Non-strict repetition system adaptive fault-tolerant control method based on echo state network

Abstract

The invention relates to an adaptive fault-tolerant control method for a non-strict repetition system based on an echo state network, which comprises the following steps of: constructing a nonlinear dynamic system with an actuator fault, and setting a hypothesis condition; introducing an expected error trajectory, constructing a dynamic error equation and defining a nonlinear function in the equation; approaching a nonlinear function by using an echo state network; a self-adaptive iterative learning fault-tolerant algorithm is proposed, a barrier Lyapunov function is combined, control input is deduced, a controller is constructed, and redundant batches are filtered; and constructing an obstacle composite energy function to verify the stability and convergence of the designed adaptive iterative learning fault-tolerant algorithm. The nonlinear dynamic system constructed by the method can still realize effective tracking and control of the system state on an expected trajectory under the conditions that an additive or multiplicative fault occurs in an actuator, the system state is limited and external disturbance exists.

Description

Technical Field

[0001] The present invention relates to the field of intelligent control technology, and in particular to an adaptive fault-tolerant control method for a non-strictly repetitive system based on an echo state network. Background Art

[0002] Traditional iterative learning control (ILC) has been widely used in repetitive tasks such as industrial control, trajectory tracking, and automated manufacturing due to its ability to continuously optimize control inputs and improve tracking accuracy during task repetition. However, existing ILC methods typically assume that the system's initial state is consistent and that the actuators are operating normally, without considering issues such as changes in initial conditions and actuator failures. Furthermore, controlled systems often exhibit nonlinear characteristics, accompanied by state disturbances and model uncertainties, which increase the complexity of controller design. In recent years, neural networks have been widely used in control system modeling and compensation due to their excellent nonlinear approximation capabilities. Among them, the echo state network (ESN) has become an important tool for modeling nonlinear terms in dynamic systems due to its simple structure, high training efficiency, and ease of online updating.

[0003] Although some research has attempted to combine neural networks with ILC control methods, there is still a lack of an effective unified solution for meeting state constraints, compensating for actuator failures, and adapting to changes in initial conditions. Therefore, how to construct an iterative learning control method that can still guarantee convergence, safety, and fault tolerance in non-strictly repetitive environments remains a key technical issue that urgently needs to be overcome in the field of intelligent control. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention provides an adaptive fault-tolerant control method for non-strictly repetitive systems based on an echo state network. This method addresses the issue of degraded control performance during trajectory tracking control of non-strictly repetitive nonlinear systems, often caused by factors such as inconsistent initial states, state constraints, actuator fault interference, and inaccurate nonlinear modeling. By incorporating an echo state network approximation mechanism and designing an adaptive iterative learning fault-tolerant algorithm to construct a control framework, the present invention can effectively track and control the system state against the desired trajectory even in the presence of additive or multiplicative actuator faults, constrained system states, and external disturbances.

[0005] To solve the above technical problems, the present invention provides the following technical solution: a method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network, comprising the following steps:

[0006] S1. Construct a nonlinear dynamic system with actuator failure and calculate the initial state x k(0), state-dependent mismatch perturbation w(x k (t),t), additive fault η k (t), multiplicative fault ρ k (t) setting assumptions;

[0007] S2. Design an error tracking method, introduce the expected error trajectory into the error dynamic system, construct a dynamic error equation and define the nonlinear function Φ in the equation. k (t), guiding nonlinear dynamic systems to achieve stable tracking in the presence of inconsistent initial states and batch variations;

[0008] S3, the parameter set X of the constructed nonlinear system k Input the echo state network and use the approximation characteristics of the echo state network to approximate the output of the network to the nonlinear function Φ k (t), update the dynamic error equation to improve the adaptability of nonlinear dynamic systems to model uncertainties;

[0009] S4. Based on the adaptive iterative learning fault tolerance algorithm, design an adaptive fault-tolerant controller, combine the obstacle Lyapunov function, and derive the control input u k , and filter redundant batches and redefine the dynamic error equation;

[0010] S5. Based on the barrier Lyapunov function of the tracking error, a barrier composite energy function is constructed to verify the stability and convergence of the designed adaptive iterative learning fault-tolerant algorithm.

[0011] Furthermore, in step S1, the specific process includes the following steps:

[0012] S11. Define a nonlinear dynamic system with actuator failure:

[0013]

[0014] where t∈[0,T d ] is the time, k is the number of iterations, Indicates the system state, satisfying |x k (t)|<|x max (t)|,x max (t) is the upper bound of the system, represents a control input with a fault, where the superscript F represents a fault, including additive fault η k (t) and multiplicative fault ρ k (t), is the input matrix of the system and is full rank, Θ k (t) is a known nonlinear function, is an unknown time-varying vector, w(x k(t),t) represents the state-dependent mismatch perturbation;

[0015] S12, for the initial state x k (0) Set the assumptions, that is, x k (0) Bounded, but iteratively changing;

[0016] S13, the state-dependent mismatch perturbation w(x k (t),t) set the assumption that w(x k (t),t) norm is bounded;

[0017] S14, additive fault η k (t) Set the assumptions, i.e. η k (t) has an upper bound, which is expressed as:

[0018]

[0019] S15, Multiplicative Fault ρ k (t) Set the assumption that ρ k (t) has a lower bound ρ min , and ρ k (t) in the interval [ρ min ,1] changes and ρ min >0, that is, ρ k (t)∈[ρ min ,1], Where T d is the desired run batch length.

[0020] Furthermore, in step S2, the specific process includes the following steps:

[0021] S21. Define the error dynamic system:

[0022]

[0023] Among them, x k (t) represents the current system state, x r (t) represents the reference trajectory, Ξ k (t) represents a known nonlinear function;

[0024] S22. Define the expected error trajectory t∈[0,T d ],satisfy:

[0025]

[0026] where z k (0) is z k (t) The initial value of each iteration, the expected error trajectory is designed as:

[0027]

[0028] Where δ is a sufficiently small number used to connect the starting position and the set time of the expected error trajectory, and κ(t) is a monotonically decreasing function in [0,δ], satisfying κ(0) = 1, κ(δ) = 0;

[0029] S23. Define the dynamic error equation based on the error dynamic system and the expected error trajectory:

[0030]

[0031] The nonlinear function The dynamic error equation is used to track the current system state to the reference trajectory.

[0032] Furthermore, in step S3, the specific process includes the following steps:

[0033] S31, define the echo state network equation, and transform the parameter set X of the nonlinear dynamic system into k Input the echo state network and obtain the network output W T φ(X k );

[0034] S32, using the approximation characteristics of the echo state network, the nonlinear function Φ of the dynamic error equation is k (t) Approximation using the output of the echo state network:

[0035] Φ k (t) = W T φ(X k )+ε(X k )

[0036] =W T φ+ε

[0037] Among them, W is the ideal output weight matrix, and the parameter set φ=φ(X k ) is the activation function of the dynamic storage layer of the echo state network, ε=ε(X k ) is an arbitrary constant, and is the upper bound of ε;

[0038] S33. Based on the approximation result of step S32, the dynamic error equation obtained in step S23 is updated:

[0039]

[0040] where ζ = w(x k (t),t)+ε, has an upper bound

[0041] Furthermore, in step S32, it specifically includes: the ideal output weight matrix W is estimated by adopting an online learning strategy in the echo state network, and is optimized by an adaptive update algorithm.

[0042] Furthermore, in step S4, the adaptive iterative learning fault tolerance algorithm specifically includes the following steps:

[0043] S41. Construct the barrier Lyapunov function based on the state constraints:

[0044]

[0045] where k b >0 is the bounded parameter to be designed;

[0046] S42, design the control law so that the error e k (t) in the interval (-k b ,k b ), then the control input u of the kth iteration k Defined as:

[0047]

[0048] in The parameter γ = 1 / ρ min The estimated value of ρ min is the lower bound of multiplicative failure, α k satisfy:

[0049] α k =α1+α2+α3

[0050] in:

[0051]

[0052] Where K is the feedback gain matrix, sign(·) is the sign function, and the parameter The settings are as follows:

[0053]

[0054] and

[0055]

[0056] in Indicates the parameters before the batch length is determined. T kis the batch length of the kth run, Γ, λ are the gain terms to be designed, where

[0057] proj W ,proj γ is the projection mechanism, defined as:

[0058]

[0059] Where W j,min , W j,max They are the upper and lower bounds of the parameter estimates, respectively. For the parameter assumption W(t)∈[W min ,W max ], where W min ={W j,min}, W max ={W j,max}, and γ min , γ max is a parameter Estimated upper and lower bounds;

[0060] S43, filter redundant batches and redefine the dynamic error equation obtained in step S33:

[0061] ∈ k (t) = υ k (t)υ k (t)+(1-υ k (t))e k (T k ),t∈[0,T d ]

[0062] in:

[0063]

[0064] Where υ k is a Bernoulli distributed random variable, taking the value 0 or 1.

[0065] Furthermore, in step S41, the state constraint specifically includes: the current system state x k (t) must satisfy ||x k (t)||<||x max (t)||, The state x of the reference trajectory r (t) must satisfy ||x r (t)||≤||x * (t)||, where x * (t) is the upper bound of the reference trajectory, and ||x max (t)||>||x * (t)||.

[0066] Furthermore, in step S5, the adaptive iterative learning fault tolerance algorithm specifically includes: constructing the following barrier composite energy function:

[0067]

[0068] Among them, trace() is the operation of solving the matrix trace. V ∈,k (t) is the obstacle Lyapunov function of the dynamic error equation obtained in step S43, V W,k (t), V γ,k (t) is the parameter estimation error under the L2 norm.

[0069] By means of the above technical solution, the present invention provides an adaptive fault-tolerant control method for a non-strictly repetitive system based on an echo state network, which has at least the following beneficial effects:

[0070] (1) The present invention effectively overcomes the impact of system initial state inconsistency and batch changes on control performance by constructing an error tracking mechanism;

[0071] (2) The present invention uses an echo state network to approximate the uncertain nonlinear terms in the system, which greatly reduces the computational complexity and improves the online learning ability compared to traditional neural networks;

[0072] (3) The present invention introduces the barrier Lyapunov function in the controller design to effectively meet the system state constraint requirements. At the same time, combined with the barrier composite energy function design, it ensures the asymptotic convergence of the system error under the influence of actuator failure and disturbance. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0074] Figure 1 This is a flow chart of an adaptive fault-tolerant control method for a non-strictly repetitive system based on an echo state network according to the present invention;

[0075] Figure 2 Position tracking curve of the adaptive iterative learning fault-tolerant algorithm designed for the present invention;

[0076] Figure 3 The speed tracking curve of the adaptive iterative learning fault tolerance algorithm designed by the present invention;

[0077] Figure 4 This is the error convergence curve of the adaptive iterative learning fault tolerance algorithm designed by the present invention. DETAILED DESCRIPTION

[0078] To make the above-mentioned objectives, features, and advantages of the present invention more clearly understood, the present invention is further described below in detail with reference to the accompanying drawings and specific embodiments. This will enable a full understanding of how this application uses technical means to solve technical problems and achieve technical effects, and to implement the invention accordingly.

[0079] Those skilled in the art will appreciate that all or part of the steps in the above-mentioned embodiment methods can be accomplished by instructing the relevant hardware through a program. Therefore, the present application may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Furthermore, the present application may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0080] Please refer to Figures 1-4 , shows a specific implementation method of this embodiment, which constructs a nonlinear dynamic system with actuator faults, constructs a dynamic error equation, uses an echo state network to approximate the uncertain nonlinear terms in the system, and proposes an adaptive iterative learning fault-tolerant algorithm combined with an obstacle Lyapunov function to construct a controller; the nonlinear dynamic system constructed by the present invention can still achieve effective tracking and control of the system state to the desired trajectory under the conditions of additive or multiplicative faults in the actuator, limited system state and external disturbances, thereby solving the problem that traditional iterative learning control methods cannot maintain control accuracy and robustness under non-ideal initial conditions and system disturbances.

[0081] Please refer to Figure 1 This embodiment proposes a method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network, the method comprising the following steps:

[0082] S1. Construct a nonlinear dynamic system with actuator failure and calculate the initial state x k (0), state-dependent mismatch perturbation w(x k (t),t), additive fault η k (t), multiplicative fault ρ k (t) setting assumptions;

[0083] As a preferred implementation of step S1, the specific process includes the following steps:

[0084] S11. Define a nonlinear dynamic system with actuator failure:

[0085]

[0086] where t∈[0,T d ] is the time, k is the number of iterations, Indicates the system state, satisfying |x k (t)|<|x max (t)|,x max (t) is the upper bound of the system, represents a control input with a fault, where the superscript F represents a fault, including additive fault η k (t) and multiplicative fault ρ k (t), is the input matrix of the system and is full rank, Θ k (t) is a known nonlinear function, is an unknown time-varying vector, w(x k (t),t) represents the state-dependent mismatch perturbation;

[0087] The control input with the fault is represented as:

[0088]

[0089] in is the normal control input, ρ k (t) represents the multiplicative fault of the actuator, satisfying 0<ρ k (t)≤1,η k (t) represents an additive fault, which mainly manifests as the uncertainty caused by the actuator fault. When ρ k (t) = 1, η k When (t) = 0, it means that the actuator is in a healthy state.

[0090] S12, for the initial state x k (0) Set the assumptions, that is, x k (0) Bounded, but iteratively changing;

[0091] S13, the state-dependent mismatch perturbation w(x k (t),t) set the assumption that w(x k (t),t) norm is bounded;

[0092] S14, additive fault η k (t) Set the assumptions, i.e. η k (t) has an upper bound, which is expressed as:

[0093]

[0094] S15, Multiplicative Fault ρ k (t) Set the assumption that ρ k (t) has a lower bound ρmin , and ρ k (t) in the interval [ρ min ,1] changes and ρ min >0, that is, ρ k (t)∈[ρ min ,1], Where T d is the desired run batch length.

[0095] S2. Design an error tracking method, introduce the expected error trajectory into the error dynamic system, construct a dynamic error equation and define the nonlinear function Φ in the equation. k (t), guiding nonlinear dynamic systems to achieve stable tracking in the presence of inconsistent initial states and batch variations;

[0096] As a preferred implementation of step S2, the specific process includes the following steps:

[0097] S21. Define the error dynamic system:

[0098]

[0099] Among them, x k (t) represents the current system state, x r (t) represents the reference trajectory, Ξ k (t) represents a known nonlinear function;

[0100] S22, according to the assumption of step S12, the initial state x k (0) may be related to x r (0) Different, the traditional method is difficult to apply because it usually considers the same initial state of the system. The present invention defines the expected error trajectory t∈[0,T d ],satisfy:

[0101]

[0102] where z k (0) is z k (t) The initial value of each iteration, the expected error trajectory is designed as:

[0103]

[0104] Where δ is a sufficiently small number used to connect the starting position and the set time of the expected error trajectory, κ(t) is a monotonically decreasing function in [0,δ], satisfying κ(0)=1,κ(δ=0; the expected error trajectory In the interval [0,T d ] is continuous, mainly composed of z k(0) and δ. The batch length T in the kth run k =T d In the case of z k (t) in the interval [0,T d ] to track the expected error trajectory When the system state x k (t) in the interval [0,T d ] Track the reference trajectory x r (t), even if the batch length changes, i.e., T k Each iteration changes, when z k (t) in the interval [0,T k ] to track the expected error trajectory When the system state x k (t) in the interval [0,T k ] Track the reference trajectory x r (t).

[0105] S23. Define the dynamic error equation based on the error dynamic system and the expected error trajectory:

[0106]

[0107] The nonlinear function The dynamic error equation is used to track the current system state to the reference trajectory.

[0108] In this embodiment, compared with most existing methods, this method relaxes the requirements on the reference trajectory x r (t), which means that for different desired tracking trajectories, the expected error trajectory can be set the same.

[0109] S3, the parameter set X of the constructed nonlinear system k Input the echo state network and use the approximation characteristics of the echo state network to approximate the output of the network to the nonlinear function Φ k (t), update the dynamic error equation to improve the adaptability of nonlinear dynamic systems to model uncertainties;

[0110] As a preferred implementation of step S3, the specific process includes the following steps:

[0111] S31. Define the Echo State Network equation. The Echo State Network (ESN) is a special type of Recurrent Neural Network (RNN) whose basic structure includes an input layer, a hidden layer, and an output layer. Compared with traditional feedforward neural networks (such as Radial Basis Function Neural Networks (RBFNNs), the ESN only needs to adjust the output weights, while the weights from the input layer to the hidden layer remain unchanged, thereby significantly reducing computational complexity and improving training efficiency. The continuous-time dynamics of the ESN are defined as follows:

[0112]

[0113] where φ(x) is the activation function of the dynamic storage layer; λ is the leakage rate of the storage neuron, and tanh(·) is the hyperbolic tangent function. are the input, internal and feedback connection weight matrices respectively, u represents the external input, and its dimension is K, we can get:

[0114] y=W T φ(x)

[0115] in is the output weight matrix; the parameter set X of the nonlinear dynamic system k Input the echo state network and obtain the network output W T φ(X k );

[0116] S32. Studies have shown that ESN has universal approximation capabilities and can approximate any continuous function. On a sufficiently large compact set On the other hand, the output of ESN satisfies the following error bounds:

[0117]

[0118] in is an arbitrary constant. And the function f(x) can be approximated as:

[0119]

[0120] Where ε is the approximation error, and the upper bound satisfies The activation function φ(t) in the dynamic storage layer is composed of multiple neural units φ(t) = [φ1(t),…φ N (t)], each neuron unit selection is given by the following sigmoid form:

[0121]

[0122] where a j , b j , p j And l j are all constants, and 0<φ j (x)<l m , upper bound l m =max{|(a j / b j )+l j |,|[a j / (b j +1)]|}. Using the approximation characteristics of the echo state network, the nonlinear function Φ of the dynamic error equation is transformed into k (t) Approximation using the output of the echo state network:

[0123] Φ k (t) = W T φ(X k )+ε(X k )

[0124] =W T φ+ε

[0125] Among them, W is the ideal output weight matrix, and the parameter set φ=φ(X k ) is the activation function of the dynamic storage layer of the echo state network, ε=ε(X k ) is an arbitrary constant, and is the upper bound of ε;

[0126] More specifically, the ideal output weight matrix W in step S32 is estimated using an online learning strategy in the echo state network and optimized using an adaptive update algorithm;

[0127] S33. Based on the approximation result of step S32, update the dynamic error equation:

[0128]

[0129] where ζ = w(x k (t),t)+ε, has an upper bound

[0130] S4. Based on the adaptive iterative learning fault tolerance algorithm, design an adaptive fault-tolerant controller, combine the obstacle Lyapunov function, and derive the control input u k , and filter redundant batches and redefine the dynamic error equation;

[0131] As a preferred implementation of step S4, the specific process includes the following steps:

[0132] S41. Construct the barrier Lyapunov function based on the state constraints:

[0133]

[0134] where k b >0 is the bounded parameter to be designed;

[0135] More specifically, the state constraints in step S41 include: the current system state x k (t) must satisfy ||x k (t)||<||x max (t)||, The state x of the reference trajectory r (t) must satisfy ||x r (t)||≤||x * (t)||, where x * (t) is the upper bound of the reference trajectory, and ||x max (t)||>||x * (t)||;

[0136] S42, when the error e k (t) tends to k b When V b,k (t) will tend to infinity, so in order to ensure that the BLF is bounded, it is necessary to design a control law so that the error e k (t) in the interval (-k b ,k b ) range; In addition, this method can ensure that the system state will not exceed the set upper limit x max (t), so as to meet the state constraint requirements. Specifically, the control input u of the kth iteration k Defined as:

[0137]

[0138] in The parameter γ = 1 / ρ min The estimated value of ρ min is the lower bound of multiplicative failure, α k satisfy:

[0139] α k =α1+α2+α3

[0140] in:

[0141]

[0142] Where K is the feedback gain matrix, sign(·) is the sign function, and the parameter The settings are as follows:

[0143]

[0144]

[0145] and

[0146]

[0147] in Indicates the parameters before the batch length is determined. T k is the batch length of the kth run, Γ, λ are the gain terms to be designed, where

[0148] proj W ,proj γ is the projection mechanism, defined as:

[0149]

[0150] Where W j,min , W j,max They are the upper and lower bounds of the parameter estimates, respectively. For the parameter assumption W(t)∈[W min ,W max ], where W min ={W j,min}, W max ={W j,max}, and γ min , γ max is a parameter The upper and lower bounds of the estimate can be guaranteed not to diverge by the projection rule;

[0151] S43. Considering the change of batch length, for t∈(T d ,T N ] This part is redundant output and has no effect on the learning of the system. This part of the batch can be discarded. Therefore, we only need to consider t∈(T min ,T d ] The output of this part of the batch length redefines the dynamic error equation obtained in step S33:

[0152] ∈ k (t) = υ k (t)υ k (t)+(1-υ k (t))e k (T k),t∈[0,T d ]

[0153] in:

[0154]

[0155] Where υ k is a Bernoulli distributed random variable, taking the value 0 or 1.

[0156] In this embodiment, the designed adaptive iterative learning fault tolerance algorithm is mainly composed of α k Dominant, α k The feedback item Ke k (t) and compensation control; the main function of the feedback term is to ensure the stability of the system and enhance the robustness, while the compensation term offsets the influence of uncertainties, faults and noise in the system, thereby improving the tracking accuracy of the system.

[0157] It should be noted that the adaptive fault-tolerant controller designed in this embodiment refers to the control law (i.e., the control input u k ) is used to ensure that the system state can effectively track the desired trajectory under the conditions of actuator failure, disturbance, uncertainty and state constraints. As a specific implementation of the "adaptive iterative learning fault tolerance algorithm", the controller is used to calculate the control input u online based on the state error dynamics and estimation model. k Effective tracking and convergence control of the reference trajectory are guaranteed in the presence of additive and multiplicative faults, model uncertainties, and disturbances. Therefore, the adaptive fault-tolerant controller designed in this invention is not a standalone module, but rather a core component of the "Adaptive Iterative Learning Fault Tolerance Algorithm," embodied through specific control law formulas and featuring algorithm-driven control implementation.

[0158] S5. Construct an obstacle composite energy function based on the obstacle Lyapunov function of the tracking error to verify the stability and convergence of the designed adaptive iterative learning fault-tolerant algorithm;

[0159] As a preferred implementation of step S5, specifically, considering that the constructed nonlinear system itself has multiple complex factors such as state constraints, actuator failures, and state disturbances, a single Lyapunov barrier is difficult to meet the stability requirements. Therefore, the following barrier composite energy function is constructed:

[0160]

[0161] Among them, trace() is the operation of solving the matrix trace. V ∈,k(t) is the obstacle Lyapunov function of the dynamic error equation obtained in step S43, V W,k (t), V γ,k (t) is the parameter estimation error under the L2 norm; the increment of the barrier composite energy function can be expressed as:

[0162] ΔE k (t) = E k (t)-E k-1 (t)

[0163] =V,k(t)-V,k-1(t)+V W,k (t)-V W,k-1 (t)+V γ,k (t)-V γ,k-1 (t)

[0164] Because of the change in batch length, the above increments are in the range 0<t≤T k and T k <t≤T d The two time periods are different and need to be discussed separately:

[0165] Case 1: For t≤T k Based on the barrier composite energy function formula, it can be deduced that

[0166]

[0167] Substituting the dynamic error equation into the above formula, we get:

[0168]

[0169] α k Substituting into the above formula we can get:

[0170]

[0171] Further deduction yields:

[0172]

[0173] Further reasoning about V in the increment formula W,k (t)-V W,k-1 (t):

[0174]

[0175] Through the above formula, we can deduce

[0176]

[0177] Substituting further, we get:

[0178]

[0179] Case 2: When T k ≤T d , for T k ≤t≤T d Need to revisit:

[0180] According to the deduction of case 1, we can get:

[0181]

[0182] Combining the parameters from the barrier composite energy function formula The properties of can be deduced:

[0183]

[0184] Then launched

[0185]

[0186] Similarly, we can deduce:

[0187]

[0188] Combining parameters The properties of , we get:

[0189]

[0190] Further, we get:

[0191]

[0192] It can be introduced:

[0193]

[0194] Where T k ∨t=min{T k ,t}, then deduce from the above formula:

[0195]

[0196] If E0(t) is in [0,T d ] is bounded, then according to the above formula and E k The positivity of (t) can be deduced when t∈[0,T d ], Then we prove the boundedness of E0(t), which can be derived from the barrier composite energy function formula:

[0197]

[0198] Discuss separately for different time t, when 0<t≤T k Taking the derivative of E0(t) we can see that:

[0199]

[0200] Because 0<ρ min ≤ρ0, we can get:

[0201]

[0202] Derivation of the right half of the above formula:

[0203]

[0204] Combined update rate parameter The properties are further obtained:

[0205]

[0206] and then can be rewritten as:

[0207]

[0208] It can be obtained that E0(t) is bounded. Similarly, when T k <t≤T d When E0(t) is bounded, E0(t) is also bounded. k The positivity of (t) can be deduced as follows:

[0209]

[0210] According to the convergence principle of series, we can get:

[0211]

[0212] Therefore, we can get

[0213] Because t is changing, when t∈P a When time t is at the minimum running time, it can be found that the newly defined error ∈ k (t) = e k (t), according to the above, we can know that under this set If t∈P b When the set is a time-varying set, when t<T k When ∈ k (t) = e k (t) still holds true when t≥T k When ∈k (t) = e k (T k ), further introduced:

[0214]

[0215] It can be obtained that when t∈P b When , we get:

[0216]

[0217] When t∈P b When p(t)>0, there is obviously υ in infinite iterations k (t)=1, further deducing Finally, it is ensured that the entire batch is run It ensures that when the number of iterations tends to infinity, the tracking error e k (t) can converge to zero point by point. So far, the stability and convergence of the designed adaptive iterative learning fault-tolerant algorithm have been verified.

[0218] This embodiment of the application also provides a simulation experiment to verify the performance of the proposed adaptive iterative learning fault tolerance algorithm to demonstrate its practicality and effectiveness. Specifically, a single-link robotic arm is used as a simulation model to verify the fault tolerance, robustness, and reliability of the proposed control algorithm under load disturbances, multiplicative actuator faults, and additive actuator faults. The noisy robotic arm model can be described as follows:

[0219]

[0220] where t∈[0,6], x k,1 (t) and x k,2 (t) represent angular position and angular velocity respectively, is the actuator input torque. m l represents the mass of the manipulator, M, g, l and c represent the tip load, gravitational acceleration, length and damping coefficient respectively. 2 +m l l 2 / 3. The physical parameters of the single-link manipulator are m = 2 kg, M = 4 kg, l = 0.5 m, g = 9.8 m / s 2 . Change interval T k Belongs to [3,6], and the expected tracking trajectory x r (t) = 0.1*sin(3*t) + 0.3. The present invention also takes into account the constraints and sets k b=0.1, this parameter is the constraint between the desired tracking trajectory and the system trajectory. This embodiment also considers the problem of initial value change and adopts the error tracking method, so the trajectory is reset to:

[0221]

[0222] The specific form of κ(t) is as follows:

[0223]

[0224] Where δ = 0.3. The designed adaptive iterative learning fault tolerance algorithm is:

[0225]

[0226] in and φ are the weight estimation and activation function of the neural network. The specific form is shown in (12), and the specific parameters are shown in the following table:

[0227]

[0228] The simulation considers state-related disturbances and actuator faults. Actuator faults consist of multiplicative faults and additive faults. The multiplicative fault is ρ k (t) = 0.3 + e -5t , additive fault η k (t)=0.2*sin(0.2πt)+rand[-0.1,0.1]. The specific parameters in the adaptive iterative learning fault tolerance algorithm are λ=10. Figure 2 and Figure 3 They represent the position and speed tracking curves of the adaptive iterative learning fault-tolerant algorithm for the 1st, 5th, 20th and 40th times respectively. Figure 4 It is the error convergence curve of the adaptive iterative learning fault-tolerant algorithm. From the figure, it can be seen that the proposed adaptive iterative learning fault-tolerant algorithm still has good robustness and tracking performance in the face of state-related disturbances, changes in system initial values, changes in batch length, and actuator failures.

[0229] In the description of this specification, the reference terms "one embodiment," "some embodiments," "example," "specific example," or "some examples" mean that the specific features, structures, materials, or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present application. Moreover, the specific features, structures, materials, or characteristics described may be combined in any appropriate manner in any one or more embodiments or examples. In addition, those skilled in the art may combine and combine different embodiments or examples described in this specification, as well as features of different embodiments or examples, unless they are mutually inconsistent.

[0230] The logic and / or steps represented in the flowchart or otherwise described herein may be considered, for example, as an ordered list of executable instructions for implementing logical functions, and may be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a system including a processor, or other system that can fetch and execute instructions from an instruction execution system, apparatus, or device).

[0231] The above embodiments provide a detailed introduction to the present invention. Specific examples are used herein 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 ideas. At the same time, for those skilled in the art, according to the ideas of the present invention, there may be changes in the specific implementation methods and application scopes. In summary, the contents of this specification should not be understood as limiting the present invention.

Claims

1. An adaptive fault-tolerant control method for non-strictly repetitive systems based on echo state networks, characterized in that: The following steps are involved: S1. Construct a nonlinear dynamic system with actuator failure and calculate the initial state x k (0), state-dependent mismatch perturbation w(x k (t),t), additive fault η k (t), multiplicative fault ρ k (t) setting assumptions; S2. Design an error tracking method, introduce the expected error trajectory into the error dynamic system, construct a dynamic error equation and define the nonlinear function Φ in the equation. k (t), guiding nonlinear dynamic systems to achieve stable tracking in the presence of inconsistent initial states and batch variations; S3, the parameter set X of the constructed nonlinear system k Input the echo state network and use the approximation characteristics of the echo state network to approximate the output of the network to the nonlinear function Φ k (t), update the dynamic error equation to improve the adaptability of nonlinear dynamic systems to model uncertainties; S4. Based on the adaptive iterative learning fault tolerance algorithm, design an adaptive fault-tolerant controller, combine the obstacle Lyapunov function, and derive the control input u k , and filter redundant batches and redefine the dynamic error equation; S5. Based on the barrier Lyapunov function of the tracking error, a barrier composite energy function is constructed to verify the stability and convergence of the designed adaptive iterative learning fault-tolerant algorithm.

2. The method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network according to claim 1, characterized in that: The specific process of step S1 includes the following steps: S11. Define a nonlinear dynamic system with actuator failure: where t∈[0,T d ] is the time, k is the number of iterations, Indicates the system state, satisfying |x k (t)|<|x max (t)|,x max (t) is the upper bound of the system, represents a control input with a fault, where the superscript F represents a fault, including additive fault η k (t) and multiplicative fault ρ k (t), is the input matrix of the system and is full rank, Θ k (t) is a known nonlinear function, is an unknown time-varying vector, w(x k (t),t) represents the state-dependent mismatch perturbation; S12, for the initial state x k (0) Set the assumptions, that is, x k (0) Bounded, but iteratively changing; S13, the state-dependent mismatch perturbation w(x k (t),t) set the assumption that w(x k (t),t) norm is bounded; S14, additive fault η k (t) Set the assumptions, i.e. η k (t) has an upper bound, which is expressed as: S15, Multiplicative Fault ρ k (t) Set the assumption that ρ k (t) has a lower bound ρ min , and ρ k (t) in the interval [ρ min ,1] changes and ρ min >0, that is Where T d is the desired run batch length.

3. The method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network according to claim 2, characterized in that: The specific process of step S2 includes the following steps: S21. Define the error dynamic system: Among them, x k (t) represents the current system state, x r (t) represents the reference trajectory, Ξ k (t) represents a known nonlinear function; S22. Define the expected error trajectory satisfy: where z k (0) is z k (t) The initial value of each iteration, the expected error trajectory is designed as: Where δ is a sufficiently small number used to connect the starting position and the set time of the expected error trajectory, and κ(t) is a monotonically decreasing function in [0,δ], satisfying κ(0) = 1, κ(δ) = 0; S23. Define the dynamic error equation based on the error dynamic system and the expected error trajectory: The nonlinear function The dynamic error equation is used to track the current system state to the reference trajectory.

4. The method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network according to claim 3, characterized in that: The specific process of step S3 includes the following steps: S31, define the echo state network equation, and transform the parameter set X of the nonlinear dynamic system into k Input the echo state network and obtain the network output W T φ(X k ); S32, using the approximation characteristics of the echo state network, the nonlinear function Φ of the dynamic error equation is k (t) Approximation using the output of the echo state network: F k (t)=W T φ(X k )+ε(X k )=W T f+e Among them, W is the ideal output weight matrix, and the parameter set φ=φ(X k ) is the activation function of the dynamic storage layer of the echo state network, ε=ε(X k ) is an arbitrary constant, and is the upper bound of ε; S33. Based on the approximation result of step S32, the dynamic error equation obtained in step S23 is updated: where ζ=w(x k (t),t)+ε, has an upper bound 5. The method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network according to claim 4, characterized in that: The ideal output weight matrix W in step S32 specifically includes: using an online learning strategy in the echo state network to estimate W, and optimizing it through an adaptive update algorithm.

6. The method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network according to claim 4, characterized in that: In step S4, the adaptive iterative learning fault tolerance algorithm specifically includes the following steps: S41. Construct the barrier Lyapunov function based on the state constraints: where k b >0 is the bounded parameter to be designed; S42, design the control law so that the error e k (t) in the interval (-k b ,k b ), then the control input u of the kth iteration k Defined as: in The parameter γ = 1 / ρ min The estimated value of ρ min is the lower bound of multiplicative failure, α k satisfy: α k ＝α1+α2+α3 in: Where K is the feedback gain matrix, sign(·) is the sign function, and the parameter The settings are as follows: and in Indicates the parameters before the batch length is determined. T k is the batch length of the kth run, Γ, λ are the gain terms to be designed, where proj W ,proj γ is the projection mechanism, defined as: Where W j,min , W j,max They are the upper and lower bounds of the parameter estimates, respectively. For the parameter assumption W(t)∈[W min ,W max ], where W min ={W j,min }, W max ={W j,max }, and γ min , γ max is a parameter Estimated upper and lower bounds; S43, filter redundant batches and redefine the dynamic error equation obtained in step S33: ∈ k (t)=υ k (t)u k (t)+(1-υ k (t))e k (T k ),t∈[0,T d ] in: Where v k is a Bernoulli distributed random variable, taking the value 0 or 1.

7. The method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network according to claim 6, characterized in that: The state constraints in step S41 specifically include: the current system state x k (t) must meet The state x of the reference trajectory r (t) must satisfy ||x r (t)||≤||x * (t)||, where x * (t) is the upper bound of the reference trajectory, and ||x max (t)||>||x * (t)||.

8. The method for adaptive fault-tolerant control of a non-strictly repetitive system based on an echo state network according to claim 6, characterized in that: Step S5 specifically includes: constructing the following barrier composite energy function: Among them, trace() is the operation of solving the matrix trace. V ∈,k (t) is the obstacle Lyapunov function of the dynamic error equation obtained in step S43, V W,k (t), V γ,k (t) is the parameter estimation error under the L2 norm.

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