DNN-based adaptive optimization control method for fractional order single-machine infinite bus system

Abstract

The invention discloses a DNN-based adaptive optimization control method for a fractional order single-machine infinite bus system, and relates to the field of single-machine infinite bus system control. The method comprises the following steps: establishing a dynamical model of a single-machine infinite bus system, and converting the dynamical model of the system into a state model; designing a function approximate DNN architecture, approaching an unknown function in the system, and designing a weight updating law based on a first-order Taylor series to reduce the mathematical difficulty; in the backstepping process, a virtual controller and an actual controller are constructed by utilizing an optimization backstepping technology, and the overall control optimization of the system is realized; an event trigger function is designed, and consumption of system communication resources is reduced; and carrying out Lyapunov analysis to ensure that each signal of the system is bounded. The adaptive optimization backstepping controller based on the DNN architecture is designed for a fractional order single-machine infinite bus system, unknown nonlinear terms in the system can be compensated, and a single-machine power angle tracks a given reference signal.

Description

Technical Field

[0001] This invention relates to the field of control technology for fractional-order single-machine infinite bus systems, and specifically to an adaptive control method for fractional-order single-machine infinite bus systems with uncertain single-machine power angles based on a DNN architecture. Background Technology

[0002] Fractional-order single-machine infinite bus systems, as one of the core models for power system stability analysis, can not only more accurately characterize the memory effect and non-exponential damping characteristics of generator rotors, but also provide a more reliable theoretical tool for system stability assessment in complex grid environments with large-scale integration of intermittent power sources such as wind and solar power. Research on fractional-order single-machine infinite bus systems has been widely applied in key areas such as subsynchronous oscillation suppression, transient stability enhancement, and smart grid adaptive control. The mathematical essence of these problems can be reduced to the stability and control problems of fractional-order nonlinear systems. However, in actual power system operation, due to factors such as measurement errors, time-varying equipment parameters, and random load fluctuations, it is difficult to obtain accurate system models and real-time state information. Therefore, research on robust control methods for fractional-order single-machine infinite bus systems facing model uncertainties and unknown disturbances has significant academic value and application prospects for improving the dynamic stability and anti-interference capabilities of power systems.

[0003] The key to solving the problem of stable power angle tracking in a single-unit power system lies in the accurate modeling and compensation of uncertain dynamics within the power system. Due to complex factors such as parameter perturbations, random load fluctuations, and intermittent power output from renewable energy sources in actual power grids, traditional linear control methods based on integer-order models struggle to guarantee accurate power angle tracking. Current technologies often approximate the unknown dynamics of a single-unit infinite bus system using shallow neural networks. However, deep neural networks (DNNs), with their powerful nonlinear mapping capabilities, can effectively reconstruct the unknown dynamics of power systems. Compared to shallow neural networks, DNNs, through multi-layer nonlinear transformations, can approximate complex nonlinear functions in fractional-order single-unit infinite bus systems with higher accuracy.

[0004] Therefore, it is urgent to design an adaptive controller based on deep neural networks to track the relative angle between equivalent generators in two regions and ensure that the power angle error eventually converges to a safe range, thereby ensuring the transient stability of the power system. Summary of the Invention

[0005] The purpose of this invention is to provide an adaptive control method for a fractional-order single-machine infinite bus system based on DNN. This method is designed for fractional-order single-machine infinite bus systems and achieves stable tracking of the relative angle between equivalent generators in two regions while allowing the existence of unknown nonlinear terms in the system. It also enables the tracking error of the system to converge to a stable range.

[0006] To achieve the above objectives, the technical solution of the present invention is as follows:

[0007] An adaptive control method for a fractional-order single-machine infinite system based on a DNN architecture, comprising the following steps:

[0008] S1. Establish a dynamic model of a single-machine infinite system and transform it into a state model;

[0009] S2. Design a DNN architecture, use the DNN architecture to approximate the unknown function, and design a weight update law;

[0010] S3. Construct virtual and actual controllers using optimized backstepping techniques;

[0011] S4. Design the event triggering function for the actual controller;

[0012] S5. Through Lyapunov analysis, it is found that all signals of the system are bounded.

[0013] Furthermore, step S1 specifically includes the following steps:

[0014] S11. Establish the dynamic model of the single-machine infinite system using formula (1):

[0015] (1);

[0016] in, This indicates finding the 0.95th derivative; The relative angle between the equivalent engines in the two regions; The angular frequency of the relative angle between the equivalent engines in the two regions; This represents the damping constant of the generator; The moment of inertia of the generator; This indicates the generator's maximum power. Indicates the mechanical power of the generator; The amplitude of a periodic power disturbance applied externally; For control input of a single-machine infinite bus system; and Represents an unknown function; A coefficient representing the running time; Runtime;

[0017] S12. Reconstruct the dynamic model and transform it into a state model, as shown in formula (2):

[0018] (2);

[0019] Among them, let the dynamic model in , ; and Both represent the system state of a single-machine infinite system; ; For the system output of a single-machine infinite system, and Both represent unknown functions; Indicates external disturbance, and , It is the upper bound of the external disturbance and is a known constant.

[0020] Furthermore, step S2 specifically includes the following steps:

[0021] S21. Design the DNN architecture, as shown in formula (3):

[0022] (3);

[0023] in, Represents the DNN architecture; Indicates function composition; , express Veuve space; Represents the original input vector. , express Veuve space; The concatenated vector representing the hidden layer weights of the DNN architecture is formed by concatenating and transposing the vectorized weight matrices of each hidden layer, and is used to represent the hidden layer weight information as a whole. , Represents a vectorized function; This represents the hidden layer weights of the DNN architecture. For dimension The matrix, , express Veuve space; Indicates the hidden layer index; Indicates the total number of hidden layers. Represents the set of positive integers; Indicates the DNN architecture's first The number of nodes in the layer; Represents the concatenated input vector. , Represents the original input vector The transpose of , where 1 is the bias term and is a constant. express Veuve space; Indicates the DNN architecture's first The smooth activation function vector of the layer, , express Veuve space, here, , Indicates the first Activation functions at each node;

[0024] S22. Using a DNN architecture to approximate unknown functions in a system. and ;

[0025] Step S22 specifically includes the following steps:

[0026] S221. Based on the generalized function approximation theorem, for any continuous function... The system is an unknown function, and it is determined that there exists an ideal weight that satisfies formula (4). and activation function vector This makes the approximation relationship valid;

[0027] (4);

[0028] in, Represents an unknown function in the system; This represents the ideal DNN output. Represents the input vector. Indicates the ideal weight; This represents the approximation error, and the error constraint satisfies... , Indicates the maximum upper bound. Indicates the upper bound of the error;

[0029] S222, Generate real-time adaptive weight estimation Using weight estimation Adaptive feedforward terms for generating DNN architecture ;

[0030] S223. Obtain the weight estimation error of the DNN architecture, and use formula (5) to approximate the hyperbolic tangent function through a first-order Taylor series:

[0031] (5);

[0032] in, This represents the ideal value for the DNN architecture; This represents the estimation of the DNN architecture; The derivative of the estimate of the DNN architecture; This represents the weight estimation error of the DNN architecture;

[0033] S224. Using formula (6), the estimated activation function vectors of each layer are obtained by recursion layer by layer, and the recursive relationship is determined:

[0034] (6);

[0035] in, Indicates the first Estimation of layer activation function vectors, Indicates the first Layer activation function vector, This represents the transpose of the weight estimate for layer 0. Represents the concatenated input vector. Indicates the first Transpose of layer weight estimation Indicates the first Estimation of layer activation function vectors;

[0036] S225, Based on the ideal value of the DNN architecture Estimation of DNN architecture And the aforementioned recursive relationship, which will be used to estimate the DNN architecture. Represented as And use equation (7) to determine the derivative of the estimated DNN architecture. :

[0037] (7);

[0038] in, Indicates the first Estimation of layer weights; Indicates the first The estimation of layer weights, here, ; Indicates the first Estimation of layer weights; Represents a vectorized function; Represents the gradient of the DNN architecture;

[0039] S226. By deriving the vectorized derivative, the gradient calculation of the DNN architecture is transformed into a matrix-vector operation that can be recursively applied layer by layer, and the specific form of the gradient is shown in formula (8):

[0040] (8);

[0041] in, Indicates the multiplication symbol; ; Here, ; Indicates the first The derivative of the layer activation function vector estimate; express 3D identity matrix; Indicates the Kronecker product; Represents the concatenated input vector;

[0042] S227. Based on the specific form of gradient, the error between the ideal output and the estimated output of the DNN architecture. We design a weight update law as shown in the following equation to approximate the unknown function:

[0043] ;

[0044] in, This represents the time derivative of the weight estimate; This represents the positive definite learning rate matrix; Indicates error.

[0045] Furthermore, step S3 specifically includes the following steps:

[0046] S31. Design a first-order performance index function as shown in formula (9):

[0047] (9);

[0048] in, This represents a first-order performance index function; Represents a time variable; Indicates runtime; Indicates tracking error. Indicates the actual state. Indicates the expected output; express Tracking error at any given moment; Represents the exponential decay coefficient. It is a positive number; Indicates a virtual controller;

[0049] S32. Use equation (10) to determine the first-order optimization performance index function of the DNN architecture:

[0050] (10);

[0051] in, The minimum performance metric among all feasible virtual controllers; Indicates the optimal virtual controller; Indicates compactness;

[0052] S33. Based on the first-order optimization performance index function, establish the first-order HJB equation as shown in formula (11):

[0053] (11);

[0054] in, Representing the first-order HJB equation, Represents the first-order performance index function pair The first-order partial derivative of the tracking error , Expressing the request Derivative order; Represents an unknown function. Indicates an unknown disturbance. express of First derivative, Indicates the expected output;

[0055] S34. Solving for the optimal virtual controller based on the first-order HJB equation. ;

[0056] S35, will Decomposition yields unknown functions and And based on the DNN architecture, for unknown functions and To approximate the representation, design the system's virtual controller;

[0057] S36. Construct the second-order optimization performance index function as shown in equation (16):

[0058] (16);

[0059] in, This represents the second-order optimization performance index. Indicates the second-order error. , Representing second order The square of the time error, Represents a second-order state variable. Represents the exponential decay coefficient. It is a positive number; This represents the optimal actual controller; express time; Indicates runtime; This represents the square of the actual controller.

[0060] S37. Based on the second-order optimization performance index function, the second-order HJB equation is established using formula (17):

[0061] (17);

[0062] in, Representing the second-order HJB equation, Represents the second-order performance index function pair The second-order partial derivative of the tracking error, , Represents an unknown function. Indicates an unknown disturbance. Virtual controller Derivative order;

[0063] S38, Yes Decomposition yields the unknown function and Based on the DNN architecture, for unknown functions and An approximate representation is made, and the actual controller of the system is designed.

[0064] Furthermore, step S4 specifically includes the following steps:

[0065] S41. Design the event triggering function of the actual controller using equation (22):

[0066] (twenty two);

[0067] in, Indicates intermediate variables. Indicates the event-triggered controller, This represents an estimate of the actual controller. This represents the controller's adjustment coefficient. This represents the sensitivity coefficient for adjusting the hyperbolic tangent function. Indicates the error threshold. This represents the error tolerance coefficient. Indicates runtime. , , , , and These represent two adjacent trigger times;

[0068] S42. Design the event triggering conditions for the actual controller, as shown in formula (23):

[0069] (twenty three);

[0070] in, Let represent the infimum function.

[0071] Furthermore, step S5 specifically includes the following steps:

[0072] S51. Construct the first candidate Lyapunov function, calculate the fractional derivative of the first candidate Lyapunov function, and design the first fractional adaptive law.

[0073] S52. A non-smooth analysis framework is used to perform variable substitution operations, and combined with the weight estimation error, the fractional derivative of the first Lyapunov function is simplified using Young's inequality.

[0074] S53. Construct a second candidate Lyapunov function, integrate the error term and weight estimation error under the event triggering mechanism, perform fractional derivative of the second candidate Lyapunov function, and design a second fractional adaptive law.

[0075] S54, Based on time-varying parameters Boundedness Using formula (34) for inequality scaling, the nonlinear term of the event-triggered controller is transformed into an analyzable bounded form, and formula (35) is used to determine the upper bound of the fractional derivative of the second candidate Lyapunov function:

[0076] (34);

[0077] in, Indicates intermediate variables. Indicates the second-order error. This represents the controller's adjustment coefficient. and Indicates two different time-varying parameters. Indicates the error threshold. Indicates runtime. , ;

[0078] (35);

[0079] in, Represents the second-order chosen Lyapunov function beg First derivative, Indicates error. This represents the sensitivity coefficient for adjusting the hyperbolic tangent function. Indicates the error threshold. This represents the error tolerance coefficient. This represents the controller's adjustment coefficient. Indicates approaching The ideal value of DNN, Indicates approaching The approximate error, Indicates a second-order perturbation. Represents approximating an unknown function The transpose of the difference between the ideal value of the weight vector and the estimated weight vector. Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The transpose of the difference between the ideal value of the weight vector and the estimated weight vector. Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents approximating an unknown function The estimated value of the weight vector, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, This represents an estimate of the actual controller;

[0080] S55, based on Using Young's inequality and formula (36), the stable form of the second candidate Lyapunov function is determined:

[0081] (35);

[0082] in, and , Indicates approaching The upper bound of higher-order terms, Indicates approaching The approximate error, Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the first-order perturbation. Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and It is a positive parameter of the design. Indicates the second-order error. Indicates the first-order control gain. Indicates the second-order control gain. Indicates approaching Ideal DNN, Indicates approaching Ideal DNN;

[0083] S56. By aggregating the first and second candidate Lyapunov functions, a global Lyapunov function is constructed, and the upper bound of its fractional derivative is derived to obtain the boundedness of all signals in the system.

[0084] Furthermore, step S34 specifically includes the following steps:

[0085] S341. Using the optimality conditions of the first-order HJB equation, for about Find the partial derivative and set it to 0:

[0086] ;

[0087] in, Representing the first-order HJB equation, Indicates tracking error. This represents the optimal virtual controller. Represents the first-order performance index function pair The first-order partial derivative of the tracking error;

[0088] S342. The optimal virtual controller is obtained by calculating using the following formula. Analytical solution:

[0089] (12);

[0090] Step S35 specifically includes the following steps:

[0091] S351, Using the following formula break down:

[0092] ,

[0093] in, Indicates control gain. ; Indicates tracking error; and Represents an unknown function. , , Represents a first-order state variable; and Indicates the input to the neural network; Represents an unknown nonlinear function; express of Derivative order; ; ;

[0094] S352. Based on the DNN architecture, the unknown function of the system is approximated using equations (13) and (14):

[0095] (13);

[0096] (14);

[0097] in, , , and They represent and The ideal value of the DNN architecture, and They represent approximation. and Ideal weights, and Indicates the approximation error;

[0098] S353. Based on the approximate representation of the unknown function, the virtual controller of the system is designed as shown in formula (15):

[0099] (15);

[0100] in, This represents an estimate of the virtual controller. for The estimated value, Indicates approaching DNN architecture, for The estimated value, Indicates approaching The DNN architecture;

[0101] Step S38 specifically includes the following steps:

[0102] S381, through solving The optimal actual controller is obtained as shown in formula (18):

[0103] (18);

[0104] S382, Using the following formula to... Decompose:

[0105] ;

[0106] in, , , and ; This represents the second-order control gain and is a positive number. Indicates the second-order error. and Represents an unknown function. and This represents the input to the neural network. It is a virtual controller;

[0107] S383. Based on the DNN architecture, the unknown function is approximated using equations (19) and (20):

[0108] (19);

[0109] (20);

[0110] in, , , and They represent approximation. and Ideal weights, and Indicates approaching and Approximate error;

[0111] S384. Based on the approximate representation of the unknown function, the actual controller of the system is designed as shown in formula (21):

[0112] (twenty one);

[0113] in, This represents an estimate of the actual controller. This represents the second-order control gain and is a positive number. Indicates the second-order error. for The estimated value, for The estimated value, Indicates approaching DNN architecture, Indicates approaching The DNN architecture.

[0114] Furthermore, step S51 specifically includes the following steps:

[0115] S511. Construct the first candidate Lyapunov function as shown in formula (24):

[0116] ;

[0117] in, Represents a first-order Lyapunov function; Indicates tracking error; Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. , Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function Weight vector estimation; Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. , Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function Weight vector estimation, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and All are positive numbers;

[0118] S512. Using formula (25), perform fractional derivative of the first candidate Lyapunov function:

[0119] ;

[0120] in, Expressing the request First derivative, Indicates the second-order error. Indicates a virtual controller. Represents an unknown function. This represents a first-order unknown perturbation. This represents taking the q-th derivative with respect to y;

[0121] S513. Design the first fractional adaptive law using formula (26):

[0122] ;

[0123] Among them, the operator Indicates the projection algorithm; Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents an unknown function. express The adjustment rate;

[0124] Step S52 specifically includes the following steps:

[0125] S521. Using a non-smooth analysis framework, the fractional derivative of the first candidate Lyapunov function is substituted using formula (27):

[0126] (27);

[0127] in, Indicates a first-order perturbation. Indicates the first-order control gain. express The estimate, Indicates approaching Ideal DNN, express The estimate, Indicates approaching Ideal DNN;

[0128] S522, Order , Transforming formula (28), we get:

[0129] (28);

[0130] in, Indicates approaching higher-order terms, Indicates approaching Higher-order terms;

[0131] S523. Using Young's inequality, simplify formula (29):

[0132] (29);

[0133] in, , , Indicates approaching The upper bound of higher-order terms, Indicates approaching Approximation error of higher-order terms, Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the first-order perturbation. This represents the sum of all constant terms in formula (29);

[0134] Step S53 specifically includes the following steps:

[0135] S531. Construct the second candidate Lyapunov function as shown in formula (30):

[0136] (30);

[0137] in, This represents a second-order chosen Lyapunov function. Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, , , express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and All are positive parameters;

[0138] S532, Define intermediate variables Constructing intermediate variables With event triggering controller Mapping relationship:

[0139] ;

[0140] in, Indicates time-varying parameters, ; Indicates intermediate variables; Indicates the event-triggered controller; This represents the controller's adjustment coefficient. Indicates the error threshold. , ;

[0141] S533. Determine the event trigger controller as shown in formula (31) through equation transformation:

[0142] (31);

[0143] S534. Using formula (32) to evaluate the second candidate Lyapunov function Perform fractional derivatives:

[0144] (32);

[0145] S535. Design a second fractional adaptive law using formula (33):

[0146] (33);

[0147] in, Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents an unknown function. Indicates approaching The estimated derivative of the DNN architecture is transposed;

[0148] Step S56 specifically includes the following steps:

[0149] S561. Based on the first and second candidate Lyapunov functions, construct the global Lyapunov function using formula (37). :

[0150] (37);

[0151] S562. Using formula (38) to analyze the global Lyapunov function Perform fractional derivative calculations and determine the upper bound of its fractional derivative:

[0152] (38);

[0153] in, and , , , Indicates approaching The upper bound of higher-order terms, Indicates approaching The approximate error, Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the second-order perturbation. Indicates the first-order control gain. Indicates the second-order control gain;

[0154] S57, By analyzing the event trigger controller With intermediate variables Given the boundedness of the fractional derivative, and considering the triggering condition, the lower bound of the event triggering interval can be determined using formula (39):

[0155] .

[0156] Beneficial effects

[0157] Compared with existing technologies, this invention is designed for fractional-order single-machine infinite-scale systems. It employs a fractional-order system adaptive control architecture based on deep neural networks (DNNs), combining DNNs with fractional-order control theory for the first time to construct an adaptive control framework suitable for single-machine infinite-scale systems. This invention uses a multi-layer fully connected neural network structure and innovatively designs a weight update law incorporating fractional-order differential operators, thereby overcoming the limitations of traditional linear control methods in handling the nonlinear dynamic characteristics of the system. In particular, it solves the problem of compensating for unmodeled dynamics and parameter uncertainties in fractional-order systems, thus achieving stable tracking of the relative angle between equivalent generators in two regions. Simultaneously, this invention uses a distributed event triggering mechanism, proposing an adaptive event triggering condition based on Lyapunov stability. It innovatively establishes a dynamic correlation between the triggering threshold and the DNN approximation error, effectively reducing the communication resource waste caused by traditional time-triggered control, resolving the contradiction between control updates and stability in sampled data systems, reducing the number of triggers, and thus reducing system resource consumption. Attached Figure Description

[0158] Figure 1 This is a flowchart of the adaptive control method for a fractional-order single-machine infinite system based on a DNN architecture in this invention.

[0159] Figure 2 This is a schematic diagram of the structure of a single-machine infinite bus system;

[0160] Figure 3 The control flowchart for a single-machine infinite bus system;

[0161] Figure 4 For the actual desired signal of the system Expected signal and the system's output signal A curve graph;

[0162] Figure 5 For the system Tracking error A curve graph;

[0163] Figure 6 For system control input A curve graph;

[0164] Figure 7 This is a diagram illustrating the event trigger interval. Detailed Implementation

[0165] To provide a better understanding of the structural features and effects achieved by the present invention, a detailed description is provided below, accompanied by preferred embodiments and accompanying drawings:

[0166] This invention addresses, for example Figure 2 The single-machine infinite bus system shown is implemented to track and control the relative angle between equivalent generators in two uncertain regions. For example... Figure 3 As shown, the adaptive control method for a fractional-order single-machine infinite system based on a DNN architecture, as described in this invention, includes the following steps:

[0167] S1. Establish a dynamic model of a single-machine infinite system and transform the system dynamic model into a state model.

[0168] Step S1 specifically includes the following steps:

[0169] S11. Establish the dynamic model of the single-machine infinite system using formula (1):

[0170] (1);

[0171] in, This indicates finding the 0.95th derivative; The relative angle between the equivalent engines in the two regions; The angular frequency of the relative angle between the equivalent engines in the two regions; This represents the damping constant of the generator; The moment of inertia of the generator; This indicates the generator's maximum power. Indicates the mechanical power of the generator; The amplitude of a periodic power disturbance applied externally; For control input of a single-machine infinite bus system; and Represents an unknown function; A coefficient representing the running time; Operating time. Externally imposed periodic power disturbances, such as load fluctuations or power oscillations caused by intermittent grid connection of renewable energy sources.

[0172] S12. Reconstruct the dynamic model and transform it into a state model;

[0173] Step S12 specifically includes the following steps:

[0174] make , The dynamic model of the single-machine infinite system is reconstructed to obtain the state model shown in formula (2):

[0175] (2);

[0176] in, and Both represent the system state of a single-machine infinite system; ; For the system output of a single-machine infinite system, and Both represent unknown functions; Indicates external disturbance, and ,in, It is the upper bound of the external disturbance and is a known constant.

[0177] S2. Design a DNN architecture for approximating functions. Use the DNN architecture to approximate unknown functions in the system, and design a weight update law based on the first-order Taylor series to reduce mathematical difficulty.

[0178] Step S2 specifically includes the following steps:

[0179] S21. Design the DNN architecture, as shown in formula (3):

[0180] (3);

[0181] in, This represents the DNN architecture. Represents function composition. , express Veuve space; Represents the original input vector. , express Veuve space; Represents the hidden layer weights of the DNN architecture The concatenated vector, formed by concatenating and transposing the vectorized weight matrices of each hidden layer, is used to represent the overall weight information of the hidden layers. , Represents a vectorized function. This represents the hidden layer weights of the DNN architecture. For dimension The matrix, , express European-style space, This represents the hidden layer index, used to distinguish different hidden layers. express subscript, using Express arrive , Indicates the total number of hidden layers. Represents the set of positive integers; Indicates the DNN architecture's first The number of nodes in the layer, and express subscript; Represents the concatenated input vector. , Represents the original input vector transpose, The 1 in the equation is a bias term, which is a constant. express Veuve space; Indicates the DNN architecture's first The smooth activation function vector of the layer is used to represent the DNN architecture's first layer. The overall vector of activation functions of each neuron in the layer, integrating the activation function information of each node in that layer. , express European-style space, express subscript, Indicates the first Activation functions at each node.

[0182] S22. Using a DNN architecture to approximate unknown functions in a system. and .

[0183] Step S22 specifically includes the following steps:

[0184] S221. Based on the generalized function approximation theorem, for any continuous function... The system is an unknown function, and it is determined that there exists an ideal weight that satisfies formula (4). and activation function vector This makes the approximation relationship valid.

[0185] (4);

[0186] in, Represents an unknown function in the system; Represents the ideal DNN output, using ideal weights. and activation function vector DNN, for input vector Perform transformation and fit ; Represents the input vector; Indicates the ideal weight; This represents the approximation error, and the error constraint satisfies... , Indicates the maximum upper bound. Indicates the upper bound of the error.

[0187] Specifically, Represents a set A continuous function space on the set, i.e., all functions defined on the set The set of continuous functions on, Indicates compactness, , express The function space of the DNN architecture is in Euclidean space. According to the generalized function approximation theorem, the function space is in Euclidean space. The middle is dense. Therefore, for any given continuous function... and any given error requirement There must exist an ideal set of weights. and activation function vector Let the DNN output and The error satisfies .in, , Indicates the first Layer weight matrix.

[0188] S222, Generate real-time adaptive weight estimation Using weight estimation Adaptive feedforward terms for generating DNN architecture .

[0189] Due to the ideal weights of DNN Since the values ​​are unknown and cannot be used directly, an approximate weight needs to be estimated to replace the ideal weight. The real-time adaptive weight estimation is as follows: , Indicates the first The layer weight estimation matrix, Indicates the first The layer weight estimation matrix, where, Indicates the first The layer weight estimation matrix, express In Euclidean space, the weight estimates are generated for developing DNN-based adaptive feedforward terms. .

[0190] S223. Obtain the weight estimation error of the DNN architecture, and use formula (5) to approximate the hyperbolic tangent function through a first-order Taylor series:

[0191] (5);

[0192] in, This represents the ideal value for the DNN architecture. ; This represents the estimation of the DNN architecture. ; The derivative of the estimate of the DNN architecture; This represents the weight estimation error of the DNN architecture. ; Represents higher-order terms. , Indicates about The higher-order terms of the second-order infinitesimal order. The higher-order terms in the Taylor expansion are bounded and have constants. , making ,and .

[0193] S224. Using formula (6), the estimated activation function vectors of each layer are obtained by recursion layer by layer, and the recursive relationship is determined:

[0194] (6);

[0195] in, Indicates the first Estimation of layer activation function vectors, Indicates the first Layer activation function vector, This represents the transpose of the weight estimate for layer 0. Represents the concatenated input vector. Indicates the first Transpose of layer weight estimation Indicates the first Estimation of layer activation function vectors.

[0196] S225, Based on the ideal value of the DNN architecture Estimation of DNN architecture And the aforementioned recursive relationship, which will be used to estimate the DNN architecture. Represented as And use equation (7) to determine the derivative of the estimated DNN architecture. :

[0197] (7);

[0198] in, The derivative of the estimate for the DNN architecture. This represents the estimation of the DNN architecture. Indicates the first Estimation of layer weights, Indicates the first Estimation of layer weights, express subscript; Indicates the first Estimation of layer weights, This represents a vectorized function.

[0199] In step S225, the overall activation function is estimated through recursion. Use the product of inter-layer weights and the activation of the previous layer. This indicates that hierarchical relationships are established; formula (7) is used to construct the activation function estimate. Vectorization of weights at each layer partial derivative matrix This is used for subsequent weight updates or sensitivity analysis to clarify the response pattern of the activation function to weight changes.

[0200] S226. By deriving vectorized derivatives, the gradient calculation of the DNN architecture is transformed into a matrix-vector operation that can be recursively applied layer by layer.

[0201] Step S226 specifically includes the following steps:

[0202] Using the properties of vectorized operators, the chain derivative rule, and derivative relations, the gradient in formula (7) is calculated through a series of multiplications and the Kronecker product. The specific form.

[0203] The derivative relationship is as follows:

[0204] and ;

[0205] in, , express 3D identity matrix Indicates the Kronecker product. Represents the concatenated input vector. Indicates the first Estimation of layer activation function vectors.

[0206] The gradient in formula (7) The specific form is shown in formula (8):

[0207] (8);

[0208] in, This represents the estimation of the DNN architecture; Indicates the first Estimation of layer weights; Indicates the multiplication symbol; ; , ; Indicates the first The derivative of the layer activation function vector estimate; express 3D identity matrix; Indicates the Kronecker product; This represents the cascaded input vector.

[0209] S227. Using the gradient derived in step S226 The error between the ideal output and the estimated output of the DNN architecture. We design a weight update law as shown in the following equation to approximate the unknown function:

[0210] ;

[0211] in, This represents the time derivative of the weight estimate, i.e., the update rate of the weight estimate; This represents the positive definite learning rate matrix, used to control the update speed; This represents the error and is used to drive the weights to be updated in a direction that reduces the error. This represents the gradient, used to quantify the sensitivity of the DNN output to the weights.

[0212] Step S227 involves using the error between the ideal output and the estimated output of the DNN architecture (i.e., the estimation deviation of the unknown function) and the gradient information of the output on the weights to drive the weight parameters of the neural network to be iteratively adjusted continuously, so that the estimated weights gradually approach the ideal weights, and finally reduce the error between the estimated output and the true output of the unknown function, thereby achieving effective approximation of the unknown function.

[0213] S3. Construct virtual and actual controllers using optimized backstepping techniques to achieve overall system control optimization.

[0214] Step S3 specifically includes the following steps:

[0215] S31. Design a first-order performance index function as shown in formula (9):

[0216] (9);

[0217] in, This represents a first-order performance index function; Represents a time variable; Indicates runtime; Indicates tracking error. express Tracking error at any time, Indicates the actual state. Indicates the expected output; Represents the exponential decay coefficient. It is a positive number; This refers to a virtual controller.

[0218] Step S31 defines the optimization objective of the first-order subsystem through formula (9), integrates tracking error and control cost, provides a mathematical basis for the design of the virtual controller for back-propagation control, and achieves a balance between small error and stable control.

[0219] S32. Use equation (10) to determine the first-order optimization performance index function of the DNN architecture:

[0220] (10);

[0221] in, The minimum performance metric among all feasible virtual controllers; Indicates the optimal virtual controller; It indicates compactness.

[0222] Step S32 defines the optimal performance index of the first-order subsystem. The optimal virtual controller is found through optimization and compact constraints, and an engineering approximation is achieved with the help of DNN.

[0223] S33. To optimize the performance of the algorithm, fractional HJB equations are used to solve for the optimal virtual controller.

[0224] Based on the first-order optimization performance index function, a first-order HJB equation is established, as shown in formula (11):

[0225] (11);

[0226] in, Representing the first-order HJB equation, Indicates tracking error. This represents the optimal virtual controller. Represents the first-order performance index function pair The first-order partial derivative of the tracking error , Expressing the request Derivative order; Represents an unknown nonlinear function. Indicates an unknown disturbance. express of First derivative, This indicates the system output.

[0227] S34. Solving for the optimal virtual controller based on the first-order HJB equation. Optimal Virtual Controller It is the only solution that satisfies the HJB equation.

[0228] Step S34 specifically includes the following steps:

[0229] S341. Using the optimality conditions of the first-order HJB equation, for about Find the partial derivative and set it to 0:

[0230] ;

[0231] in, Representing the first-order HJB equation, Indicates tracking error. This represents the optimal virtual controller. Represents the first-order performance index function pair The first-order partial derivative of the tracking error;

[0232] S342. The optimal virtual controller is obtained by calculating using the following formula. Analytical solution:

[0233] (12).

[0234] S35, will Decomposition yields unknown functions and Based on the DNN architecture, for unknown functions and An approximate representation is made, and a virtual controller for the system is designed.

[0235] Step S35 specifically includes the following steps:

[0236] S351, because Since it cannot be solved directly, in order to achieve the control objective, the following formula is used: break down:

[0237] ,

[0238] in, Indicates the first-order control gain. ; Indicates tracking error; and Represents an unknown function. , , Represents a first-order state variable; and Indicates the input to the neural network; Represents an unknown nonlinear function; express of Derivative order; ; .

[0239] S352, Based on DNN architecture, utilizing formula Japanese style An approximate representation of the unknown function of the system:

[0240] (13);

[0241] (14);

[0242] in, , , and They represent and The ideal value of the DNN architecture, and They represent approximation. and Ideal weights, and This indicates the approximation error.

[0243] S353. Based on the approximate representation of the unknown function, design the virtual controller of the system as shown in the formula. As shown:

[0244] (15);

[0245] in, This represents an estimate of the virtual controller. for The estimated value, Indicates approaching DNN architecture, for The estimated value, Indicates approaching The DNN architecture.

[0246] S36. Construct the second-order optimization performance index function as shown in equation (16):

[0247] (16);

[0248] in, This represents the second-order optimization performance index. Indicates the second-order error. , Representing second order The square of the time error, Represents a second-order state variable. Represents the exponential decay coefficient. It is a positive number; This represents the optimal actual controller; express time, Indicates runtime. This represents the square of the actual controller.

[0249] S37. Based on the second-order optimization performance index function, the second-order HJB equation is established using formula (17):

[0250] (17);

[0251] in, Representing the second-order HJB equation, Indicates the second-order error. This represents the optimal actual controller. Represents the second-order performance index function pair The second-order partial derivative of the tracking error, , Represents an unknown nonlinear function. Indicates an unknown disturbance. Virtual controller The first derivative.

[0252] S38, Yes Decomposition yields the unknown function and Based on the DNN architecture, for unknown functions and An approximate representation is made, and the actual controller of the system is designed.

[0253] Step S38 specifically includes the following steps:

[0254] S381, through solving The optimal actual controller is obtained as shown in equation (18):

[0255] (18);

[0256] S382, Using the following formula to... Decompose:

[0257] ;

[0258] in, , , and ; It is a second-order control gain and is positive. Indicates the second-order error. and Represents an unknown function. and This represents the input to the neural network. It is a virtual controller.

[0259] S383. Based on the DNN architecture, the unknown function is approximated using equations (19) and (20):

[0260] (19);

[0261] (20);

[0262] in, , , and They represent approximation. and Ideal weights, and Indicates approaching and Approximate error;

[0263] S384. Based on the approximate representation of the unknown function, the actual controller of the system is designed as shown in formula (21):

[0264] (twenty one);

[0265] in, This represents an estimate of the actual controller. It is a second-order control gain and is positive. Indicates the second-order error. for The estimated value, for The estimated value, Indicates approaching DNN architecture, Indicates approaching The DNN architecture.

[0266] S4. Design the event trigger function of the actual controller of the system to reduce the consumption of system communication resources.

[0267] Step S4 specifically includes the following steps:

[0268] S41. In order to reduce the consumption of system communication resources, it is necessary to use event-triggered functions in the actual controller design process.

[0269] Design the event trigger function of the actual controller using equation (22):

[0270] (twenty two);

[0271] in, Indicates intermediate variables. Indicates the event-triggered controller, This represents an estimate of the actual controller. This represents the controller's adjustment coefficient. This represents the sensitivity coefficient for adjusting the hyperbolic tangent function. Indicates the error threshold. This represents the error tolerance coefficient. Indicates runtime. , , , , and This indicates two adjacent trigger times.

[0272] S42. Design the event triggering conditions for the actual controller, as shown in formula (23):

[0273] (twenty three);

[0274] in, Denotes the infimum function. This indicates the error threshold.

[0275] S5. Perform Lyapunov analysis on the tracking error and weight estimation error to ensure that each signal in the system is bounded.

[0276] Step S5 specifically includes the following steps:

[0277] S51. Construct the first candidate Lyapunov function, calculate the fractional derivative of the first candidate Lyapunov function, and design the first fractional adaptive law.

[0278] Step S51 specifically includes the following steps:

[0279] S511. Construct the first candidate Lyapunov function as shown in formula (24):

[0280] (twenty four);

[0281] in, Represents a first-order Lyapunov function; Indicates tracking error; Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. , Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function Weight vector estimation; Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. , Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function Weight vector estimation, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and All are positive numbers.

[0282] S512. Using formula (25), perform fractional derivative of the first candidate Lyapunov function:

[0283] (25);

[0284] in, Expressing the request First derivative, Indicates tracking error. Indicates the second-order error. Indicates a virtual controller. Represents an unknown function. This represents a first-order unknown perturbation. This represents finding the q-th derivative with respect to y. express learning rate, express learning rate, and All are positive numbers. Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Formula (25) is the difference between the ideal value of the weight vector and the estimated weight vector. Regarding time requests Derivative.

[0285] S513. Design the first fractional adaptive law using formula (26):

[0286] (26);

[0287] Among them, the operator Indicates the projection algorithm; Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents a first-order unknown function. express The adjustment rate, Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents a second-order unknown function. express The adjustment rate.

[0288] The first row in formula (26) The adaptive law, the second action The adaptive law. Formula Involving The item must be declared. For all , Indicates transpose. and Established, among which, The parameters representing the adjustment of the adaptive law, This represents the parameter update law. Describes a positive definite matrix. express 3D Euclidean space, express Euclidean space of 3D.

[0289] Step S51 is used to implement fractional-order adaptive control. Multiple errors are fused using Lyapunov functions, and the system's stability under unknown disturbances and model uncertainties is ensured using fractional derivatives and projective adaptive laws. Step S51 uses... The operator adapts to the fractional-order dynamic characteristics of power systems (such as synchronous generators), while considering both tracking error and weighted approximation error, ensuring synergy between control performance and DNN approximation accuracy. Projection operator... This is used to avoid weight divergence and make the algorithm more reliable.

[0290] S52. Using a non-smooth analysis framework, variable substitution is performed, and combined with weight estimation error, the fractional derivative of the first Lyapunov function is simplified using Young's inequality.

[0291] Step S52 specifically includes the following steps:

[0292] S521. Using a non-smooth analysis framework, perform variable substitution operations on the fractional derivatives of the first candidate Lyapunov function:

[0293] (27);

[0294] in, Indicates a first-order perturbation. Indicates the first-order control gain. express The estimate, Indicates approaching Ideal DNN, express The estimate, Indicates approaching Ideal DNN, Represents an unknown function. express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and All are positive numbers.

[0295] S522, Order , Transforming formula (27), we get:

[0296] (28);

[0297] in, Indicates approaching Ideal DNN, Indicates approaching Ideal DNN, Indicates approaching higher-order terms, Indicates approaching The higher-order term.

[0298] S523. Using Young's inequality, the formula... Simplify:

[0299] (29);

[0300] in, , . Indicates approaching The upper bound of higher-order terms, Indicates approaching Approximation error of higher-order terms, Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the first-order perturbation. express The adjustment rate, express learning rate, express The adjustment rate, This represents the sum of all constant terms in formula (29).

[0301] Step S52, based on the projection operator properties established in Step S51 (satisfying positive definite matrix constraints for all weight estimation errors), uses a non-smooth analysis framework to substitute variables, transforming the fractional derivative expression of the Lyapunov function into a form containing tracking error, second-order error, weight estimation error, perturbation, and higher-order terms. In Steps S521 and S522, by substituting variables and utilizing the definition of weight estimation error, the expression is simplified to include the ideal DNN output, estimated output, and higher-order terms. In Step S523, Yang's inequality is applied to scale the cross-terms, transforming the complex nonlinear terms into a simplified form of negative definite error terms and bounded constant terms, where the constant terms include upper bounds on perturbation, higher-order terms, and approximation errors. Step S52 provides a simplified mathematical basis for system stability analysis, clarifying the quantitative relationship between tracking error, weight estimation error, perturbation, and higher-order terms, supporting the subsequent progressive proof of global stability.

[0302] S53. Construct a second candidate Lyapunov function, integrate the error term and weight estimation error under the event triggering mechanism, perform fractional derivative of the second candidate Lyapunov function, and design a second fractional adaptive law.

[0303] Step S53 specifically includes the following steps:

[0304] S531. Construct the second candidate Lyapunov function as shown in formula (30):

[0305] (30);

[0306] in, This represents a second-order chosen Lyapunov function. Indicates the second-order error. Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, , , express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and All are positive parameters.

[0307] S532, Define intermediate variables Constructing intermediate variables With event triggering controller Mapping relationship:

[0308] ;

[0309] in, Indicates time-varying parameters, ; Indicates intermediate variables; Indicates the event-triggered controller; This represents the controller's adjustment coefficient. This indicates the error threshold. , .

[0310] S533. Determine the event trigger controller as shown in formula (31) through equation transformation:

[0311] (31);

[0312] S534. Using formula (32) to evaluate the second candidate Lyapunov function Perform fractional derivatives:

[0313] (32);

[0314] in, This indicates finding the q-th derivative.

[0315] S535. Design a second fractional adaptive law using formula (33):

[0316] (33);

[0317] in, Represents the projection operator. Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents an unknown function. Indicates approaching The estimated derivative transpose of the DNN architecture. Represents an unknown function. express learning rate, express The adjustment rate, express learning rate, express The adjustment rate.

[0318] S54, Utilizing Time-Varying Parameters Boundedness Inequality scaling is performed to transform the nonlinear terms of the event-triggered controller into an analyzable bounded form, and the upper bound of the fractional derivative of the second candidate Lyapunov function is determined.

[0319] The inequality scaling is shown in formula (34):

[0320] (34);

[0321] in, Indicates intermediate variables. Indicates the second-order error. This represents the controller's adjustment coefficient. and Indicates two different time-varying parameters. Indicates the error threshold. Indicates runtime. , .

[0322] Based on the above analysis, according to the formula Combined with formula and formula The formula can be Convert to formula That is, the upper bound of the fractional derivative of the second candidate Lyapunov function.

[0323] The upper bound of the fractional derivative of the second candidate Lyapunov function is shown in formula (35):

[0324] (35);

[0325] in, Represents the second-order chosen Lyapunov function beg First derivative, Indicates error. This represents the sensitivity coefficient for adjusting the hyperbolic tangent function. Indicates the error threshold. This represents the error tolerance coefficient. This represents the controller's adjustment coefficient. Indicates approaching The ideal value of DNN, Indicates approaching The approximate error, Indicates a second-order perturbation. Represents approximating an unknown function The transpose of the difference between the ideal value of the weight vector and the estimated weight vector. Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The transpose of the difference between the ideal value of the weight vector and the estimated weight vector. Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents approximating an unknown function The estimated value of the weight vector, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, This represents an estimate of the actual controller.

[0326] S55, based on Using Young's inequality and formula (36), the stable form of the second candidate Lyapunov function is determined:

[0327] (36);

[0328] in, and , Indicates approaching The upper bound of higher-order terms, Indicates approaching The approximate error, Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the second-order perturbation. This represents the sensitivity coefficient for adjusting the hyperbolic tangent function. , Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and It is a positive parameter of the design. Indicates the second-order error. Indicates the first-order control gain. Indicates the second-order control gain. Indicates approaching Ideal DNN, Indicates approaching The ideal DNN.

[0329] S56. By aggregating the first and second candidate Lyapunov functions, construct the global Lyapunov function and derive the upper bound of its fractional derivative, verifying that the system satisfies either input-state stability or global uniform eventual boundedness.

[0330] Step S56 specifically includes the following steps:

[0331] S561. Based on the first and second candidate Lyapunov functions, construct the global Lyapunov function using formula (37). :

[0332] (37);

[0333] S562. Using formula (38) to analyze the global Lyapunov function Perform fractional derivative calculations and determine the upper bound of its fractional derivative:

[0334] (38);

[0335] in, and , , Indicates approaching The upper bound of higher-order terms, Indicates approaching The approximate error, Indicates approaching The upper bound of higher-order terms, Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector; Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, This represents the upper bound of the first-order perturbation. , Indicates approaching The upper bound of higher-order terms, Indicates approaching The approximate error, This represents the sensitivity coefficient for adjusting the hyperbolic tangent function. , Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the first-order perturbation. Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and It is a positive parameter of the design. Indicates tracking error. Indicates the second-order error. Indicates the first-order control gain. Indicates the second-order control gain. Indicates approaching Ideal DNN, Indicates approaching The ideal DNN.

[0336] Step S562 Calculate the q-th order fractional derivative, and combine the results of the layered derivatives from steps S51 to S55 to perform negative definite term aggregation and bounded term merging.

[0337] Based on the above analysis of the Lyapunov equations, formula (38) shows that it satisfies the Lyapunov stability theorem, thus yielding the optimal controller. The second-order optimization performance index function shown in formula (16) is satisfied, thus achieving optimal control. Based on the constructed... and Two Lyapunov functions, and in The formula proves that these two formulas are stable, and the conclusion can be easily drawn from the formula. and The variables designed in the code are all bounded, that is... , , , , and Both are bounded.

[0338] S57, By analyzing the event trigger controller With intermediate variables The fractional derivative of the event is bounded, and combined with the triggering condition, the event triggering interval is determined to have a lower bound.

[0339] (39);

[0340] Based on the boundedness proof and formula (22), the result shows that... Keep it bounded, and arrive It is differentiable of order 1. Therefore, it has a positive constant. Make Furthermore, given the conditions and Therefore, the event triggering interval satisfies Therefore, it can be concluded that Zeno behavior was effectively avoided.

[0341] Step S1: Establish a dynamic model of the single-machine infinite bus system and transform the system dynamic model into a state model. Step S2: Design a function approximation DNN architecture to approximate the unknown functions in the system, and design a weight update law based on first-order Taylor series to reduce mathematical complexity. Step S3: Use backstepping control, constructing a virtual controller and an actual controller using optimized backstepping techniques to achieve adaptive control optimization of the single-machine infinite bus system. Step S4: Design an event trigger function to reduce the consumption of system communication resources. Finally, Step S5: Perform Lyapunov analysis to ensure that every signal in the system is bounded.

[0342] The system parameters of the single-machine infinite bus system used in this embodiment are as follows:

[0343] , , , .

[0344] in, The moment of inertia of the generator; This represents the damping constant of the generator; This indicates the generator's maximum power. Indicates mechanical power; It is the amplitude of the power disturbance.

[0345] The parameters of the DNN architecture used to approximate the unknown term in this embodiment are selected as follows:

[0346] The DNN has 5 hidden layers, with 10 nodes in each layer, and a learning rate of [missing information]. , , , ( The initial values ​​of the weights in the hidden and output layers follow a set order. . a random uniform distribution.

[0347] The controller parameters selected in this embodiment are as follows:

[0348] , .

[0349] The event triggering parameters selected in this embodiment are as follows:

[0350] , , , .

[0351] In this embodiment, the desired unknown power angle is: The initial values ​​of the system state variables and adaptive signals are: , .

[0352] Numerical simulations were performed based on the above controller parameters, and the simulation results are as follows: Figures 4-6 As shown. Figure 4 The horizontal axis in the graph represents time in seconds, and the vertical axis represents the system output in degrees. The curve in the graph represents the actual desired signal of the system. and the system's output signal The curve; Figure 5 The horizontal axis represents time in seconds, and the vertical axis represents the magnitude of the error. The curve in the graph represents the system's tracking error. The closer the curve is to zero, the better the tracking performance. Figure 6 The horizontal axis represents time in seconds, and the vertical axis represents the actual size of the controller, which is the system's control input. The curve; Figure 7 The horizontal axis in the graph represents time in seconds, and the vertical axis represents the event triggering interval. The curve in the graph represents the event triggering interval. According to the simulation results, from... Figure 4 The two curves fit well, indicating good tracking performance; from Figure 5 It can be seen that the curve approaches zero, indicating that the tracking error tends to zero; from Figure 6 The curve oscillates around zero, indicating that the control input... Gradually stabilizing; from Figure 7 As can be seen from the curve, compared to a trigger frequency of once every 0.001 seconds, the curve is relatively sparse, indicating a lower number of event triggers and effectively reducing the consumption of control resources. Therefore, for a single-machine infinite bus system with an uncertain power angle, the control method designed in this invention can achieve convergence of the single-machine power angle. Simultaneously, the boundedness of the adaptive signal ensures the stability of the system, which is beneficial for practical engineering applications.

[0353] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention. The scope of protection claimed by the appended claims and their equivalents is defined.

Claims

1. An adaptive control method for a fractional-order single-machine infinite loop system based on a DNN architecture, characterized in that, The method includes the following steps: S1. Establish a dynamic model of a single-machine infinite system and transform it into a state model; S2. Design a DNN architecture, use the DNN architecture to approximate the unknown function, and design a weight update law; S3. Construct virtual and actual controllers using optimized backstepping techniques; S4. Design the event triggering function for the actual controller; S5. Through Lyapunov analysis, it is found that all signals of the system are bounded.

2. The adaptive control method for a fractional-order single-machine infinite bus system based on a DNN architecture as described in claim 1, characterized in that, Step S1 specifically includes the following steps: S11. Establish the dynamic model of the single-machine infinite system using formula (1): （1）； in, This indicates finding the 0.95th derivative; The relative angle between the equivalent engines in the two regions; The angular frequency of the relative angle between the equivalent engines in the two regions; This represents the damping constant of the generator; The moment of inertia of the generator; This indicates the generator's maximum power. Indicates the mechanical power of the generator; The amplitude of a periodic power disturbance applied externally; For control input of a single-machine infinite bus system; and Represents an unknown function; A coefficient representing the running time; Runtime; S12. Reconstruct the dynamic model and transform it into a state model, as shown in formula (2): （2）； Among them, let the dynamic model in , ; and Both represent the system state of a single-machine infinite system; ; For the system output of a single-machine infinite system, and Both represent unknown functions; Indicates external disturbance, and , It is the upper bound of the external disturbance and is a known constant.

3. The adaptive control method for a fractional-order single-machine infinite bus system based on a DNN architecture as described in claim 1, characterized in that, Step S2 specifically includes the following steps: S21. Design the DNN architecture, as shown in formula (3): （3）； in, Represents the DNN architecture; Indicates function composition; , express Veuve space; Represents the original input vector. , express Veuve space; The concatenated vector representing the hidden layer weights of the DNN architecture is formed by concatenating and transposing the vectorized weight matrices of each hidden layer, and is used to represent the hidden layer weight information as a whole. , Represents a vectorized function; This represents the hidden layer weights of the DNN architecture. For dimension The matrix, , express Veuve space; Indicates the hidden layer index; Indicates the total number of hidden layers. Represents the set of positive integers; Indicates the DNN architecture's first The number of nodes in the layer; Represents the concatenated input vector. , Represents the original input vector The transpose of , where 1 is the bias term and is a constant. express Veuve space; Indicates the DNN architecture's first The smooth activation function vector of the layer, , express Veuve space, here, , Indicates the first Activation functions at each node; S22. Using a DNN architecture to approximate unknown functions in the system. and ; Step S22 specifically includes the following steps: S221. Based on the generalized function approximation theorem, for any continuous function... The system is an unknown function, and it is determined that there exists an ideal weight that satisfies formula (4). and activation function vector This makes the approximation relationship valid; （4）； in, Represents an unknown function in the system; This represents the ideal DNN output. Represents the input vector. Indicates the ideal weight; This represents the approximation error, and the error constraint satisfies... , Indicates the maximum upper bound. Indicates the upper bound of the error; S222, Generate real-time adaptive weight estimation Using weight estimation Adaptive feedforward terms for generating DNN architecture ; S223. Obtain the weight estimation error of the DNN architecture, and use formula (5) to approximate the hyperbolic tangent function through a first-order Taylor series: （5）； in, This represents the ideal value for the DNN architecture; This represents the estimation of the DNN architecture; The derivative of the estimate of the DNN architecture; This represents the weight estimation error of the DNN architecture; S224. Using formula (6), the estimated activation function vectors of each layer are obtained by recursion layer by layer, and the recursive relationship is determined: （6）； in, Indicates the first Estimation of layer activation function vectors Indicates the first Layer activation function vector, This represents the transpose of the weight estimate for layer 0. Represents the concatenated input vector. Indicates the first Transpose of layer weight estimation Indicates the first Estimation of layer activation function vectors; S225, Based on the ideal value of the DNN architecture Estimation of DNN architecture And the aforementioned recursive relationship, which will be used to estimate the DNN architecture. Represented as And use equation (7) to determine the derivative of the estimated DNN architecture. : （7）； in, Indicates the first Estimation of layer weights; Indicates the first The estimation of layer weights, here, ; Indicates the first Estimation of layer weights; Represents a vectorized function; Represents the gradient of the DNN architecture; S226. By deriving the vectorized derivative, the gradient calculation of the DNN architecture is transformed into a matrix-vector operation that can be recursively applied layer by layer, and the specific form of the gradient is shown in formula (8): （8）； in, Indicates the multiplication symbol; ; Here, ; Indicates the first The derivative of the layer activation function vector estimate; express 3D identity matrix; Indicates the Kronecker product; Represents the cascaded input vector; S227. Based on the specific form of gradient, the error between the ideal output and the estimated output of the DNN architecture. We design a weight update law as shown in the following equation to approximate the unknown function: ； in, This represents the time derivative of the weight estimate; This represents the positive definite learning rate matrix; Indicates error.

4. The adaptive control method for a fractional-order single-machine infinite bus system based on a DNN architecture according to claim 1, characterized in that, Step S3 specifically includes the following steps: S31. Design a first-order performance index function as shown in formula (9): （9）； in, This represents a first-order performance index function; Represents a time variable; Indicates runtime; Indicates tracking error. Indicates the actual state. Indicates the expected output; express Tracking error at any given moment; Represents the exponential decay coefficient. It is a positive number; Indicates a virtual controller; S32. Use equation (10) to determine the first-order optimization performance index function of the DNN architecture: （10）； in, The minimum performance metric among all feasible virtual controllers; Indicates the optimal virtual controller; Indicates compactness; S33. Based on the first-order optimization performance index function, establish the first-order HJB equation as shown in formula (11): （11）； in, Representing the first-order HJB equation, Represents the first-order performance index function pair The first-order partial derivative of the tracking error , Expressing the request Derivative order; Represents an unknown function. Indicates an unknown disturbance. express of First derivative, Indicates the expected output; S34. Solving for the optimal virtual controller based on the first-order HJB equation. ; S35, will Decomposition yields unknown functions and And based on the DNN architecture, for unknown functions and To approximate the representation, design the system's virtual controller; S36. Construct the second-order optimization performance index function as shown in equation (16): （16）； in, This represents the second-order optimization performance index. Indicates the second-order error. , Representing second order The square of the time error, Represents a second-order state variable. Represents the exponential decay coefficient. It is a positive number; This represents the optimal actual controller; express time; Indicates runtime; This represents the square of the actual controller. S37. Based on the second-order optimization performance index function, the second-order HJB equation is established using formula (17): （17）； in, Representing the second-order HJB equation, Represents the second-order performance index function pair The second-order partial derivative of the tracking error, , Represents an unknown function. Indicates an unknown disturbance. Virtual controller Derivative order; S38, Yes Decomposition yields the unknown function and Based on the DNN architecture, for unknown functions and An approximate representation is made, and the actual controller of the system is designed.

5. The adaptive control method for a fractional-order single-machine infinite bus system based on a DNN architecture according to claim 1, characterized in that, Step S4 specifically includes the following steps: S41. Design the event triggering function of the actual controller using equation (22): （22）； in, Indicates intermediate variables. Indicates the event-triggered controller, This represents an estimate of the actual controller. This represents the controller's adjustment coefficient. This represents the sensitivity coefficient for adjusting the hyperbolic tangent function. Indicates the error threshold. This represents the error tolerance coefficient. Indicates runtime. , , , , and These represent two adjacent trigger times; S42. Design the event triggering conditions for the actual controller, as shown in formula (23): （23）； in, Let represent the infimum function.

6. The adaptive control method for a fractional-order single-machine infinite bus system based on a DNN architecture according to claim 1, characterized in that, Step S5 specifically includes the following steps: S51. Construct the first candidate Lyapunov function, calculate the fractional derivative of the first candidate Lyapunov function, and design the first fractional adaptive law. S52. A non-smooth analysis framework is used to perform variable substitution operations, and combined with the weight estimation error, the fractional derivative of the first Lyapunov function is simplified using Young's inequality. S53. Construct a second candidate Lyapunov function, integrate the error term and weight estimation error under the event triggering mechanism, perform fractional derivative of the second candidate Lyapunov function, and design a second fractional adaptive law. S54, Based on time-varying parameters Boundedness Using formula (34) for inequality scaling, the nonlinear term of the event-triggered controller is transformed into an analyzable bounded form, and formula (35) is used to determine the upper bound of the fractional derivative of the second candidate Lyapunov function: （34）； in, Indicates intermediate variables. Indicates the second-order error. This represents the controller's adjustment coefficient. and Indicates two different time-varying parameters. Indicates the error threshold. Indicates runtime. , ; （35）； in, Represents the second-order chosen Lyapunov function beg First derivative, Indicates error. This represents the sensitivity coefficient for adjusting the hyperbolic tangent function. Indicates the error threshold. This represents the error tolerance coefficient. This represents the controller's adjustment coefficient. Indicates approaching The ideal value of DNN, Indicates approaching The approximate error, This represents a second-order perturbation. Represents approximating an unknown function The transpose of the difference between the ideal value of the weight vector and the estimated weight vector. Indicates approaching The transpose of the derivative of the estimate of the DNN architecture. Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The transpose of the difference between the ideal value of the weight vector and the estimated weight vector. Indicates approaching The transpose of the estimated derivative of the DNN architecture. Represents approximating an unknown function The estimated value of the weight vector, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, This represents an estimate of the actual controller; S55, based on Using Young's inequality and formula (36), the stable form of the second candidate Lyapunov function is determined: （35）； in, and , Indicates approaching The upper bound of higher-order terms, Indicates approaching The approximate error, Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the first-order perturbation. Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and It is a positive parameter of the design. Indicates the second-order error. Indicates the first-order control gain. Indicates the second-order control gain. Indicates approaching Ideal DNN, Indicates approaching Ideal DNN; S56. By aggregating the first and second candidate Lyapunov functions, a global Lyapunov function is constructed, and the upper bound of its fractional derivative is derived to obtain the boundedness of all signals in the system.

7. The adaptive control method for a fractional-order single-machine infinite bus system based on a DNN architecture according to claim 4, characterized in that, Step S34 specifically includes the following steps: S341. Using the optimality conditions of the first-order HJB equation, for about Find the partial derivative and set it to 0: ； in, Representing the first-order HJB equation, Indicates tracking error. This represents the optimal virtual controller. Represents the first-order performance index function pair The first-order partial derivative of the tracking error; S342. The optimal virtual controller is obtained by calculating using the following formula. Analytical solution: （12）； Step S35 specifically includes the following steps: S351, Using the following formula break down: ， in, Indicates control gain. ; Indicates tracking error; and Represents an unknown function. , , Represents a first-order state variable; and Indicates the input to the neural network; Represents an unknown nonlinear function; express of Derivative order; ; ; S352. Based on the DNN architecture, the unknown function of the system is approximated using equations (13) and (14): （13）； （14）； in, , , and They represent and The ideal value of the DNN architecture, and They represent approximation. and Ideal weights, and Indicates the approximation error; S353. Based on the approximate representation of the unknown function, the virtual controller of the system is designed as shown in formula (15): （15）； in, This represents an estimate of the virtual controller. for The estimated value, Indicates approaching DNN architecture, for The estimated value, Indicates approaching The DNN architecture; Step S38 specifically includes the following steps: S381, through solving The optimal actual controller is obtained as shown in formula (18): （18）； S382, Using the following formula to... Decompose: ； in, , , and ; This represents the second-order control gain and is a positive number. Indicates the second-order error. and Represents an unknown function. and This represents the input to the neural network. It is a virtual controller; S383. Based on the DNN architecture, the unknown function is approximated using equations (19) and (20): （19）； （20）； in, , , and They represent approximation. and Ideal weights, and Indicates approaching and Approximate error; S384. Based on the approximate representation of the unknown function, the actual controller of the system is designed as shown in formula (21): （21）； in, This represents an estimate of the actual controller. This represents the second-order control gain and is a positive number. Indicates the second-order error. for The estimated value, for The estimated value, Indicates approaching DNN architecture, Indicates approaching The DNN architecture.

8. The adaptive control method for a fractional-order single-machine infinite bus system based on a DNN architecture as described in claim 6, characterized in that, Step S51 specifically includes the following steps: S511. Construct the first candidate Lyapunov function as shown in formula (24): ； in, Represents a first-order Lyapunov function; Indicates tracking error; Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. , Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function Weight vector estimation; Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. , Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function Weight vector estimation, express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and All are positive numbers; S512. Using formula (25), perform fractional derivative of the first candidate Lyapunov function: ； in, Expressing the request First derivative, Indicates the second-order error. Indicates a virtual controller. Represents an unknown function. This represents a first-order unknown perturbation. This represents finding the q-th derivative with respect to y; S513. Design the first fractional adaptive law using formula (26): ； Among them, the operator Indicates the projection algorithm; Indicates approaching The transpose of the estimated derivative of the DNN architecture. Indicates approaching The transpose of the estimated derivative of the DNN architecture. Represents an unknown function. express The adjustment rate; Step S52 specifically includes the following steps: S521. Using a non-smooth analysis framework, the fractional derivative of the first candidate Lyapunov function is substituted using formula (27): （27）； in, Indicates a first-order perturbation. Indicates the first-order control gain. express The estimate, Indicates approaching Ideal DNN, express The estimate, Indicates approaching Ideal DNN; S522, Order , Transforming formula (28), we get: （28）； in, Indicates approaching higher-order terms, Indicates approaching Higher-order terms; S523. Using Young's inequality, simplify formula (29): （29）； in, , , Indicates approaching The upper bound of higher-order terms, Indicates approaching Approximation error of higher-order terms, Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the first-order perturbation. This represents the sum of all constant terms in formula (29); Step S53 specifically includes the following steps: S531. Construct the second candidate Lyapunov function as shown in formula (30): （30）； in, This represents a second-order chosen Lyapunov function. Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, Represents approximating an unknown function The difference between the ideal value of the weight vector and the estimated weight vector. Represents approximating an unknown function The ideal value of the weight vector, Represents approximating an unknown function The estimated value of the weight vector, , , express learning rate, express The adjustment rate, express learning rate, express The adjustment rate, , , and All are positive parameters; S532, Define intermediate variables Constructing intermediate variables With event-triggered controller Mapping relationship: ； in, Indicates time-varying parameters, ; Indicates intermediate variables; Indicates the event-triggered controller; This represents the controller's adjustment coefficient. Indicates the error threshold. , ; S533. Determine the event trigger controller as shown in formula (31) through equation transformation: （31）； S534. Using formula (32) to evaluate the second candidate Lyapunov function Perform fractional derivatives: （32）； S535. Design a second fractional adaptive law using formula (33): （33）； in, Indicates approaching The transpose of the estimated derivative of the DNN architecture. Represents an unknown function. Indicates approaching The estimated derivative of the DNN architecture is transposed; Step S56 specifically includes the following steps: S561. Based on the first and second candidate Lyapunov functions, construct the global Lyapunov function using formula (37). : （37）； S562. Using formula (38) to analyze the global Lyapunov function Perform fractional derivative calculations and determine the upper bound of its fractional derivative: （38）； in, and , , , Indicates approaching The upper bound of higher-order terms, Indicates approaching The approximate error, Indicates approaching The upper bound of higher-order terms, This represents the upper bound of the second-order perturbation. Indicates the first-order control gain. Indicates the second-order control gain; S57, By analyzing the event trigger controller With intermediate variables Given the boundedness of the fractional derivative, and considering the triggering condition, the lower bound of the event triggering interval can be determined using formula (39): 。

Images