Adaptive prescribed performance control method for flexible manipulator with actuator faults

Abstract

The application discloses a flexible manipulator adaptive prescribed performance control method with actuator failure, and the method comprises the following steps: firstly, an improved prescribed performance control method is proposed to enhance the robustness of the system to input disturbance, ensure the convergence of tracking error within a predetermined time, and convert the constrained system into an unconstrained system; secondly, an adaptive passive fault-tolerant controller is constructed to offset the actuator failure in the system; then, the uncertainty in the flexible joint manipulator system is solved by combining a radial basis function neural network and an adaptive technique into a fixed-time backstepping framework; finally, a fixed-time second-order filter is constructed to well solve the "complexity explosion" problem, and a designed compensation mechanism eliminates the filtering error.

Description

Technical Field

[0001] This invention belongs to the field of flexible joint robotic arm control technology, specifically an adaptive performance control method for a flexible robotic arm with actuator failure. Background Technology

[0002] In recent years, flexible joint robotic arms (FJRs) have been widely used due to their advantages such as high flexibility, strong adaptability to complex environments, and safe human-machine interaction. FJRs typically exhibit uncertain dynamic characteristics, including flexible actuation, inaccurate system parameters, discontinuous friction, and unknown disturbances. Furthermore, unexpected failures may frequently occur in complex environments, affecting system stability. Therefore, achieving high-performance fault-tolerant control of FJRs is a prerequisite for safe and reliable applications. Although many tracking control methods for FJRs have been proposed in recent years, such as passive control, singular disturbance control, sliding mode control, and adaptive control, few schemes consider fixed-time performance-specific fault-tolerant control for multi-link FJRs, which warrants further investigation.

[0003] For uncertainties in FJR systems, adaptive intelligent techniques (neural networks or fuzzy logic systems) are effective solutions. For example, to approximate the model uncertainty of FJR, an observer employing an adaptive self-recurrent wavelet neural network was constructed in the paper "Yoo, SJ, Park, JB, Choi, YH: Adaptive output feedback control for flexible joint robots using neural networks: a dynamic surface design method. IEEE Trans. Neural Networks. 19, 1712–1726 (2008). https: / / doi.org / 10.1109 / TNN.2008.2001266". In the paper “Diao, S., Sun, W., Su, SF, Xia, J.: Actuator Faults in a Single-Link Flexible Joint Robot with Adaptive Fuzzy Event-Triggered Control. IEEE Trans. Cybern. 1–11 (2021). https: / / doi.org / 10.1109 / TCYB.2021.3049536”, a fuzzy logic system is used to identify unknown nonlinearities in a single-link FJR. The paper “Yang, Y., Li, J., Hua, C., Guan, X.: Adaptive Synchronization Control Design for a Flexible Remote Robot with Actuator Faults and Input Saturation. Int. J. Robust Nonlinear Control. 28, 1016–1034 (2018). https: / / doi.org / 10.1002 / rnc.3922” employs adaptive techniques to identify unknown parameters in a single-link FJR system. These adaptive intelligent techniques are often integrated into backpropagation control frameworks. However, the classic backstepping method suffers from the problem of "complexity explosion." Therefore, dynamic surface techniques have been proposed to address this issue. For example, in the paper "Yoo, SJ, Park, JB, Choi, YH: Adaptive output feedback control for flexible joint robots using neural networks: a dynamic surface design method. IEEE Trans. Neural Networks. 19, 1712–1726 (2008). https: / / doi.org / 10.1109 / TNN.2008.2001266", a first-order filter is introduced to bypass the repetitive differentiation of the virtual control law. To achieve faster filtering performance, a second-order filter for robotic arms was proposed in the paper “Pan,Y.,Wang,H.,Li,X.,Yu,H.: Inverse step control for adaptive command filtering of robotic arms with compliant actuators. IEEE Trans.Control Syst.Technol.26,1149–1156(2018).https: / / doi.org / 10.1109 / TCST.2017.2695600”.To eliminate filter errors, a compensation mechanism was introduced in the literature “Diao, S., Sun, W., Su, SF: Adaptive event-triggered tracking control of the neural network of a flexible joint robot with random noise. Int. J. Robust Nonlinear Control. 32, 2722–2740 (2022). https: / / doi.org / 10.1002 / rnc.5382”. Another problem is that in the classical back-calculation method, the control system is asymptotically stable, that is, the system's settling time tends to infinity. To achieve faster system stability, a finite-time control method for FJRs systems was proposed in the paper “Wang, H., Zhang, Y., Zhao, Z., Tang, X., Yang, J., Chen, I.: Trajectory tracking control of flexible joint robots based on finite-time disturbance observers. Nonlinear Dyn. 106, 459–471 (2021). https: / / doi.org / 10.1007 / s11071-021-06868-4”. Unfortunately, the finite-time strategy will not work once the initial value of the system is previously unknown or the initial state is relatively far from the equilibrium point. In contrast, the fixed-time scheme can stabilize the system within a predetermined time without obtaining information about the initial state. The fixed-time method was first proposed in the paper “Polyakov, A.: Nonlinear feedback design for fixed-time stability of linear control systems. IEEE Trans. Automat. Contr. 57, 2106–2110 (2011)” and has been applied to many nonlinear systems. Therefore, the primary concern of this invention is how to integrate fixed-time techniques into the adaptive reverse-propagation dynamic surface framework to achieve fixed-time convergence, effectively handle unknown nonlinearities in the system, and avoid the "complexity explosion" problem.

[0004] Furthermore, FJRs (Flexible Joint Robot Manipulators) have high requirements for tracking performance when performing complex tasks in unstructured spaces. Because it can guarantee both transient and steady-state performance of tracking errors, Prescribed Performance Control (PPC) is considered a promising tool. Recently, PPC technology has also been applied to FJR systems. For example, in the paper "Ma, H., Zhou, Q., Li, H., Lu, R.: Adaptive Prescribed Performance Control for Flexible Joint Robot Manipulators with Dynamic Uncertainty. IEEE Trans. Cybern. 1–11 (2021). https: / / doi.org / 10.1109 / TCYB.2021.3091531", an adaptive fuzzy prescribed performance control scheme is proposed for single-link FJRs. In the paper "Kostarigka, AK, Doulgeri, Z., Rovithakis, GA: Prescribed performance tracking of a flexible joint robot with unknown dynamics and variable elasticity. Automatica. 49, 1137–1147 (2013). https: / / doi.org / 10.1016 / j.automatica.2013.01.042", a pre-defined performance for link position error is considered for a flexible joint FJR. In the paper "Lin, Z., Du Wang, H., Karkoub, M., Shah, UH, Li, M.: Prescribed performance-based sliding mode path tracking control of a flexible joint UVMS using a sliding mode disturbance observer based on an extended state observer. Ocean Eng. 240, 109915 (2021)", a prescribed performance tracking method is proposed for a flexible joint underwater vehicle manipulator system. These methods achieve good tracking performance while strictly limiting the error to a preset range, and the traditional Prescribed Performance Control (CPPC) method can further improve the transient performance of the tracking error. More importantly, when the target signal changes drastically, if the system fails to track effectively and in a timely manner, it can lead to a large tracking error that violates the predefined fixed limits. Therefore, it is necessary to develop an improved Prescribed Performance Controller (MPPC) for FJR to overcome these limitations of CPPC.

[0005] The above discussion assumes the system is in good working order. In practice, complex environments and variable tasks can lead to system failures, with actuator failures causing the greatest damage. To provide control systems with the ability to cope with drive errors, fault-tolerant control (FTC) has emerged, typically divided into passive and active FTC. The literature “Li, L., Yao, L., Wang, H.: Model Predictive Fault-Tolerant Tracking Control for PDF Control Systems with Packet Loss. IEEE Trans. Syst. Man, Cybern. Syst. 1–11 (2021). https: / / doi.org / 10.1109 / TSMC.2021.3103814” describes active FTC, which obtains fault estimation signals by introducing a fault diagnosis observer. Using additional observers can increase the computational burden, and delays in fault feedback can lead to system instability. In contrast, the passive fault-tolerant control (FTC) method in the literature "Lebreton, C., Damour, C., Benne, M., Grondin-Perez, B., Chabriat, JP: Passive fault-tolerant control for PEMFC intake systems. Int. J. Hydrogen Energy. 41, 15615–15621 (2016). https: / / doi.org / 10.1016 / j.ijhydene.2016.06.210" addresses faults by designing a compensating controller and does not require a fault observer. An FTC strategy has also been proposed for faulty FJR systems. For example, a sliding mode observer is proposed in the paper "Mao, Z., Jiang, B., Ding, SX: A fault-tolerant control framework for a class of nonlinear networked control systems. Int. J. Syst. Sci. 40, 449–460 (2009). https: / / doi.org / 10.1080 / 00207720802556260" to obtain necessary fault information. By switching supervision, a series of pre-computed candidate controllers are designed in the paper "Yang, H., Jiang, B., Staroswiecki, M.: Supervised fault-tolerant control for a class of uncertain nonlinear systems. Automatica. 45, 2319–2324 (2009)" to implement FTC. In the paper “Diao, S., Sun, W., Su, SF, Xia, J.: Actuator failure of a single-link flexible joint robot with adaptive fuzzy event-triggered control. IEEE Trans. Cybern. 1–11 (2021). https: / / doi.org / 10.1109 / TCYB.2021.3049536”, actuators with partial failure modes are considered for single-link FJR systems.However, none of these fault-tolerant methods for FJRs consider abrupt and gradual scenarios in the fault model. Furthermore, to the authors' knowledge, fault-tolerant control with fixed-time specified performance is rarely reported. Therefore, investigating a passive FTC scheme for FJRs and incorporating it into fixed-time adaptive specified performance control is the final research focus. Summary of the Invention

[0006] The technical problem solved by this invention is to provide an adaptive performance control method for a flexible robotic arm with actuator failure, so as to solve the technical problems existing in the prior art.

[0007] The technical solution adopted in this invention is: an adaptive performance control method for a flexible robotic arm with actuator failure, the method comprising the following steps:

[0008] (1) Establish the n-link FJR dynamic model as follows:

[0009]

[0010] Where q = [q1, q2, ..., q n ] T , u = [u1, u2, ..., u n ] T These represent the connection position vector, motor position vector, and input torque vector, respectively; J∈R n×n K∈R n×n and B∈R n×n M(q) represents the matrix of motor inertia, motor damping, and elasticity coefficients, and R represents the symmetric matrix of inertia. n×n Represents an n x n real matrix. Indicates the acceleration of the connecting rod. Indicates the speed of the connecting rod. Indicates the angular velocity of the motor. This represents the angular acceleration of the motor. Represents the Coriolis centripetal force matrix, G(q)∈R n It is the gravitational torque. Represents the friction term;

[0011] Introducing variables Simplifying the dynamic model formula (1), we get:

[0012]

[0013] In the formula, Indicates the speed of the robotic arm, acceleration, Indicates the angular velocity of the motor, Motor angular acceleration, M -1(x1) represents the inertial symmetric matrix, C(x1,x2) represents the Coriolis centripetal force matrix, and J represents the inertial symmetric matrix. -1 (u-Bx4-K(x3-x1)) represents the Lagrange equation for the output torque of the motor;

[0014] The control input of the model is constructed as follows:

[0015]

[0016] Where u c ∈R n It is the actual input, u∈R n The expected input is Δu∈R. n Indicates error. It is the time when the fault occurred. and These represent the time intervals in which the drive is generated. Refers to the evolution of failures:

[0017]

[0018] In the formula, This indicates a fault in the i-th drive. For a robotic arm with n joints, there are n drives, i = 1, 2, ..., n.

[0019] By inserting equation (3) into equation (2), we obtain the n-link FJR model for actuator failure:

[0020]

[0021] in Disturbances representing actuator malfunctions;

[0022] (2) Design AFTMPPC

[0023] First, we make the following assumptions, definitions, and lemmas:

[0024] Assumption 1: Reference value x d Its nth-order time derivative is continuous and bounded;

[0025] Assumption 2: The actuator faults of FJR are bounded, where the i-th actuator fault satisfies |Π i |≤Λ i ,|Π i | represents the fault disturbance of the i-th actuator, Λ i This represents the upper limit of the fault disturbance of the i-th actuator;

[0026] Definition 1: Consider a smooth nonlinear dynamic system Assuming the system is stable at the origin, if for any time t ≥ t *For all x(t) = 0, t * It is a finite time constant, the system It is stable over a finite time interval if the time constant t * There is an upper bound, the system It is stable over a specified period of time;

[0027] Lemma 1: If there exists a smooth function V(x) such that V(x)≥0 and satisfies the following inequalities:

[0028]

[0029] Where the function If a and b are positive real constants, 0 < β1 < 1, 1 < β2 < ∞, then the nonlinear system... The stability is semi-global with a real fixed time, and the convergence time satisfies:

[0030]

[0031] Lemma 2: For real variables and If σ, g, and ρ are positive constants, the following relationship holds:

[0032]

[0033] Lemma 3: If constants β1 and β2 satisfy 0 < β1 ≤ 1 and 1 ≤ β2 < +∞, and x r ∈R + ∪{0}, r=1,...,N, then we get:

[0034]

[0035] In the formula, R + Let N represent the set of positive real numbers; let N represent natural numbers, and let there be N variables, the r-th variable being x. r where r ranges from 1 to N;

[0036] Lemma 4: Let f ≥ d, ω > 1, then the inequality is as shown in equation (10):

[0037]

[0038] In the formula, f and d represent variables, and ω represents positive constants greater than 0;

[0039] Lemma 5: If the continuous function f(X):R is unknown n →R is defined in a compact set Ω X f(X) is estimated by a radial basis function neural network (RBFNN) as shown in equation (11):

[0040] f(X) = WT ψ(X) (11)

[0041] Where X∈R n Represents the input vector, W = [W1, W2, ..., W...]. l ] T W represents the weight vector. T Represents the transpose of the weight vector, ψ(·)=[ψ1(·),ψ2(·),...,ψ l (·)] T It is a basis function vector, derived from the Gaussian function ψ i (X) yields:

[0042]

[0043] Where z i and b i These are the center point and the width, respectively. Based on the universal approximation capability of neural networks, f(X) = W. *T ψ(X)+δ is approximated online with arbitrary precision by RBFNN, where the error δ is minimized to the greatest extent by selecting an ideal weight vector.

[0044]

[0045] In the formula, Ω X The number X representing the neural network converges to the compact set Ω. X Inside;

[0046] Secondly, an adaptive fault-tolerant modified performance control (AFTMPPC) is designed using the backstepping method, with steps shown in A, B, and C:

[0047] A. Fixed-time dynamic surface

[0048] First, construct a fixed-time filter:

[0049]

[0050] in and ξ i ∈R n ξ represents the input vector, output vector, and intermediate vector of the filter. i Indicates intermediate variables. α represents the output of the filter. i Indicates the input of the filter. γ² > 1 represents the power term, ω i1 ω i2 , and It is a positive design constant.

[0051] Lemma 6: The virtual control law α is obtained in a fixed time using the differentiator formula (14). i The derivative of , with a fixed time T1, is:

[0052]

[0053] Where ρ=λ min (Q) / λ max (P), ρ1=λ min (Q1) / λ max (P1) and w≤λ min (P1) is a positive constant, and P, Q, P1, and Q1 are derived from formulas (16)-(18), where λ min (Q) represents the minimum eigenvalue of matrix Q, and λ max (P1) represents the largest eigenvalue of P1;

[0054] Positive definite symmetric matrices P, Q, P1 and Q1∈R n×n The following equations must be satisfied:

[0055]

[0056] The definitions of A and A1 are as follows:

[0057]

[0058]

[0059] Where, ω i1 ω i2 , and It is the quantity that makes A and A1 satisfy Hurwitz;

[0060] The compensation mechanism is designed as follows:

[0061]

[0062] Among them, l ij These are positive constants, i = 1, ... 4, j = 1, 2; Y i This represents the filter error, where i ranges from 1 to 3; χ i Indicates the signal, i ranges from 1 to 4; Y i The formula is as follows:

[0063]

[0064] The error surface is then defined as follows:

[0065]

[0066] In the formula, υ1 represents the first error surface, υ i+1 Let represent the (i+1)th error surface, and e1 represent the system output tracking error;

[0067] B. Design Improvement of Specified Performance Functions

[0068] The performance function is typically selected from the following formula.

[0069] φ(t)=(φ0-φ ∞ )exp(-βt)+φ ∞ (twenty two)

[0070] Where β>0 represents the attenuation ratio, φ0 and φ ∞ Let φ(t) represent the initial conditions and final state, where φ0 > φ ∞ >0;

[0071] Design a new fixed-time performance function based on formula (22).

[0072]

[0073] in, and yes The initial and final states, T represents the preset convergence time. The expression represents the change in the desired signal, where c, π1, and π2 are positive design constants, and c ≥ 1; e1(0) represents the initial time error, i.e., x1(0) - x d ;

[0074] Subsequently, the tracking error e1 is strictly limited to a preset area:

[0075]

[0076] in and It is a positive number;

[0077] Lemma 7: For all t≥0, It is continuous, bounded, and differentiable, and its derivative is... Bounded;

[0078] Next, we introduce the logarithmic transformation function as follows:

[0079]

[0080] in, Let ν denote ν(t);

[0081] The time derivative of Ψ(ν):

[0082]

[0083] in,

[0084] C. Reverse Design Process

[0085] Introducing coordinate transformation:

[0086]

[0087] Step 1: Select candidate Lyapunov functions as

[0088]

[0089] In the formula, θ1 represents the first variable in (32), and χ1 represents the first compensation signal in (19); the time derivative of V1 is obtained by taking the time derivative of formula (33):

[0090]

[0091] Combining the first equation of formulas (30), (31) and (32), we get:

[0092]

[0093] Incorporating this into formula (34) yields:

[0094]

[0095] According to formula (21), the time derivative of e1 is obtained as follows: By combining the first equation of integral formula (5) and the second equation of formula (21), we obtain:

[0096]

[0097] Substituting the first equation of formula (19) and formula (37) into formula (36) yields...

[0098]

[0099] In the formula, α1 represents the first virtual control law to be designed, i.e., the α mentioned earlier. i The case where i is 1;

[0100] By using Young's inequality and Lemma 2, we have:

[0101]

[0102] Inserting formula (39) into formula (38) yields:

[0103]

[0104] Design the virtual control law α1 as follows:

[0105]

[0106] Where k1, λ1 are positive design constants;

[0107] Substituting (41) into (40) gives

[0108]

[0109] Step 2: Introduce RBFNN to handle the unknown nonlinear function f1(X1) given later. Then, introduce... Estimate the ideal weight W1 * ,in, and These are the estimation error and its derivative. W1 * The estimated value, It represents its derivative;

[0110] Candidate Lyapunov functions were selected as

[0111]

[0112] Where μ1 is a positive constant. This indicates the estimation error of the weights;

[0113] The derivative of V2 with respect to time is:

[0114]

[0115] From formula (32), we get:

[0116]

[0117] Substituting the second equation of formula (2) into formula (45) yields...

[0118]

[0119] According to formulas (20), (21) and (32):

[0120] x3=θ3+χ3+Y2+α2 (47)

[0121] By combining the second equation of formula (19) and formula (47) into formula (46), we obtain

[0122]

[0123] Define f1(X1) = [f11 ,f 12 ,…,f 1n ] T for

[0124] f1(X1)=M -1 (x1)(-C(x1,x2)x2-G(x1)-F(x2)-Kx1) (49)

[0125] Where X1 = [x1, x2] T ;

[0126] Then, formula (48) is rewritten as:

[0127]

[0128] Substituting formulas (19) and (50) into formula (44), we get:

[0129]

[0130] According to formula (49), f1(X1) is a compact set. A continuous function on f1(X1) can be approximated by Lemma 5 as follows:

[0131]

[0132] The estimation error δ1(X1) satisfies

[0133] Combining Young's inequality and Lemma 2, we get:

[0134]

[0135] Combining formula (53) with formula (51), we get:

[0136]

[0137] Subsequently, the virtual control law α2 and the adaptive law are constructed. for:

[0138]

[0139]

[0140] Where k2 and λ2 are positive constants, and M(x1) represents the inertial symmetric matrix;

[0141] Substituting formulas (55) and (56) into formula (54), we get:

[0142]

[0143] Based on Young's inequality and get:

[0144]

[0145] make And γ² = ω, according to Lemma 4, the following relationship holds:

[0146]

[0147] Substituting into formula (42) and formula (58)-(59) into formula (57), we get:

[0148]

[0149] Where κ1=k1-l 11 / 2γ1,κ2=k2-l 21 / 2γ1, and Step 3: The candidate Lyapunov function is designed as follows:

[0150]

[0151] Taking its derivative, we get:

[0152]

[0153] From formula (32), we know Its derivative is:

[0154]

[0155] The third equation of integral formula (5), formula (20), and formula (20) to formula (63) are:

[0156]

[0157] Substituting formulas (63) and (62) into formula (64) yields...

[0158]

[0159] Similarly, according to formula (53), we have:

[0160]

[0161] Inserting formula (66) into formula (65) yields:

[0162]

[0163] Then, the virtual control law α3 is designed as follows:

[0164]

[0165] Where k3 and λ3 are positive constants;

[0166] Combining formulas (60) and (68) to formula (67), become:

[0167]

[0168] Where κ3=k3-l 31 / 2γ1 and

[0169] Step 4: Use RBFNN to approximate the unknown function f2(X2) defined later. in It is the ideal weight W2 * The estimated vector, The time derivative is

[0170] According to hypothesis 2, the disturbance range of the i-th actuator failure is |Π i |≤Λ i An adaptive technique is used to identify the value of Λ, and the estimation error is defined as follows: Its time derivative is

[0171] Then, the candidate Lyapunov function is constructed as follows:

[0172]

[0173] Where μ2 and It is a positive number;

[0174] Its time derivative is calculated as follows:

[0175]

[0176] From formula (32), we get Then, take its time derivative as

[0177]

[0178] Inserting the fourth subsystem of formula (5) and formula (19) into formula (72) yields:

[0179]

[0180] Substituting formulas (19) and (73) into formula (71), we get:

[0181]

[0182] Define the function f2(X2) = [f 21 ,f 22 ,…,f 2n ] T for:

[0183]

[0184] in R4=[θ 41 -1 ,θ 42 -1 ,…,θ 4n -1 ] T and X2 = [x i ,Y i ,χ i ,θ4] T , i = 1, 2, ..., 3, j = 1, 2, ..., 4, R4 represents the column vector formed by the reciprocals of each element of θ4;

[0185] Therefore, according to Lemma 5, f2(X2) is approximated as:

[0186]

[0187] Where the estimation error It represents a positive constant;

[0188] According to Young's inequality and Lemma 2, we have:

[0189]

[0190] Inserting formulas (76) and (77) into formula (74) yields:

[0191]

[0192] In the formula and Let represent the estimation error of the actuator fault, and be the derivative of the estimated value; Represents a positive constant;

[0193] The FTC will then be designed as follows:

[0194] u = J(u n -u as (79)

[0195] in

[0196]

[0197] Compensation item u as yes:

[0198]

[0199] Where ζ = [ζ1, ζ2, ... ζ] n ] T and ζ i >0, i = 1, 2, ..., n express The nth term;

[0200] Constructing Adaptive Laws Adaptive estimation of the upper limit of driving error for:

[0201]

[0202]

[0203] Where Θ=[|θ 41 |,|θ 42 |,…,|θ 4n |] T and It is a positive constant;

[0204] Substituting formulas (79), (82), and (83) into formula (78), we get:

[0205]

[0206] Because |Π i |≤Λ i and So:

[0207]

[0208] Based on Lemma 4 and Young's inequality, we obtain:

[0209]

[0210] Inserting the results of formula (69), formulas (85), and (86) into the result of formula (84), we have:

[0211]

[0212] Where κ4=k4-l 41 / 2γ1 and but:

[0213]

[0214] The advantages of this invention are:

[0215] 1) Compared with the results in the existing literature "Lin, Z., Du Wang, H., Karkoub, M., Shah, UH, Li, M. Using a sliding mode disturbance observer based on an extended state observer for sliding mode path tracking control of a UVMS with flexible joints based on specified performance. Ocean Eng. 240, 109915 (2021)," the convergence time of the decreasing smooth performance function tends to infinity, and the steady-state boundary is fixed and cannot be adjusted. This invention proposes an improved specified performance controller (MPPC) to improve the tolerance of the specified boundary to the reference input oscillation and achieve tracking error convergence within a specified time, which greatly improves the tracking performance and the robustness of the system.

[0216] 2) Unlike the existing literature "Diao, S., Sun, W., Su, SF, Xia, J.: Actuator Faults in a Single-Link Flexible Joint Robot with Adaptive Fuzzy Event Triggered Control. IEEE Trans. Cybern. 1–11 (2021). https: / / doi.org / 10.1109 / TCYB.2021.3049536", which only considers fixed faults in the actuator, this invention proposes a novel mathematical model for faults to study the gradual evolution and sudden changes of actuator faults. Then, by introducing adaptive identification technology to estimate fault information, an Adaptive Passive Fault-Tolerant Controller (APFTC) is proposed to effectively overcome the impact of faults.

[0217] 3) Adaptive radial basis neural networks (RBFNNs) are used to approximate the nonlinear uncertain dynamics within the system. Then, MPPC and APFTC are integrated into a fixed-time backpropagation framework to design an adaptive fixed-time fault-tolerant improved specified performance controller (AFTMPPC), in which a fixed-time second-order filter and compensation mechanism are constructed to bypass the "complexity explosion" of the problem and eliminate filter errors. Attached Figure Description

[0218] Figure 1 This is a schematic diagram of the 1n-link flexible robotic arm.

[0219] Figure 2 Control chart for the FJR system;

[0220] Figure 3 The comparison diagram is shown under the traditional reverse tracking method; in the diagram, (a) the tracking trajectory of joint 1; (b) the tracking error of joint 1; (c) the tracking trajectory of joint 2; and (d) the tracking error of joint 2.

[0221] Figure 4 The figure shows the tracking performance of joint 1; in the figure, (a) is the tracking trajectory of joint 1; and (b) is the tracking error of joint 1.

[0222] Figure 5 The figure shows the tracking performance of joint 2; in the figure, (a) is the tracking trajectory of joint 2; and (b) is the tracking error of joint 2.

[0223] Figure 6 The control input u1 diagram for joint 1;

[0224] Figure 7 The control input diagram for joint 2 is shown in diagram u2.

[0225] Figure 8 The figure shows the tracking performance of joint 1; in the figure, (a) trajectory condition q1; (b) conversion error e1;

[0226] Figure 9 The figure shows the tracking performance of joint 2; in the figure, (a) trajectory condition q2; (b) conversion error e2;

[0227] Figure 10 For the control input of joint 1 u1 picture;

[0228] Figure 11 For the control input of joint 2 u2 picture;

[0229] Figure 12 For RBFNN 1 and Norm graph;

[0230] Figure 13 For RBFNN 2 and Norm graph. Detailed Implementation

[0231] The present invention will now be described in more detail with reference to specific embodiments:

[0232] Example 1: As Figure 1-13 As shown, the adaptive performance control method for a flexible robotic arm with actuator failure includes the following steps:

[0233] (1) Establish the n-link FJR dynamic model as follows:

[0234]

[0235] Where q = [q1, q2, ..., q n ] T , u = [u1, u2, ..., u n ] T These represent the connection position vector, motor position vector, and input torque vector, respectively; J∈Rn×n K∈R n×n and B∈R n×n M(q) represents the matrix of motor inertia, motor damping, and elasticity coefficients, and R represents the symmetric matrix of inertia. n×n Represents an n x n real matrix. Indicates the acceleration of the connecting rod. Indicates the speed of the connecting rod. Indicates the angular velocity of the motor. This represents the angular acceleration of the motor. Represents the Coriolis centripetal force matrix, G(q)∈R n It is the gravitational torque. Represents the friction term;

[0236] Introduce the variable x1 = q, Simplifying the dynamic model formula (1), we get:

[0237]

[0238] In the formula, Indicates the speed of the robotic arm, acceleration, Indicates the angular velocity of the motor, Motor angular acceleration, M -1 (x1) represents the inertial symmetric matrix, C(x1,x2) represents the Coriolis centripetal force matrix, and J represents the inertial symmetric matrix. -1 (u-Bx4-K(x3-x1)) represents the Lagrange equation for the output torque of the motor;

[0239] Subsequently, considering the faults present in the actuator, the control input of the model is constructed as follows:

[0240]

[0241] Where u c ∈R n It is the actual input, u∈R n The expected input (considering driving errors, u in Equation 1 becomes u) c (that is, the two are consistent), Δu∈R n Indicates error. It is the time when the fault occurred. and These represent the time intervals in which the drive is generated. Refers to the evolution of failures:

[0242]

[0243] In the formula, This indicates a fault in the i-th drive. Since this patent considers a robotic arm with n joints, there are n drives, i = 1. , 2,…,n;

[0244] From equation (3), it can be seen that the FJR system has three operating modes: 1) When Δu = 0, the actuator operates normally. 2) When Δu > 0, the actuator operates in partial failure mode. 3) When... The actuator operates in a completely loss-of-efficiency manner.

[0245] More importantly, the types of actuator failures are described by an evolution function that shows an increase from occurrence to gradual increase. Definition. If ι i It is a very small constant, and actuator failures grow slowly. Conversely, actuator failures can occur and rapidly increase to a certain value.

[0246] Inserting (3) into (2) yields the n-link FJR model for actuator failure:

[0247]

[0248] in Disturbances representing actuator malfunctions;

[0249] (2) Design AFTMPPC

[0250] First, we make the following assumptions, definitions, and lemmas:

[0251] Assumption 1: Reference value x d Its nth-order time derivative is continuous and bounded;

[0252] Assumption 2: The actuator faults of FJR are bounded, where the i-th actuator fault satisfies |Π i |≤Λ i ,|Π i | represents the fault disturbance of the i-th actuator, Λ i This represents the upper limit of the fault disturbance of the i-th actuator;

[0253] Definition 1: Consider a smooth nonlinear dynamic system x∈R n Assuming the system is stable at the origin, if for any time t ≥ t * For all x(t) = 0, t * It is a finite time constant, the system It is stable over a finite time interval if the time constant t * There is an upper bound, the system It is stable over a specified period of time;

[0254] Lemma 1: If there exists a smooth function V(x) such that V(x)≥0 and satisfies the following inequalities:

[0255]

[0256] Where the function If a and b are positive real constants, 0 < β1 < 1, 1 < β2 < ∞, then the nonlinear system... The stability is semi-global with a real fixed time, and the convergence time satisfies:

[0257]

[0258] Lemma 2: For real variables and If σ, g, and ρ are positive constants, the following relationship holds:

[0259]

[0260] Lemma 3: If constants β1 and β2 satisfy 0 < β1 ≤ 1 and 1 ≤ β2 < +∞, and x r ∈R + ∪{0}, r=1,...,N, then we get:

[0261]

[0262] In the formula, R + Let N represent the set of positive real numbers; let N represent natural numbers, and let there be N variables, the r-th variable being x. r where r ranges from 1 to N;

[0263] Lemma 4: Let f ≥ d, ω > 1, then:

[0264]

[0265] In the formula, f and d represent variables, and ω represents positive constants greater than ω.

[0266] Lemma 5: If the continuous function f(X):R is unknown n →R is defined in a compact set Ω X f(X) is estimated by RBFNN (Radial Basis Neural Network):

[0267] f(X) = W T ψ(X) (11)

[0268] Where X∈R n Represents the input vector, W = [W1, W2, ..., W...]. l ] T W represents the weight vector. TRepresents the transpose of the weight vector, ψ(·)=[ψ1(·),ψ2(·),...,ψ l (·)] T It is a basis function vector, derived from the Gaussian function ψ i (X) yields:

[0269]

[0270] Where z i and b i These are the center point and the width, respectively. Based on the universal approximation capability of neural networks, f(X) = W. *T ψ(X)+δ is approximated online with arbitrary precision by RBFNN, where the error δ is minimized to the greatest extent by selecting an ideal weight vector.

[0271]

[0272] In the formula, Ω X The number X representing the neural network converges to the compact set Ω. X Inside;

[0273] Secondly, AFTMPPC (Adaptive Fault-Tolerant Modified Performance Control) is designed using the backstepping method, with the following steps:

[0274] A. Fixed-time dynamic surface

[0275] First, construct a fixed-time filter:

[0276]

[0277] in and ξ i ∈R n ξ represents the input vector, output vector, and intermediate vector of the filter. i Indicates intermediate variables. α represents the output of the filter. i Indicates the input of the filter. γ² > 1 represents the power term, ω i1 ω i2 , and It is a positive design constant.

[0278] Lemma 6: The virtual control law α is obtained in a fixed time using the differentiator formula (14). i The derivative of , with a fixed time T1, is:

[0279]

[0280] Where ρ=λmin (Q) / λ max (P), ρ1=λ min (Q1) / λ max (P1) and w≤λ min (P1) is a positive constant. P, Q, P1, and Q1 can be obtained from formulas (16)-(18), where λ min (Q) represents the minimum eigenvalue of matrix Q, and λ max (P1) represents the largest eigenvalue of P1;

[0281] Positive definite symmetric matrices P, Q, P1 and Q1∈R n×n The following equations must be satisfied:

[0282]

[0283] The definitions of A and A1 are as follows:

[0284]

[0285]

[0286] Where, ω i1 ω i2 , and It is the quantity that makes A and A1 satisfy Hurwitz;

[0287] To eliminate filtering errors, the compensation mechanism is designed as follows:

[0288]

[0289] Among them, l ij These are positive constants, i = 1, ..., 4, j = 1, 2; Y i This represents the filter error, where i ranges from 1 to 3; χ i This represents the compensation signal, where i ranges from 1 to 4; Y i The formula is as follows:

[0290]

[0291] The error surface is then defined as follows:

[0292]

[0293] In the formula, υ1 represents the first error surface, υ i+1 Let represent the (i+1)th error surface, and e1 represent the system output tracking error;

[0294] Unlike the common first-order linear filter used in the literature “Liu, L., Yao, W., Guo, Y.: Prescribed performance tracking control of a free-flying flexible-joint space robot with disturbances under inputsaturation. J. Franklin Inst. 358, 4571–4601 (2021)”, a second-order nonlinear filter formula (14) is constructed to obtain better filtering performance and a fixed-time convergence filter error Y. i ;

[0295] The controller involves multiple filters, and filter errors can affect control performance. Therefore, a filter error compensation mechanism was designed to effectively reduce filter errors and ensure that the compensated signal meets fixed-time stability.

[0296] B. Design Improvement of Specified Performance Functions

[0297] The performance function is typically selected from the following formula.

[0298] φ(t)=(φ0-φ ∞ )exp(-βt)+φ ∞ (twenty two)

[0299] Where β>0 represents the attenuation ratio, φ0 and φ ∞ Let φ(t) represent the initial conditions and final state, where φ0 > φ ∞ >0;

[0300] To enable the specified performance function to adaptively compensate for the target signal disturbance and converge to the equilibrium state within a specified time, a new fixed-time specified performance function is designed based on formula (22).

[0301]

[0302] in, and yes The initial and final states, T represents the preset convergence time. The expression represents the change in the desired signal, where c, π1, and π2 are positive design constants, and c ≥ 1; e1(0) represents the initial time error, i.e., x1(0) - xd ;

[0303] Subsequently, the tracking error e1 is strictly limited to a preset area:

[0304]

[0305] in and It is a positive number;

[0306] Lemma 7: For all t≥0, It is continuous, bounded, and differentiable, and its derivative is... Bounded;

[0307] Prove the computation function For the left and right limits of T, we get

[0308]

[0309] In the formula, T + and T - express;

[0310] According to hypothesis 1, we have express Established, obviously It is continuous at t=T; The time intervals t∈[0,T) and t∈[T,∞) in formula (23) are continuous respectively. at last If the condition is continuous and bounded on t≥0, then:

[0311]

[0312]

[0313] function It is differentiable on t≥0, so It is differentiable on t≥0, and then we have:

[0314]

[0315] Therefore, it is concluded that It is differentiable at time T, according to By definition, it is differentiable over time intervals t∈[0,T) and t∈[T,∞). Therefore... It is differentiable for all t ≥ 0;

[0316] Next, take The generated time derivative:

[0317]

[0318] In the formula, and These represent the expected inputs. xd First and second derivatives; because and When t≥0, it is bounded. Bounded, from the second equation of formula (29), we know The function is bounded in t∈[T,∞). Furthermore, the function... It is bounded in t∈[0,T), which means Since it is bounded in t∈[0,T), the result is... It is bounded at t≥0;

[0319] The proof of Lemma 7 is now complete.

[0320] By constructing the modified performance function as formula (23), it can be guaranteed that This means that the initial error of the system is always included in the performance function. Then, by importing ((Tt) / T) c The function is defined to converge to a stable state within T. Simultaneously, when the desired input x... d Sudden changes can be compensated for through design elements.

[0321] Next, we introduce the logarithmic transformation function as follows:

[0322]

[0323] in, For simplification, let ν denote ν(t);

[0324] The time derivative of Ψ(ν):

[0325]

[0326] in,

[0327] Based on formula (30), it is clear that Ψ(ν): Therefore, Equation (5) for a constrained system is transformed into an equivalent unconstrained system. Furthermore, The term was used to circumvent the singularity problem.

[0328] C. Reverse Design Process

[0329] Introducing coordinate transformation:

[0330]

[0331] Step 1: Select candidate Lyapunov functions as

[0332]

[0333] In the formula, θ1 represents the first variable in (32), and χ1 represents the first compensation signal in (19). Taking the time derivative of formula (33) yields the time derivative of V1:

[0334]

[0335] Combining the first equation of formulas (30), (31) and (32), we get:

[0336]

[0337] Incorporating this into formula (34) yields:

[0338]

[0339] According to formula (21), the time derivative of e1 is obtained as follows:

[0340] By combining the first equation of integral formula (5) and the second equation of formula (21), we obtain:

[0341]

[0342] Substituting the first equation of formula (19) and formula (37) into formula (36) yields...

[0343]

[0344] In the formula, α1 represents the first virtual control law to be designed, i.e., the α mentioned earlier. i The case where i is 1;

[0345] By using Young's inequality and Lemma 2, we have:

[0346]

[0347] Inserting formula (39) into formula (38) yields:

[0348]

[0349] Design the virtual control law α1 as follows:

[0350]

[0351] Where k1 and λ1 are positive design constants.

[0352] Substituting (41) into (40), we get:

[0353]

[0354] Step 2: In this step, RBFNN will be introduced to handle the unknown nonlinear function f1(X1) given later. Then, [the following steps will be introduced]. Estimate the ideal weight W1 * ,in, and These are the estimation error and its derivative. W1 * The estimated value, It represents its derivative;

[0355] Candidate Lyapunov functions were selected as

[0356]

[0357] Where μ1 is a positive constant. This indicates the estimation error of the weights;

[0358] The derivative of V2 with respect to time is:

[0359]

[0360] From formula (32), we get:

[0361]

[0362] Substituting the second equation of formula (2) into formula (45) yields...

[0363]

[0364] According to formulas (20), (21) and (32):

[0365] x3=θ3+χ3+Y2+α2 (47)

[0366] Combining the second equation of formula (19) and formula (47) into formula (46), we get:

[0367]

[0368] Define f1(X1) = [f 11 ,f 12 ,…,f 1n ] T for:

[0369] f1(X1)=M -1 (x1)(-C(x1,x2)x2-G(x1)-F(x2)-Kx1) (49)

[0370] Where X1 = [x1, x2] T ;

[0371] Then, formula (48) is rewritten as:

[0372]

[0373] Substituting formulas (19) and (50) into formula (44), we get:

[0374]

[0375] According to formula (49), f1(X1) is a compact set. A continuous function on , by Lemma 5, f1(X1) is approximated as:

[0376]

[0377] The estimation error δ1(X1) satisfies

[0378] Combining Young's inequality and Lemma 2, we get:

[0379]

[0380] Combining formula (53) with formula (51), we get:

[0381]

[0382] Subsequently, a virtual control law α2 (α2 is designed to make the system stable at a fixed time) and an adaptive law are constructed. for:

[0383]

[0384] Where k2 and λ2 are positive constants, and M(x1) represents the inertial symmetric matrix;

[0385] Substituting formulas (55) and (56) into formula (54), we get:

[0386]

[0387] Based on Young's inequality and get:

[0388]

[0389] make And γ² = ω, according to Lemma 4, the following relationship holds:

[0390]

[0391] Substituting into formula (42) and formula (58)-(59) into formula (57), we get:

[0392]

[0393] Where κ1=k1-l 11 / 2γ1,κ2=k2-l 21 / 2γ1, and

[0394] Step 3: The candidate Lyapunov function is designed as follows:

[0395]

[0396] Taking its derivative, we get:

[0397]

[0398] From formula (32), we know Its derivative is:

[0399]

[0400] The third equation of integral formula (5), formula (19), and formulas (20) to (63) are:

[0401]

[0402] Substituting formulas (63) and (62) into formula (64) yields...

[0403]

[0404] Similarly, according to formula (53), we have:

[0405]

[0406] Inserting formula (66) into formula (65) yields:

[0407]

[0408] Then, the virtual control law α3 (with the same meaning as the same variables in (64)) is designed as follows:

[0409]

[0410] Where k3 and λ3 are positive constants;

[0411] Combining formulas (60) and (68) to formula (67), become:

[0412]

[0413] Where κ3=k3-l 31 / 2γ1 and

[0414] Step 4: Use RBFNN to approximate the unknown function f2(X2) defined later. in It is the ideal weight W2 * The estimated vector, The time derivative is

[0415] Furthermore, according to hypothesis 2, the disturbance range of the i-th actuator failure is |Π i |≤Λ i Since it is difficult to obtain the actual value of unknown actuator faults in practical applications, an adaptive technique is used to identify the value of Λ. The estimation error is defined as follows: Its time derivative is

[0416] Then, the candidate Lyapunov function is constructed as follows:

[0417]

[0418] Where μ2, It is a positive number;

[0419] Its time derivative is calculated as follows:

[0420]

[0421] From formula (32), we get Then, take its time derivative as

[0422]

[0423] Inserting the fourth subsystem of formula (5) and formula (19) into formula (72) yields:

[0424]

[0425] Substituting formulas (19) and (73) into formula (71), we get:

[0426]

[0427] Define the function f2(X2) = [f 21 ,f 22 ,…,f 2n ] T for:

[0428]

[0429] in R4=[θ 41 -1 ,θ 42 -1 ,…,θ 4n -1 ] T and X2 = [x i ,Y i ,χ i ,θ4] T , i = 1, 2, ..., 3, j = 1, 2, ..., 4, R4 represents the column vector formed by the reciprocals of each element of θ4;

[0430] Therefore, according to Lemma 5, f2(X2) is approximated as:

[0431]

[0432] Where the estimation error Represent a positive integer;

[0433] According to Young's inequality and Lemma 2, we have:

[0434]

[0435] Inserting formulas (76) and (77) into formula (74) yields:

[0436]

[0437] In the formula and Let represent the estimation error of the actuator fault and the derivative of the estimated value, respectively. Represents a positive constant;

[0438] The FTC will then be designed as follows:

[0439] u = J(u n -u as (79)

[0440] in

[0441]

[0442] Compensation item u as yes:

[0443]

[0444] Where ζ = [ζ1, ζ2, ... ζ] n ] T and ζ i>0, i = 1, 2, ..., n express The nth term;

[0445] Constructing Adaptive Laws (representing the adaptive estimation of neural network weights) and (The adaptive estimate representing the upper limit of the driving error) is:

[0446]

[0447]

[0448] Where Θ=[|θ 41 |,|θ 42 |,…,|θ 4n |] T and It is a positive constant;

[0449] This invention employs an adaptive method to estimate the unknown upper limit of actuator faults and constructs u. as To compensate for them, as in formulas (81) and (83). This overcomes the difficulty of obtaining unknown faults and effectively eliminates the impact of actuator faults.

[0450] Substituting formulas (79), (82), and (83) into formula (78), we get:

[0451]

[0452] Because |Π i |≤Λ i and So:

[0453]

[0454] Based on Lemma 4 and Young's inequality, we obtain:

[0455]

[0456] Inserting the results of formula (69), formulas (85), and (86) into the result of formula (84), we have:

[0457]

[0458] Where κ4=k4-l 41 / 2γ1 and but:

[0459]

[0460] This invention also performs a stability analysis of the control method, the analysis method is as follows:

[0461] Theorem 1: For the FJR system, combining the virtual control law formulas (41), (55), and (68) and the adaptive law formulas (56) and (82), the controller formulas (79) and (83) are constructed. Then, it can be concluded that the closed-loop control system is fixed-time stable, and the output x1 can track the reference signal x. d The tracking error e1 is limited to the specified boundary, and all signals of the closed-loop control system are bounded.

[0462] Proof: Construct the entire Lyapunov function as

[0463]

[0464] From formula (87), we have

[0465]

[0466] According to Lemma 2, let σ=γ1, ρ = 1 - γ1 and g = γ1 -1 ,get

[0467]

[0468] Similarly, the following inequality holds.

[0469]

[0470]

[0471] From formulas (91)-(93), we can draw the following conclusion:

[0472]

[0473] Substituting into formula (94) into formula (90), we get:

[0474]

[0475] Subsequently, formula (95) can be further rewritten as

[0476]

[0477] in

[0478] From Lemma 3, we have:

[0479]

[0480] in,

[0481] Let β1 = γ1, β2 = γ2. According to Lemma 1, all signals V of the Lyapunov function are semi-global, practically fixed-time stable, and their settling times satisfy:

[0482]

[0483] According to formula (30), we can obtain

[0484]

[0485] Based on the bounded sum of θ1 = Ψ1 - χ1 Knowing that e1 is within a predefined range Bounded.

[0486] From the definition of formula (41), the function α1 is determined by the bounded variables θ1,x d ,e1, The composition, i.e., α1, is bounded. Similarly, the boundaries of α2, α3, and u can be easily obtained from equations (55), (68), and (79). Therefore, all signals in the closed-loop control system are bounded. The proof is now complete.

[0487] Simulation results: In this invention, the simulation was conducted on a double-link FJR, and the mathematical model is as follows:

[0488]

[0489] in

[0490] The literature “Wang, H., Peng, W., Tan, X., Sun, J., Tang, X., Chen, I.-M.: Robust output feedback tracking control for flexible-joint robots based on CTSMC technique. IEEE Trans. Circuits Syst. II Express Briefs. 68, 1982–1986 (2020)” selects the parameters of the double-link FJR as m1=m2=1kg, l1=l2=2m, J1=J2=1Kg·m 2 , B1=B2=0.9N·m·s / rad, K1=K2=100N·m / rad, g=9.8m / s 2 The initial values ​​of the system were chosen as x1 = [0.01; 0.01], x2 = x3 = x4 = [0; 0]. d1 =0.6sin(0.5πt) and x d2 =0.5sin(0.2πt)+0.5sin(0.4πt) is the desired trajectory of joint 1 and joint 2. Then, the design constants of the proposed controller are chosen as k1=k3=10, k2=k4=80, λ1=λ2=λ3=λ4=50. μ1 = 0.8, μ2 = 0.01. The controller uses three filters with the following design parameters: and The parameter of the compensation mechanism formula (19) is chosen to be l. 11 =l 31 =10, l 21 =l 41 =80, l 12 =l 22 =l 32 =l 42 =50. The RBFNN of this invention includes 11 nodes, and the center and width of the Gaussian function are z and z respectively. i ∈[-21,21] and b=10.

[0491] According to the fixed-time Lyapunov theory, the parameter κ... i =k i -l i1 / 2γ1 and i = 1, ..., 4 must be positive. Therefore, k i , λ i l ij γ1 and γ2 must be suitable to ensure κ i and is correct.

[0492] Case 1: In this case, the proposed MPPC is compared with the classical prescribed performance control (CPPC) in the literature "Ma,H.,Zhou,Q.,Li,H.,Lu,R.:Adaptive Prescribed Performance Control of A Flexible-Joint Robotic Manipulator With Dynamic Uncertainties.IEEE Trans.Cybern.1–11(2021).https: / / doi.org / 10.1109 / TCYB.2021.3091531(Ma,H.,Zhou,Q.,Li,H.,Lu,R.:Adaptive Prescribed Performance Control of A Flexible-Joint Robotic Manipulator With Dynamic Uncertainties.IEEE Trans.Cybern.1–11(2021).https: / / doi.org / 10.1109 / TCYB.2021.3091531)". The performance functions of CPPC and MPPC using the same convergence rate, initial value, and final value are:

[0493] φ(t)=3exp(-3t)+0.1 (101)

[0494]

[0495] Then, the specified convergence time for MPPC is set to T = 0.8.

[0496] The comparison between MPPC and CPPC is conducted within the traditional backstepping framework. From Figure 3 From (b) and 3(d), it can be seen that the tracking error e1 of joint 1 and the tracking error e2 of joint 2 in MPPC can reach a stable state within a predetermined time T. More importantly, when the expected input x d1 x d2 When a mutation occurs, the boundary of MPPC can be adaptively compensated for. In contrast, the convergence time of CPPC is longer than T, and the maximum overshoot of CPPC is greater than that of MPPC. Figure 3In (d), the tracking error e2 of joint 2 in the CPPC even significantly exceeds the boundary of the MPPC. Furthermore, the CPPC lacks self-adjustment capability to handle input oscillations. Therefore, it can be concluded that the proposed MPPC has better tracking performance than the CPPC.

[0497] Case 2: In this case, a comparison is made between fixed-time MPPC (MPPC γ1 = 97 / 101, γ2 = 5 / 3 under the fixed-time back-calculation framework), finite-time MPPC (MPPC γ1 = 97 / 101, γ2 = 1 under the finite-time back-calculation framework), and conventional MPPC (MPPC γ1 = 1, γ2 = 1 under the conventional back-calculation framework). The results are as follows... Figure 4-7 As shown. Figure 4 and Figure 5 This shows that the fixed-time backstepping method converges faster than the traditional finite-time MPPC. Figure 6 and Figure 7 This indicates that the control laws of finite-time and conventional schemes exhibit significant oscillations near the initial state of the system. In contrast, the fixed-time MPPC provides a relatively smooth and stable control input throughout the control process, verifying the effectiveness of the fixed-time MPPC of this invention.

[0498] Case 3: This case verifies the effectiveness of the proposed FTC strategy for dual-link FJR. Two different types of actuator failures were considered in the simulation. The first type of failure is that the actuators of joint 1 and joint 2 suddenly lose 50% of their effective drive within the time interval t∈[5j,5j+3), j=1,3,5…

[0499]

[0500] Fault 2 is that the actuators of joint 1 and joint 2 experience gradient driver errors during time intervals t∈[5j,5j+3), j=1,3,5…

[0501]

[0502] Simulation results are as follows Figure 8-13 As shown. Figure 8 and Figure 9 This indicates that despite the presence of faults 1 and 2, tracking performance and stability were still guaranteed. Figure 10 and Figure 11 This represents the actual control inputs for joints 1 and 2. Clearly, the proposed method is robust to drive faults, meaning the controller reacts rapidly when a fault occurs (see...). Figure 11-12 This is to offset adverse effects, thereby limiting tracking errors within specified boundaries (see...). Figure 8-9 ). Figure 12 and 13represents the norms of the estimated weights in RBFNN1 and RBFNN2, respectively. Clearly, they are bounded regardless of external disturbances.

[0503] From the above analysis, it can be concluded that the proposed MPPC has better performance than CPPC. Furthermore, compared to finite-time and conventional methods, the fixed-time MPPC exhibits the best performance. The designed FTC can effectively handle sudden and gradual actuator failures, ensuring that the system does not violate the modified performance boundaries and effectively tracks the target trajectory. Therefore, the effectiveness of the proposed AFTMPPC is verified.

[0504] Simulation Conclusions: This invention proposes an AFTMPPC scheme to achieve high-performance tracking control of n-link FJRs with actuator faults. Through the proposed MPPC, the system achieves convergence of the tracking error within a specified time, enhancing robustness against target signal oscillations. Then, an adaptive passive FTC is designed to compensate for actuator faults. Subsequently, an adaptive RBFNN is introduced to approximate the system's uncertainty function, and the "complexity explosion" problem in the back-calculation framework is overcome by combining a fixed-time second-order filter with an error compensation mechanism. Stability analysis proves that the closed-loop system is fixed-time stable, and all signals in the system are bounded. Finally, simulations of dual-link FJRs further demonstrate the effectiveness of the proposed AFTMPPC.

[0505] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of protection of the claims.

Claims

1. A method for adaptive performance control of a flexible robotic arm with actuator failure, characterized in that: The method includes the following steps: (1) Establish the n-link FJR dynamic model as follows: （1）， in, , , These represent the connection position vector, motor position vector, and input torque vector, respectively. , and This represents the matrix of motor inertia, motor damping, and elasticity coefficients. Represents an inertial symmetric matrix. express OK A real matrix of columns, Indicates the acceleration of the connecting rod. Indicates the speed of the connecting rod. Indicates the angular velocity of the motor. This represents the angular acceleration of the motor. Represents the Coriolis centripetal force matrix. It is the gravitational torque. Represents the friction term; Introducing variables Simplifying the dynamic model formula (1), we get: （2）， In the formula, Indicates the speed of the robotic arm, Indicates acceleration, Indicates the angular velocity of the motor, Indicates the angular acceleration of the motor, Representing an inertial symmetric matrix, Representing the Coriolis centripetal force matrix and The Lagrange equation representing the output torque of a motor; The control input of the model is constructed as follows: （3）， in, It's real input. It is the expected input. Indicates error. , It is the time when the fault occurred. and These represent the time intervals in which the drive is generated. Refers to the evolution of failures: （4）， In the formula, This indicates a fault in the i-th drive. For a robotic arm with n joints, there are n drives. ; By inserting equation (3) into equation (2), we obtain the n-link FJR model for actuator failure: （5）， in, Disturbances representing actuator malfunctions; (2) Design AFTMPPC First, we make the following assumptions, definitions, and lemmas: Assumption 1: Reference value Its nth-order time derivative is continuous and bounded; Assumption 2: The actuator faults of FJR are bounded, where the i-th actuator fault satisfies , This represents the fault disturbance of the i-th actuator. This represents the upper limit of the fault disturbance of the i-th actuator; Definition 1: Consider a smooth nonlinear dynamic system , Assuming the system is stable at the origin, if for any time... They all , It is a finite time constant, the system It is stable over a finite time interval, if the time constant is... There is an upper bound, the system It is stable over a specified period of time; Lemma 1: If there exists a smooth function and And satisfy the following inequalities: （6）， Among them, the function , and It is a positive real constant. , Then nonlinear system The stability is semi-global stability, and the time required for the system to stabilize is a fixed actual time, which satisfies the following: （7）， Lemma 2: For real variables and ,if , and For a positive integer, the following relations hold: （8）， Lemma 3: If the constant and satisfy and ,and Then we get: （9）， In the formula, Represents the set of positive real numbers; Represent natural numbers, and there exist The variable, the first The variables are ,in Choose from 1 to N; Lemma 4: Let , Then the inequality is as shown in equation (10): （10）， In the formula, and Represents variables, Represents positive constants greater than 0; Lemma 5: If the unknown continuous function Defined in a compact set , The radial basis function neural network (RBFNN) estimates the result as shown in equation (11): （11）， in, Represents the input vector. Represents the weight vector. This represents the transpose of the weight vector. It is a basis function vector, derived from the Gaussian function. get: （12）， in, and These are the center point and the width, respectively. Based on the general approximation capability of neural networks, Online approximation with arbitrary precision using RBFNN Among them, error To be minimized to the greatest extent possible by selecting an ideal weight vector. : （13）， In the formula, Number representing a neural network Converging in compact Inside; Secondly, an adaptive fault-tolerant modified performance control (AFTMPPC) is designed using the backstepping method, with steps shown in A, B, and C: A. Fixed-time dynamic surface First, construct a fixed-time filter: （14）， in, , and Represents the input vector, output vector, and filter intermediate vector. Indicates the output of the filter. Indicates the input of the filter. , Represents the power term. , , and It is a positive design constant. Lemma 6: The virtual control law is obtained in a fixed time using the differentiator formula (14). The derivative of the fixed time for: （15）， in, , and It is a positive number. , , , From formulas (16)-(18), we can derive that, Representation matrix Minimum eigenvalue, and express The largest eigenvalue; Positive definite symmetric matrix The following equations must be satisfied: （16）， in, and The definition is as follows: （17）， （18）， in, Is to make and The amount that satisfies Hurwitz; The compensation mechanism is designed as follows: （19）， in, It is a positive number. , ; This represents the filter error, where i ranges from 1 to 3. This indicates a signal, where i ranges from 1 to 4; The formula is as follows: （20）， The error surface is then defined as follows: （21）， In the formula, Indicates the first error surface. Indicates the first One error surface, This indicates the system output tracking error; B. Design Improvement of Specified Performance Functions The performance function is typically selected from the following formula. （22）， in, Indicates the attenuation ratio. and express Initial conditions and final state, ; Design a new fixed-time performance function based on formula (22). : （23）， in, and yes The initial and final states, , This indicates the preset convergence time. Indicates the change in the desired signal. , and It is a positive design constant, where ; This represents the initial time error, i.e. ; Then, the tracking error Strictly limited to the preset area: （24）， in, and It is a positive number; Lemma 7: For all , It is continuous, bounded, and differentiable, and its derivative is... Bounded; Next, we introduce the logarithmic transformation function as follows: （30）， in, ,use express ; Time derivative: （31）， in, ; C. Reverse Design Process Introducing coordinate transformation: （32）， Step 1: Select candidate Lyapunov functions as （33）， In the formula, Represents the first variable in (32), This represents the first compensation signal in (19); Taking the time derivative of formula (33) yields Time derivative: （34）， Combining the first equation of formulas (30), (31) and (32), we get: （35）， Incorporating it into formula (34) yields: （36）， According to formula (21). The time derivative is obtained as ; By combining the first equation of integral formula (5) and the second equation of formula (21), we obtain: （37）， Substituting the first equation of formula (19) and formula (37) into formula (36) yields... （38）， In the formula, This represents the first virtual control law to be designed, i.e., the one mentioned earlier. middle The case where the value is 1; By using Young's inequality and Lemma 2, we have: （39）， Inserting formula (39) into formula (38) yields: （40）， Design virtual control law for （41）， in, and It is a positive design constant; Substituting (41) into (40) gives （42）， Step 2: Introduce RBFNN to handle the unknown nonlinear function given later. Then, introduce Estimating ideal weights ,in, and These are the estimation error and its derivative. express The estimated value, It represents its derivative; The candidate Lyapunov function was selected as （43）， in, It is a positive constant. This indicates the estimation error of the weights; The derivative with respect to time is: （44）， From formula (32), we get: （45）， Substituting the second equation of formula (2) into formula (45) yields... （46）， According to formulas (20), (21) and (32): （47）， Combining the second equation of formula (19) and formula (47) into formula (46) yields （48）， definition for （49）， in, ; Then, formula (48) is rewritten as: （50）， Substituting formulas (19) and (50) into formula (44), we get: （51）， According to formula (49). It is compact A continuous function on, by Lemma 5, Approximately: （52）， Among them, the estimation error satisfy ; Combining Young's inequality and Lemma 2, we get: （53）， Combining formula (53) with formula (51), we get: （54）， Subsequently, a virtual control law was constructed. And adaptive law for: （55）， （56）， in, and It is a positive number. Represents an inertial symmetric matrix; Substituting formulas (55) and (56) into formula (54), we get: （57）， Based on Young's inequality and ,get: （58）， make , and According to Lemma 4, the following relation holds: （59）， Substituting into formula (42) and formula (58)-(59) into formula (57), we get: （60）， in, , , and ; Step 3: The candidate Lyapunov function is designed as follows: （61）， Taking its derivative, we get: （62）， From formula (32), we know Its derivative is: （63）， The third equation of integral formula (5), formula (20), and formulas (61) to (63) are: （64）， Substituting formulas (63) and (62) into formula (64) yields... （65）， Similarly, according to formula (53), we have: （66）， Inserting formula (66) into formula (65) yields: （67）， Then, design the virtual control law. for: （68）， in, and It is a positive number; Combine formulas (60) and (68) to form formula (67). become: （69）， in, and ; Step 4: Use RBFNN to approximate the unknown function defined later. , ,in Ideal weight The estimated vector, The time derivative is ; According to hypothesis 2, the disturbance range of the i-th actuator failure is Adaptive technology is used to identify The value, the estimation error is defined as Its time derivative is ; Then, the candidate Lyapunov function is constructed as follows: （70）， in, It is a positive number; Its time derivative is calculated as follows: （71）， From formula (32), we get Then, take its time derivative as （72）， Inserting the fourth subsystem of formula (5) and formula (19) into formula (72) yields: （73）， Substituting formulas (19) and (73) into formula (71), we get: （74）， Define function for: （75）， in, , and , , express A column vector consisting of the reciprocals of each element; therefore, According to Lemma 5, it is approximately: （76）， Among them, the estimation error , It represents a positive constant; According to Young's inequality and Lemma 2, we have: （77）， Inserting formulas (76) and (77) into formula (74) yields: （78）， In the formula, Let represent the estimation error of the actuator fault and the derivative of the estimated value, respectively. Represents a positive constant; The FTC will then be designed as follows: （79）， in （80）， Compensation yes: （81）， in, and , express The nth term; Constructing Adaptive Laws Adaptive estimation of the upper limit of driving error for: （82）， （83）， in, , It is a positive constant; Substituting formulas (79), (82), and (83) into formula (78), we get: （84）， because and ,So: （85）， Based on Lemma 4 and Young's inequality, we obtain: （86）， Inserting the results of formula (69), formulas (85) and (86) into the result of formula (84), we have: （87）， in, and ,but: （88）。

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