A data-driven based van der pol oscillator system cost preserving event-triggered control method

Abstract

The application discloses a data-driven Van der Pol oscillator system cost preserving event-triggered control method. The method first establishes a Taylor expansion linearization equation of a Van der Pol oscillator system, then proposes a data-based state feedback cost preserving control strategy, then designs a data-based cost preserving event-triggered mechanism for reducing communication transmission, and finally demonstrates the stability of a closed-loop system. The method combines a data-driven strategy and a cost preserving triggering mechanism, effectively reduces network communication load while guaranteeing system stability and control performance, and solves the problems of model uncertainty, disturbance influence and communication constraints in traditional control methods in the Van der Pol oscillator system.

Description

Technical Field

[0001] This invention relates to a control method for a van der Bohr oscillator system, specifically a data-driven cost-preservation event-triggered control method for a van der Bohr oscillator system. Background Technology

[0002] With the widespread application of van der Bohr oscillators in electronic oscillating circuits, biological rhythm modeling, and electromechanical coupling, the nonlinear characteristics, state-dependent damping, and external disturbances of the systems make accurate modeling difficult. Traditional control methods based on accurate models often fail to guarantee system stability and performance, and in networked control environments, the load problems caused by continuous sampling and frequent communication are becoming increasingly prominent. Therefore, there is an urgent need for a control strategy that can ensure control performance while reducing communication burden.

[0003] Data-driven control methods avoid dependence on precise system models and can adaptively respond to dynamic changes in the system. Simultaneously, event-triggered control mechanisms can reduce the sampling frequency while maintaining system performance. Existing methods often fail to balance data-driven modeling, network constraints, and robustness requirements, especially in van der Bohr oscillator systems, where a method is lacking to construct a dynamic event-triggered mechanism under unknown model conditions while ensuring system stability. Therefore, this paper proposes a data-driven, cost-preserving event-triggered control method for van der Bohr oscillator systems to achieve efficient, robust, and low-communication-consumption control performance. Summary of the Invention

[0004] The purpose of this invention is to propose a data-driven, cost-protection event-triggered control method for van der Bohr oscillator systems, enabling efficient and robust control of van der Bohr oscillator systems in the absence of an accurate system model. This method combines data-driven control with a cost-protection triggering mechanism, effectively reducing network communication load while ensuring system stability and control performance. It solves the problems of model uncertainty, disturbance effects, and communication constraints faced by traditional model-dependent control methods in van der Bohr oscillator systems.

[0005] The specific technical solution of the present invention is as follows: A cost-preservation event-triggered control method for a data-driven van der Bohr oscillator system, characterized by comprising the following steps:

[0006] The Taylor expansion linearization equations for the van der Bohr oscillator system were established, and the specific steps are as follows:

[0007] According to the principles of Lagrange mechanics, neglecting the control input u(τ), the standard van der Bohr oscillator equations are:

[0008] Where τ is the time index, It is the first derivative of v(τ). is the second derivative of v(τ), μ is the nonlinear damping coefficient, and v(τ) is the voltage in the circuit;

[0009] When μ = 1, the equation simplifies to:

[0010] To control or regulate the system, an external control input u(τ) is added to the equations, resulting in the controlled van der Bohr equations, as follows:

[0011] Transforming the above second-order differential equation into state-space form, we define the state variables x1(τ) = v(τ) and... The first-order state equation of the van der Bohr oscillator system is:

[0012] A suboptimal cost-preservation control strategy was designed, and the specific steps are as follows:

[0013] Linearizing the first-order state equations of the van der Bohr oscillator system using Taylor expansion, we obtain the following equation:

[0014] in, r(x(τ), u(τ)) are linearized residuals satisfying ||r(x(τ), u(τ))||≤δ(x(τ)), where δ(x(τ)) is a known function;

[0015] A data-driven state feedback-based cost control strategy is proposed, with the following specific steps:

[0016] Run the van der Bohr oscillator system offline and collect a set of data matrix U0=[u(0) u(τ) Δ ) … u((T-1)τ Δ )],X0=[x(0) x(τ Δ ) … x((T-1)τ Δ )], Therefore, these data satisfy the equation: X1=AX0+BU0+R0, where R0 is the matrix formed by the residual terms. Δ is a known matrix;

[0017] For a state feedback controller u(τ) = Kx(τ), the closed-loop van der Bohr oscillator system can be represented as:

[0018] Where K is the controller gain, and matrix G is an equation Any solution;

[0019] If there exist decision matrices S and Y, and a positive scalar ∈ such that the following linear matrix inequality holds:

[0020] Then u(τ) = U0Gx(τ) is a cost-preserving suboptimal controller, and the upper bound of the cost is x. T (0)S -1 x(0), where, V = M(MM) T ) -1 , W is a given positive definite matrix, I is the identity matrix, * denotes the symmetric element of the matrix, and the positive definite matrices Q and R are indices. The weights are proven as follows:

[0021] A001: Choose the Lyapunov function as V(x(τ))=x T If (τ)Px(τ), then its derivative with respect to τ is:

[0022] Where P = S -1 ;

[0023] A002: Expand We can obtain:

[0024] A003: Will Substituting A002, we arrive at:

[0025] A004: Multiply both sides of A003 by P. -1 ,have:

[0026] A005: Applying the first linear matrix inequality condition proposed by Schul complement, we obtain:

[0027] A006: Combining A005 and A004, the following is introduced:

[0028] A007: Applying the second linear matrix inequality condition proposed by Schul complement, we obtain:

[0029] A008: Because so:

[0030] A009: Substituting A008 into A007, we have:

[0031] A010: Combining A009 and A006, we get:

[0032] A011: Because the linearized residuals satisfy ||r(x(τ), u(τ))||≤δ(x(τ)), there exists a positive scalar λ such that when ||x(τ)||≤λ, -x T (τ)Yx(τ)+2x T (τ)Pr(x(τ), u(τ))≤0, then A010 can be deduced as follows:

[0033] A012: Performing integration on both sides of A011 yields:

[0034] A013: Because Therefore, V(x(∞))=0, then A012 indicates End of proof;

[0035] A data-driven cost-preservation event triggering mechanism is proposed, with the following specific steps:

[0036] The designed triggering mechanism is as follows: τ l+1 =inf{τ|τ>τ l ,βe T (τ)e(τ)>σ 2 x T (τ)x(τ)+x T (τ)Qx(τ)+u T (τ)Ru(τ)},

[0037] Where, τ l It is the trigger time sequence, e(τ) = x(τ) - x(τ) l ) represents the triggering error, and β and σ are the triggering parameters to be solved;

[0038] Under the triggering mechanism, the corresponding controller is u(τ)=Kx(τ) l ), and the closed-loop system is:

[0039] because Since the row rank is full, there exists a matrix L that satisfies... Based on this, the above closed-loop system is represented as:

[0040] If there exist positive scalars σ, β, and μ such that the following linear matrix inequality holds:

[0041] Then u(τ)=U0Gx(τ) can guarantee the asymptotic stability of the closed-loop system under the proposed triggering mechanism, where, V = M(MM) T ) -1 , The proof is as follows:

[0042] B001: Choose the Lyapunov function as V(x(τ))=x T If (τ)Px(τ), then its derivative with respect to τ is:

[0043] Where P = S -1 ;

[0044] B002: Expand We can obtain:

[0045] B003: During the trigger interval, there is σ 2 x T (τ)x(τ)+x T (τ)Qx(τ)+u T (τ)Ru(τ)-βe T (τ)e(τ)>0, therefore:

[0046] B004: Based on A005, we can obtain: 2x T (τ)P(X1-R0)Gx(τ)≤-x T (τ)Qx(τ)-u T (τ)Ru(τ)-x T (τ)S -1 WS -1 x(τ),

[0047] B005: Substituting B004 into B003, we derive:

[0048] B006: Applying the linear matrix inequality conditions proposed by Schur complement, we have:

[0049] B007: Applying Young's inequality, the following equation holds:

[0050] Where Ψ = [P 0],

[0051] B008: Combining B007 and B006, the following is generated:

[0052] B009: Substituting B008 into B005, we get:

[0053] B010: Because the linearized residuals satisfy ||r(x(τ), u(τ))||≤δ(x(τ)), there exists a positive scalar λ such that when ||x(τ)||≤λ, Therefore, B009 can be used to deduce:

[0054] B011: Therefore, the closed-loop system is asymptotically stable, and the proof is complete. Attached Figure Description

[0055] Figure 1 This is a flowchart illustrating the data-driven cost-preservation event-triggered control method for the van der Bohr oscillator system described in an embodiment of the present invention.

[0056] Figure 2 The example illustrates the state response of the van der Bohr oscillator under the data-driven cost-preservation event-triggered control strategy proposed in this invention. Figure 3 Release interval curve; Detailed Implementation

[0057] 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.

[0058] like Figure 1 As shown, a data-driven van der Bohr oscillator system cost-preservation event-triggered control method includes the following steps:

[0059] Step 1: Let u(τ) randomly take values ​​in [-1, 1] and feed them into the system. Collect data offline to obtain the data matrix U0 = [u(0) u(τΔ) … u((T-1)τ] Δ )],X0=[x(0) x(τ Δ ) … x((T-1)τ Δ )],

[0060] Step 2: Choose Δ = 0.1I, T = 10, R = I, Q = 25I, W = I, and use the proposed linear matrix inequality conditions to solve for matrices S and Y, as well as the positive scalars ε, σ, β, and μ.

[0061] Step 3: Configure the controller using u(τ) = U0Gx(τ);

[0062] Step 4: Verify the trigger conditions. If the conditions are met, transmit the status data to the controller.

[0063] Step 5: Update the input u(τ) with the data after triggering to control the inverted pendulum system;

[0064] Step 6: Repeat step 3 to enter the next cycle.

[0065] An embodiment of the present invention is described below:

[0066] Consider the cost-preservation event-triggered control problem of a data-driven van der Bohr oscillator system that cannot be modeled. Its state-space model is as follows:

[0067] Figure 1 This is a flowchart of a method according to an embodiment of the present invention; applying the proposed method, Figure 2 The system state curves generated after applying the proposed method are shown. Figure 3 The release interval is shown in these curves. It can be seen that the proposed method can still achieve satisfactory tracking performance even when the van der Bohr oscillator system model is unknown.

[0068] References

[0069] [1]Peng Li,Shizhan Wang,Hongjiu Yang,Hai Zhao.Trajectory tracking andobstacle avoidance for wheeled mobile robots based on EMPC with an adaptiveprediction horizon[J].IEEE Transactions on Cybemetics,vol.52,pp.13536-13545,2022.

[0070] [2]Mouquan Shen,Xianming Wang,Song Zhu,Zhengguang Wu,TingwenHuang.Data-driven event-triggered adaptive dynamic programming control fornonlinear systems with input saturation[J].IEEE Transactions on Cybernetics,vol.54,pp.1178-1188,2023。

Claims

1. A data-driven, cost-preservation event-triggered control method for a van der Bohr oscillator system, characterized in that, Includes the following steps: The Taylor expansion linearization equations for the van der Bohr oscillator system were established, and the specific steps are as follows: According to the principles of Lagrange mechanics, neglecting the control input u(τ), the standard van der Bohr oscillator equations are: Where τ is the time index, It is the first derivative of v(τ). is the second derivative of v(τ), μ is the nonlinear damping coefficient, and v(τ) is the voltage in the circuit; When μ = 1, the equation simplifies to: To control or regulate the system, an external control input u(τ) is added to the equations, resulting in the controlled van der Bohr equations, as follows: Transforming the above second-order differential equation into state-space form, we define the state variables x1(τ) = v(τ) and... The first-order state equation of the van der Bohr oscillator system is: A suboptimal cost-preservation control strategy was designed, and the specific steps are as follows: Linearizing the first-order state equations of the van der Bohr oscillator system using Taylor expansion, we obtain the following equation: in, r(x(τ), u(τ)) are linearized residuals satisfying ||r(x(τ), u(τ))||≤δ(x(τ)), where δ(x(τ)) is a known function; A data-driven state feedback-based cost control strategy is proposed, with the following specific steps: Run the van der Bohr oscillator system offline and collect a set of data matrix U0=[u(0) u(τ) Δ ) … u((T-1)τ Δ )],X0=[x(0) x(τ Δ ) … x((T-1)τ Δ )], Therefore, these data satisfy the equation: X1=AX0+BU0+R0, where R0 is the matrix formed by the residual terms. Δ is a known matrix; For a state feedback controller u(τ) = Kx(τ), the closed-loop van der Bohr oscillator system can be represented as: Where K is the controller gain, and matrix G is an equation Any solution; If there exist decision matrices S and Y, and a positive scalar ∈ such that the following linear matrix inequality holds: Then u(τ) = U0Gx(τ) is a cost-preserving suboptimal controller, and the upper bound of the cost is x. T (0)S -1 x(0), where, V = M(MM) T ) -1 , W is a given positive definite matrix, I is the identity matrix, * denotes the symmetric element of the matrix, and the positive definite matrices Q and R are indices. The weights are proven as follows: A001: Choose the Lyapunov function as V(x(τ))=x T If (τ)Px(τ), then its derivative with respect to τ is: Where P = S -1 ; A002: Expand We can obtain: A003: Will Substituting A002, we arrive at: A004: Multiply both sides of A003 by P. -1 ,have: A005: Applying the first linear matrix inequality condition proposed by Schul complement, we obtain: A006: Combining A005 and A004, the following is introduced: A007: Applying the second linear matrix inequality condition proposed by Schul complement, we obtain: A008: Because so: A009: Substituting A008 into A007, we have: A010: Combining A009 and A006, we get: A011: Because the linearized residuals satisfy ||r(x(τ), u(τ))||≤δ(x(τ)), there exists a positive scalar λ such that when ||x(τ)||≤λ, -x T (τ)Yx(τ)+2x T (τ)Pr(x(τ), u(τ))≤0, then A010 can be deduced as follows: A012: Performing integration on both sides of A011 yields: A013: Because Therefore, V(x(∞))=0, then A012 indicates End of proof; A data-driven cost-preservation event triggering mechanism is proposed, with the following specific steps: The designed triggering mechanism is as follows: t l+1 =inf{τ|τ>τ l ,be T (τ)e(τ)>s 2 x T (τ)x(τ)+x T (τ)Qx(τ)+u T (τ)Ru(τ)}, Where, τ l It is the trigger time sequence, e(τ) = x(τ) - x(τ) l ) represents the triggering error, and β and σ are the triggering parameters to be solved; Under the triggering mechanism, the corresponding controller is u(τ)=Kx(τ) l ), and the closed-loop system is: because Since the row rank is full, there exists a matrix L that satisfies... Based on this, the above closed-loop system is represented as: If there exist positive scalars σ, β, and μ such that the following linear matrix inequality holds: Then u(τ)=U0Gx(τ) can guarantee the asymptotic stability of the closed-loop system under the proposed triggering mechanism, where, V = M(MM) T ) -1 , The proof is as follows: B001: Choose the Lyapunov function as V(x(τ))=x T If (τ)Px(τ), then its derivative with respect to τ is: Where P = S -1 ; B002: Expand We can obtain: B003: During the triggering interval, there is σ 2 x T (τ)x(τ)+x T (τ)Qx(τ)+u T (τ)Ru(τ)-βe T (τ)e(τ) > 0, thus: B004: Based on A005, we can obtain: 2x T (τ)P(X1-R0)Gx(τ)≤-x T (τ)Qx(τ)-u T (τ)Ru(τ)-x T (t)S -1 WS -1 x(τ), B005: Substituting B004 into B003, we derive: B006: Applying the linear matrix inequality conditions proposed by Schur complement, we have: B007: Applying Young's inequality, the following equation holds: Where Ψ = [P 0], B008: Combining B007 and B006, the following is generated: B009: Substituting B008 into B005, we get: B010: Because the linearized residuals satisfy ||r(x(τ), u(τ))||≤δ(x(τ)), there exists a positive scalar λ such that when ||x(τ)||≤λ, Therefore, B009 can be used to deduce: B011: Therefore, the closed-loop system is asymptotically stable, and the proof is complete.

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