A cost-performance-based dynamic optimization control method for switching Chua circuit system

Abstract

The application discloses a cost performance-based dynamic optimization control method for a switched Chua circuit system. The method constructs a cost function of the switched Chua circuit system, gives a corresponding Hamiltonian equation, and solves the optimization control input corresponding to the equation through a Critic neural network. In addition, based on the Lyapunov stability theorem, the consistent ultimate bounded condition of the switched Chua circuit system is ensured.

Description

Technical Field

[0001] This invention relates to a dynamic optimization control method, specifically a dynamic optimization control method for a switching Chua's circuit system based on cost-saving performance. Background Technology

[0002] Many dynamic models for switching Chua's circuit systems only consider how to design a controller to achieve the desired performance. However, in practical applications, the system not only needs to achieve the preset performance target, but also needs to guarantee system performance while maintaining optimal cost. Based on this idea, researchers have conducted extensive research on the dynamic optimization control of switching Chua's circuit systems under cost-effective performance conditions, and have achieved a series of results. Summary of the Invention

[0003] The purpose of this invention is to propose a dynamic optimization control method for switching Chua's circuit systems based on cost-effectiveness, which can effectively reduce the control cost of switching Chua's circuit systems.

[0004] The specific technical solution of the present invention is as follows: A dynamic optimization control method for a switching Chua's circuit system based on cost-saving performance, comprising the following steps:

[0005] Construct the cost function of the switching Chua's circuit system, derive the corresponding Hamiltonian equation, and solve the optimal control input corresponding to the equation using a Critic neural network.

[0006] Give the uniform eventual bounded condition for switched Chua's circuit systems based on Lyapunov's stability theorem.

[0007] The Chua's circuit system model is switched as follows:

[0008]

[0009] f(x) = 1.5tanh(x), α 1 =1.2, α 2 =0.4,

[0010] In the formula Let A, B, C, H, and E represent the system state, and let E represent the known parameter matrix. Indicates Let α be a composite function of the independent variable, u(t) represent the control input, R(t) represent the system fault, τ(t) represent the external input, and α be the independent variable. σ(t) >0 indicates coupling strength, G σ(t) Γ represents the external coupling matrix. σ(t) Represents the inner coupling matrix. I represents the Kronecker integral operator. N Represents an N-order identity matrix;

[0011] Cost function corresponding to control input u(t) Designed as

[0012]

[0013] Where Q and R are two positive definite matrices;

[0014] Cost function Satisfies the following Hamiltonian equation:

[0015]

[0016] in Represented as the gradient operator;

[0017] The optimized cost function is expressed as follows: Its Hamiltonian equation satisfies

[0018]

[0019] And optimize the control input u corresponding to the cost function. * (t) is represented as

[0020]

[0021] Where R -1 Let T be the inverse of matrix R, and let T be the transpose of matrix R.

[0022] In optimizing control input u * Under (t), the system can be rewritten as

[0023]

[0024] To solve Hamiltonian equations containing an optimization cost function, Critic neural networks are used to estimate the unknown cost function. And its corresponding gradient is estimated as

[0025]

[0026]

[0027] Where W is the unknown ideal weight. For activation function, To estimate the error

[0028] The output and gradient of the Critic neural network are represented as follows:

[0029]

[0030]

[0031] W passes through The estimated pass The estimated pass The estimated

[0032] So, the ideal optimal control input u * (t) and its exact value Described as

[0033]

[0034]

[0035] Define ζ(t) as a Hamiltonian equation and Residual error, i.e.

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042] in

[0043]

[0044] Squared residual e ζ (t) is defined as By minimizing the squared residual e ζ (t), parameter satisfy

[0045]

[0046] Where β is the positive learning law

[0047] This control scheme can guarantee the uniform and bounded stability of the system. The proof is as follows:

[0048] C001: Select the energy function in the following form:

[0049]

[0050] in

[0051] C002: The derivative of V(t) is

[0052]

[0053] C003: For neural networks, the following assumptions are given:

[0054] (g) The unknown ideal weight W satisfies ||W||≤W M Where |||| denotes the norm, W M Represents positive integers;

[0055] (h) Activation function and its Jacobian derivative It is bounded, that is... Where σ M and d σM Represents positive integers;

[0056] (i) Gradient of estimation error It is bounded, that is... Where ε M Represents positive integers;

[0057] (j)ρ(t) is bounded, i.e., ρ m ≤ρ(t)≤ρ M , where ρ m and ρ M Represents a constant;

[0058] (k) It is bounded, that is... Where e M Represents positive integers;

[0059] (f)α1||x|| 2 ≤x T Qx≤α2||x|| 2 α3||x|| 2 ≤||u * (t)||≤α4||x|| 2 , where α1, α2, α3, α4 represent positive constants;

[0060] C004: Based on the assumptions in C003, we can obtain:

[0061]

[0062] Where η is The upper bound;

[0063] C005: Based on Lyapunov stability theory, if... have Therefore, the closed-loop system achieves consistent eventual boundedness under dynamic programming optimization control. Attached Figure Description

[0064] Figure 1-2 These are two topological diagrams of the system;

[0065] Figure 3-7 To optimize the dynamic trajectory diagram of the system under control input;

[0066] Figure 8-12 To optimize the dynamic trajectory of neural network weights under control input; Detailed Implementation

[0067] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading the present invention, any modifications of the present invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0068] A dynamic optimization control method for a switching Chua's circuit system based on cost-saving performance includes the following steps:

[0069] Step 1: Set the various system parameters;

[0070] Step 2: Construct the cost function and Hamiltonian equation;

[0071] Step 3: Solve the corresponding optimal control input for the equation using a Critic neural network;

[0072] Step 4: Verify whether the estimation error system achieves consistent eventual boundedness.

[0073] An embodiment of the present invention is described below:

[0074] Consider a dynamic optimization control method for a switching Chua's circuit system based on cost-saving performance, with the corresponding dynamic models as follows:

[0075]

[0076] f(x) = 1.5tanh(x), α 1 =1.2, α 2 =0.4,

[0077] The two topology diagrams of the system are shown in Figure 1-2. The dynamic trajectory diagram of the system under optimized control input is shown in Figure 1-2. Figure 3-7 As shown, the dynamic trajectory diagram of the neural network weights under optimized control input is as follows: Figure 8-12 As shown.

Claims

1. A dynamic optimization control method for a switching Chua's circuit system based on cost-saving performance, characterized in that, Includes the following steps: The cost function of the switching Chua's circuit system is constructed, and the corresponding Hamiltonian equation is given. The optimal control input corresponding to the equation is solved using a Critic neural network. The specific process is as follows: The Chua's circuit system model is switched as follows: ； in , , , , , ， ， ， ， ， ， ， In the formula Indicates the system status. , , , , Represents a known parameter matrix. Indicates A composite function of the independent variable. Indicates control input, This indicates a system malfunction. Indicates external input. Indicates coupling strength. Represents the external coupling matrix. Represents the inner coupling matrix. This represents the Kronecker integral operator. express An identity matrix of order; Control input Corresponding cost function Designed for ； in and They are two positive definite matrices; Cost function The corresponding Hamiltonian function is as follows: ； in Represented as the gradient operator; The optimized cost function is expressed as follows: Its Hamiltonian equation satisfies ； And optimize the control input corresponding to the cost function. Represented as ； in Represented as a matrix The inverse matrix, This is represented as transpose; Optimize control input The system can be further written as follows: Using Critic Neural Networks to Estimate Unknown Cost Functions ; And its corresponding gradient is estimated as ； ； in It is an unknown ideal weight. For activation function, To estimate the error; The output and gradient of the Critic neural network are represented as follows: ； ； in pass The estimated pass The estimate; Ideal Optimal Control Input and its precise value Described as ； ； definition Hamiltonian function and Residual error, i.e. ； in ； Squared residuals Defined as By minimizing the squared residual ,parameter satisfy ； in It is a positive learning law.

2. The dynamic optimization control method for a switching Chua's circuit system based on cost-saving performance according to claim 1, characterized in that, The uniform eventual bounded condition for switched Chua's circuit systems based on Lyapunov's stability theorem is given, including the following steps: B001: Select the energy function in the following form: ， in ; B002: The derivative is ； B003: For neural networks, the following assumptions are given: (a) Unknown ideal weight satisfy ,in Represents the norm, Represents positive integers; (b) Activation function and its Jacobian derivative It is bounded, that is... , ,in and Represents positive integers; (c) Gradient of estimation error It is bounded, that is... ,in Represents positive integers; (d) It is bounded, that is... ,in and Represents a constant; (e) It is bounded, that is... ,in Represents positive integers; (f) ,in Represents positive integers; B004: Based on the assumptions in B003, we can obtain: in yes The upper bound; B005: Based on Lyapunov stability theory, if... ,have Therefore, the closed-loop system achieves consistent eventual boundedness under dynamic programming optimization control.

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