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

Abstract

The invention discloses a fractional order nonlinear system fuzzy fault-tolerant controller design method for a flexible mechanical arm. The method comprises the steps that a fractional order nonlinear system with external disturbance and actuator faults is constructed; estimating a nonlinear part in the fractional order nonlinear system based on a fuzzy logic system; and defining coordinate transformation related to tracking errors, 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 invention, the output of the fractional order nonlinear system tracks the reference signal within the finite time, and all signals in the closed-loop system are ensured to be bounded within the finite time. And finally, the controller is applied to a flexible mechanical arm system.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible manipulator control, and in particular to a fuzzy fault-tolerant control method of a fractional-order nonlinear system for a flexible manipulator. Background Art

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

[0003] In actual engineering systems, increasingly complex industrial processes often lead to unexpected failures in factories. 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, considered the actuator failure problem with completely unknown parameters and modes in fractional-order systems, and used the fractional-order command filtering method to avoid the "computational explosion" problem [Liu H, Pan Y, Cao J, et al. Adaptive neural network backstepping control offractional-order nonlinear systems with actuator faults [J]. IEEE Transactions on Neural Networks and Learning Systems, 2020, 31 (12): 5166-5177]. However, when using the backstepping method to solve the controller, it is necessary to design a virtual control signal (which is a composite function containing multiple functions), and in this process, it is necessary to repeatedly solve the fractional-order derivatives of the virtual control signal. Due to the redefinition of fractional-order calculus, some traditional derivative chain rules cannot be directly applied to fractional-order systems, which increases the difficulty of solving and also increases the amount of calculation required for solving. Yang W et al. studied observer-based event-triggered adaptive fuzzy control for fractional-order time-varying delayed MIMO systems against actuator faults, taking into account actuator stuck and failure, and used a compensation function with adjustable parameters for compensation [Yang W, Zheng WX, 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 an actuator fails, in addition to ensuring that the system has acceptable performance indicators, the system should also respond quickly within a finite time. Therefore, how to use finite-time stability theory to solve the actuator faults of fractional-order nonlinear systems and achieve fast tracking and convergence is a difficult problem that needs to be solved urgently and deserves in-depth 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 technology, which is a finite-time sliding mode fault-tolerant control scheme suitable 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 offractional-order nonlinear systemswith actuator faults via command-filtering and sliding mode technique[J]. Information Sciences, 2022, 600: 189-208]; however, Lemma 2 adopted in this scheme is incorrect; therefore, the finite-time stability of fractional-order nonlinear systems obtained using this lemma in some literature may be unreliable.

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

[0005] Moreover, existing research and practical applications have shown that the flexible manipulator has complex rigid-flexible coupling, numerous sensing and actuation parts / components that are prone to failure, and difficulty in mechanism modeling, which makes conventional fault-tolerant control methods difficult to work, seriously reducing the operating efficiency of the flexible manipulator at work. The use of flexible manipulators in a faulty state or without a fault-tolerant mechanism will cause damage to the robot itself or greatly shorten its service life. Actuator failure is one of the most common types of failures and is also a challenging problem that a flexible manipulator system needs to solve. Summary of the invention

[0006] In view of the above-mentioned deficiencies in the prior art, the present invention provides a fuzzy fault-tolerant control system and method for a fractional-order nonlinear system for a flexible robotic arm, and constructs a finite-time adaptive fuzzy fault-tolerant controller for the fractional-order nonlinear system, so that the output of the fractional-order nonlinear system can track the reference signal within a finite time, and ensures that all signals in the fractional-order system are also bounded within a finite time. Finally, the controller is applied in a flexible robotic arm system.

[0007] In order to achieve the above-mentioned object of the invention, the present invention adopts the following technical solutions to achieve it.

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

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

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

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

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

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

[0014] As an option, the fractional-order nonlinear system is specifically:

[0015]

[0016] in, represents the Caputo fractional derivative, α represents the order of the fractional nonlinear system, x i =[x1,x2,...,x i ] T 、x= x n =[x1,x2,...,x n ] T represents the state variable of the system, T represents the transpose symbol of the vector or matrix, g i represents the virtual control coefficient, f i ( x i ): represents an unknown smooth nonlinear function, d i (t) represents the external disturbance of the system, represents the known control gain constant vector, represents the control input vector of the fractional-order nonlinear system, and y(t) represents the output of the fractional-order nonlinear system.

[0017] Alternatively, the actuator fault includes a stuck fault and a failure fault.

[0018] As an option, the mathematical model of the stuck fault is specifically:

[0019]

[0020] Among them, u j (t) indicates that the jth actuator is stuck. Indicates the setting constant, t jrepresents the time when the jth actuator is stuck, and p represents the total number of actuator stuck failures;

[0021] The mathematical model of failure is as follows:

[0022] u j (t) = ψ j μ j (t),t≥t j ,

[0023] ψ j ∈( ψ j ,1],0< ψ j <1

[0024] Among them, u j (t) indicates that the jth actuator fails, ψ j Indicates the proportion of actuators that are still effective after failure, μ j (t) represents the jth control input of the fractional-order nonlinear system, t j represents the time when the jth actuator fails, q represents the total number of actuators, represents the complement of q actuators stuck, ψ j Represents ψ j The lower bound of μ j (t) Adoption The control structure of and express The lower and upper bounds of u r0 Represents the actual control input signal of the actuator.

[0025] Alternatively, the unknown nonlinear function in the system can be estimated using a fuzzy logic system. The fractional-order nonlinear system can be further described as:

[0026]

[0027] in, represents the optimal parameter vector of the fuzzy logic system, T represents the transposed symbol, φ i ( x i ) represents the fuzzy basis function vector, δ i Represents the optimal estimation error of the 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, y represents the output of the fractional-order nonlinear system, and y d represents the reference signal for tracking, s i represents the tracking error introduced by the system state variables, x i represents the state variable of the system, ξ i represents the output of the α-order filter, represents the error surface, ν i-1 Indicates a virtual control signal.

[0031] As an option, the following virtual control signal ν is designed in n steps using the fractional order dynamic surface control method combined with the backstepping method: i , parameter adaptation law θ i , Λ i And the actual control input signal u r0 , and then construct a finite-time adaptive fuzzy fault-tolerant controller, which includes:

[0032] According to the coordinate transformation, the Lyapunov function of step 1 is set as:

[0033]

[0034] Among them, Γ1 and γ1 are both positive parameters. Both represent the error estimation parameters of the parameter adaptation law; Λ1 is The estimated value of Λ1 * represents the upper bound of the sum of the optimal fuzzy approximation error δ1 and the external disturbance d1(t), θ1 is The estimated value of .

[0035] The α-order derivative of V1(t) is calculated, and the virtual control signal ν1 and parameter adaptive laws θ1 and Λ1 are constructed by combining Young's inequality and fractional-order dynamic surface control method:

[0036]

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

[0038] According to the coordinate transformation, the Lyapunov function of the i-th step is set as:

[0039]

[0040] Among them, Γ i , γ iAll represent positive parameters, Both represent the error estimation parameters of the parameter adaptation law, represents the error surface, Λ i yes The estimated value of i * Denotes the optimal fuzzy approximation error δ i and external disturbances i (t) is the upper bound of the sum, θ i yes Estimated value of , i=2,…,n-1.

[0041] V i (t) Calculate the α-order derivative and construct the virtual control signal ν by combining Young's inequality and fractional-order dynamic surface control method i and parameter adaptation law θ i ,Λ i for:

[0042]

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

[0044] According to the coordinate transformation, the Lyapunov function of the nth step is set as:

[0045]

[0046] Among them, Γ n , γ n All represent positive parameters, Both represent the error of the parameter adaptation law, θ n yes The estimated value of ; n yes The estimated value of n * Denotes the optimal fuzzy approximation error δ n and external disturbances n (t) The upper bound of the sum;

[0047] V n (t) Calculate the α-order derivative and construct the actual control input signal u by combining Young's inequality and fractional-order dynamic surface control method r0 and parameter adaptation law θ n ,Λ n for:

[0048]

[0049] Among them, 0<ρ<1, k n 、c n , σ n , τ n,1 , τ n,2 , κ represents the positive parameters of the design.

[0050] Substituting the designed virtual control signal, parameter adaptive law and actual control input signal into the fractional derivative of the Lyapunov function of the nth step, we can get:

[0051]

[0052] in,

[0053] l1, l2, ε1, ε2, Q j They represent designed normal numbers respectively.

[0054] from It can be deduced that:

[0055]

[0056] Where E α (·) denotes the Mittag-Leffler function, and η denotes a positive constant;

[0057] The final tracking error will converge to

[0058] The finite time adaptive fuzzy fault-tolerant controller converges within a finite time T ft satisfy:

[0059]

[0060] Where Γ(·) represents the gamma function. That is, it can be concluded that all signals in the closed-loop system are ft Everything is bounded.

[0061] The present invention 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: Construct the fractional-order state space equation of the flexible manipulator based on the integer-order dynamic model of the flexible manipulator;

[0063] Step 2: Introduce global coordinate transformation, decouple the constructed fractional-order state space equation of the flexible manipulator, and obtain the fractional-order nonlinear system of the flexible manipulator;

[0064] Step 3: Apply the fractional-order nonlinear system fuzzy fault-tolerant controller designed by the method 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.

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

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

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

[0068] (3) The present invention first introduces global coordinate transformation, decouples the fractional-order dynamic system of the underactuated single-link flexible manipulator, and transforms it into a lower triangular system. Under this coordinate transformation, there is no need to linearize or approximate the original nonlinear single-link flexible manipulator system, and finally realizes the trajectory tracking control of the angle of the manipulator end in the event of actuator failure. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 It is a flow chart of the fuzzy fault-tolerant control method of the fractional-order nonlinear system for the flexible manipulator in the present invention;

[0070] Figure 2 This is a schematic diagram of the structure of a single-link flexible mechanical arm in the present invention;

[0071] Figure 3 The system output y tracks the reference signal y for the single-link flexible manipulator fractional-order system and the virtual experimental platform d And a schematic diagram of the time response trajectory of the tracking error s1;

[0072] Figure 4 It is a schematic diagram of the time response trajectory of the state variables x2, x3 of the fractional-order model of the single-link flexible manipulator and the state variables x2, x3 of the virtual experimental platform;

[0073] Figure 5 It is a schematic diagram of the time response trajectory of the state variable x4 of the fractional-order model of the single-link flexible manipulator, the state variable x4 of the virtual experimental platform, and the fault-tolerant control input signal u(t);

[0074] Figure 6Schematic diagram of the time response trajectories of adaptive parameters Λ1, Λ2, Λ3, Λ4, ||θ1, ||θ2||, ||θ3||, ||θ4||. DETAILED DESCRIPTION

[0075] The specific implementation modes of the present invention are described below to facilitate those skilled in the art to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0076] like Figure 1 As shown, an embodiment of the present invention provides a method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system of a flexible manipulator, comprising the following steps:

[0077] Step 1: Construct a fractional-order nonlinear system with external disturbances and actuator faults;

[0078] Step 2: Estimate the nonlinear part of 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, a finite-time adaptive fuzzy fault-tolerant controller is constructed.

[0080] The present invention aims at the problem of how to realize fast tracking control of a fractional-order nonlinear system with unknown external disturbances and multiple actuator faults. By constructing a fractional-order nonlinear system with external disturbances and actuator faults of a flexible manipulator, and adopting a fuzzy logic system to identify the unknown uncertainty of the system as an auxiliary function for online estimation of the external disturbance of the fractional-order nonlinear system and the unknown upper bound of the fuzzy approximation error, the output of the fractional-order nonlinear system can track a reference signal within a finite time, and all signals in the fractional-order system are also bounded within a finite time. Therefore, not only the unknown external disturbances and uncertainties of the fractional-order nonlinear system can be identified online, but also the problem of decreased system tracking performance caused by the actuator faults of the system can be effectively overcome.

[0081] In an optional embodiment of the present invention, the fractional-order nonlinear system of the flexible manipulator with external disturbance and actuator failure constructed in step 1 is specifically:

[0082]

[0083] in, represents the Caputo fractional derivative, α represents the order of the fractional nonlinear system, x i =[x1,x2,...,x i] T ,x=x n =[x1,x2,...,x n ] T represents the state variable of the system, T represents the transpose symbol of the vector or matrix, g i represents the virtual control coefficient, f i ( x i ) represents an unknown smooth nonlinear function, d i (t) represents the external disturbance of the system, represents the known control gain constant vector, represents the control input vector of the fractional-order nonlinear system, and y(t) represents the output of the fractional-order nonlinear system.

[0084] In this embodiment, the actuator fault includes a stuck fault and a failure fault.

[0085] Among them, the mathematical model of the stuck fault is specifically:

[0086]

[0087] Among them, u j (t) indicates that the jth actuator is stuck. Indicates the setting constant, t j It represents the time when the jth actuator is stuck, and p represents the total number of actuator stuck failures.

[0088] Among them, the mathematical model of failure is specifically:

[0089]

[0090] Among them, u j (t) indicates that the jth actuator fails, ψ j Indicates the proportion of actuators that are still effective after failure, μ j (t) represents the jth control input of the fractional-order nonlinear system, t j represents the time when the jth actuator fails, q represents the total number of actuators, represents the complement of q actuators stuck, ψ j Represents ψ j The lower bound of μ j (t) can be used The control structure of and express The lower and upper bounds of 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 fault to meet the following conditions:

[0092] Assumption 1: Reference signal y d , and is 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: Unknown external disturbance d i (t) is bounded, that is, in, is an unknown constant.

[0095] In an optional embodiment of the present invention, the fuzzy logic system has a universal approximation property and can be used to estimate the unknown nonlinear function f in the system. i ( x i ). In general, 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, 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] Among them, U i ,Ξ i represents a compact set, f i ( x i ) Fuzzy logic system can be used i ( x i |θ i )=θ i T φ i ( x i ) to estimate, represents the optimal parameter vector of the fuzzy logic system, T represents the transposed symbol, φ i ( x i ) represents the fuzzy basis function vector.

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

[0102]

[0103] Among them, δ i Represents the optimal estimation error of the fuzzy logic system, satisfying

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

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

[0106] Step 3.2 uses the fractional-order dynamic surface control method combined with the backstepping method to design a finite-time adaptive fuzzy fault-tolerant controller for the system.

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

[0108]

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

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

[0111]

[0112] in, is a constant.

[0113] In this embodiment, the fractional order dynamic surface control method is combined with the backstepping method to design the following virtual control signal v in n steps: i , parameter adaptation law θ i , Λ i And the actual control input signal u r0, and then construct a finite-time adaptive fuzzy fault-tolerant controller, which includes:

[0114] According to the coordinate transformation, the Lyapunov function of step 1 is set as:

[0115]

[0116] Among them, Γ1 and γ1 are both positive parameters. Both represent the error estimation parameters of the parameter adaptation law; Λ1 is The estimated value of Λ1 * represents the upper bound of the sum of the optimal fuzzy approximation error δ1 and the external disturbance d1(t), θ1 is The estimated value of .

[0117] The α-order derivative of V1(t) is calculated, and the virtual control signal ν1 and parameter adaptive laws θ1 and Λ1 are constructed by combining Young's inequality and fractional-order dynamic surface control method:

[0118]

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

[0120]

[0121] Among them, Γ i , γ i All represent positive parameters, Both represent the error estimation parameters of the parameter adaptation law, represents the error surface, Λ i express The estimated value of i * Denotes the optimal fuzzy approximation error δ i and external disturbances i (t) is the upper bound of the sum, θ i express The estimated value of .

[0122] V i (t) Calculate the α-order derivative and construct the virtual control signal ν by combining Young's inequality and fractional-order dynamic surface control method i and parameter adaptation law θ i ,Λ i for:

[0123]

[0124] Among them, 0<ρ<1, k i 、c i , σ i , τ i,1 , τ i,2 , κ represents the positive parameters of the design.

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

[0126]

[0127] Among them, Γ n , γ n All represent positive parameters, Both represent the error of the parameter adaptation law, θ n yes The estimated value of ; n yes The estimated value of n * Denotes the optimal fuzzy approximation error δ n and external disturbances n (t) is an upper bound on the sum.

[0128] V n (t) Calculate the α-order derivative and construct the actual control input signal u by combining Young's inequality and fractional-order dynamic surface control method r0 and parameter adaptation law θ n ,Λ n for:

[0129]

[0130] where 0<ρ<1, k n 、c n , σ n , τ n,1 , τ n,2 , κ represents the positive parameters of the design.

[0131] Next, we conduct a stability analysis on the designed controller and estimate the convergence time and convergence range of the system.

[0132] Substituting the virtual control signal, parameter adaptive law and actual control input signal designed above into the fractional derivative of the Lyapunov function of the nth step, we can get:

[0133]

[0134] in,

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

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

[0137]

[0138] Where E α (·) denotes the Mittag-Leffler function, and η denotes a positive constant.

[0139] The final tracking error will converge to From the above equation (20), we can also get that all the signals s in the fractional-order nonlinear system are i , Λ i , It can achieve finite time convergence, and the finite time of convergence is T ft satisfy:

[0140]

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

[0142] Experimental Examples

[0143] A single-link flexible robotic arm is used as an example for experimental verification. Consider the fractional-order dynamics model of a single-link flexible robotic arm, and verify the validity of the model through experiments. In this experimental example, the product provided by Quanser is used to verify the validity of the method proposed in the embodiment. The structure of the single-link flexible robotic arm is as follows: Figure 2 As shown. Based on the fractional-order model, the MATLAB Simulink module is used to design a finite-time adaptive fuzzy fault-tolerant fractional-order nonlinear system controller as the input voltage signal, and the implementation of the QLabs virtual flexible platform verifies the effectiveness of the method. The DC motor (actuator) is coupled to the flexible connecting rod. A strain gauge is installed at one end of the motor to measure the deflection angle of the connecting rod head. The final system output is an analog signal proportional to the connecting rod deflection. The Quarc interface software is used to provide a real-time interface.

[0144] The fuzzy fault-tolerant control method for fractional-order nonlinear systems of flexible manipulators provided in this experiment includes the following steps:

[0145] Step 1: Construct the fractional-order state space equation of the flexible manipulator based on the integer-order dynamic model of the flexible manipulator;

[0146] Step 2: Introduce global coordinate transformation, decouple the constructed fractional-order state space equation of the flexible manipulator, and obtain the fractional-order nonlinear system of the flexible manipulator;

[0147] Step 3: Apply the fractional-order nonlinear system fuzzy fault-tolerant controller designed by the above method 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.

[0148] In step 1, without considering external disturbances, the integer-order dynamic model of the single-link flexible manipulator can be described as:

[0149]

[0150] Among them, β(t) represents the servo motor angle, η(t) represents the rod end deflection angle, y(t) represents the output angle of the single-link flexible robotic arm, and U τ represents the torque, and B eq represents the equivalent friction coefficient of the system, J eq Indicates the motor's moment of inertia, K s Represents the total stiffness of the model, J l Represents the moment of inertia of the connecting rod.

[0151] The MATLAB oscilloscope can observe the angular position signal represented by real-time β(t) and deflection η(t) measured by the encoder and 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 robotic arm

[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 Indicates the motor input torque.

[0157] In step 2, the global coordinate transformation is introduced: x1=z1+z3, x2=z2+z4, x3=z3, x4=z4. Then, we have

[0158]

[0159] In the formula, b represents the control gain constant. When b = 1, formula (25) is the general form of system (24). Through the above coordinate transformation, the system decoupling is completed. Let f1(x1) = f2( x 2)=f3( x 3)=0,f4( x 4) = ω1(x2-x4)-ω4x3, formula (25) is a special form of the fractional-order nonlinear system (5).

[0160] In step 3, for this simulation, only the actuator failure is considered, that is, when t>5s, u(t)=0.7μ(t). And the 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, τ 1,1 =τ 2,1 =τ 3,1 =τ 4,1 =1.3,τ 1,2 =τ 2,2 =τ 3,2 =τ 4,2 =1.5,κ=2, The fuzzy membership function is Some 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 of the system d To track a period of 6.25 seconds, the amplitude is A square wave signal with an angle of .

[0161] The experimental simulation results are as follows Figure 3-Figure 6 shown.

[0162] The output angle y of the single-link flexible manipulator of the fractional-order model and the experimental model includes the sum of the servo motor angle β(t) and the tip deflection angle η(t), as well as their relationship with the reference signal y d The trajectory of the tracking error s1 is as follows: Figure 3 According to the simulation results, we can know that the output y of the experimental model and the fractional-order model can track the given reference signal y well. d , and the tracking error s1 can converge to 0 in a finite 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 fractional-order model are described respectively. Figure 6 The response curves of adaptive parameters Λ1, Λ2, Λ3, Λ4, ||θ1||, ||θ2||, ||θ3||, ||θ4|| are shown, which further demonstrate the response to external disturbances, fuzzy approximation errors, and the approximation properties of the fuzzy logic system.

[0163] Therefore, the present invention introduces global coordinate transformation, decouples the fractional-order dynamic system for the underactuated single-link flexible manipulator, and transforms it into a lower triangular system, thereby designing a fault-tolerant scheme for the system, improving the tracking performance of the system, and achieving system state convergence within a finite time. Under this coordinate transformation, there is no need to linearize or approximate the original nonlinear single-link flexible manipulator system.

[0164] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, 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 device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0165] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0166] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0167] The present invention uses specific embodiments to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

[0168] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A design method for a fuzzy fault-tolerant controller of a fractional-order nonlinear system for a flexible manipulator, characterized in that: The following steps are involved: Step 1: Construct a fractional-order nonlinear system with external disturbances and actuator faults; Step 2: Estimate the nonlinear part of the fractional-order nonlinear system based on the fuzzy logic system; Step 3: Based on the fractional-order nonlinear system obtained in step 2, a finite-time adaptive fuzzy fault-tolerant controller is constructed. This step includes the following sub-steps: Step 3.1 defines the coordinate transformation related to the tracking error; Step 3.2 uses the fractional-order dynamic surface control method combined with the backstepping method to design a finite-time adaptive fuzzy fault-tolerant controller.

2. The method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system for a flexible manipulator according to claim 1, characterized in that: The fractional-order nonlinear system is specifically: in, represents the Caputo fractional derivative, α represents the order of the fractional nonlinear system, x i =[x1,x2,...,x i ] T 、x=x n =[x1,x2,...,x n ] T represents the state variable of the system, T represents the transpose symbol of the vector or matrix, g i represents the virtual control coefficient, represents an unknown smooth nonlinear function, d i (t) represents the external disturbance of the system, represents the known control gain constant vector, represents the control input vector of the fractional-order nonlinear system, and y(t) represents the output of the fractional-order nonlinear system.

3. The method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system for a flexible manipulator according to claim 1, characterized in that: Actuator faults u(t) include stuck faults and failure faults.

4. The method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system for a flexible manipulator according to claim 3, characterized in that: The mathematical model of the stuck fault is as follows: Among them, u j (t) indicates that the jth actuator is stuck. Indicates the setting constant, t j represents the time when the jth actuator is stuck, and p represents the total number of actuator stuck failures; The mathematical model of failure is as follows: ψ j ∈(ψ j ,1],0<ψ j <1 Among them, u j (t) indicates that the jth actuator fails, ψ j Indicates the proportion of actuators that are still effective after failure, μ j (t) represents the jth control input of the fractional-order nonlinear system, t j represents the time when the jth actuator fails, q represents the total number of actuators, represents the complement of q actuators stuck, ψ j Represents ψ j The lower bound of μ j (t) Adoption The control structure of and express The lower and upper bounds of u r0 Represents the actual control input signal of the actuator.

5. The method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system for a flexible manipulator according to claim 2, characterized in that: By using the fuzzy logic system to estimate the unknown nonlinear function in the system, the fractional-order nonlinear system is further described as: in, represents the optimal parameter vector of the fuzzy logic system, T represents the transposed symbol, φ i (x i ) represents the fuzzy basis function vector, δ i Represents the optimal estimation error of the fuzzy logic system.

6. The method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system for a flexible manipulator according to claim 5, characterized in that: The global coordinate transformation is as follows: Where s1 represents the tracking error of the fractional-order nonlinear system, y represents the output of the fractional-order nonlinear system, and y d represents the reference signal for tracking, s i represents the tracking error introduced by the system state variables, x i represents the state variable of the system, ξ i represents the output of the α-order filter, represents the error surface, ν i-1 Indicates a virtual control signal.

7. The method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system of a flexible manipulator according to claim 6, characterized in that: The following virtual control signal ν is designed in n steps by using the fractional-order dynamic surface control method combined with the backstepping method: i , parameter adaptation law θ i , Λ i And the actual control input signal u r0 , and then construct a finite-time adaptive fuzzy fault-tolerant controller, which includes: According to the coordinate transformation, the Lyapunov function of step 1 is set as: Among them, Γ1 and γ1 are both positive parameters. Both represent the error estimation parameters of the parameter adaptation law; Λ1 represents The estimated value of Λ1 * represents the upper bound of the sum of the optimal fuzzy approximation error δ1 and the external disturbance d1(t), θ1 is An estimated value of The α-order derivative of V1(t) is calculated, and the virtual control signal ν1 and parameter adaptive laws θ1 and Λ1 are constructed by combining Young's inequality and fractional-order dynamic surface control method: Among them, 0<ρ<1, k1, c1, κ, σ1, τ 1,1 , τ 1,2 All represent positive parameters of the design; According to the coordinate transformation, the Lyapunov function of the i-th step is set as: Among them, Γ i , γ i All represent positive parameters, Both represent the error estimation parameters of the parameter adaptation law, represents the error surface, Λ i express The estimated value of i * Denotes the optimal fuzzy approximation error δ i and external disturbances i (t) is the upper bound of the sum, θ i yes The estimated value of , i = 2,…,n-1; V i (t) Calculate the α-order derivative and construct the virtual control signal ν by combining Young's inequality and fractional-order dynamic surface control method i and parameter adaptation law θ i ,Λ i for: Among them, 0<ρ<1, k i 、c i , σ i , τ i,1 , τ i,2 ,κ represents the positive parameter of the design; According to the coordinate transformation, the Lyapunov function of the nth step is set as: Among them, Γ n , γ n All represent positive parameters, Both represent the error of the parameter adaptation law, θ n yes The estimated value of ; n yes The estimated value of n * Denotes the optimal fuzzy approximation error δ n and external disturbances n (t) The upper bound of the sum; V n (t) Calculate the α-order derivative and construct the actual control input signal u by combining Young's inequality and fractional-order dynamic surface control method r0 and parameter adaptation law θ n ,Λ n for: Among them, 0<ρ<1, k n 、c n , σ n , τ n,1 , τ n,2 , κ represents the positive parameters of the design.

8. The method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system for a flexible manipulator according to claim 7, characterized in that: Substitute the designed virtual control signal, parameter adaptive law and actual control input signal into the Lyapunov function V of the nth step n From the fractional derivative of (t), we can get: in, l1, l2, ε1, ε2, Q j They represent the designed normal constants respectively; from It can be deduced that: In the formula, E α (·) denotes the Mittag-Leffler function, and η denotes a positive constant; The final tracking error will converge to 9. The method for designing a fuzzy fault-tolerant controller for a fractional-order nonlinear system for a flexible manipulator according to claim 8, characterized in that: The finite time adaptive fuzzy fault-tolerant controller converges in a finite time T ft satisfy: Here, Γ(·) represents the gamma function.

10. A fuzzy fault-tolerant control method for a fractional-order nonlinear system of a flexible manipulator, characterized in that: The following steps are involved: Step 1: Construct the fractional-order state space equation of the flexible manipulator based on the integer-order dynamic model of the flexible manipulator; Step 2: Introduce global coordinate transformation, decouple the constructed fractional-order state space equation of the flexible manipulator, and obtain the fractional-order nonlinear system of the flexible manipulator; Step three: Apply the fractional-order nonlinear system fuzzy fault-tolerant controller designed by the method described in any one of claims 1 to 9 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.

Images