Dynamic optimization control method for switching Chua's circuit system based on guaranteed cost performance

Abstract

The invention discloses a dynamic optimization control method for a switching Chua's circuit system based on guaranteed cost performance. According to the method, a cost function of a switching Chua's circuit system is constructed, a corresponding Hamiltonian equation is given, and optimization control input corresponding to the equation is solved through a Critic neural network. In addition, based on the Lyapunov stability theorem, it is ensured that the switching Chua's circuit system is consistent with the final bounded condition.

Description

Technical Field

[0001] The present invention relates to a dynamic optimization control method, in particular to a dynamic optimization control method for a switching Chua's circuit system based on guaranteed cost performance. Background Art

[0002] Many dynamic models for switching Chua circuit systems only consider how to design a controller to achieve performance. In practical applications, the system not only needs to consider achieving the preset performance goals, but also needs to ensure system performance under the optimal cost. Based on this idea, researchers have conducted a lot of research on the dynamic optimization control of switching Chua circuit systems under cost-performance guarantee, and have achieved a series of results. Summary of the invention

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

[0004] The specific technical solution of the present invention is as follows: A method for dynamically optimizing and controlling a switching Chua's circuit system based on guaranteed cost performance comprises the following steps:

[0005] Construct the cost function of the switching Chua circuit system, give the corresponding Hamiltonian equation, and solve the optimal control input corresponding to the equation through the Critic neural network

[0006] The uniform final boundedness condition of the switched Chua circuit system based on Lyapunov stability theorem is given.

[0007] The switching Chua circuit system model is as follows:

[0008] f(x)=1.5tanh(x),α 1 =1.2,α 2 =0.4,

[0009] In the formula represents the system state, A, B, C, H, E represent the known parameter matrix, Indicates is a composite function of independent variables, u(t) represents the control input, R(t) represents the system fault, τ(t) represents the external input, α σ(t) >0 indicates coupling strength, G σ(t) represents the external coupling matrix, Γ σ(t) represents the internal coupling matrix, represents the Kronecker product operator, I N represents the N-order unit matrix;

[0010] The cost function corresponding to the control input u(t) Designed to Where Q and R are two positive definite matrices;

[0011] Cost Function Satisfies the following Hamiltonian equation: in Represented as a gradient operator;

[0012] The optimization cost function is expressed as Its Hamiltonian equation satisfies

[0013] And the optimal control input u corresponding to the optimization cost function * (t) is expressed as Where R -1 It is represented as the inverse matrix of matrix R, and T is represented as the transpose

[0014] In the optimal control input u * (t), the system can be rewritten as

[0015] In order to solve the Hamiltonian equation with the optimization cost function, the Critic neural network is used to estimate the unknown cost function and its corresponding gradient is estimated as Where W is the unknown ideal weight, is the activation function, Estimation error

[0016] The output of the critic neural network and its gradient are expressed as Where W is It is estimated that pass It is estimated that pass Estimated

[0017] Then the ideal optimal control input u * (t) and its exact value Described as

[0018] Define ζ(t) as the Hamiltonian equation and The residual error is in

[0019] Squared residual e ζ (t) is defined as By minimizing the squared residual e ζ (t), parameters satisfy where β is a positive learning law

[0020] This control scheme can ensure that the system is uniformly bounded and stable. The proof process is as follows:

[0021] C001: Select the energy function in the following form:

[0022] in

[0023] C002: The derivative of V(t) is

[0024] C003: For neural networks, the following assumptions are made: (g) The unknown ideal weight W satisfies ||W||≤W M , where |||| represents the norm, W M represents a positive constant; (h) Activation function and its Jacobian derivative is bounded, that is where σM and d σM represents a positive constant; (i) Gradient of the estimated error is bounded, that is where ε M represents a positive constant; (j)ρ(t) is bounded, that is, ρ m ≤ρ(t)≤ρ M , where ρ m and ρ M represents a constant; (k) is bounded, that is where e M represents a positive constant; (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;

[0025] C004: Based on the assumptions in C003, we can obtain: where η is The upper bound of

[0026] C005: Based on Lyapunov stability theory, if have Therefore, the closed-loop system is uniformly eventually bounded under dynamic programming optimization control. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1-2 Two topological diagrams of the system;

[0028] Figure 3-7 Dynamic trajectory diagram of the system under optimal control input;

[0029] Figure 8-12 Dynamic trajectory diagram of neural network weights under optimized control input; DETAILED DESCRIPTION

[0030] The present invention is further explained below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent forms of modifications to the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0031] A method for dynamic optimization control of a switching Chua's circuit system based on guaranteed cost performance comprises the following steps:

[0032] Step 1: Set various system parameters;

[0033] Step 2: Construct the cost function and Hamiltonian equation;

[0034] Step 3: Solve the optimal control input corresponding to the equation through the Critic neural network;

[0035] Step 4: Verify whether the estimated error system is consistently ultimately bounded;

[0036] An embodiment of the present invention is described below:

[0037] Consider a dynamic optimization control method for a switching Chua circuit system based on guaranteed cost performance, and the corresponding dynamic models are:

[0038] f(x)=1.5tanh(x),α 1 =1.2,α 2 =0.4,

[0039] The two topological structures of the system are shown in Figure 1-2, and the dynamic trajectory of the system under the optimized control input is shown in Figure 1-2. Figure 3-7 As shown in the figure, the dynamic trajectory of the neural network weight under the optimized control input is as follows Figure 8-12 shown.

Claims

1. A method for dynamic optimization control of a switching Chua circuit system based on guaranteed cost performance, characterized in that: The following steps are involved: Construct the cost function of the switching Chua circuit system, give the corresponding Hamiltonian equation, and solve the optimal control input corresponding to the equation through the Critic neural network; The uniformly eventually bounded conditions for switched Chua circuit systems are given based on the Lyapunov stability theorem.

2. The method for dynamic optimization control of a switching Chua's circuit system based on guaranteed cost performance according to claim 1, characterized in that: Construct the cost function of the Chua circuit system, give the corresponding Hamiltonian equation, and solve the optimal control input corresponding to the equation through the Critic neural network. The specific steps are as follows: The switching Chua circuit system model is as follows: where f(x) = 1.5tanh(x), α 1 = 1.2, αα 2 = 0.4, In the formula represents the system state, A, B, C, H, E represent the known parameter matrix, Indicates is a composite function of independent variables, u(t) represents the control input, R(t) represents the system fault, τ(t) represents the external input, α σ(t) >0 indicates coupling strength, G σ(t) represents the external coupling matrix, Γ σ(t) represents the internal coupling matrix, represents the Kronecker product operator, I N represents the N-order unit matrix; The cost function corresponding to the control input u(t) Designed to Where Q and R are two positive definite matrices; Cost Function Satisfies the following Hamiltonian equation: in Represented as a gradient operator; The optimization cost function is expressed as Its Hamiltonian equation satisfies And the optimal control input u corresponding to the optimization cost function * (t) is expressed as Where R -1 It is represented as the inverse matrix of matrix R, and T is represented as the transpose; In the optimal control input u * (t), the system can be rewritten as In order to solve the Hamiltonian equation with the optimization cost function, the Critic neural network is used to estimate the unknown cost function and its corresponding gradient is estimated as Where W is the unknown ideal weight, is the activation function, is the estimation error; The output of the critic neural network and its gradient are expressed as Where W is It is estimated that pass It is estimated that pass estimated; Then the ideal optimal control input u * (t) and its exact value Described as Define ζ(t) as the Hamiltonian equation and The residual error, i.e. in Squared residual e ζ (t) is defined as By minimizing the squared residual e ζ (t), parameters satisfy where β is a positive learning law.

3. The method for dynamic optimization control of a switching Chua's circuit system based on guaranteed cost performance according to claim 1, characterized in that: The uniform final boundedness condition of the switched Chua circuit system based on Lyapunov stability theorem is given: B001: Select the energy function in the following form: in B002: The derivative of V(t) is B003: For the neural network, the following assumptions are given: (a) The unknown ideal weight W satisfies ||W||≤W M , where || || represents the norm, W M represents a positive constant; (b) Activation function and its Jacobian derivative is bounded, that is where σ M and d σM represents a positive constant; (c) Gradient of the estimated error is bounded, that is where ε M represents a positive constant; (d)ρ(t) is bounded, that is, ρ m ≤ρ(t)≤ρ M , where ρ m and ρ M represents a constant; (e) is bounded, that is where e M represents a positive constant; (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; B004: Based on the assumptions in B003, we can obtain: where η is The upper bound of B005: Based on Lyapunov stability theory, if have Therefore, the closed-loop system is uniformly eventually bounded under dynamic programming optimization control.

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