A fractional-order PID controller and parameter tuning method based on KAN

Abstract

The invention discloses a fractional-order PID controller and parameter tuning method based on KAN, which relates to the technical field of PID control and comprises the following steps: step 1, determining the structures of two KAN subnetworks and initializing the networks, wherein the two KAN subnetworks are a tuning network KAN1 and an identification network KAN2; step 2, sampling to obtain the actual output of a controlled object, comparing it with the target output, and calculating the error; step 3, inputting , , into KAN1, and KAN1 outputting five adjustable parameters , , , , of the fractional-order PID controller, wherein is a proportional coefficient, is an integral coefficient, and is a differential coefficient; step 4, calculating the control output of the fractional-order PID controller and inputting it to the controlled object; step 5, KAN1 and KAN2 adjusting activation functions of different nodes to realize the adaptive tuning of the fractional-order PID controller; step 6, setting , and returning to step 2; step 7, pruning KAN1 and KAN2.

Description

Technical Field

[0001] The present invention relates to the technical field of PID control, and is applicable to parameter setting in the field of automatic control, and in particular to a fractional-order PID controller and a parameter setting method based on KAN. Background Art

[0002] In the parameter setting of automatic control field, the classic PID controller is widely used, and its application effect is highly dependent on , , Three parameters. It is difficult to achieve ideal results when facing complex controlled objects with nonlinear, large inertia, large hysteresis and other characteristics. Among the classic PID controllers, the Ziegler-Nichols (ZN) engineering tuning method is the most commonly used and influential parameter tuning method. However, the goal of this method is to achieve a decay rate of one-fourth, which is easy to cause excessive overshoot and difficult to achieve robustness under complex working conditions.

[0003] The fractional-order PID controller is an extended design based on the classic integer-order PID controller. It has better robustness and control accuracy, but due to the introduction of additional integral orders and differential order , its computational cost is higher and the numerical range of its parameters is larger, so its parameter tuning is also more challenging.

[0004] Compared with traditional tuning methods, some PID parameter tuning methods combined with neural networks can achieve better dynamic performance, such as BP network, wavelet network, RBF network, etc. The network adjusts the weights through iterative training and dynamically optimizes the PID parameters, which can achieve more precise control. However, it is difficult for such networks to balance parameter scale and fitting ability, and more often they need to sacrifice more real-time performance in exchange for improved accuracy.

[0005] In summary, how to design a PID control and parameter tuning method with higher precision and faster real-time performance is an industry problem that needs to be solved urgently.

[0006] Therefore, technicians in this field are committed to developing a new fractional-order PID controller and parameter tuning method to solve the above-mentioned problems existing in the prior art. Summary of the invention

[0007] In view of the above-mentioned defects of the prior art, the technical problem to be solved by the present invention is how to design a PID controller and parameter setting method with higher accuracy and faster real-time performance.

[0008] To achieve the above object, the present invention provides a fractional-order PID controller and parameter tuning method based on KAN, the method comprising the following steps:

[0009] Step 1: Determine the structure of the two KAN sub-networks and initialize the network: determine the number of input layer nodes and the number of middle layer nodes, and select the learning rate and the coefficient of inertia ,frequency , wherein the two KAN sub-networks are a tuning network KAN1 and an identification network KAN2;

[0010] Step 2: Sampling to obtain the actual output of the controlled object and with the target output Compare and calculate the error ;

[0011] Step 3: , , Input KAN1, KAN1 outputs the five adjustable parameters of the fractional-order PID controller , , , , ,in, is the proportionality coefficient, is the integration coefficient, is the differential coefficient, is the integration order, and , is the differential order, and ;

[0012] Step 4: Calculate the control output of the fractional-order PID controller , and input it to the controlled object;

[0013] Step 5, KAN1 and KAN2 adjust the activation functions of different nodes to achieve adaptive tuning of the fractional-order PID controller;

[0014] Step 6: , return to step 2;

[0015] Step 7: Prune KAN1 and KAN2.

[0016] Furthermore, in step 1, the output layer of the tuning network KAN1 is The nodes can be represented as:

[0017]

[0018] in, is the number of input layer state parameters, is the proportionality coefficient , is the integration coefficient , is the differential coefficient , yes , yes ; Indicates The input nodes and the middle layer A unary function between nodes, Indicates the middle layer Nodes to the output layer A unary function between nodes.

[0019] Furthermore, the activation function of KAN1 is given by -Spline representation.

[0020] Furthermore, the activation function of KAN1 can be expressed as a one-dimensional function :

[0021]

[0022]

[0023] in, are the spline function weight coefficients optimized during training; is defined on the grid -Spline basis functions, the grid points define the intervals where each basis function is active and significantly affects the shape and smoothness;

[0024] Initialization command =1, , , Set to 0.1;

[0025] Determine the number of input layer nodes and the number of middle-layer nodes , set the initial activation function, where for the number of input layer nodes , each input layer node is a state parameter, that is It is the number of state parameters of the input layer;

[0026] make is 4, then the input vector is:

[0027]

[0028] The performance indicator function of KAN1 is:

[0029] .

[0030] Furthermore, the structure of the identification network KAN2 is similar to the structure of the setting network KAN1;

[0031] The output layer of KAN2 can be expressed as:

[0032]

[0033] in, is the number of input layer state parameters.

[0034] Furthermore, in KAN2, is 2, then the two inputs are :

[0035]

[0036] The output is , the performance indicator function of KAN2 is:

[0037] .

[0038] Furthermore, the step 3 , are all set to 1, the fractional-order PID controller becomes an adaptive PID controller:

[0039] .

[0040] Furthermore, the control output of step 4 is for:

[0041]

[0042] in, , , , is the number of time steps.

[0043] Furthermore, in step 5, KAN1 and KAN2 use the gradient descent method to adjust the learning rate and activation function by comparing the parameters of the input value and the output value.

[0044] Furthermore, the adjustment formula is as follows:

[0045]

[0046] make , then the spline function weights of the first and second layers are The update rules are as follows:

[0047]

[0048]

[0049]

[0050]

[0051] in, is the learning rate, for the accused Information, obtained through KAN2 identification.

[0052] The KAN-based fractional-order PID controller and parameter tuning method provided by the present invention have at least the following technical effects:

[0053] 1. Compared with the traditional network whose activation function cannot be learned, the technical solution proposed in this invention can timely adjust the parameters of the fractional-order PID controller according to the set value. Through the collaborative framework of identification network + tuning network, combined with KAN, it can achieve more streamlined parameters and more complex function approximation capabilities. It is also an advanced control algorithm that expands the traditional PID controller. In some complex system controls, the fractional-order PID controller has better dynamic robustness.

[0054] 2. The technical solution proposed in the present invention helps to realize a neural network fractional-order PID parameter tuning method with both high precision and fast real-time performance, which has the following advantages: a) adaptive tuning of PID parameters; b) more streamlined parameters leading to better real-time performance; c) better nonlinear fitting ability leading to higher control accuracy; d) better interpretability.

[0055] The concept, specific structure and technical effects of the present invention will be further described below in conjunction with the accompanying drawings to fully understand the purpose, characteristics and effects of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 is a flow chart of a parameter setting method of a preferred embodiment of the present invention;

[0057] Figure 2 It is a structural diagram of a fractional-order PID controller of a preferred embodiment of the present invention;

[0058] Figure 3 It is a structural diagram of a setting network KAN1 of a preferred embodiment of the present invention;

[0059] Figure 4 It is a schematic diagram of the structure of the identification network KAN2 of a preferred embodiment of the present invention. DETAILED DESCRIPTION

[0060] The following describes several preferred embodiments of the present invention with reference to the drawings in the specification, so that the technical content is clearer and easier to understand. The present invention can be embodied in many different forms of embodiments, and the protection scope of the present invention is not limited to the embodiments mentioned in the text.

[0061] The Kolmogorov-Arnold theorem states that any multivariable continuous function can be represented as a combination of a finite number of continuous functions. Kolmogorov-Arnold Networks (KAN) is a new neural network architecture proposed in 2024. Theorem, compared with traditional networks such as multilayer perceptron (MLP) that update linear weight matrices with fixed activation functions, the activation function of each layer is learnable.

[0062] Each neuron in the KAN network uses a learnable nonlinear activation function. Through multiple layers and multiple neurons in each layer, the network can learn arbitrarily complex mappings. KAN fundamentally eliminates the dependence on the linear weight matrix by replacing the fixed activation function with a learnable function at the edge of the network, thereby improving the nonlinear fitting ability of the network. Each weight parameter in KAN can be replaced by a single variable function, which is usually parameterized in the form of a spline function, thus providing extremely high flexibility and the ability to simulate complex functions with fewer parameters, and also increasing the interpretability of the network. KAN can have better generalization ability, more complex function approximation ability, and better interpretability based on a smaller number of parameters. Therefore, the KAN network brings new possibilities for high-precision and high-real-time PID parameter tuning methods.

[0063] Example 1

[0064] like Figure 1 and Figure 2 As shown, a fractional-order PID controller and parameter tuning method based on KAN is provided in an embodiment of the present invention, comprising the following steps:

[0065] Step 1: Determine the structure of the two KAN sub-networks and initialize the network: determine the number of input layer nodes and the number of middle layer nodes, and select the learning rate and the coefficient of inertia ,frequency , where the two KAN sub-networks are the tuning network KAN1 and the identification network KAN2;

[0066] Step 2: Sampling to obtain the actual output of the controlled object and with the target output Compare and calculate the error ;

[0067] Step 3: , , Input KAN1, KAN1 outputs five adjustable parameters of the fractional-order PID controller , , , , ,in, is the proportionality coefficient, is the integration coefficient, is the differential coefficient, is the integration order, and , is the differential order, and ;

[0068] Step 4: Calculate the control output of the fractional-order PID controller , and input it to the object being controlled;

[0069] Step 5: KAN1 and KAN2 adjust the activation functions of different nodes to achieve adaptive tuning of the fractional-order PID controller;

[0070] Step 6: , return to step 2;

[0071] Step 7: Prune KAN1 and KAN2 to remove less important nodes or connections after network training, making the network lighter and improving the real-time performance in practical applications.

[0072] Example 2

[0073] like Figure 3 As shown, based on Example 1, in step 1, the output layer of the network KAN1 is adjusted. The nodes can be represented as:

[0074]

[0075] in, It is the number of input layer state parameters, which can be adjusted according to actual needs. Generally, the more inputs there are, the higher the upper limit of the dynamic accuracy of the system will be. is the proportionality factor , is the integration coefficient , is the differential coefficient , yes , yes ; Indicates The input nodes and the middle layer A unary function between nodes, Indicates the middle layer Nodes to the output layer A unary function between nodes.

[0076] In particular, the activation function of KAN1 is given by -Spline representation. The flexibility of splines enables them to adaptively model complex relationships in the data by adjusting their shape to minimize approximation errors, enhancing the network's ability to learn subtle patterns from high-dimensional datasets.

[0077] In particular, the activation function of KAN1 can be expressed as a one-dimensional function :

[0078]

[0079]

[0080] in, are the spline function weight coefficients optimized during training; is defined on the grid -Spline basis functions; the grid points define the intervals where each basis function is active and significantly affects the shape and smoothness, which are hyperparameters that affect the accuracy of the network. More grids mean more learnable parameters, more complex control, and higher accuracy.

[0081] In particular, the initialization command =1, , , It is usually set very small, for example, it can be set to 0.1.

[0082] In particular, determine the number of input layer nodes and the number of middle-layer nodes , set the initial activation function, where for the number of input layer nodes , each input layer node is a state parameter, that is It is the number of state parameters of the input layer;

[0083] make is 4, then the input vector is:

[0084]

[0085] The performance indicator function of KAN1 is:

[0086] .

[0087] Example 3

[0088] Based on the embodiment 1 or 2, the structure of the identification network KAN2 is similar to the structure of the setting network KAN1, such as Figure 4 shown.

[0089] The output layer of KAN2 can be expressed as:

[0090]

[0091] in, It is the number of input layer state parameters, which can be adjusted according to actual needs. Generally, the more inputs there are, the higher the upper limit of the dynamic accuracy of the system.

[0092] In particular, In, take is 2, then the two inputs are :

[0093]

[0094] The output is , the performance indicator function of KAN2 is:

[0095] .

[0096] In particular, in step 3 , If both are set to 1, the fractional-order PID controller becomes an adaptive PID controller:

[0097] .

[0098] Example 4

[0099] Based on Embodiment 1, 2 or 3, the control output of step 4 for:

[0100]

[0101] in, , , , is the number of time steps.

[0102] In particular, in step 5, the neural network learns according to the state changes, adjusts the activation functions of different nodes in the network, and realizes the adaptive tuning of the fractional-order PID controller.

[0103] Specifically, KAN1 and KAN2 use the gradient descent method to adjust the learning rate and activation function by comparing the parameters of the input value and the output value. The adjustment formula is as follows:

[0104]

[0105] make , then the spline function weights of the first and second layers are The update rules are as follows:

[0106]

[0107]

[0108]

[0109]

[0110] in, is the learning rate, For the accused Information, obtained through KAN2 identification.

[0111] The KAN network is based on Theorem has better interpretability. Compared with other methods combined with machine learning, the method proposed in the embodiment of the present invention has better interpretability and can understand the tuning law of the network through the learned KAN network. This tuning law based on machine learning can complement traditional methods or manual experience.

[0112] Based on the above steps, the embodiment of the present invention also provides a KAN network online learning architecture, which can be used in offline learning to apply the fixed weight network after training to the following situations: Figure 2 In the architecture shown, the structure of KAN1 is as follows Figure 3 As shown, the structure of KAN2 is as follows Figure 4 shown.

[0113] The KAN2 module can also be removed if the controlled object is known.

[0114] The preferred specific embodiments of the present invention are described in detail above. It should be understood that ordinary technicians in the field can make many modifications and changes based on the concept of the present invention without creative work. Therefore, all technical solutions that can be obtained by technicians in the technical field based on the concept of the present invention through logical analysis, reasoning or limited experiments on the basis of the prior art should be within the scope of protection determined by the claims.

Claims

1. A fractional-order PID controller and parameter tuning method based on KAN, characterized in that: The method comprises the following steps: Step 1: Determine the structure of the two KAN sub-networks and initialize the network: determine the number of input layer nodes and the number of middle layer nodes, and select the learning rate and the coefficient of inertia ,frequency , wherein the two KAN sub-networks are a tuning network KAN1 and an identification network KAN2; Step 2: Sampling to obtain the actual output of the controlled object and with the target output Compare and calculate the error ; Step 3: , , Input KAN1, KAN1 outputs the five adjustable parameters of the fractional-order PID controller , , , , ,in, is the proportionality coefficient, is the integration coefficient, is the differential coefficient, is the integration order, and , is the differential order, and ; Step 4: Calculate the control output of the fractional-order PID controller , and input it to the controlled object; Step 5, KAN1 and KAN2 adjust the activation functions of different nodes to achieve adaptive tuning of the fractional-order PID controller; Step 6: , return to step 2; Step 7: Prune KAN1 and KAN2.

2. The fractional-order PID controller and parameter tuning method based on KAN as claimed in claim 1, characterized in that: In step 1, the output layer of the tuning network KAN1 is The nodes can be represented as: in, is the number of input layer state parameters, is the proportionality coefficient , is the integration coefficient , is the differential coefficient , yes , yes ; Indicates The input nodes and the middle layer A unary function between nodes, Indicates the middle layer Nodes to the output layer A unary function between nodes.

3. The KAN-based fractional-order PID controller and parameter tuning method according to claim 2, characterized in that: The activation function of KAN1 is given by -Spline representation.

4. The fractional-order PID controller and parameter tuning method based on KAN as claimed in claim 3, characterized in that: The activation function of KAN1 can be expressed as a one-dimensional function : in, are the spline function weight coefficients optimized during training; is defined on the grid -Spline basis functions, the grid points define the intervals where each basis function is active and significantly affects the shape and smoothness; Initialization command =1, , , Set to 0.1; Determine the number of input layer nodes and the number of middle-layer nodes , set the initial activation function, where for the number of input layer nodes , each input layer node is a state parameter, that is It is the number of state parameters of the input layer; make is 4, then the input vector is: The performance indicator function of KAN1 is: 。 5. The KAN-based fractional-order PID controller and parameter tuning method according to claim 4, characterized in that: The structure of the identification network KAN2 is similar to that of the setting network KAN1; The output layer of KAN2 can be expressed as: in, is the number of input layer state parameters.

6. The fractional-order PID controller and parameter tuning method based on KAN as claimed in claim 5, characterized in that: In KAN2, is 2, then the two inputs are : The output is , the performance indicator function of KAN2 is: 。 7. The KAN-based fractional-order PID controller and parameter tuning method according to claim 6, characterized in that: Step 3 , are all set to 1, the fractional-order PID controller becomes an adaptive PID controller: 。 8. The KAN-based fractional-order PID controller and parameter tuning method according to claim 6, characterized in that: The control output of step 4 for: in, , , , is the number of time steps.

9. The KAN-based fractional-order PID controller and parameter tuning method according to claim 6, characterized in that: In step 5, KAN1 and KAN2 use the gradient descent method to adjust the learning rate and activation function by comparing the parameters of the input value and the output value.

10. The KAN-based fractional-order PID controller and parameter tuning method according to claim 6, characterized in that: The adjustment formula is as follows: make , then the spline function weights of the first and second layers are The update rules are as follows: in, is the learning rate, for the accused Information, obtained by KAN2 identification.

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