An echo state network-based adaptive fault-tolerant control method for non-strictly repetitive systems

Abstract

The application relates to an echo state network-based adaptive fault-tolerant control method for a non-strict repetitive system, which comprises the following steps: constructing a nonlinear dynamic system with an actuator fault, and setting a hypothesis condition; introducing a desired error trajectory, constructing a dynamic error equation, and defining a nonlinear function in the equation; approximating the nonlinear function by using an echo state network; proposing an adaptive iterative learning fault-tolerant algorithm, combining a barrier Lyapunov function, deducing a control input, constructing a controller, and filtering redundant batches; and constructing a barrier 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 application can still realize effective tracking and control of the system state to the desired trajectory under the conditions that the actuator has additive or multiplicative faults, the system state is limited, and external disturbances exist.

Description

Technical Field

[0001] This invention relates to the field of intelligent control technology, and in particular to an adaptive fault-tolerant control method for non-strictly repeating systems based on echo state networks. Background Technology

[0002] In repetitive tasks such as industrial control, trajectory tracking, and automated manufacturing, traditional Iterative Learning Control (ILC) has been widely used due to its ability to continuously optimize control inputs and improve tracking accuracy during task repetition. However, existing ILC methods typically assume a consistent initial system state and normal actuator operation, neglecting issues such as changes in initial conditions and actuator failures. Furthermore, controlled systems often exhibit nonlinear characteristics, accompanied by state disturbances and model uncertainties, increasing the complexity of controller design. In recent years, neural networks have been widely applied to control system modeling and compensation due to their excellent nonlinear approximation capabilities. Among them, Echo State Networks (ESNs), with their advantages of simple structure, high training efficiency, and ease of online updates, have become an important tool for modeling nonlinear terms in dynamic systems.

[0003] Although existing research has attempted to combine neural networks with ILC control methods, there is still a lack of effective unified solutions for satisfying state constraints, actuator fault compensation, and adaptation 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 problem that urgently needs to be solved in the field of intelligent control. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides an adaptive fault-tolerant control method for non-strictly repeating nonlinear systems based on echo state networks. This method solves the problem of performance degradation in trajectory tracking control of non-strictly repeating nonlinear systems caused by factors such as inconsistent initial states, state constraints, actuator faults, and inaccurate nonlinear modeling. This invention introduces an echo state network approximation mechanism and combines it with an adaptive iterative learning fault-tolerant algorithm to construct a control framework. Even under conditions of additive or multiplicative actuator faults, constrained system states, and external disturbances, it can still achieve effective tracking and control of the desired trajectory.

[0005] To address the aforementioned technical problems, this invention provides the following technical solution: an adaptive fault-tolerant control method for a non-strictly repeating system based on echo state networks, comprising the following steps:

[0006] S1. Construct a nonlinear dynamic system with actuator failure, and consider 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) Set the assumptions;

[0007] S2. Design an error tracking method to introduce the desired error trajectory into the error dynamic system, construct the dynamic error equation, and define the nonlinear function Φ in the equation. k (t), which guides the nonlinear dynamic system to achieve stable tracking under conditions of inconsistent initial states and batch variations;

[0008] S3. The parameter set X of the constructed nonlinear system k By inputting an echo-state network, and utilizing the approximation properties of the echo-state network, the network's output is approximated to the nonlinear function Φ. k (t), update the dynamic error equation to improve the adaptability of the nonlinear dynamic system to model uncertainty;

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

[0010] S5. Based on the obstacle 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 [ ] represents time, and k is the number of iterations. Represents the system state, satisfying |x k (t)|<|x max (t)|,x max (t) is the upper bound of the system. This indicates a control input with a fault, where the superscript F indicates a fault, including additive faults η. k (t) and multiplicative fault ρ k (t), It is the input matrix of the system and is of full rank, Θ k (t) is a known nonlinear function. It is an unknown time-varying vector, w(x) k(t),t) represents a mismatch perturbation in state dependency;

[0015] S12, Regarding the initial state x k (0) Set the assumptions, i.e., x k (0) is bounded, but iterates through changes;

[0016] S13, Mismatch perturbation w(x) to state dependencies k (t),t) sets the assumptions, i.e., w(x) k The norm of (t),t) is bounded;

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

[0018]

[0019] S15, Multiplicative fault ρ k (t) Set the assumptions, i.e., ρ k (t) has a lower bound ρ min , and ρ k (t) in the interval [ρ min The value changes on ,1] and ρ min >0, i.e., ρ k (t)∈[ρ min ,1], Where T d This is the expected batch length.

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

[0021] S21. Define the error dynamic system:

[0022]

[0023] Where, 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 for each iteration, the expected error trajectory is designed as follows:

[0027]

[0028] Where δ is a sufficiently small number used to connect the starting position with the set time of the desired 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 desired error trajectory:

[0030]

[0031] Nonlinear function The current system state is tracked to the reference trajectory using this dynamic error equation.

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

[0033] S31. Define the echo state network equations, and apply them to 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 );

[0034] S32. Utilizing the approximation properties of echo state networks, 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] Where W is the ideal output weight matrix, and the parameter set is... 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 It is the upper bound of ε;

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

[0039]

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

[0041] Further, step S32 specifically includes: estimating the ideal output weight matrix W in the echo state network using an online learning strategy, and optimizing it using an adaptive update algorithm.

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

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

[0044]

[0045] Where k b >0 represents a bounded parameter to be designed;

[0046] S42. Design a control law to minimize the error e. k (t) in the interval (-k) b ,k b Within the range of ), the control input u for the k-th iteration is... k Defined as:

[0047]

[0048] in The parameter γ = 1 / ρ min The estimated value, ρ min α is the lower bound for multiplicative faults. 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 parameters are... The settings are as follows:

[0053]

[0054] and

[0055]

[0056] in This refers to the parameters before the batch length determination. T kThe length of the batch in the k-th run is Γ, and λ is the gain term to be designed.

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

[0058]

[0059] Among them W j,min W j,max These are the upper and lower bounds of the parameter estimation, 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 It is a parameter The 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] In the formula υ k Let be 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] Further, in step S5, the adaptive iterative learning fault-tolerant algorithm specifically includes: constructing the following obstacle composite energy function:

[0067]

[0068] The `trace()` function is used to find the trace of a matrix. V ∈,k (t) is the barrier Lyapunov function of the dynamic error equation obtained in step S43, V W,k (t), V γ,k (t) represents the parameter estimation error under the L2 norm.

[0069] By employing the above technical solution, the present invention provides an adaptive fault-tolerant control method for non-strictly repeating systems based on echo state networks, which has at least the following beneficial effects:

[0070] (1) By constructing an error tracking mechanism, this invention effectively overcomes the impact of inconsistent initial states and batch variations on control performance.

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

[0072] (3) The present invention introduces the obstacle Lyapunov function in the controller design, which effectively satisfies the system state constraint requirements. At the same time, combined with the obstacle composite energy function design, it ensures the asymptotic convergence of system error under the influence of actuator failure and disturbance. Attached Figure Description

[0073] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings:

[0074] Figure 1 This is a flowchart of an adaptive fault-tolerant control method for a non-strictly repeating system based on an echo state network, according to the present invention.

[0075] Figure 2 The position tracking curve of the adaptive iterative learning fault-tolerant algorithm designed in this invention;

[0076] Figure 3 The speed tracking curve of the adaptive iterative learning fault-tolerant algorithm designed in this invention;

[0077] Figure 4 The error convergence curve is shown for the adaptive iterative learning fault-tolerant algorithm designed in this invention. Detailed Implementation

[0078] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. This will allow for a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects, and to facilitate its implementation.

[0079] Those skilled in the art will understand that all or part of the steps in the methods of the above embodiments can be implemented by a program instructing related hardware. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, this application can 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 This illustration shows a specific implementation of the present invention. This embodiment constructs a nonlinear dynamic system with actuator faults, builds dynamic error equations, approximates uncertain nonlinear terms in the system using echo state networks, and proposes an adaptive iterative learning fault-tolerant algorithm combined with a barrier Lyapunov function to construct a controller. The nonlinear dynamic system constructed by this invention can still achieve effective tracking and control of the system state to the desired trajectory under conditions of additive or multiplicative actuator faults, system state constraints, and external disturbances. This solves 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 an adaptive fault-tolerant control method for non-strictly repeating systems based on echo state networks. The method includes the following steps:

[0082] S1. Construct a nonlinear dynamic system with actuator failure, and consider 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) Set the assumptions;

[0083] As a preferred embodiment 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 [ ] represents time, and k is the number of iterations. Represents the system state, satisfying |x k (t)|<|x max (t)|,x max (t) is the upper bound of the system. This indicates a control input with a fault, where the superscript F indicates a fault, including additive faults η. k (t) and multiplicative fault ρ k (t), It is the input matrix of the system and is of full rank, Θ k (t) is a known nonlinear function. It is an unknown time-varying vector, w(x) k (t),t) represents a mismatch perturbation in state dependency;

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

[0088]

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

[0090] S12, Regarding the initial state x k (0) Set the assumptions, i.e., x k (0) is bounded, but iterates through changes;

[0091] S13, Mismatch perturbation w(x) to state dependencies k (t),t) sets the assumptions, i.e., w(x) k The norm of (t),t) is bounded;

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

[0093]

[0094] S15, Multiplicative fault ρ k (t) Set the assumptions, i.e., ρ k (t) has a lower bound ρmin , and ρ k (t) in the interval [ρ min The value changes on ,1] and ρ min >0, i.e., ρ k (t)∈[ρ min ,1], Where T d This is the expected batch length.

[0095] S2. Design an error tracking method to introduce the desired error trajectory into the error dynamic system, construct the dynamic error equation, and define the nonlinear function Φ in the equation. k (t), which guides the nonlinear dynamic system to achieve stable tracking under conditions of inconsistent initial states and batch variations;

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

[0097] S21. Define the error dynamic system:

[0098]

[0099] Where, 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. Based on the assumptions in step S12, the initial state x is known. k (0) may be related to x r (0) Unlike traditional methods, which typically consider the case where the initial state of the system is the same, this invention defines the expected error trajectory. t∈[0,T d ],satisfy:

[0101]

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

[0103]

[0104] Where δ is a sufficiently small number used to connect the starting position and the set time of the desired error trajectory, and κ(t) is a monotonically decreasing function on [0,δ], satisfying κ(0)=1, κ(δ)=0; desired error trajectory In the interval [0, T] d ] is continuous and mainly consists of z kThe batch length T in the k-th run is determined by two factors: (0) and δ. k =T d In the case of z k (t) in the interval [0,T) d Tracking the expected error trajectory At that time, the system state x k (t) in the interval [0,T) d Tracked 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 Tracking the expected error trajectory At that time, the system state x k (t) in the interval [0,T) k Tracked the reference trajectory x r (t).

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

[0106]

[0107] Nonlinear function The current system state is tracked to the reference trajectory using this dynamic error equation.

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

[0109] S3. The parameter set X of the constructed nonlinear system k By inputting an echo-state network, and utilizing the approximation properties of the echo-state network, the network's output is approximated to the nonlinear function Φ. k (t), update the dynamic error equation to improve the adaptability of the nonlinear dynamic system to model uncertainty;

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

[0111] S31. Define the Echo State Network equation. An Echo State Network (ESN) is a special type of Recurrent Neural Network (RNN). Its basic structure includes an input layer, hidden layers, and an output layer. Compared to traditional feedforward neural networks (such as Radial Basis Function Neural Networks, RBFNNs), Echo State Networks only need to adjust the output weights, while the weights from the input layer to the hidden layer remain unchanged, thus significantly reducing computational complexity and improving training efficiency. The continuous-time dynamics of the Echo State Network 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 neurons; and tanh(·) is the hyperbolic tangent function. Let be the input, internal, and feedback connection weight matrices, respectively, and let u represent the external input, which has dimension K. We can obtain:

[0114] y = W T φ(x)

[0115] in 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. Existing research has shown that ESN has a universal approximation capability and can approximate any continuous function. On a sufficiently large compact set Above, the output of ESN satisfies the following error bounds:

[0117]

[0118] in It 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), ... φ1(t)]. N [t], where the selection of each neuron unit is given by the following sigmoid form:

[0121]

[0122] Where a j b j p j And l j Both are constants, and 0 < φ j (x)<l m Upper boundary l m =max{|(a j / b j )+l j |,|[a j / (b j +1)]|}。 Utilizing the approximation properties of echo state networks, the nonlinear function Φ of the dynamic error equation is transformed. 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] Where W is the ideal output weight matrix, and the parameter set is... 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 It is the upper bound of ε;

[0126] More specifically: the ideal output weight matrix W mentioned in step S32 is estimated in the echo state network using an online learning strategy, and then 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-tolerant algorithm, design an adaptive fault-tolerant controller, and derive the control input u by combining the obstacle Lyapunov function. k And filter out redundant batches and redefine the dynamic error equation;

[0131] As a preferred embodiment 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 represents a bounded parameter to be designed;

[0135] More specifically, the state constraints described 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 towards k b When, then V b,k (t) will tend to infinity. Therefore, to ensure that BLF is bounded, a control law needs to be designed to make the error e k (t) in the interval (-k) b ,k b Within the specified range; furthermore, this method ensures that the system state will not exceed the set upper bound x. max (t), thus satisfying the state constraint requirements. Specifically, the control input u in the k-th iteration... k Defined as:

[0137]

[0138] in The parameter γ = 1 / ρ min The estimated value, ρ min α is the lower bound for multiplicative faults. 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 parameters are... The settings are as follows:

[0143]

[0144]

[0145] and

[0146]

[0147] in This refers to the parameters before the batch length determination. T k The length of the batch in the k-th run is Γ, and λ is the gain term to be designed.

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

[0149]

[0150] Among them W j,min W j,max These are the upper and lower bounds of the parameter estimation, 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 It is a parameter The upper and lower bounds of the estimate can be guaranteed not to diverge using the projection rule;

[0151] S43. Considering the variation in batch length, for t∈(T) d ,T N This part is redundant output and has no effect on the system's learning; therefore, this batch can be discarded. Thus, we only need to consider t∈(T) min ,T d The output of this 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] In the formula υ k Let be a Bernoulli distributed random variable, taking the value 0 or 1.

[0156] In this embodiment, the designed adaptive iterative learning fault-tolerant algorithm mainly consists of α k Dominant, α k Then the feedback item Ke k (t) and compensation terms Control; the main function of the feedback term is to ensure the stability of the system and enhance its robustness, while the compensation term offsets the effects 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 a controller based on a control law (i.e., control input u). k The execution mechanism, consisting of [missing information], is used to ensure that the system state can effectively track the desired trajectory even under conditions of actuator failure, disturbance, uncertainty, and state constraints. This controller, as a specific implementation of the "adaptive iterative learning fault-tolerant algorithm," calculates the control input u online based on the dynamic and estimation model of the state error. k This invention ensures effective tracking and convergence control of the reference trajectory even in the presence of additive and multiplicative faults, model uncertainties, and disturbances. Therefore, the adaptive fault-tolerant controller designed in this invention is not an independent module, but rather a core component of the "adaptive iterative learning fault-tolerant algorithm," embodied in specific control law formulas, and possesses algorithm-driven control implementation characteristics.

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

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

[0160]

[0161] The `trace()` function is used to find the trace of a matrix. V ∈,k(t) is the barrier Lyapunov function of the dynamic error equation obtained in step S43, V W,k (t), V γ,k (t) represents 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] Due to variations in batch length, the above increments are in the range 0 < t ≤ T. k And T k <t≤T d They are different and need to be discussed separately for these two time periods:

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

[0166]

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

[0168]

[0169] α k Substituting into the above equation, we get:

[0170]

[0171] Further derivation yields:

[0172]

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

[0174]

[0175] From 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 Needs to be discussed again:

[0180] Based on the derivation in case 1, we can similarly obtain:

[0181]

[0182] From the formula of the composite energy function of the obstacle, combined with parameters The properties can be deduced as follows:

[0183]

[0184] Then it was introduced

[0185]

[0186] Similarly, it can be deduced that:

[0187]

[0188] Combined parameters From the properties, we obtain:

[0189]

[0190] Furthermore, we obtain:

[0191]

[0192] It can be deduced that:

[0193]

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

[0195]

[0196] If E0(t) is in [0,T] d ] is bounded, therefore, according to the above formula and E k The positive definiteness of (t) implies that when t∈[0,T] d ], Next, we prove that the boundedness of the analysis E0(t) can be derived from the formula for the barrier composite energy function:

[0197]

[0198] We will discuss this separately for different time periods t, when 0 < t ≤ T k Taking the derivative of E0(t), we know that:

[0199]

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

[0201]

[0202] The derivation of the right half of the above equation is as follows:

[0203]

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

[0205]

[0206] and then It can be rewritten as:

[0207]

[0208] We can conclude that E0(t) is bounded. Similarly, when T... k <t≤T d At that time, E0(t) is also bounded. Because E0(t) and E k The positive definiteness of (t) implies that:

[0209]

[0210] By the convergence principle of series, we can obtain:

[0211]

[0212] Therefore, we can obtain

[0213] Because t is variable, 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), based on the above, we can know that under this set... If t∈P b When setting a set, this 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 developments include:

[0214]

[0215] We can obtain when t∈P b When, we get:

[0216]

[0217] When t∈P b When p(t) > 0, it is obvious that υ exists in infinite iterations. k (t) = 1, further leading to... Ultimately, this ensured that throughout the entire batch of operations... This ensures that the tracking error e decreases as the number of iterations approaches infinity. k (t) converges to zero point by point. This verifies the stability and convergence of the designed adaptive iterative learning fault-tolerant algorithm.

[0218] This application also provides a simulation experiment to verify the performance of the proposed adaptive iterative learning fault-tolerant algorithm, demonstrating its practicality and effectiveness. Specifically, it includes using a single-link robotic arm as a simulation model to verify the fault-tolerant performance, robustness, and reliability of the proposed control algorithm under load disturbances, multiplicative actuator failures, and additive actuator failures. 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. It is the actuator input torque. (m) l The mass of the robotic arm is represented by M, g, l, and c, which represent the tip load, gravitational acceleration, length, and damping coefficient, respectively. Where J = Ml 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 / s². 2 Variation range T k Belongs to [3,6], expected to track trajectory x r (t) = 0.1*sin(3*t) + 0.3. This invention also considers constraint conditions, setting 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 changes and adopts an error tracking method, therefore the trajectory is reset to:

[0221]

[0222] The specific form of κ(t) is shown below:

[0223]

[0224] Where δ = 0.3. The designed adaptive iterative learning fault-tolerant algorithm is as follows:

[0225]

[0226] in φ and φ are the weight estimation and activation functions of the neural network, respectively, as shown in (12), and the specific parameters are shown in the table below:

[0227]

[0228] The simulation considered state-related disturbances and actuator faults. Actuator faults consist of multiplicative and additive faults, with multiplicative faults being 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-tolerant algorithm are... λ = 10. Figure 2 and Figure 3 The figures represent the position and velocity tracking curves for the 1st, 5th, 20th, and 40th iterations of the adaptive iterative learning fault-tolerant algorithm, respectively. Figure 4 The figure shows the error convergence curve of the adaptive iterative learning fault tolerance algorithm. As can be seen from the figure, the proposed adaptive iterative learning fault tolerance algorithm still has good robustness and tracking performance when facing state-related disturbances, changes in initial system values, changes in batch length, and actuator failures.

[0229] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of those different embodiments or examples.

[0230] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can 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 processor-included system or other system that can fetch and execute instructions from, an instruction execution system, apparatus or device).

[0231] The above embodiments provide a detailed description of the present invention. Specific examples have been used to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of the present invention. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. An adaptive fault-tolerant control method for a non-strictly repeating system based on echo state networks, characterized in that, Includes the following steps: S1. Construct a nonlinear dynamic system with actuator failure, and consider the initial state. Mismatch perturbations due to state dependencies Additive faults Multiplicative fault Set up assumptions; S2. Design an error tracking method, introduce the desired error trajectory into the error dynamic system, construct the dynamic error equation, and define the nonlinear function in the equation. This guides nonlinear dynamic systems to achieve stable tracking under conditions of inconsistent initial states and batch variations. S3. The parameter set of the constructed nonlinear system By inputting an echo-state network, and utilizing its approximation properties, the network's output is approximated to a nonlinear function. The dynamic error equation is updated to improve the adaptability of the nonlinear dynamic system to model uncertainties; step S3 specifically includes the following steps: S31. Define the echo state network equations and the parameter set of the nonlinear dynamic system. Input the echo state network and obtain the network output. ; S32. Utilizing the approximation properties of echo state networks, the nonlinear function of the dynamic error equation is... Approximation using the output of an echo-state network: ； in, For the ideal output weight matrix, the parameter set , For the activation function of the dynamic storage layer of the echo-state network, Let be any constant, and , yes The upper bound; S33. Based on the approximation result of step S32, update the dynamic error equation obtained in step S23: ； in There is an upper realm , This represents a mismatch perturbation in state dependencies. It is the number of iterations. Indicates the system status. Indicates a control input with a fault, where the superscript... Indicates a fault. It is the system's input matrix and is of full rank; S4. Based on the adaptive iterative learning fault-tolerant algorithm, design an adaptive fault-tolerant controller, and derive the control input by combining the obstacle Lyapunov function. And filter redundant batches, and redefine the dynamic error equation; 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: ； in For the bounded parameters to be designed, For time; S42. Design a control law to minimize the error. In the interval Within the range, then the first Control input for the next iteration Defined as: ； in , It is a parameter The estimated value, This is the lower bound for multiplicative faults. satisfy: ； in: ； ； ； in, It is the feedback gain matrix. Additive fault The upper realm, It is a symbolic function, with parameters , The settings are as follows: ； ； ； and ； ； ； in , This refers to the parameters before the batch length determination. , , It is the first Batch length for each run This is the expected batch length. , It is the gain term to be designed, where ; , The projection mechanism is defined as follows: ； ； ； in , These are the upper and lower bounds of the parameter estimation, respectively, for the parameter assumptions. ,in , ,and , It is a parameter The estimated upper and lower bounds; S43. Filter redundant batches and redefine the dynamic error equation obtained in step S33: ； in: ； In the formula Let be a Bernoulli distributed random variable, taking the value 0 or 1; S5. Based on the obstacle 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 adaptive fault-tolerant control method for a non-strictly repeating system based on an echo state network according to claim 1, characterized in that: Step S1 specifically includes the following steps: S11. Define a nonlinear dynamic system with actuator failure: ； in For time, It is the number of iterations. Represents the system state, satisfying , The upper bound of the system, Indicates a control input with a fault, where the superscript... Indicates faults, including additive faults. Multiplicative fault , It is the system's input matrix and is of full rank. Given a known nonlinear function, It is an unknown time-varying vector. This indicates a mismatch perturbation in state dependencies; S12, Regarding the initial state Set the assumptions, that is Bounded, but iteratively changing; S13. Mismatch perturbation of state dependency Set the assumptions, that is Norm is bounded; S14, Additive Faults Set the assumptions, that is There exists an upper bound, represented as: ； S15, Multiplicative fault Set the assumptions, that is There is a lower world ,and In the interval The above changes and ,Right now ,in This is the expected batch length.

3. The adaptive fault-tolerant control method for a non-strictly repeating system based on an echo state network according to claim 2, characterized in that: Step S2 specifically includes the following steps: S21. Define the error dynamic system: ； in, Indicates the current system status. Indicates the reference trajectory. Represents a known nonlinear function; S22. Define the expected error trajectory , ,satisfy: ； in yes The initial value for each iteration is designed with the expected error trajectory as follows: ； in It is a sufficiently small number to connect the starting position with the set time of the desired error trajectory. exist It is a monotonically decreasing function that satisfies , ; S23. Define the dynamic error equation based on the error dynamic system and the desired error trajectory: ； Nonlinear function This dynamic error equation is used to track the current system state to the reference trajectory.

4. The adaptive fault-tolerant control method for a non-strictly repeating system based on an echo state network according to claim 3, characterized in that: The ideal output weight matrix described in step S32 Specifically, this includes: employing online learning strategies in echo state networks to estimate... And it is optimized using an adaptive update algorithm.

5. The adaptive fault-tolerant control method for a non-strictly repeating system based on an echo state network according to claim 3, characterized in that: The state constraints mentioned in step S41 specifically include: the current system state. Must meet The state of the reference trajectory Must meet ,in It is the upper bound of the reference trajectory, and .

6. The adaptive fault-tolerant control method for a non-strictly repeating system based on an echo state network according to claim 3, characterized in that: Step S5 specifically includes: constructing the following barrier composite energy function: ； in The operation for solving the trace of a matrix, , , It is the barrier Lyapunov function of the dynamic error equation obtained in step S43. , That is in Error in parameter estimation under norm.

Images