Circuit system data driving event trigger control method based on Koopman operator

Abstract

The invention discloses a circuit system data driving event trigger control method based on a Koopman operator. According to the method, a circuit system linearization method based on the Koopman operator is designed, then a data-based event trigger control method is established, the effectiveness of the proposed method is demonstrated, and the problem of data-driven event trigger control of the circuit system based on the Koopman operator and incapable of modeling is effectively relieved.

Description

Technical Field

[0001] The present invention relates to a control method for a circuit system, and particularly to a data-driven event-triggered control method for a circuit system based on the Koopman operator. Background Art

[0002] Circuit systems are widely used in important industries such as automobiles, robots, and intelligent manufacturing. However, due to the inherent unmodeled dynamic characteristics, it is difficult or even impossible to obtain an accurate model of the system. How to design a controller only using the measured input and output data is a very meaningful work. Based on this idea, researchers have carried out a large number of studies on data-driven controller design and achieved a series of results.

[0003] However, the above research work mainly focuses on the control methods of linear circuit systems, without considering the influence of nonlinear systems and finite bandwidth, which has certain limitations. In order to reduce the limitations, it is necessary to study the data-driven event-triggered control method for circuit systems based on the Koopman operator. Summary of the Invention

[0004] The object of the present invention is to propose a data-driven event-triggered control method for a circuit system based on the Koopman operator, which can effectively solve the problem of data-driven event-triggered control for non-linear circuit systems that cannot be modeled.

[0005] The specific technical solution of the present invention is as follows: A data-driven event-triggered control method for a circuit system based on the Koopman operator, characterized by comprising the following steps:

[0006] A linearization method for a circuit system based on the Koopman operator is designed, and the specific steps are as follows:

[0007] For a parallel resistor-inductor-capacitor circuit system, its characteristics are: x(κ + 1) = f(x(κ)) + Hu(κ)

[0008] In the formula, x1(κ) is the capacitor voltage, x2(κ) is the inductor current, u(κ) is the power supply voltage, H is the signal amplifier, and f(·) is the unknown non-linear circuit structure;

[0009] Using the above characteristics, a linearized state space model of the parallel resistor-inductor-capacitor circuit system is established using the Koopman operator as: Φ(x(κ + 1)) = AΦ(x(κ)) + Bu(κ) + Ed(κ)

[0010] Where, Φ(x(κ)) is a known observation function, A and B are unknown system matrices, E is an unknown disturbance matrix satisfying ‖E‖≤d1, d1 is a known positive constant, d(κ) is an unknown linearization error satisfying ‖d(κ)‖≤d2, d2 is a known positive constant. So far, the linearization method of the circuit system based on the Koopman operator is designed;

[0011] A data-based event-triggered control method is established and the effectiveness of the proposed method is demonstrated. The specific steps are as follows:

[0012] Define the event-triggered sequence κ l ={1, 2, …}, and construct the following event-triggered condition: e(κ) T e(κ) > σ 2 Φ(x(κ)) T Φ(x(κ))}

[0013] Where, e(κ)=Φ(x(κ l )) - Φ(x(κ)) is the triggering error, Φ(x(κ l )) is the observation function at the triggering moment, and σ is the threshold parameter to be solved;

[0014] Run the resistor-inductor-capacitor parallel circuit system offline, and collect a set of data set matrices X = [x(1) x(2) x(3) … x(T + 1)] and U0 = [u(1) u(2) u(3) … u(T)], where T is the sampling time; calculate the matrices Φ0 = [Φ(x(1)) Φ(x(2)) … Φ(x(T))] and Φ1 = [Φ(x(2)) Φ(x(3)) … Φ(x(T + 1))], define the matrix W0 = [d(1) d(2) d(3) … d(T)], W0 is unknown. Obviously, Φ1 = AΦ0 + BU0 + EW0; for the control law u(κ) = KΦ(x(κ)), K is the control gain. According to the properties of these measured data, it can be obtained that G is an unknown matrix, I N×N is the identity matrix. Therefore, the closed-loop system equation is shown as follows:

[0015] If the closed-loop system is to be stable, there should exist matrices Q and L, positive constants σ and ∈ such that the following conditions are satisfied: and U0L = U0Q(Φ0Q) -1

[0016] Where, 0 N×p ,0 p×N ,0p×T and 0 N×T is the zero matrix, I T×T is the identity matrix, * represents the matrix in the symmetric position, and K = U0Q(Φ0Q) under this condition -1 can ensure the stability of the resistor - inductor - capacitor parallel circuit system. The proof process is as follows:

[0017] A001: Select the Lyapunov function V(κ) as follows: V(κ) = Φ(x(κ)) T P -1 Φ(x(κ))

[0018] where P -1 is a positive definite symmetric matrix;

[0019] A002: According to the proposed triggering rule, we can get: e(κ) T e(κ) ≤ σ 2 Φ(x(κ)) T Φ(x(κ))

[0020] A003: Combining A002 and A001, we get:

[0021] where Ω 11 =(A + BK) T P -1 (A + BK)-P -1 +(1 + σ 2 )I N×N , Ω 12 =(A + BK) T P -1 E, Ω 13 =(A + BK) T P -1 BK, Ω 22 =E T P -1 E, Ω 23 =E T P -1 BK, Ω 33 =(BK) T P -1 BK - I N×N ;

[0022] A004: Note that is equivalent to the following formula:

[0023] A005: Using the Schur complement lemma for A004, we can get the following formula:

[0024] A006: Multiply by the diagonal matrix diag{P I on both sides of A006 p×p I N×N P I N×N}, and the following equation can be obtained:

[0025] A007: There exists a matrix such that Therefore, it can be obtained that:

[0026] A008: Substitute A007 into A006, and the following can be obtained:

[0027] In the formula,

[0028] A009: Because ‖E‖ ≤ d1, so

[0029] A010: Using the Young's inequality, the following equation can be obtained:

[0030] A011: Substitute A010 and A009 into A008, and the following can be obtained:

[0031] A012: Note that the constraints and can be decomposed into the following: K = U0G, 0 N×N = Φ0G, K = U0L, 0 N×N = Φ0L

[0032] A013: By defining Q = GP, A012 is equivalent to P = Φ0Q, K = U0Q(Φ0Q) -1 , U0L = U0Q(Φ0Q) -1 , 0 N×N = Φ0L. Based on these equivalent relationships, it can be concluded that the proposed conditions can deduce A011, that is, K = U0Q(Φ0Q) -1 can ensure the stability of the resistor-inductor-capacitor parallel circuit system;

[0033] A014: So far, the design of the data-based event-triggered control method and the demonstration of the effectiveness of the proposed method are completed. Description of the Drawings

[0034] Figure 1It is the flowchart of the method of the embodiment of the present invention;

[0035] Figure 2 It is the state curve diagram of the embodiment adopting the method proposed by the present invention;

[0036] Figure 3 It is the number of trigger times of the trigger condition adopted by the embodiment of the present invention; Detailed implementation manners

[0037] The present invention will be further clarified below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and not to limit the scope of the present invention. After reading the present invention, various equivalent forms of modification by those skilled in the art fall within the scope defined by the appended claims of this application.

[0038] As Figure 1 shown, a data-driven event-triggered control method for a circuit system based on the Koopman operator includes the following steps:

[0039] Step 1: Let u(κ)=rand(1) be input into the system, collect data offline, that is, U, X, and calculate Φ0, Φ1;

[0040] Step 2: Solve Q and σ using the collected data;

[0041] Step 3: Configure the controller using K = U0Q(Φ0Q) -1 ;

[0042] Step 4: Verify the trigger condition, and if it is satisfied, transmit the data into the controller;

[0043] Step 5: Update the input u(κ) using the triggered data to control the circuit system;

[0044] Step 6: Repeat Step 3 to enter the next cycle.

[0045] The following introduces an embodiment of the present invention:

[0046] Consider the data-driven control problem of a performance-triggered circuit system that cannot be modeled, and its discretized state-space model is: x1(κ + 1)=x1(κ)+x2(κ)+u(κ) x2(κ + 1)=+x2(κ)+x1(κ)x2(κ)+u(κ)

[0047] Figure 1 It is the flowchart of the method of the embodiment of the present invention; Applying the proposed method, Figure 2 the state curve diagram after applying the proposed method to the system is shown. It can be seen from this diagram that the proposed method can obtain a satisfactory event-triggered control performance without the need for system model information.

[0048] References

[0049] [1]De Persis C,Tesi P.Formulas for data-driven control:Stabilization,optimality,and robustness[J].IEEE Transactions on Automatic Control,2019,65(3):909-924.

[0050] [2]Berberich J, J,Müller M A,et al.Data-driven model predictivecontrol with stability and robustness guarantees[J].IEEE Transactions onAutomatic Control,2020,66(4):1702-1717.

Claims

1. A data-driven event-triggered control method for a circuit system based on the Koopman operator, characterized in that It includes the following steps: A linearization method of the circuit system based on the Koopman operator is designed, and the specific steps are as follows: For the parallel circuit system of resistor, inductor and capacitor, its characteristics are: x(κ + 1) = f(x(κ)) + Hu(κ) wherein, x1(κ) is the capacitor voltage, x2(κ) is the inductor current, u(κ) is the power supply voltage, H is the signal amplifier, and f(·) is the unknown non-linear circuit structure; Using the above characteristics, the linearized state-space model of the parallel circuit system of resistor, inductor and capacitor is established by using the Koopman operator as: Φ(x(κ + 1)) = AΦ(x(κ)) + Bu(κ) + Ed(κ) In the formula, Φ(x(κ)) is a known observation function, A and B are unknown system matrices, E is an unknown disturbance matrix satisfying ‖E‖ ≤ d1, d1 is a known positive constant, d(κ) is an unknown linearization error satisfying ‖d(κ)‖ ≤ d2, d2 is a known positive constant. So far, the linearization method of the circuit system based on the Koopman operator is designed; A data-based event-triggered control method is established and the effectiveness of the proposed method is demonstrated. The specific steps are as follows: Define the event trigger sequence κ l = {1, 2, …}, and construct the following event trigger conditions: e(κ) T e(κ) > σ 2 Φ(x(κ)) T Φ(x(κ))} where \(e(\kappa)=\varPhi(x(\kappa l )) - \varPhi(x(\kappa))\) is the triggering error, \(\varPhi(x(\kappa l ))\) is the observation function at the triggering moment, and \(\sigma\) is the threshold parameter to be solved; Offline running resistor-inductor-capacitor parallel circuit system, collect a set of dataset matrices \(X = [x(1)x(2)x(3)\cdots x(T + 1)]\) and \(U_0=[u(1)u(2)u(3)\cdots u(T)]\), where \(T\) is the sampling time; calculate matrices \(\varPhi_0=[\varPhi(x(1))\varPhi(x(2))\cdots\varPhi(x(T))]\) and \(\varPhi_1=[\varPhi(x(2))\varPhi(x(3))\cdots\varPhi(x(T + 1))]\), define matrix \(W_0 = [d(1)d(2)d(3)\cdots d(T)]\), \(W_0\) is unknown, obviously, \(\varPhi_1=A\varPhi_0 + BU_0+EW_0\); for the control law \(u(\kappa)=K\varPhi(x(\kappa))\), \(K\) is the control gain, according to the properties of these measured data, it can be obtained that \(G\) is an unknown matrix, \(I\) N×N is the identity matrix, therefore, the closed-loop system equation is as follows: If the closed-loop system is to be stable, there must exist matrices Q and L, and positive constants σ and ∈ such that the following conditions are satisfied: and U0L = U0Q(Φ0Q) -1 In the formula, 0 N×p , 0 p×N , 0 p×T and 0 N×T are zero matrices, I T×T is the identity matrix, * represents the matrix in the symmetric position, and K = U0Q(Φ0Q) under this condition -1 can ensure the stability of the resistor-inductor-capacitor parallel circuit system. The proof process is as follows: A001: Select the Lyapunov function V(κ) as follows: V(κ) = Φ(x(κ)) T P -1 Φ(x(κ)) where P -1 is a positive definite symmetric matrix; A002: According to the proposed triggering rule, it can be obtained that: e(κ) T e(κ) ≤ σ 2 Φ(x(κ)) T Φ(x(κ)) A003: Combining A002 and A001, it can be obtained that: where, Ω 11 =(A + BK) T P -1 (A + BK)-P -1 +(1 + σ 2 )I N×N , Ω 12 =(A + BK) T P -1 E, Ω 13 =(A + BK) T P -1 BK, Ω 22 =E T P -1 E, Ω 23 =E T P -1 BK, Ω 33 =(BK) T P -1 BK - I N×N ; A004: Notice that is equivalent to the following formula: A005: Using the Schur complement lemma for A004, the following formula can be obtained: A006: Multiply the diagonal matrices diag{P I} on the left and right of A006 respectively, and the following equation can be obtained: p×p I N×N P I N×N}, and the following equation can be obtained: A007: There exists a matrix such that Therefore, it can be obtained that: A008: Substituting A007 into A006, it can be obtained that: Wherein, A009: Since ‖E‖ ≤ d1, so A010: Using the young inequality, the following formula can be obtained: A011: Substituting A010 and A009 into A008, it can be obtained that: A012: Note the constraint and can be decomposed into the following formula: K = U0G, 0 N×N = Φ0G, K = U0L, 0 N×N = Φ0L A013: By defining Q = GP, A012 is equivalent to P = Φ0Q, K = U0Q(Φ0Q) -1 , U0L = U0Q(Φ0Q) -1 , 0 N×N = Φ0L, based on these equivalence relations, it can be concluded that the proposed conditions can deduce A011, that is, K = U0Q(Φ0Q) -1 can ensure the stability of the resistor-inductor-capacitor parallel circuit system; A014: So far, the design of the data-based event-triggered control method and the demonstration of the effectiveness of the proposed method are completed.

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