abstract

The invention relates to the technical field of industrial robot control, and discloses a single-connecting-rod mechanical arm system fault-tolerant asynchronous control method based on double-variable decision, which comprises the following steps: establishing a semi-Markov jump system dynamic model for describing the operating characteristics of a single-connecting-rod mechanical arm; based on the dynamic model, designing a fault-tolerant asynchronous controller structure based on a double-variable decision and an experience-machine cooperative operation strategy; establishing sufficient conditions which enable the closed-loop system to be randomly stable and meet H infinity performance indexes; and the numerical value of the gain of the machine controller is obtained by solving the linear matrix inequality set, and a final control law is generated to conduct fault-tolerant asynchronous control on the single-connecting-rod mechanical arm. According to the fault-tolerant asynchronous control method for the mechanical arm, fault-tolerant asynchronous control of the mechanical arm is achieved through a double-variable decision and an experience-machine cooperation strategy in combination with modal transfer rate boundary processing, and the random stability and the H-infinity performance of the system are guaranteed under the modal mismatching and fault working conditions.

Description

Technical Field

[0001] This invention relates to the field of industrial robot control technology, specifically to a fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making. Background Technology

[0002] Single-link robotic arms, as fundamental actuators in industrial automation, are widely used in material handling, welding, and assembly operations. During actual operation, due to the diversity of tasks, the effective payload mass and overall moment of inertia at the robotic arm's end effector often undergo abrupt changes or switching. This abrupt change in physical parameters is typically modeled using Markov jump-linear systems. However, traditional Markov jump-linear system models are based on the memoryless exponential distribution assumption, assuming that the transition rate between system modes is constant. In the actual physical processes of robotic arms, mode switching is often related to the system's dwell time in the current state; for example, parameter changes caused by mechanical wear, aging, or specific processes are more consistent with non-exponential distribution characteristics. Directly using a constant transition rate model to describe the actual dynamics exhibiting semi-Markov characteristics leads to a decrease in model accuracy, thus affecting the stability analysis results of the control system.

[0003] In networked control environments or under conditions limited by sensor bandwidth, the controller's acquisition of the actual system modes often involves time delays or errors. Existing control strategies assume that the controller can acquire system mode information in real time and synchronously, requiring the controller's modes to be consistent with the system modes at all times. When signal transmission delays or packet loss occur in the actual system, this synchronization assumption fails, causing the controller to operate in an asynchronous state that does not match the actual physical state of the system. Ignoring this modal asynchrony and directly applying synchronous control laws can easily lead to performance degradation or even instability of the closed-loop system.

[0004] Furthermore, single-link robotic arms inevitably experience actuator performance loss or bias failures during long-term operation. Most existing fault-tolerant control methods rely solely on online optimization algorithms to calculate feedback gains to offset the effects of these failures. This purely mathematical optimization approach often ignores long-accumulated human operational experience or expert control strategies. The lack of effective utilization of prior experience data results in long adjustment times for control systems facing complex faults, making it difficult to ensure robustness while simultaneously considering the expected trajectory of the system's dynamic response. Currently, there is a lack of a fault-tolerant control scheme that can simultaneously handle semi-Markov dwell time dependence, modal asynchronous mismatch, and the integration of human experience data. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a fault-tolerant asynchronous control method for single-link robotic arm systems based on bivariate decision-making. This method solves the problems of poor system stability and low control accuracy in complex operating conditions of single-link robotic arm systems, such as modal dwell time following a non-exponential distribution, asynchronous mismatch between the controller and system modes, and actuator failure, due to model mismatch in traditional control methods and the lack of utilization of prior human experience.

[0006] To achieve the above objectives, the present invention provides the following technical solution: The first aspect of this invention provides a fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making, comprising the following steps: S1. To address the mode switching behavior of a single-link robotic arm caused by changes in effective load and moment of inertia, as well as actuator failures and external disturbances during operation, and based on the semi-Markov jump theory, a dynamic model of a semi-Markov jump system describing the operating characteristics of a single-link robotic arm is established. S2. Based on the dynamic model, design a fault-tolerant asynchronous controller structure based on bivariate decision-making and experience-machine cooperative operation strategy; the controller structure is composed of the machine controller gain to be solved and the pre-set human experience injection gain, and asynchronous mismatch is allowed between the operating mode of the controller and the actual mode of the dynamic system. S3. Apply the fault-tolerant asynchronous controller structure to the dynamic model to construct a closed-loop system, and perform stochastic stability analysis on the closed-loop system based on Lyapunov stability theory to establish sufficient conditions for the closed-loop system to be stochastically stable and to satisfy the H∞ performance index. The sufficient conditions are expressed in the form of a matrix inequality containing the mode transition rate parameter. S4. Introduce boundary information for the mode transition rate parameter in the sufficient condition, transform the sufficient condition into a solvable set of linear matrix inequalities, obtain the value of the machine controller gain by solving the set of linear matrix inequalities, and synthesize the calculated machine controller gain with the human experience-injected gain to generate the final control law for fault-tolerant asynchronous control of the single-link robotic arm.

[0007] In one specific embodiment, step S1, the semi-Markov jump theory specifically includes: The operation of a single-link robotic arm system is regarded as a random process in a continuous time domain, which switches between a finite number of discrete operating modes; The dwell time of the operating mode is defined to follow a probability density function that is not exponentially distributed, but rather the memoryless exponential distribution of a traditional Markov process. The transition probability between modes is determined by a transition rate function, which is a time-varying function of the current mode dwell time. This function describes the dynamic characteristics of the probability change of the system switching to the next mode as the current mode dwell time increases.

[0008] In one specific embodiment, step S1, the dynamic model of the semi-Markov transition system specifically includes: The system's state vector is defined as consisting of the joint angles and joint angular velocities of the single-link robotic arm; Establish a state-space equation that includes a state matrix, a control input matrix, and an external disturbance matrix, wherein the state matrix, control input matrix, and external disturbance matrix are all time-varying matrices that depend on the current system mode; The system modes correspond to different physical parameter configurations of a single-link robotic arm, and the physical parameters include the payload mass and moment of inertia that change with the load task. The state-space equations describe the linear dynamic relationship between the derivative of the system state vector with time and the current state vector, the control input affected by the fault, and the external disturbance in the current mode.

[0009] In one specific embodiment, step S2, the machine controller gain to be solved specifically includes: The machine controller gain is defined as a feedback gain matrix that depends on two variables, namely the actual modal variables of the system and the observed modal variables of the controller. The machine controller gain is a function of the decision variables in a system of linear matrix inequalities, and the value of the function is calculated using an optimization algorithm.

[0010] In one specific embodiment, step S2, the pre-set human experience specifically includes: Quantify the historical human operation data or expert control strategy of the single-link robotic arm under a specific operating mode, construct a feedback gain matrix with a fixed value, and define the human experience injection gain as the matrix. The human experience injection gain is embedded as a known constant parameter into the mathematical expression of the fault-tolerant asynchronous controller structure, so that it is multiplied with the system state vector to form a deterministic empirical control component; During the calculation of the machine controller gain, the value of the human experience-injected gain is kept constant and used as a constraint term in the operation of the linear matrix inequality system, so that the final generated control law is formed by the linear superposition of the experience control component and the compensation component calculated by the machine.

[0011] In one specific embodiment, step S3, the construction of the closed-loop system specifically includes: An actuator failure model is established, which includes a multiplicative failure factor matrix describing the degree of actuator performance loss and an additive bias failure vector describing the actuator output deviation. Substituting the output of the fault-tolerant asynchronous controller structure into the actuator fault model, the actual control input expression affected by the fault is obtained; The actual control input expression is substituted into the state-space equation of the dynamic model of the semi-Markov jump system to replace the original control input term. After substituting the equations, we obtain the closed-loop state-space equations that describe the relationship between the derivative of the closed-loop system state vector and the current system state, external disturbances, and additive bias faults.

[0012] In one specific embodiment, step S3, establishing sufficient conditions for the closed-loop system to be stochastically stable and to satisfy the H∞ performance index, specifically includes: Construct a stochastic quadratic Lyapunov function that depends on the system modes, wherein the Lyapunov function contains a positive definite symmetric matrix to be solved; The infinitesimal generator operator is applied to the Lyapunov function to calculate the differential expectation value under a semi-Markov jump process. The calculation process introduces a transition rate parameter that describes the dynamics of mode switching. The H∞ performance index is defined as the ratio constraint of the energy of the controlled output vector to the sum of the energies of the external disturbance vector and the bias fault vector. Combining the differential expectation value and the H∞ performance index, a matrix inequality condition is derived. This matrix inequality condition requires that the linear combination of the differential expectation value and the perturbation energy term be less than zero, in order to ensure the trajectory convergence of the closed-loop system in the presence of a perturbation.

[0013] In one specific embodiment, step S4, the introduction of boundary information into the mode transition rate parameter specifically includes: Given that the modal transfer rate parameter is a variable that varies with modal residence time, determine the upper and lower bounds of the parameter within its domain; By utilizing the principle of convex combination, the original single matrix inequality condition that depends on the time-varying transition rate is relaxed into two sets of definite matrix inequality conditions. The two sets of determined matrix inequality conditions correspond to the cases where the mode transfer rate parameter takes an upper bound and the cases where it takes a lower bound, respectively.

[0014] In one specific embodiment, step S4, which involves transforming the sufficient condition into a solvable system of linear matrix inequalities, specifically includes: Perform congruent transformation on the matrix inequalities in the sufficient conditions, and use the inverse matrix of the positive definite symmetric matrix to be solved to perform left and right multiplication operations on the matrix inequalities to eliminate nonlinear coupling terms in the original inequalities. New decision variables are introduced for variable substitution. The product of the machine controller gain and the inverse of the positive definite symmetric matrix is ​​defined as the first intermediate variable, and the inverse of the positive definite symmetric matrix itself is defined as the second intermediate variable. By reconstructing the elements of the matrix inequalities using the first and second intermediate variables, the original non-convex constraints concerning the machine controller gain and the positive definite symmetric matrix are transformed into linear convex constraints concerning the first and second intermediate variables, thus forming a set of linear matrix inequalities that can be directly solved.

[0015] A second aspect of the present invention provides a fault-tolerant asynchronous control system for a single-link robotic arm system based on bivariate decision-making, comprising: The dynamic modeling module is used to establish a dynamic model of a semi-Markov transition system for a single-link robotic arm. The dynamic model of the semi-Markov transition system includes a parameter matrix describing mode switching, actuator failure and external disturbance, and provides the current system state vector and model parameters to the subsequent modules. The bivariate decision module, connected to the dynamic modeling module, is used to acquire the actual modal information of the system in real time and receive the controller's observed modal information. Based on the bivariate pair of actual modal variables of the system and observed modal variables of the controller, it determines the dimension and mapping relationship of the gain matrix used for controller calculation at the current moment. The collaborative gain calculation module, connected to the bivariate decision module, is used to store the pre-set human experience injection gain, solve the linear matrix inequalities according to the boundary information of the mode transition rate to obtain the machine controller gain, and synthesize the machine controller gain with the human experience injection gain to generate the total feedback gain. An asynchronous control execution module, connected to the cooperative gain calculation module, is used to receive the total feedback gain and the system state vector, calculate the final control law, and apply control torque to the single-link robotic arm via the actuator to achieve fault-tolerant asynchronous control. The stability monitoring module is connected to the dynamic modeling module and the asynchronous control execution module respectively. It is used to monitor the state convergence and H∞ performance index of the closed-loop system in real time based on Lyapunov stability theory, and to trigger an alarm or parameter reset when the system state deviates from the preset random stability conditions.

[0016] This invention provides a fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making. It offers the following advantages: 1. This invention transforms the time-varying stability conditions of a semi-Markov jump system, which depend on dwell time, into a deterministic set of linear matrix inequalities by introducing numerical upper and lower bounds for the modal transition rate. This approach enables the controller design to adapt to modal switching processes where the dwell time follows a non-exponential distribution, ensuring the stochastic stability of the single-link robotic arm under conditions where physical parameters change with load and the transition probability fluctuates dynamically.

[0017] 2. This invention employs a bivariate decision-making mechanism based on the actual system mode and the controller's observed mode, considering the potential asynchronous mismatch between the two during the controller design phase. By solving the gain matrix covering these asynchronous situations, the control system can maintain the trajectory convergence and disturbance suppression capabilities of the closed-loop system even when delays or errors in mode detection cause inconsistencies between the controller's operating mode and the system's current physical mode.

[0018] 3. This invention integrates a pre-set human experience-injected gain with the machine controller gain to be solved into the fault-tolerant control law, quantifying the empirical data into fixed parameters embedded in the closed-loop model. This strategy ensures compensation for actuator multiplicative failures and bias faults while utilizing prior experience to determine the fundamental components of the control signal, generating a synthetic control command that combines historical operating characteristics with fault adjustment capabilities. Attached Figure Description

[0019] Figure 1 This is a schematic diagram of the method flow of the present invention; Figure 2 This is a system architecture diagram of the present invention; Figure 3 This is a schematic diagram of the single-link robotic arm model of the present invention; Figure 4 The modal changes of the semi-Markov process of this invention; Figure 5 This is the open-loop state response diagram of the single-link robotic arm system of the present invention; Figure 6 This is a closed-loop state response diagram of the single-link robotic arm system of the present invention.

[0020] Among them, 10 is the dynamic modeling module; 20 is the bivariate decision-making module; 30 is the collaborative gain calculation module; 40 is the asynchronous control execution module; and 50 is the stability monitoring module. Detailed Implementation

[0021] The technical solutions in the embodiments 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, and not all embodiments. 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.

[0022] Reference Figures 1-6 , Figure 1 This is an overall flowchart of a fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making, according to an embodiment of the present invention. The present invention provides a fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making, the specific implementation of which includes the following steps: Step S100: Establish a semi-Markov jump dynamic model of the single-link robotic arm system.

[0023] To address the multimodal characteristics of a single-link robotic arm during actual operation caused by changes in effective load mass (such as switching between no-load and full-load) and changes in rotational inertia, its operation process is modeled as a semi-Markov jump process in the continuous time domain.

[0024] The operational process modeling considers the probabilistic characteristics of modal residence time following a non-exponential distribution, rather than the exponential distribution assumption of traditional Markov models. Furthermore, it introduces multiplicative fault parameters describing actuator performance loss, additive bias fault parameters describing output deviation, and external disturbance parameters describing environmental interference into the Markov model, thereby establishing a semi-Markov jumping linear system model that can reflect the true physical dynamics of the system.

[0025] Step S200: Construct a controller structure based on bivariate decision-making and experience-machine collaborative operation.

[0026] Based on a dynamic model, a fault-tolerant asynchronous controller structure is designed. This controller employs a bivariate decision-making mechanism, meaning its output depends not only on the current system state feedback but also on two independent modal variables: one is the actual system modal variable representing the system's true physical state, and the other is the controller's observed modal variable. This mechanism allows these two modal variables to be numerically inconsistent, allowing for an asynchronous mismatch between the controller's operating mode and the actual system mode.

[0027] In the design of the feedback gain, an experience-machine collaborative operation strategy is adopted. The total controller gain is synthesized from two parts: one part is a pre-set human experience-injected gain, which is determined offline based on historical operating data or expert strategies and is used as a known constant; the other part is the machine controller gain to be solved, which is calculated by subsequent optimization algorithms and is used to compensate for the uncertainty and fault effects of the system.

[0028] Step S300: Establish sufficient conditions for the closed-loop system to be stochastically stable and satisfy H∞ performance.

[0029] Substituting the constructed fault-tolerant asynchronous controller structure into the dynamic model of the single-link robotic arm, the closed-loop system state equation is obtained. Based on Lyapunov stability theory, a Lyapunov functional containing a mode-dependent positive definite matrix is ​​constructed.

[0030] By calculating the infinitesimal generators of this functional under a semi-Markov process and combining this with the H∞ performance index (used to measure the system's ability to suppress external disturbances and faults), sufficient conditions that guarantee the stochastic stability of the closed-loop system are derived. These sufficient conditions are initially expressed as a set of nonlinear matrix inequalities that include time-varying mode transfer rate parameters.

[0031] Step S400: Solve the system of linear matrix inequalities and generate the final control law.

[0032] For the matrix inequalities containing time-varying parameters obtained in step S300, numerical boundary information (upper and lower bounds) of the mode transfer rate is introduced. Using the convex combination principle and variable substitution technique, the original nonlinear constraints are transformed into a set of deterministic, numerically solvable linear matrix inequalities.

[0033] The system of linear matrix inequalities is solved using mathematical calculation software to obtain the machine controller gain value that satisfies the conditions. Finally, the calculated machine controller gain is linearly superimposed with the human experience-injected gain preset in step S200 to generate the final control gain matrix, which is then deployed in the single-link robotic arm control system to achieve fault-tolerant asynchronous control of the robotic arm joint angles and angular velocities.

[0034] Reference Figures 1-6 In step S100, considering the mode switching behavior of the single-link manipulator caused by changes in effective load and moment of inertia, as well as actuator faults and external disturbances during operation, a dynamic model of the semi-Markov transition system describing the operating characteristics of the single-link manipulator is established based on the semi-Markov transition theory. The specific implementation process is as follows: Considering the dynamic equations of a single-link robotic arm, according to the principles of rigid body dynamics, its motion is affected by gravity, damping, and control torque, as described below: ; In the formula, Indicates the joint angle of the robotic arm; Indicates joint angular velocity; Indicates joint angular acceleration; For the effective payload mass; The length of the link; It is the moment of inertia; It is the viscous damping coefficient; It is the acceleration due to gravity; This is the control input torque applied at the joint; This represents the operations of sine and trigonometric functions; This is an external disturbance.

[0035] To facilitate controller design, the above nonlinear dynamic equations are linearized near the equilibrium point. The system state vector is defined. ,in Corresponding joint angles Corresponding joint angular velocity.

[0036] Introducing a semi-Markov process This describes the multimodal switching behavior of a robotic arm caused by load changes. In finite sets The values ​​are selected from the range, each representing a specific operating mode (e.g., "light load mode" or "heavy load mode"). The physical parameters of the system differ under different modes. and The values ​​are different.

[0037] Considering the external disturbances that exist in actual operation In addition to actuator failure, the following continuous-time semi-Markov jump linear system (S-MJLS) ​​model is established: ; In the formula, This represents the actual control input after a fault occurs; Represents the controlled output vector; , , , , All depend on the current mode 0 real matrix; External disturbance; This represents the state vector of a single-link robotic arm system. Represents the system state vector Regarding time The first derivative.

[0038] For semi-Markov processes Its state transition characteristics are determined by the transition probability matrix. Description. The system starts from modal... Switch to mode The transition probability depends on the dwell time of the current mode. The specific transfer rate function Defined as: ; In the formula, This indicates that when the system is in the current mode The length of stay is At that time, from the modal Jump to modality The transfer rate function; The mathematical limit operator represents the time increment. Approaching ; Symbols representing probability measures; Indicating a semi-Markov process in The state at any given moment; Represents the index value of the target mode, and satisfies ; Indicating a semi-Markov process in The state at any given moment; Indicates the index value of the current mode; This indicates that the system is already in the current mode. The length of time spent in the middle; This indicates an extremely small time interval increment.

[0039] When building the model, the specific form of the state matrix is ​​shown in the following example: ; In the formula, The equilibrium point angle selected for linearization; and Modal The mass and moment of inertia parameters are as follows; This indicates when the system is in a modal state. The system state matrix at that time; It is the viscous damping coefficient; This indicates when the system is in a modal state. Control input matrix at time; It is the acceleration due to gravity; The length of the link; This represents the operation of cosine and trigonometric functions. This modeling method accurately transforms the complex nonlinear multimodal dynamics of the robotic arm into a piecewise linear semi-Markov jump system model, providing an accurate mathematical foundation for subsequent controller design.

[0040] Reference Figure 1 In step S200, this embodiment constructs a fault-tolerant asynchronous controller structure based on a bivariate decision-making and experience-machine cooperative operation strategy, and establishes an actuator fault model. The specific implementation process is as follows: To address the inconsistency in modal recognition caused by network latency, packet loss, or detection noise in real-world industrial environments, this invention designs a bivariate decision-making mechanism. When the system is in modal... At that time, the mode that the controller may observe is At this point, the conditional probability It is used to describe the mapping relationship between these modes, where This mechanism explicitly allows This means that it achieves tolerance for modal asynchronous mismatch.

[0041] Based on the aforementioned bivariate decision-making mechanism, a state feedback controller with experience-machine cooperative operation characteristics is designed. The mathematical expression of this controller is constructed as follows: ; When the controller observes the mode When, it is abbreviated as: ; In the formula, This is the control input torque applied at the joint; This represents the state vector of a single-link robotic arm system. This indicates dependence on the current observation mode. The total state feedback control gain matrix; This indicates dependence on the current observation mode. The gain section of the machine controller; This indicates dependence on the current observation mode. Human experience is injected into the gain component; Represents the observation mode The specific numerical index currently being retrieved; It is a pre-defined gain matrix infused with human experience. This matrix is ​​a constant matrix obtained offline based on historical human operation data or expert control laws of a single-link robotic arm under specific working conditions. It represents the prior control knowledge humans have about the physical system. This is the gain matrix of the machine controller to be designed. It will be calculated online or offline using subsequent optimization algorithms and is used to compensate for system uncertainties, suppress disturbances, and handle the effects of modal asynchrony. This superposition structure allows the final control law to retain the baseline performance of human experience while possessing the adaptive robustness of machine computation.

[0042] Furthermore, to describe potential actuator failures under harsh environments, a comprehensive fault model incorporating multiplicative failures and additive biases is established. This assumes that the actual control input acting on the system... The ideal output calculated by the controller The following relationship exists between them: ; In the formula, This is the control input torque applied at the joint; This represents the control input vector that actually acts on the single-link robotic arm system after being affected by an actuator malfunction. The distribution matrix representing additive bias faults; Represents the time-varying additive bias fault vector; Represents the fault matrix of a time-varying multiplicative actuator; This is the fault matrix for time-varying multiplicative actuators. Diagonal elements. Indicates the first The health factor of each actuator is set to a value range of [value range]. .For example, This indicates that the actuator is completely healthy. This indicates a 50% loss in actuator performance.

[0043] Combining the above controller structure and fault model, and substituting them into the system equations in step S100, the final closed-loop system dynamic equations are obtained: ; In the formula, Represents the system state vector Regarding time The first derivative; This represents the state vector of a single-link robotic arm system. External disturbance; The distribution matrix representing additive bias faults; Represents the time-varying additive bias fault vector; This indicates when the system is in a modal state. The system state matrix at that time; This indicates when the system is in a modal state. Control input matrix at time; It is a pre-set gain matrix for injecting human experience; It is the gain matrix of the machine controller to be designed; This represents the time-varying multiplicative actuator fault matrix, used to describe the fluctuation of the actuator performance factor; This indicates when the system is actually in a modal state. The external perturbation input matrix at that time.

[0044] The closed-loop equation explicitly includes the system modes. Controller Mode Machine gain Experience gain and fault parameters and This provides a complete mathematical description object for subsequent stability analysis.

[0045] Reference Figures 1-6 Step S300: In this embodiment, the stochastic stability of the closed-loop system is analyzed based on Lyapunov stability theory, and sufficient conditions for satisfying the H∞ performance index are established. The specific implementation process is as follows: To analyze the stochastic stability of the above closed-loop semi-Markov jumping system, a mode-dependent stochastic quadratic Lyapunov functional is constructed as follows: ; In the formula, This represents the scalar value of the constructed mode-dependent stochastic quadratic Lyapunov functional at the current time, used to evaluate the energy state of the system; Represents the system state vector The transpose of ; This represents the current mode of the semi-Markov system; This represents the state vector of a single-link robotic arm system. It is a positive definite symmetric matrix, and this matrix depends not only on the current system modes. It also depends on the dwell time in the current mode. In specific calculations, it is usually assumed that... That is, it simplifies to a positive definite matrix that depends only on the mode. This is to reduce computational complexity.

[0046] definition This is an infinitesimal generator operator. For a semi-Markov jump process, This describes the expected rate of change of the Lyapunov function along the system's trajectory. Based on the properties of semi-Markov processes, The calculation formula can be expanded as follows: ; Substituting the closed-loop system equations into the above equation, we obtain: ; In the formula, It is a transfer rate function; The mathematical limit operator represents the time increment. Approaching ; This represents the expected derivative of the Lyapunov functional along the system trajectory; Represents the mathematical expectation operator; Indicates in The system state vector at time t; Indicates in System modes at time points; This represents the state vector of a single-link robotic arm system. Indicating a semi-Markov process in The state at any given moment; Indicates the index value of the current mode; This indicates that the system is already in the current mode. The length of time spent in the middle; This represents an extremely small increment in time interval; This indicates that when the system state is And the mode is Lyapunov functional value at time; Represents the system state vector The transpose of ; Represents the Hermitian operator; This indicates when the system is in a modal state. Positive definite symmetric matrix at time; This indicates that when the system mode is And the controller mode is The closed-loop system state matrix at time; Represents a set All possible next modes Perform a summation operation; This indicates that when the system is in the current mode The length of stay is At that time, from the modal Jump to modality The transfer rate function; This indicates when the system transitions to the next mode. Positive definite symmetric matrix at time; This indicates when the system is in a modal state. Positive definite symmetric matrix at time; This indicates when the system is in a modal state. Control input matrix at time; The distribution matrix representing additive bias faults; Represents the time-varying additive bias fault vector; This indicates when the system is actually in a modal state. External disturbance input matrix at time; This is an external disturbance.

[0047] To evaluate the system's fault tolerance and disturbance rejection capabilities, the H∞ performance index is introduced. Define controlled output. External disturbances and bias fault The energy constraint relationship between them is: ; In the formula, Represents the mathematical expectation operator; Indicates in Integral operations over an infinite time interval; This represents the controlled output vector, used to evaluate the system's performance. Represents the controlled output vector The transpose of ; This represents a given H∞ performance index; This represents the generalized perturbation vector, which is typically composed of external perturbations. and bias fault The augmented vector formed; Represents the generalized perturbation vector The transpose of ; The differential variable representing time.

[0048] Combination The expression and H∞ performance index are obtained by applying... Given the constraints, we derive the system's stochastic stability and H∞ performance. A sufficient condition for this is expressed as a matrix inequality of the following form: ; In the formula, Represents the Hermitian operator; This indicates when the system is in a modal state. Positive definite symmetric matrix at time; This indicates that when the system mode is The closed-loop system state matrix; Represents a set All possible next modes Perform a summation operation; This indicates that when the system is in the current mode The length of stay is At that time, from the modal Jump to modality The transfer rate function; This indicates when the system transitions to the next mode. Positive definite symmetric matrix at time; This indicates when the system is actually in a modal state. External disturbance input matrix at time; This indicates when the system is in a modal state. Control input matrix at time; The distribution matrix representing additive bias faults; Represents the closed-loop controlled output matrix The transpose of the matrix; Represents the perturbation coefficient matrix in the controlled output equation The transpose of the matrix; Denotes the terms on the diagonal, where H∞ is the performance index (the level of disturbance suppression). It is the identity matrix; This represents the negative identity matrix.

[0049] This matrix inequality is a nonlinear constraint because it contains the variables to be solved. and The product term, and the transfer rate within it. It is over time The continuously changing parameters constitute the main challenge in the subsequent solution process, requiring specific mathematical transformations to handle them.

[0050] Reference Figures 1-6 In step S400, this embodiment addresses the sufficient condition of the matrix inequality containing time-varying parameters established in step S300 by introducing boundary information and variable substitution techniques, transforming it into a solvable system of linear matrix inequalities, and calculating the final fault-tolerant asynchronous control law. The specific implementation process is as follows: Boundary treatment and relaxation of the modal transfer rate parameter: Given that the modal transfer rate parameter included in the aforementioned matrix inequality is a variable that changes continuously with residence time, directly solving for the condition satisfied in the infinite time domain is computationally infeasible. Therefore, we first determine the upper and lower bounds of the transfer rate parameter within its domain. Specifically, for the process of the system transferring from the current mode to any other target mode, we analyze the mathematical characteristics of its transfer rate function and find its maximum and minimum values.

[0051] By utilizing the principle of convex combinations in mathematics, the original single matrix inequality condition, which depends on the time-varying transition rate, is relaxed. Specifically, this involves constructing two sets of deterministic matrix inequalities: the first set corresponds to the case where the modal transition rate parameter takes an upper bound, and the second set corresponds to the case where the modal transition rate parameter takes a lower bound. According to convexity theory, if the system parameters simultaneously satisfy these two sets of boundary-value-based matrix inequalities, then for any transition rate value between the upper and lower bounds, the original time-varying matrix inequality must also hold.

[0052] Variable substitution and linear matrix inequality transformation: The matrix inequalities after boundary treatment still contain nonlinear coupling terms, that is, the positive definite symmetric matrix to be solved and the machine controller gain matrix not only exist as variables individually, but also appear in the form of a product, which makes conventional convex optimization algorithms unsuitable for direct application. To solve this non-convex problem, this embodiment adopts a method combining congruent transformation and variable substitution.

[0053] The matrix inequality is transformed using a congruent transformation operation, multiplying the left and right sides of the original inequality by the inverse of the positive definite symmetric matrix to be solved. Then, new decision variables are introduced for substitution. Specifically, two intermediate variables are defined: The first intermediate variable is defined as the product of the machine controller gain matrix and the inverse of the positive definite symmetric matrix. The second intermediate variable is defined as the inverse of a positive definite symmetric matrix itself.

[0054] The elements of the matrix inequalities are reconstructed using these two new intermediate variables. For example, the original term "positive definite symmetric matrix multiplied by the system dynamic matrix" is transformed into a linear combination of "system dynamic matrix multiplied by the second intermediate variable" and "input matrix multiplied by the first intermediate variable." After this reconstruction, the original nonlinear constraints concerning the machine controller gain and the positive definite symmetric matrix are completely transformed into linear convex constraints concerning the first and second intermediate variables.

[0055] Furthermore, for the uncertainty in the actuator fault matrix, the interval decomposition technique is used to decompose it into a nominal part and an uncertain part, and the scaling inequality is used to transform the uncertain part into a deterministic linear constraint term, which is then added to the above set of linear matrix inequalities.

[0056] Gain calculation and control law generation: Using existing semidefinite programming software, the pre-constructed system of linear matrix inequalities is solved. The goal is to find numerical solutions for the first and second intermediate variables that satisfy all constraints.

[0057] Once a numerical solution is obtained, the machine controller gain matrix can be calculated in reverse. The specific calculation method is to right-multiply the first intermediate variable obtained by the solution by the inverse matrix of the second intermediate variable.

[0058] Finally, the control law synthesis operation is performed. The calculated machine controller gain matrix is ​​linearly added to the human experience-injected gain matrix preset in step S200 to obtain the final total feedback gain matrix. This total feedback gain matrix is ​​deployed to the execution module of the control system to collect the state vector (joint angle and angular velocity) of the single-link robotic arm in real time and calculate the control torque command, thereby driving the robotic arm to achieve fault-tolerant asynchronous control.

[0059] Reference Figures 1-6 , Figure 2 This is a structural block diagram of a fault-tolerant asynchronous control system for a single-link robotic arm system based on bivariate decision-making, according to an embodiment of the present invention. The present invention provides a fault-tolerant asynchronous control system for a single-link robotic arm system based on bivariate decision-making, comprising: a dynamic modeling module 10, a bivariate decision-making module 20, a cooperative gain calculation module 30, an asynchronous control execution module 40, and a stability monitoring module 50.

[0060] The dynamic modeling module 10 is used to establish a dynamic model of a semi-Markov jump system for a single-link manipulator. This module stores a multimodal parameter set describing the single-link manipulator under different load and moment of inertia configurations, as well as mathematical models describing actuator multiplicative efficiency losses and additive bias faults. The dynamic modeling module 10 collects joint angle and angular velocity data of the single-link manipulator in real time through a sensor interface, encapsulates this data into a system state vector, and determines the system model parameter matrix by combining it with the current load configuration information. The output of the dynamic modeling module 10 is connected to the bivariate decision module 20 and the stability monitoring module 50, providing them with the current system state vector and model parameter matrix.

[0061] The bivariate decision module 20, connected to the dynamic modeling module 10, is used to determine the modal logic required for controller calculation. This bivariate decision module 20 receives actual system modal information from the dynamic modeling module 10 and simultaneously receives observed modal information from the controller side via a communication interface. Internally, the bivariate decision module 20 runs bivariate mapping logic, processing the actual system modal variables and the controller's observed modal variables as a set of bivariates to identify whether a modal asynchrony mismatch exists. Based on the identification result, the bivariate decision module 20 determines the gain matrix dimension index that should be invoked at the current moment and sends this index information to the cooperative gain calculation module 30.

[0062] The collaborative gain calculation module 30, connected to the bivariate decision module 20, is used to generate the total feedback gain of the controller. This collaborative gain calculation module 30 internally includes a storage unit and a calculation unit. The storage unit stores a pre-set human experience-injected gain matrix, which is a constant matrix obtained offline based on historical operating data. The calculation unit is used to solve a pre-set set of linear matrix inequalities based on the upper and lower bounds of the mode transition rate to obtain the machine controller gain matrix. The collaborative gain calculation module 30 linearly superimposes the obtained machine controller gain matrix with the stored human experience-injected gain matrix to generate the total feedback gain matrix, which is then sent to the asynchronous control execution module 40.

[0063] The asynchronous control execution module 40, connected to the cooperative gain calculation module 30, is used to implement the final control action. The asynchronous control execution module 40 receives the total feedback gain matrix from the cooperative gain calculation module 30 and the real-time system state vector from the dynamic modeling module 10. The asynchronous control execution module 40 performs matrix multiplication, multiplying the total feedback gain matrix by the system state vector to calculate the ideal control torque command. Subsequently, the asynchronous control execution module 40 converts the control torque command into a voltage or current signal, which is applied to the joint motor of the single-link robotic arm via a drive circuit, driving the robotic arm to move.

[0064] The stability monitoring module 50, connected to both the dynamic modeling module 10 and the asynchronous control execution module 40, ensures the safe operation of the system. This module receives real-time system status data and control input data, and calculates the energy function value of the closed-loop system based on Lyapunov stability theory. The stability monitoring module 50 compares this energy function value with a preset stochastic stable convergence boundary in real time, while simultaneously monitoring whether the H∞ performance index meets a preset threshold. If the system state deviates from the stable region or the performance index exceeds limits, the stability monitoring module 50 immediately triggers an alarm signal and sends a parameter reset command to the cooperative gain calculation module 30, forcing the system to recalculate the gain or switch to a safe mode.

Claims

1. A fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making, characterized in that, Includes the following steps: S1. To address the mode switching behavior of a single-link robotic arm caused by changes in effective load and moment of inertia, as well as actuator failures and external disturbances during operation, and based on the semi-Markov jump theory, a dynamic model of a semi-Markov jump system describing the operating characteristics of a single-link robotic arm is established. S2. Based on the dynamic model, design a fault-tolerant asynchronous controller structure based on bivariate decision-making and experience-machine cooperative operation strategy; the controller structure is composed of the machine controller gain to be solved and the pre-set human experience injection gain, and asynchronous mismatch is allowed between the operating mode of the controller and the actual mode of the dynamic system. S3. Apply the fault-tolerant asynchronous controller structure to the dynamic model to construct a closed-loop system, and perform stochastic stability analysis on the closed-loop system based on Lyapunov stability theory to establish sufficient conditions for the closed-loop system to be stochastically stable and to satisfy the H∞ performance index. The sufficient conditions are expressed in the form of a matrix inequality containing the mode transition rate parameter. S4. Introduce boundary information for the mode transition rate parameter in the sufficient condition, transform the sufficient condition into a solvable set of linear matrix inequalities, obtain the value of the machine controller gain by solving the set of linear matrix inequalities, and synthesize the calculated machine controller gain with the human experience-injected gain to generate the final control law for fault-tolerant asynchronous control of the single-link robotic arm.

2. The fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making as described in claim 1, characterized in that, In step S1, the semi-Markov jump theory specifically includes: The operation of a single-link robotic arm system is regarded as a random process in a continuous time domain, which switches between a finite number of discrete operating modes; The dwell time of the operating mode is defined to follow a probability density function that is not exponentially distributed, but rather the memoryless exponential distribution of a traditional Markov process. The transition probability between modes is determined by a transition rate function, which is a time-varying function of the current mode dwell time. This function describes the dynamic characteristics of the probability change of the system switching to the next mode as the current mode dwell time increases.

3. The fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making as described in claim 1, characterized in that, In step S1, the dynamic model of the semi-Markov transition system specifically includes: The system's state vector is defined as consisting of the joint angles and joint angular velocities of the single-link robotic arm; Establish a state-space equation that includes a state matrix, a control input matrix, and an external disturbance matrix, wherein the state matrix, control input matrix, and external disturbance matrix are all time-varying matrices that depend on the current system mode; The system modes correspond to different physical parameter configurations of a single-link robotic arm, and the physical parameters include the payload mass and moment of inertia that change with the load task. The state-space equations describe the linear dynamic relationship between the derivative of the system state vector with time and the current state vector, the control input affected by the fault, and the external disturbance in the current mode.

4. The fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making as described in claim 1, characterized in that, In step S2, the machine controller gain to be solved specifically includes: The machine controller gain is defined as a feedback gain matrix that depends on two variables, namely the actual modal variables of the system and the observed modal variables of the controller. The machine controller gain is a function of the decision variables in a system of linear matrix inequalities, and the value of the function is calculated using an optimization algorithm.

5. The fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making as described in claim 1, characterized in that, In step S2, the pre-set human experience specifically includes: Quantify the historical human operation data or expert control strategy of the single-link robotic arm under a specific operating mode, construct a feedback gain matrix with a fixed value, and define the human experience injection gain as the matrix. The human experience injection gain is embedded as a known constant parameter into the mathematical expression of the fault-tolerant asynchronous controller structure, so that it is multiplied with the system state vector to form a deterministic empirical control component; During the calculation of the machine controller gain, the value of the human experience-injected gain is kept constant and used as a constraint term in the operation of the linear matrix inequality system, so that the final generated control law is formed by the linear superposition of the experience control component and the compensation component calculated by the machine.

6. The fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making as described in claim 1, characterized in that, In step S3, the construction of the closed-loop system specifically includes: An actuator failure model is established, which includes a multiplicative failure factor matrix describing the degree of actuator performance loss and an additive bias failure vector describing the actuator output deviation. Substituting the output of the fault-tolerant asynchronous controller structure into the actuator fault model, the actual control input expression affected by the fault is obtained; The actual control input expression is substituted into the state-space equation of the dynamic model of the semi-Markov jump system to replace the original control input term. After substituting the equations, we obtain the closed-loop state-space equations that describe the relationship between the derivative of the closed-loop system state vector and the current system state, external disturbances, and additive bias faults.

7. The fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making as described in claim 1, characterized in that, In step S3, establishing sufficient conditions for the closed-loop system to be stochastically stable and to satisfy the H∞ performance index specifically includes: Construct a stochastic quadratic Lyapunov function that depends on the system modes, wherein the Lyapunov function contains a positive definite symmetric matrix to be solved; The infinitesimal generator operator is applied to the Lyapunov function to calculate the differential expectation value under a semi-Markov jump process. The calculation process introduces a transition rate parameter that describes the dynamics of mode switching. The H∞ performance index is defined as the ratio constraint of the energy of the controlled output vector to the sum of the energies of the external disturbance vector and the bias fault vector. Combining the differential expectation value and the H∞ performance index, a matrix inequality condition is derived. This matrix inequality condition requires that the linear combination of the differential expectation value and the perturbation energy term be less than zero, in order to ensure the trajectory convergence of the closed-loop system in the presence of a perturbation.

8. The fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making according to claim 1, characterized in that, In step S4, the introduction of boundary information for the mode transfer rate parameter specifically includes: Given that the modal transfer rate parameter is a variable that varies with modal residence time, determine the upper and lower bounds of the parameter within its domain; By utilizing the principle of convex combination, the original single matrix inequality condition that depends on the time-varying transition rate is relaxed into two sets of definite matrix inequality conditions. The two sets of determined matrix inequality conditions correspond to the cases where the mode transfer rate parameter takes an upper bound and the cases where it takes a lower bound, respectively.

9. The fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making according to claim 1, characterized in that, In step S4, transforming the sufficient condition into a solvable system of linear matrix inequalities specifically includes: Perform congruent transformation on the matrix inequalities in the sufficient conditions, and use the inverse matrix of the positive definite symmetric matrix to be solved to perform left and right multiplication operations on the matrix inequalities to eliminate nonlinear coupling terms in the original inequalities. New decision variables are introduced for variable substitution. The product of the machine controller gain and the inverse of the positive definite symmetric matrix is ​​defined as the first intermediate variable, and the inverse of the positive definite symmetric matrix itself is defined as the second intermediate variable. By reconstructing the elements of the matrix inequalities using the first and second intermediate variables, the original non-convex constraints concerning the machine controller gain and the positive definite symmetric matrix are transformed into linear convex constraints concerning the first and second intermediate variables, thus forming a set of linear matrix inequalities that can be directly solved.

10. A fault-tolerant asynchronous control system for a single-link robotic arm system based on bivariate decision-making, applied to the fault-tolerant asynchronous control method for a single-link robotic arm system based on bivariate decision-making as described in any one of claims 1-9, characterized in that, include: The dynamic modeling module is used to establish a dynamic model of a semi-Markov transition system for a single-link robotic arm. The dynamic model of the semi-Markov transition system includes a parameter matrix describing mode switching, actuator failure and external disturbance, and provides the current system state vector and model parameters to the subsequent modules. The bivariate decision module, connected to the dynamic modeling module, is used to acquire the actual modal information of the system in real time and receive the controller's observed modal information. Based on the bivariate pair of actual modal variables of the system and observed modal variables of the controller, it determines the dimension and mapping relationship of the gain matrix used for controller calculation at the current moment. The collaborative gain calculation module, connected to the bivariate decision module, is used to store the pre-set human experience injection gain, solve the linear matrix inequalities according to the boundary information of the mode transition rate to obtain the machine controller gain, and synthesize the machine controller gain with the human experience injection gain to generate the total feedback gain. An asynchronous control execution module, connected to the cooperative gain calculation module, is used to receive the total feedback gain and the system state vector, calculate the final control law, and apply control torque to the single-link robotic arm via the actuator to achieve fault-tolerant asynchronous control. The stability monitoring module is connected to the dynamic modeling module and the asynchronous control execution module respectively. It is used to monitor the state convergence and H∞ performance index of the closed-loop system in real time based on Lyapunov stability theory, and to trigger an alarm or parameter reset when the system state deviates from the preset random stability conditions.

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