Fractional order nonlinear system fuzzy fault-tolerant controller design and control method for flexible robot arm

Abstract

The application discloses a design method of a fractional order nonlinear system fuzzy fault-tolerant controller for a flexible mechanical arm, and comprises the following steps: constructing a fractional order nonlinear system with external disturbance and actuator fault; estimating a nonlinear part in the fractional order nonlinear system based on a fuzzy logic system; defining a coordinate transformation related to a tracking error; and designing a finite time adaptive fuzzy fault-tolerant controller by combining a fractional order dynamic surface control method with a backstepping method. According to the application, the output of the fractional order nonlinear system can track a reference signal in a finite time, and all signals in the closed loop system are also bounded in the finite time. Finally, the controller is applied to a flexible mechanical arm system.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of flexible robot arm control, and particularly relates to a fractional order nonlinear system fuzzy fault-tolerant control method for a flexible robot arm. BACKGROUND

[0002] In recent years, the tracking control problem of fractional order nonlinear systems has attracted extensive attention from scholars due to its successful application in engineering fields such as circuit systems and chaotic systems. However, in practical applications, it is difficult to obtain an accurate system model due to measurement and modeling inaccuracies, the influence of time-varying parameters, and the influence of unknown uncertain factors such as load disturbances. Intelligent control methods, such as fuzzy logic systems or neural networks, are often used to overcome the uncertainties in the model due to their good approximation. Scholars often use a backstepping method combined with neural networks / fuzzy logic systems to design the controller of the system to ensure the performance of the system. However, the use of the backstepping method to design the controller of the fractional order nonlinear system will inevitably introduce the "computational explosion" problem.

[0003] In practical engineering systems, the increasingly complex industrial processes often lead to unexpected failures in the plant. For a long time, many scholars have devoted themselves to the research of fault diagnosis and fault-tolerant control of fractional-order nonlinear systems. Liu H et al. studied the adaptive neural network backstepping control of fractional-order nonlinear systems with actuator faults, considering the problem of actuator failure with completely unknown parameters and modes in fractional-order systems, and using the fractional-order command filter method to avoid the problem of "computational explosion" [Liu H, Pan Y, Cao J, et al. Adaptive neural network backstepping control of fractional-order nonlinear systems with actuator faults [J]. IEEE Transactions on Neural Networks and Learning Systems, 2020, 31(12): 5166-5177]. However, when solving the controller using backstepping method, a virtual control signal (which is a composite function containing multiple functions) needs to be designed, and in this process, the fractional derivative of the virtual control signal needs to be solved repeatedly. Since the fractional calculus is redefined, some traditional derivation chain rules cannot be directly applied to fractional-order systems, increasing the difficulty of solving and also leading to an increase in the amount of calculation. Yang W et al. studied the observer-based adaptive fuzzy control of fractional-order time-varying delayed MIMO systems with actuator faults, considering the faults such as actuator stuck and failure, and using a parameter-adjustable compensation function for compensation [Yang W, Zheng W X, Yu W. Observer-based event-triggered adaptive fuzzy control for fractional-order time-varying delayed MIMO systems against actuator faults [J / OL]. IEEE Transactions on Fuzzy Systems, 2022. 30(12): 5445-5459]. However, when the actuator fails, in addition to ensuring that the system has acceptable performance indicators, the system should also respond quickly within a limited time. Therefore, how to use the finite-time stability theory to solve the actuator fault of fractional-order nonlinear systems and achieve fast tracking and convergence is a difficult problem to be solved and is worth further analysis.Xue G et al. studied the adaptive fuzzy finite-time backstepping control of fractional-order nonlinear systems based on command filtering and sliding mode technique, which is a finite-time sliding mode fault-tolerant control scheme for uncertain strict-feedback fractional-order nonlinear systems with actuator faults [Xue G, Lin F, Li S, et al. Adaptive fuzzy finite-time backstepping control of fractional-order nonlinear systems with actuator faults via command-filtering and sliding mode technique[J]. Information Sciences, 2022, 600: 189-208]; however, Lemma 2 used in this scheme is incorrect; therefore, the finite-time stability of fractional-order nonlinear systems obtained in some literatures using this lemma may be unreliable.

[0004] Therefore, it is an open and challenging problem to study the finite-time fault-tolerant control of fractional-order nonlinear systems.

[0005] Moreover, existing research and practical applications have shown that the flexible manipulator is rigid-flexible coupled, has many sensors and actuators that are prone to failure, and is difficult to model, making it difficult for conventional fault-tolerant control methods to work effectively, which seriously reduces the operating efficiency of the flexible manipulator in operation. The application of a flexible manipulator in a fault state or without a fault-tolerant mechanism can cause the robot to be damaged or its service life to be greatly shortened, and actuator failure is one of the most common fault types and a challenging problem that needs to be solved for a flexible manipulator system. SUMMARY

[0006] In view of the above problems in the prior art, the present application provides a fuzzy fault-tolerant controller design method for a fractional-order nonlinear system of a flexible manipulator, which constructs a finite-time adaptive fuzzy fault-tolerant controller for a fractional-order nonlinear system, so that the output of the fractional-order nonlinear system can track the reference signal in a finite time, and ensures that all signals in the fractional-order system are also bounded in a finite time, and finally the controller is applied in a flexible manipulator system.

[0007] In order to achieve the above invention purposes, the technical solutions adopted by the present application are as follows.

[0008] The fuzzy fault-tolerant controller design method for a fractional-order nonlinear system of a flexible manipulator provided by the present application comprises the following steps:

[0009] Step 1: Construct a fractional-order nonlinear system with external disturbance and actuator failure;

[0010] Step 2: Estimate the nonlinear component in the fractional-order nonlinear system based on fuzzy logic;

[0011] Step 3: Based on the fractional-order nonlinear system obtained in Step 2, construct a finite-time adaptive fuzzy fault-tolerant controller. This step includes the following sub-steps:

[0012] Step 3.1 Define the coordinate transformation related to the tracking error;

[0013] Step 3.2 Design a finite-time adaptive fuzzy fault-tolerant controller using the fractional-order dynamic surface control method combined with the backstepping method.

[0014] Alternatively, the fractional-order nonlinear system is specifically:

[0015]

[0016] in, This represents the Caputo fractional derivative, where α represents the order of the fractional nonlinear system. x i =[x1,x2,...,x i ] T x = x n =[x1,x2,...,x n ] T Let T represent the state variables of the system, and let g represent the transpose of a vector or matrix. i f represents the virtual control coefficient. i ( x i ): The expression represents an unknown smooth nonlinear function, d i (t) represents the external disturbance of the system. This represents a known vector of control gain constants. Let y(t) represent the control input vector of the fractional-order nonlinear system, and let y(t) represent the output of the fractional-order nonlinear system.

[0017] Alternatively, actuator failures include jamming failures and malfunctions.

[0018] Alternatively, the mathematical model for the stuck fault is as follows:

[0019]

[0020] Among them, u j (t) indicates that the j-th actuator has encountered a jamming failure. Indicates a constant, t jψj(t) represents the time when the jth actuator occurs stuck fault, p represents the total number of stuck faults of actuators;

[0021] The mathematical model of failure fault is specifically:

[0022] u j (t) = ψ j μ j (t), t ≥ t j ,

[0023] ψ j ∈( Ψ j , 1], 0 Ψ j <1

[0024] Wherein, u j (t) represents the jth actuator occurs failure fault, ψ j represents the proportion of actuators still effective after failure, μ j (t) represents the jth control input of fractional order nonlinear system, t j represents the time when the jth actuator occurs failure fault, q represents the total number of actuators, represents the complement of q actuators stuck fault, Ψ j represents the lower bound of ψ j , μ j (t) adopts the control structure of , wherein and represent the lower bound and upper bound of , u r0 represents the actual control input signal of the actuator.

[0025] As an option, the fractional order nonlinear system can be further described by using fuzzy logic system to estimate unknown nonlinear function in the system:

[0026]

[0027] Wherein, represents the optimal parameter vector of fuzzy logic system, T represents the transpose symbol, φ i ( x i ) represents the fuzzy basis function vector, δ i represents the optimal estimation error of fuzzy logic system.

[0028] As an option, the global coordinate transformation is specifically:

[0029]

[0030] Where s1 represents the tracking error of the fractional-order nonlinear system, and y represents the output of the fractional-order nonlinear system. d The reference signal s represents the tracking signal. i x represents the tracking error introduced by the system state variables. i ξ represents the state variable of the system. i This represents the output of the α-order filter. Represents the error surface, ν i-1 This indicates a virtual control signal.

[0031] Alternatively, the following virtual control signal ν can be designed in n steps using a fractional-order dynamic surface control method combined with a backstepping method. i Parameter adaptive law θ i Λ i and the actual control input signal u r0 This leads to the construction of a finite-time adaptive fuzzy fault-tolerant controller, specifically including:

[0032] Based on the coordinate transformation, set the Lyapunov function in step 1 as follows:

[0033]

[0034] Where Γ1 and γ1 both represent positive parameters, Both represent the error estimation parameters of the adaptive law; Λ1 is The estimated value, Λ1 * Let θ1 be the upper bound of the sum of the optimal fuzzy approximation error δ1 and the external disturbance d1(t), where θ1 is... The estimated value.

[0035] Taking the α-order derivative of V1(t), and combining Young's inequality and fractional dynamic surface control method, we construct the virtual control signal ν1 and the parameter adaptive laws θ1 and Λ1 as follows:

[0036]

[0037] Among them, 0<ρ<1, k1, c1, κ, σ1, τ 1,1 τ 1,2 All of these are positive parameters of the design.

[0038] Based on the coordinate transformation, set the Lyapunov function in step i as follows:

[0039]

[0040] Among them, Γ i γ iAll represent positive parameters. Both represent the error estimation parameters of the adaptive law. Indicates the error surface. Λ i yes The estimated value, Λ i * The optimal fuzzy approximation error δ is represented by i and external interference d i The upper bound of the sum of (t), θ i yes The estimated values ​​are i = 2, ..., n-1.

[0041] For V i (t) Find the α-th derivative and construct the virtual control signal ν using Young's inequality and fractional dynamic surface control method. i And parameter adaptive law θ i Λ i for:

[0042]

[0043] Where 0 < ρ < 1, k i c i σ i τ i,1 τ i,2 κ represents the positive parameters of the design.

[0044] Based on the coordinate transformation, set the Lyapunov function in step n as follows:

[0045]

[0046] Among them, Γ n γ n All represent positive parameters. Both represent the error of the parameter adaptive law, θ n yes The estimated value; Λ n yes The estimated value, Λ n * The optimal fuzzy approximation error δ is represented by n and external interference d n The upper bound of the sum of (t);

[0047] For V n (t) Find the α-th derivative and construct the actual control input signal u using Young's inequality and fractional dynamic surface control method. r0 And parameter adaptive law θ n ,Λ n for:

[0048]

[0049] where 0 < p < 1, k n , c n , sigma n , tau n,1 , tau n,2 , kappa represent the design of positive parameters.

[0050] The designed virtual control signal, the parameter adaptive law and the actual control input signal are substituted into the fractional order derivative of the Lyapunov function of the nth step to obtain:

[0051]

[0052] where,

[0053] l1, l2, epsilon1, epsilon2, Q j respectively represent the design of normal numbers.

[0054] From , it can be deduced that:

[0055]

[0056] In the formula, E α (·) represents the Mittag-Leffler function, and eta represents a normal number.

[0057] The final tracking error converges to

[0058] The finite time adaptive fuzzy fault-tolerant controller converges in a finite time T ft , which satisfies:

[0059]

[0060] where Gamma (·) represents the gamma function, that is, it can be concluded that all signals in the closed-loop system are bounded within a finite time T ft .

[0061] The application also provides a fuzzy fault-tolerant control method for a fractional order nonlinear system of a flexible manipulator, comprising the following steps:

[0062] Step 1: constructing a fractional order state space equation of the flexible manipulator according to an integer order dynamic model of the flexible manipulator;

[0063] Step 2: introducing a global coordinate transformation, decoupling the constructed fractional order state space equation of the flexible manipulator to obtain a fractional order nonlinear system of the flexible manipulator;

[0064] Step three: the method is designed to apply the fractional nonlinear system fuzzy fault-tolerant controller to the flexible manipulator system, and the actual control input signal of the actuator is obtained, so as to control the flexible manipulator to move along the reference trajectory.

[0065] The present application has the following beneficial effects:

[0066] (1) The present application proposes a fractional nonlinear finite-time adaptive fuzzy fault-tolerant tracking controller by using fractional dynamic surface control method, backstepping method, fuzzy logic system and Lyapunov stability theory, effectively solves the problems of external disturbance, uncertainty and actuator failure, and ensures the tracking performance of the system.

[0067] (2) The controller proposed in the present application can not only ensure that the output of the fractional nonlinear system tracks the reference signal in a limited time, but also ensure that all signals in the fractional closed-loop system are bounded in a limited time.

[0068] (3) The present application first introduces global coordinate transformation, decouples the fractional dynamic system of the underactuated single-link flexible manipulator, and transforms it into a lower triangular system. Under this coordinate transformation, without linearization or approximation of the original nonlinear single-link flexible manipulator system, the trajectory tracking control of the manipulator end angle is finally realized under the condition of actuator failure. BRIEF DESCRIPTION OF DRAWINGS

[0069] Figure 1 The flowchart of the fractional nonlinear system fuzzy fault-tolerant control method for flexible manipulator in the present application is shown in the figure;

[0070] Figure 2 The structure diagram of the single-link flexible manipulator in the present application is shown in the figure;

[0071] Figure 3 The time response trajectory diagram of the single-link flexible manipulator fractional system and the system output y tracking reference signal y d and tracking error s1 under the virtual experiment platform is shown in the figure;

[0072] Figure 4 The time response trajectory diagram of the state variables x2, x3 of the single-link flexible manipulator fractional model and the state variables x2, x3 of the virtual experiment platform is shown in the figure;

[0073] Figure 5 The time response trajectory diagram of the state variables x4 of the single-link flexible manipulator fractional model and the state variables x4 of the virtual experiment platform and the fault-tolerant control input signal u(t) is shown in the figure;

[0074] Figure 6Time response trajectory diagrams of adaptive parameters Λ1, Λ2, Λ3, Λ4, ||θ1||, ||θ2||, ||θ3||, ||θ4||. DETAILED DESCRIPTION

[0075] The specific embodiments of the present application are described below to facilitate the understanding of the present application for those skilled in the art, but it should be clear that the present application is not limited to the scope of the specific embodiments, and for those skilled in the art, as long as various changes are within the spirit and scope of the present application defined and determined by the appended claims, all the inventions utilizing the concept of the present application are within the scope of protection.

[0076] As shown in Figure 1 The embodiment of the present application provides a design method of a fractional order nonlinear system fuzzy fault-tolerant controller for a flexible manipulator, comprising the following steps:

[0077] Step 1: Construct a fractional order nonlinear system with external disturbance and actuator fault;

[0078] Step 2: Estimate the nonlinear part in the fractional order nonlinear system based on the fuzzy logic system;

[0079] Step 3: Based on the fractional order nonlinear system obtained in step 2, construct a finite time adaptive fuzzy fault-tolerant controller.

[0080] The present application aims at how to realize the fast tracking control of the fractional order nonlinear system under the condition of unknown external disturbance and multiple actuator faults, by constructing a fractional order nonlinear system with external disturbance and actuator fault for a flexible manipulator, and using a fuzzy logic system to identify the unknown uncertainty of the system as an auxiliary function for online estimation of the unknown upper bound of the external disturbance and fuzzy approximation error of the fractional order nonlinear system, so that the output of the fractional order nonlinear system can track the reference signal in a finite time, and all signals in the fractional order system are also bounded in a finite time, so as to not only identify the unknown external disturbance and uncertainty of the fractional order nonlinear system online, but also effectively overcome the problem of tracking performance degradation caused by actuator fault of the system.

[0081] In an optional embodiment of the present application, the fractional order nonlinear system with external disturbance and actuator fault constructed in step 1 is specifically:

[0082]

[0083] Wherein, Caputo fractional derivative, α represents the order of the fractional order nonlinear system, x i x = [x1, x2,..., x i] T x = x n = [x1, x2,..., x n ] T x = x i g i ( x i ) f i (x, t) = x d j (t) = x u j (t) = x

[0084] In this embodiment, the actuator faults include stuck faults and failure faults.

[0085] wherein the mathematical model of the stuck fault is specifically:

[0086]

[0087] wherein u j (t) represents that the jth actuator has a stuck fault, is a set constant, t j represents the time when the jth actuator has a stuck fault, and p represents the total number of stuck faults of the actuator.

[0088] wherein the mathematical model of the failure fault is specifically:

[0089]

[0090] wherein u j (t) represents that the jth actuator has a failure fault, ψ j represents the proportion of the actuator still effective after failure, μ j (t) represents the jth control input of the fractional nonlinear system, t j represents the time when the jth actuator has a failure fault, and q represents the total number of actuators, represents the complement of the q actuators having stuck faults, Ψ j represents the lower bound of ψ j , and μ j (t) can adopt the control structure of , wherein and represent the lower bound and the upper bound of ψ , and u r0represents the actual control input signal of the actuator.

[0091] In this embodiment, step 1 constructs a fractional order nonlinear system with external disturbance and actuator faults satisfying the following conditions:

[0092] Assumption 1: The reference signal y d , and are smooth and bounded.

[0093] Assumption 2: The fractional order nonlinear system can achieve the control objective if and only if no more than q-1 actuators are allowed to fail simultaneously.

[0094] Assumption 3: The unknown external disturbance d i (t) is bounded, i.e. where is an unknown constant.

[0095] In an optional embodiment of the present invention, a fuzzy logic system has the universal approximation property and can be used to estimate the unknown nonlinear function f i ( x i ) in the system. Generally for a continuous function f(x) defined on Ξ, there exists a fuzzy logic system such that:

[0096]

[0097] where θ represents the parameter vector of the fuzzy logic system, δ represents the estimation error of the fuzzy logic system, δ>0, represents the fuzzy basis function vector, represents the fuzzy membership function, and L represents the number of fuzzy rules, L>1.

[0098] The optimal parameter vector of the fuzzy logic system is defined as follows:

[0099]

[0100] where U i , Ξ i represents a compact set, and f i ( x i ) can be estimated by the fuzzy logic system f i ( x i | θ i ) = θ i T φ i ( x i ) is estimated, represents the optimal parameter vector of the fuzzy logic system, T represents the transpose symbol, and φ i x i represents the fuzzy basis function vector.

[0101] In step 2, the unknown nonlinear function in the system is estimated by using the fuzzy logic system, and the fractional order nonlinear system can be further described as:

[0102]

[0103] wherein δ i represents the optimal estimation error of the fuzzy logic system, and satisfies

[0104] In an optional embodiment of the present application, step 3 of constructing the finite-time adaptive fuzzy fault-tolerant controller specifically comprises:

[0105] Step 3.1 defines the coordinate transformation related to the tracking error;

[0106] Step 3.2 designs the finite-time adaptive fuzzy fault-tolerant controller of the system by using the fractional order dynamic surface control method combined with the backstepping method.

[0107] In the present embodiment, the global coordinate transformation is specifically:

[0108]

[0109] wherein s1 represents the tracking error of the fractional order nonlinear system, y represents the output quantity of the fractional order nonlinear system, y d represents the reference signal of the tracking, s i represents the tracking error introduced by the state variable of the system, x i represents the state variable of the system, ξ i represents the output of the α-order filter, represents the error surface, and v i-1 represents the virtual control signal.

[0110] The output of the α-order filter can be represented as:

[0111]

[0112] wherein, is a constant.

[0113] In the present embodiment, the virtual control signal v i , the parameter adaptive law θ i , Λ i and the actual control input signal u r0 ​, and then a finite-time adaptive fuzzy fault-tolerant controller is constructed, which specifically includes:

[0114] According to the coordinate transformation, the Lyapunov function of the first step is set as:

[0115]

[0116] wherein Γ1 and γ1 represent positive parameters, represent error estimation parameters of the parameter adaptive law; Λ1 is an estimation value of * , and Λ1 represents an upper bound of the sum of the optimal fuzzy approximation error δ1 and the external disturbance d1(t), and θ1 is an estimation value of .

[0117] The α-order derivative of V1(t) is taken, and the virtual control signal v1 and the parameter adaptive law θ1 and Λ1 are constructed by combining Young's inequality and the fractional order dynamic surface control method as:

[0118]

[0119] wherein 0 < ρ < 1, k1, c1, κ, σ1, τ 1,1 , τ 1,2 all represent positive parameters designed. According to the coordinate transformation, the Lyapunov function of the i-th step (i = 2, …, n-1) is set as:

[0120]

[0121] wherein Γ i , γ i all represent positive parameters, all represent error estimation parameters of the parameter adaptive law, represents an error surface, Λ i represents an estimation value of , and Λ i * represents an upper bound of the sum of the optimal fuzzy approximation error δ i and the external disturbance d i (t), and θ i represents an estimation value of .

[0122] The α-order derivative of V i (t) is taken, and the virtual control signal v i and the parameter adaptive law θ i , Λ i are constructed by combining Young's inequality and the fractional order dynamic surface control method as:

[0123]

[0124] where 0 < p < 1, k i , c i , s i , t i,1 , t i,2 , k represent the positive parameters designed.

[0125] According to the coordinate transformation, the Lyapunov function of the nth step is set as:

[0126]

[0127] where, G n , g n all represent positive parameters, all represent the errors of the parameter adaptive law, and q n is the estimated value of ; A n is the estimated value of , A n * represents the upper bound of the sum of the optimal fuzzy approximation error d n and the external disturbance d n (t).

[0128] The a-order derivative of V n (t) is taken, and the actual control input signal u r0 and the parameter adaptive law q n , A n are constructed by combining Young’s inequality and the fractional order dynamic surface control method as:

[0129]

[0130] where 0 < p < 1, k n , c n , s n , t n,1 , t n,2 , k represent the positive parameters designed.

[0131] The stability of the designed controller is analyzed below, and the convergence time and convergence range of the system are estimated.

[0132] The fractional order derivative of the Lyapunov function of the nth step is obtained by substituting the virtual control signal, the parameter adaptive law, and the actual control input signal designed above into the fractional order derivative of the Lyapunov function of the nth step:

[0133]

[0134] where,

[0135] l1, l2, ε1, ε2, Q j respectively represent the designed normal numbers.

[0136] According to the basic theory of fractional calculus, from the above formula (20), it can be derived that:

[0137]

[0138] In the formula, E α (·) represents the Mittag-Leffler function, and η represents a normal number.

[0139] The final tracking error converges to From the above formula (20), it can also be concluded that all signals s i , Λ i , The finite time convergence can be achieved, and the finite time T ft satisfies:

[0140]

[0141] Wherein, Γ(·) represents the gamma function.

[0142] Experimental example

[0143] A single-link flexible robot arm is taken as an example for experimental verification. The fractional dynamics model of the single-link flexible robot arm is considered, and the effectiveness of the model is verified by experiment. In this experimental example, the product provided by Quanser is used to verify the effectiveness of the method. The structure of the single-link flexible robot arm is shown in Figure 2 Based on the fractional model, the MATLAB Simulink module is used to design a finite-time adaptive fuzzy fault-tolerant fractional nonlinear system controller as an input voltage signal, and the QLabs virtual flexible platform implementation verifies the effectiveness of the method. The DC motor (actuator) is coupled with the flexible link. A strain gauge is installed at one end of the motor to measure the deflection angle of the link. The final system output is an analog signal proportional to the deflection of the link. Real-time interface is provided by Quarc interface software.

[0144] The fractional nonlinear system fuzzy fault-tolerant control method provided in this experiment for the flexible robot arm includes the following steps:

[0145] Step 1: According to the integer-order dynamics model of the flexible robot arm, the fractional-order state space equation of the flexible robot arm is constructed;

[0146] Step two: Introducing global coordinate transformation, decoupling the fractional order state space equation of the constructed flexible manipulator, obtaining the fractional order nonlinear system of the flexible manipulator;

[0147] Step three: applying the fuzzy fault-tolerant controller of the fractional order nonlinear system designed by the above method to the flexible manipulator system, obtaining the actual control input signal of the actuator, so as to control the flexible manipulator to move along the reference trajectory.

[0148] In step one, the integer order dynamics model of the single-link flexible manipulator can be described as follows without considering external disturbance:

[0149]

[0150] Wherein, β(t) represents the angle of servo motor, η(t) represents the deflection angle of the rod end, y(t) represents the output angle of the single-link flexible manipulator, U τ represents the torque, and B eq represents the equivalent friction coefficient of the system, J eq represents the moment of inertia of the motor, K s represents the total stiffness of the model, J l represents the moment of inertia of the connecting rod.

[0151] The oscilloscope of MATLAB can observe the angle position signals represented by real-time β(t) and deflection η(t) measured by the encoder and the strain gauge, and finally obtain the final output signal β(t)+η(t). The specific parameters are shown in Table 1.

[0152] Table 1 Typical parameters of single-link manipulator

[0153]

[0154] According to the results in the literature [A. Mujumdar, B. Tamhane, and S. Kurode, “Observer-based sliding mode control for a class of noncommensurate fractional-order systems,” IEEE / ASME Transactions on Mechatronics, vol. 20, pp. 2504-2512, Oct. 2015], the fractional order state space equation of the single-link flexible manipulator can be written as follows:

[0155]

[0156] In the formula, z1=β(t), z2=η(t), u(t) = U τ represents the motor input torque.

[0157] In step two, the global coordinate transformation is introduced: x1 = z1 + z3, x2 = z2 + z4, x3 = z3, x4 = z4. Then, we have

[0158]

[0159] where b represents the control gain constant, and when b = 1, equation (25) is the general form of system (24). Through the above coordinate transformation, the decoupling of the system is completed, and let f1(x1) = f2(x2) = f3(x3) = 0, f4(x4) = ω1(x2 - x4) - ω4x3, equation (25) is a special form of fractional order nonlinear system (5). x 2) = f3(x3) = 0, f4(x4) = ω1(x2 - x4) - ω4x3, equation (25) is a special form of fractional order nonlinear system (5). x 3) = 0, f4(x4) = ω1(x2 - x4) - ω4x3, equation (25) is a special form of fractional order nonlinear system (5). x

[0160] In step three, for this simulation, only actuator failure is considered, that is, when t > 5 s, u(t) = 0.7μ(t). And other parameters are set as ω1 = 39.5176, ω2 = 500, ω3 = 64.2267, ω4 = 750; c1 = 60, c2 = 35, c3 = 25, c4 = 40, k1 = 5, k2 = 15, k3 = 20, k4 = 10, ρ = 0.98, Γ1 = Γ2 = Γ3 = Γ4 = 30.2, σ1 = σ2 = σ3 = σ4 = 0.1, γ1 = γ2 = γ3 = γ4 = 0.2, τ1 = τ2 = τ3 = τ4 = 1.3, τ5 = τ6 = τ7 = τ8 = 1.5, κ = 2. 1,1 2,1 3,1 4,1 1,2 2,2 3,2 4,2 The fuzzy membership function is The partial initial conditions are x1(0) = 0.1, x2(0) = 0.3, x3(0) = -0.1, x4(0) = -0.2, Λ1(0) = 0.5, Λ2(0) = -0.2, Λ3(0) = 0.1, Λ4(0) = -0.1, and the other initial conditions are 0. The reference signal y d is to track a square wave signal with a period of 6.25 seconds and an amplitude of .

[0161] The experimental simulation results are shown in Figures 3-6 .

[0162] ​​​​​​​​​The output angle y of the single-link flexible robot arm of the fractional order model and the experimental model includes the sum of the servo motor angle β(t) and the needle tip deflection angle η(t), and their trajectories with respect to the reference signal y d The tracking error s1 is shown as Figure 3 According to the simulation results, it can be known that the outputs y of the experimental model and the fractional order model can track the given reference signal y d well, and the tracking error s1 can converge to 0 in a limited time. Figure 4 and Figure 5 The trajectories of the system states x2, x3, x4 and the control input signal u(t) of the experimental model and the fractional order model are described respectively. Figure 6 The response curves of the adaptive parameters Λ1, Λ2, Λ3, Λ4, ||θ1||, ||θ2||, ||θ3||, ||θ4|| are shown, which further indicates the approximation properties of the fuzzy logic system to the external disturbance and the fuzzy approximation error.

[0163] Therefore, the fractional order dynamics system is decoupled for the underactuated single-link flexible robot arm by introducing the global coordinate transformation, and is converted into a lower triangular system, so that the fault-tolerant scheme of the system is designed, the tracking performance of the system is improved, and the system state convergence is realized in a limited time. Under this coordinate transformation, the original nonlinear single-link flexible robot arm system does not need to be linearized or approximated.

[0164] The present application is described with reference to flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, and the combination of flows and / or blocks in the flowcharts and / or block diagrams can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing apparatus to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing apparatus generate a means for implementing the functions specified in one or more flows or one or more blocks in the flowcharts and / or block diagrams. Figure 1 The computer program instructions can also be stored in a computer-readable memory capable of guiding the computer or other programmable data processing apparatus to work in a specific manner, so that the instructions stored in the computer-readable memory produce a product including instruction means, which implements the functions specified in one or more flows or one or more blocks in the flowcharts and / or block diagrams. Figure 1 The computer program instructions can also be stored in a computer-readable memory capable of guiding the computer or other programmable data processing apparatus to work in a specific manner, so that the instructions stored in the computer-readable memory produce a product including instruction means, which implements the functions specified in one or more flows or one or more blocks in the flowcharts and / or block diagrams.

[0165] The computer program instructions can also be stored in a computer-readable memory capable of guiding the computer or other programmable data processing apparatus to work in a specific manner, so that the instructions stored in the computer-readable memory produce a product including instruction means, which implements the functions specified in one or more flows or one or more blocks in the flowcharts and / or block diagrams. Figure 1 The computer program instructions can also be stored in a computer-readable memory capable of guiding the computer or other programmable data processing apparatus to work in a specific manner, so that the instructions stored in the computer-readable memory produce a product including instruction means, which implements the functions specified in one or more flows or one or more blocks in the flowcharts and / or block diagrams. Figure 1 The computer program instructions can also be stored in a computer-readable memory capable of guiding the computer or other programmable data processing apparatus to work in a specific manner, so that the instructions stored in the computer-readable memory produce a product including instruction means, which implements the functions specified in one or more flows or one or more blocks in the flowcharts and / or block diagrams.

[0166] These computer program instructions can also be loaded into a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. Figure 1 Figure 1

[0167] The principles and implementations of the present application are described in the specific embodiments, the above description of the embodiments is only for helping to understand the method of the present application and its core idea; at the same time, for those skilled in the art, according to the idea of the present application, there will be changes in the specific implementation and application range, and the above description should not be understood as a limitation of the present application.

[0168] Those skilled in the art will realize that the embodiments described herein are for the purpose of understanding the principles of the present application and should be understood as not limited to such specific embodiments and examples. Those skilled in the art can make various other specific modifications and combinations according to the technical spirit of the present application disclosed herein without departing from the scope of the present application.​​

Claims

1. A design method for a fuzzy fault-tolerant controller for a fractional-order nonlinear system of a flexible robotic arm, characterized in that, Includes the following steps: Step 1: Construct a fractional-order nonlinear system with external disturbances and actuator failures; the fractional-order nonlinear system is specifically as follows: ； in, This represents the Caputo fractional derivative, where α represents the order of the fractional nonlinear system. Let T represent the state variables of the system, and let T denote the transpose of a vector or matrix. Represents virtual control coefficients. Represents an unknown smooth nonlinear function. This indicates external disturbances to the system. This represents a known vector of control gain constants. This represents the control input vector of a fractional-order nonlinear system. This represents the output of a fractional-order nonlinear system. Step 2: Estimate the nonlinear component of the fractional-order nonlinear system based on fuzzy logic; use fuzzy logic to estimate the unknown nonlinear function in the system, and further describe the fractional-order nonlinear system as follows: ； in, This represents the optimal parameter vector of the fuzzy logic system, where T denotes the transpose sign. Represents a fuzzy basis function vector. This represents the optimal estimation error of a fuzzy logic system. Step 3: Based on the fractional-order nonlinear system obtained in Step 2, construct a finite-time adaptive fuzzy fault-tolerant controller. This step includes the following sub-steps: Step 3.1 Define the coordinate transformation related to tracking error; the global coordinate transformation is as follows: ； in, This represents the tracking error of a fractional-order nonlinear system. This represents the output of a fractional-order nonlinear system. The reference signal for tracking, This represents the tracking error introduced by the system state variables. Represents the system's state variables. This represents the output of the α-order filter. Indicates the error surface. Indicates virtual control signals; Step 3.2 Design a finite-time adaptive fuzzy fault-tolerant controller using the fractional-order dynamic surface control method combined with the backstepping method; specifically, design the following virtual control signal in n steps using the fractional-order dynamic surface control method combined with the backstepping method. Parameter adaptive law , and actual control input signals This leads to the construction of a finite-time adaptive fuzzy fault-tolerant controller, specifically including: Based on the coordinate transformation, set the Lyapunov function in step 1 as follows: ； in, , All represent positive parameters. , Both represent the error estimation parameters of the adaptive law; , , express The estimated value, Represents the optimal fuzzy approximation error and external interference The upper bound of the sum, yes The estimated value; right Find the α-order derivative and construct a virtual control signal by combining Young's inequality and fractional dynamic surface control method. Adaptive Law of Parameters for: ； ； ； in, , , , , , , All represent positive parameters of the design; Based on the coordinate transformation, set the Lyapunov function in step i as follows: ； in, , All represent positive parameters. , Both represent the error estimation parameters of the adaptive law. Indicates the error surface. , , express The estimated value, Represents the optimal fuzzy approximation error and external interference The upper bound of the sum, yes The estimated value, ; right Find the α-order derivative and construct a virtual control signal by combining Young's inequality and fractional dynamic surface control method. Adaptive Law of Parameters for: ； ； ； in, , Indicates the positive parameters of the design; Based on the coordinate transformation, set the Lyapunov function in step n as follows: ； in, , All represent positive parameters. , Both represent the error of the parameter adaptive law. yes The estimated value; yes The estimated value, Represents the optimal fuzzy approximation error and external interference The upper bound of the sum; right Find the α-order derivative and construct the actual control input signal by combining Young's inequality and fractional dynamic surface control method. Adaptive Law of Parameters for: ； ； ； in, , Indicates the positive parameters of the design.

2. The design method for a fuzzy fault-tolerant controller for a fractional-order nonlinear system of a flexible robotic arm according to claim 1, characterized in that, Actuator failure This includes both stuck faults and failure faults.

3. The design method for a fuzzy fault-tolerant controller for a fractional-order nonlinear system of a flexible robotic arm according to claim 2, characterized in that, The mathematical model for the stuck fault is as follows: ； in, This indicates that the j-th actuator has encountered a stuck failure. This indicates a set constant. This indicates the time when the j-th actuator experiences a jamming failure. This indicates the total number of actuator jamming failures; The mathematical model for failure is as follows: ； in, Indicates the first One actuator failed. This indicates the percentage of actuators that remain effective after failure. This represents the j-th control input to a fractional-order nonlinear system. Indicates the first The time when an actuator fails. Indicates the total number of actuators. express The complement of the actuator jamming failure. express The lower bound, use The control structure, in which , and express The lower and upper bounds, This represents the actual control input signal of the actuator.

4. The design method for a fuzzy fault-tolerant controller for a fractional-order nonlinear system of a flexible robotic arm according to claim 1, characterized in that, Substitute the designed virtual control signal, parameter adaptive law, and actual control input signal into the Lyapunov function in step n. From the fractional derivative, we can obtain: ； in, , , ， l1, l2, ε1, ε2, Q j These represent the positive constants of the design; from It can be deduced that: ； In the formula, This refers to the Mittag-Leffler function. Represents positive numbers; The final tracking error will converge to .

5. The design method for a fuzzy fault-tolerant controller for a fractional-order nonlinear system of a flexible robotic arm according to claim 4, characterized in that, The finite-time adaptive fuzzy fault-tolerant controller converges within a finite time. satisfy: ； in, This represents the gamma function.

6. A fuzzy fault-tolerant control method for fractional-order nonlinear systems used in flexible robotic arms, characterized in that, Includes the following steps: Step 1: Construct the fractional-order state-space equations of the flexible robotic arm based on its integer-order dynamic model; Step 2: Introduce global coordinate transformation to decouple the fractional-order state-space equations of the constructed flexible robotic arm, thus obtaining the fractional-order nonlinear system of the flexible robotic arm; Step 3: Apply the fractional-order nonlinear system fuzzy fault-tolerant controller designed by the method described in any one of claims 1 to 5 to the flexible robotic arm system to obtain the actual control input signal of the actuator, thereby controlling the flexible robotic arm to move along the reference trajectory.

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