A circuit system iterative learning control method for processing local lipshitz nonlinearity under unknown state

Abstract

The application discloses a circuit system iterative learning control method for processing local Lipshitz nonlinearity under unknown state. The method aims at the actual problem that the circuit system current is unknown, and constructs an adaptive gain state observer based on a reference model. An iterative length selection index is constructed, so that the output of the system and the observer does not violate the given limit range. An iterative learning controller is constructed to control the circuit system with local Lipshitz nonlinearity. Compared with the traditional control method which needs an accurate mathematical model, the control algorithm is more novel and the application condition is simple. When the method is used for repeated task operation, the control precision is effectively improved, and the method has good engineering application value.

Description

TECHNICAL FIELD

[0001] The present application relates to a control method for circuit system, in particular to a circuit system iterative learning control method for processing local Lipschitz nonlinearities under unknown states. BACKGROUND

[0002] Circuit system is the cornerstone of modern communication. For example, in mobile phone communication, from signal transmission, transmission to reception, complex radio frequency circuits, digital signal processing circuits are involved to realize information modulation, demodulation and long distance transmission, to ensure that people can communicate at any time and anywhere. Circuit system has many different complexities, mainly including multivariable and multi-physical quantity, nonlinear characteristics and complex dynamic behavior caused by various circuit devices, time-varying, high control difficulty caused by large scale, randomness caused by noise interference and the like. In view of the above several aspects of complexity, the control of circuit system is very challenging, and advanced technologies and methods need to be researched and adopted to deal with it. SUMMARY

[0003] The purpose of the present application is to provide a circuit system iterative learning control method for processing local Lipschitz nonlinearities under unknown states, which can effectively solve the problems of multivariable and multi-physical quantity, nonlinearity, time-varying, high control difficulty and randomness existing in circuit system, and achieve good control effect.

[0004] The specific technical scheme of the present application is as follows: a circuit system iterative learning control method for processing local Lipschitz nonlinearities under unknown states, comprising the following steps:

[0005] For the following circuit system:

[0006]

[0007] In the formula, x k (t)=[x 1，k (t)x 2，k (t)] T Indicates the loop current, x 1,k (t), x 2,k (t) is the substate of x k (t), k represents the iteration number, R1, R2 represents resistance, L1, L2, M represents inductance, u k (t) represents the controller, θ(t) represents the unknown matrix time-varying function, ζ(x k (t)) represents the local Lipschitz nonlinear function related to the current, A, B is consistent with the coefficient in the circuit model, and the system output is y k (t)=Cx k (t), wherein C is the system output matrix;

[0008] The reference trajectory of the system is generated by the following reference model:

[0009]

[0010] where x r (t) denotes the reference state, u r (t) denotes the reference input, y r (t) denotes the reference output;

[0011] An adaptive gain state observer based on the reference model is constructed for the practical problem that the circuit system current is unknown:

[0012] where x r,k (t) denotes the current estimation, k denotes the bounded adaptive gain, y r,k (t) denotes the observer output;

[0013] The iteration length selection index is constructed as follows:

[0014]

[0015] where T k denotes the actual running time of the system in each iteration, t∈[0, T] denotes the time variable, where T denotes the ideal running length of the system in each iteration, β1, denotes the limit of the system output and the observer output, ||.|| denotes the Euclidean norm, denotes the maximum value of ||y r (t)|| on t∈[0, T];

[0016] The state estimation error e x,k (t) and the tracking error e y，k (t) are constructed as follows:

[0017] e x，k (t) = x k (t) - x r，k (t)

[0018] e y，k (t) = y k (t) - y r，k (t)

[0019] The constructed controller is as follows:

[0020] t ≤ T k

[0021] where Let ξ(x) be an estimated value of θ(t) and be bounded. r，k (t) represents the local Lipschitz nonlinear function related to the current estimate;

[0022] The current estimates have the following relationship:

[0023]

[0024] In the formula, ψ represents the transformation matrix, and from the above, we can derive x r，k (t) is bounded, and therefore we can conclude that u k (t) is bounded;

[0025] The loop current and the controller have the following relationship:

[0026]

[0027] In the formula, α1 and α2 represent k types of functions, because u k (t) is bounded, so x k (t) is bounded;

[0028] The local Lipschitz nonlinearity has the following relationship:

[0029]

[0030] z k (t)=h(t)x k (t)+(1-h(t))x r,k (t)

[0031]

[0032] In the formula, Let α denote the local Lipschitz function, α3 denote the k-type function, and h(t) denote the function greater than zero and less than one. It is a function greater than or equal to zero, because x k (t), x r，k (t) is bounded, therefore z k (t) is bounded, thus leading to... It is bounded and satisfies Where β z It is an unknown constant greater than zero;

[0033] The design parameter update law is in the following form:

[0034]

[0035] In the formula, proj θ (.), proj L (.) indicates parameter update projection. Denotes a gain greater than zero, (.) T Denotes the transpose of a vector or matrix;

[0036] To ensure continuity during analysis, define a virtual tracking error ω in the following form x，k (t):

[0037]

[0038] Construct a composite energy function to analyze the convergence of the tracking error. The proof process is as follows:

[0039] C001: Select a composite energy function in the following form:

[0040]

[0041] C002: Wherein, where β θ ≥||θ(t)|| is an unknown constant greater than zero, λ1 is an eigenvalue greater than zero, and tr(.) denotes the trace of a matrix;

[0042] C003: Consider the case of t ≤ T k for E k (t) and E k-1 [[ID=三十七]](t) the difference ΔE k (t), assuming PB = C, where P is a positive definite matrix, where:

[0043]

[0044] C004: According to C003, the following formula can be obtained:

[0046]

[0047] C005: Consider the case of T k <t ≤ T for E k (t) and E k-1 (t) the difference ΔE k (t), where:

[0048] [[ID=六十五]]

[0049] C006: According to C005, the following formula can be obtained: <

[0050]

[0051] C007: According to C004 and C006, obtain ΔE for the case of 0 < t ≤ T k (t): ​

[0052]

[0053] C008: In the formula,

[0054] C009: Considering t ≤ T k the boundedness of E1(t):

[0055]

[0056] C010: According to C009, the boundedness of E1(t) can be obtained when t ≤ T k ;

[0057] C011: Considering T k <t ≤ T, the boundedness of E1(t):

[0058]

[0059] C012: According to C011, the boundedness of E1(t) can be obtained when T k <t ≤ T;

[0060] C013: According to C009 and C011, the boundedness of E1(t) can be obtained when 0 < t ≤ T;

[0061] C014: Give the proof of the convergence of the tracking error:

[0062]

[0063] C015: From the positive definiteness of E k (t) and the boundedness of E1(t), the convergence form of the virtual estimation error can be obtained as follows:

[0064] ​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

[0066] Figure 1 This is a flowchart of a method according to an embodiment of the present invention;

[0067] Figure 2 This is a system output tracing diagram using the method proposed in this invention;

[0068] Figure 3 The observer output tracking graph using the method proposed in this invention;

[0069] Figure 4 The tracking error convergence graph is shown using the method proposed in this invention. Detailed Implementation

[0070] The present invention will be further illustrated below with reference to the 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.

[0071] like Figure 1 As shown, an iterative learning control method for circuit systems with local Lipschitz nonlinearity under unknown states includes the following steps:

[0072] Step 1: Set initial parameter values;

[0073] Step 2: Update algorithm parameters;

[0074] Step 3: Generate control input u in real time based on the updated algorithm parameters. k (t);

[0075] Step 4: Based on the control input u k The tracking error generated by (t) is used to synchronously update the algorithm parameters;

[0076] Step 5: Repeat steps 3 and 4 until the current iteration ends and the next iteration begins;

[0077] Step Six: An embodiment of the present invention is described below:

[0078] Consider an iterative learning control method for circuit systems with local Lipschitz nonlinearity under unknown states. The corresponding mathematical model is as follows:

[0079]

[0080] The system parameters are as follows:

[0081] R1 = 1Ω, R2 = 1Ω are circuit resistors, L1 = 0.36H, L2 = 0.5H are inductors, M = 0.15H is a mutual inductor, θ(t)ξ(x) k (t))=x 2，ksin 3 t+0.8sin 2 tsinx 1，k ;

[0082] The expected reference input is: The ideal operating length of the system is T, which is 3 seconds. The system output limit and the observer output limit are β1 = 2 and β2 = 1.65, respectively.

[0083] Figure 1 This is a flowchart of a method according to an embodiment of the present invention; applying the proposed method, Figure 2 , 3 These represent the system output and observer output tracking graphs, respectively. Figure 4 The graphs show the convergence of tracking error. These three graphs demonstrate that the proposed method performs well in circuit systems, achieving satisfactory tracking performance after the 10th iteration.

[0084] References

[0085] [1]

[0086] [2] D.Shen, J.-X.Xu. Adaptive learning control for nonlinear systems with randomly varying iteration lengths. IEEE Transactions on Neural Networks and Learning Systems, vol.30, pp.1119-1132, 2019.

Claims

1. An iterative learning control method for circuit systems handling local Lipschitz nonlinearity under unknown states, characterized in that, Includes the following steps: To address the problem of unknown current in a circuit system, an adaptive gain state observer based on a reference model is constructed, as follows: For the following circuit systems: In the formula, x k (t)=[x 1,k (t) x 2,k (t)] T Represents the loop current, x 1,k (t),x 2,k (t) is x k The substates of (t), k represents the iteration number, R1 and R2 represent resistances, L1, L2, and M represent inductances, and u k (t) represents the controller, θ(t) represents the time-varying function of the unknown matrix, and ξ(x) represents the time-varying function of the unknown matrix. k (t) represents the local Lipschitz nonlinear function related to the current, A and B are consistent with the coefficients in the circuit model, and the system output is y. k (t)=Cx k (t), where C is the system output matrix; The system's reference orbit is generated by the following reference model: In the formula, x r (t) represents the reference state, u r (t) represents the reference input, y r (t) represents the reference output; To address the practical problem of unknown current in circuit systems, an adaptive gain state observer based on a reference model is constructed: In the formula, x r,k (t) represents the estimated current value. Let y represent the bounded adaptive gain. r,k (t) represents the observer output; An iteration length selection metric is constructed to ensure that the outputs of the system and the observer do not violate a given constraint, as follows: The selection criteria for the iteration length are as follows: In the formula, T k Let t represent the actual running time of each system iteration, where t∈[0,T] represents the time variable, and T represents the ideal running length of each system iteration. To constrain the system output and observer output, ||.|| denotes the Euclidean norm. It represents ||y| on t∈[0,T] r The maximum value of (t)||; Construct an iterative learning controller to control a circuit system with local Lipschitz nonlinearity.

2. The iterative learning control method for a circuit system with local Lipschitz nonlinearity under unknown state conditions, as described in claim 1, is characterized in that... Construct an iterative learning controller to control a circuit system with local Lipschitz nonlinearity: Construct the state estimation error e in the following form x,k (t) and tracking error e y,k (t): e x,k (t)=x k (t)-x r,k (t) e y,k (t)=y k (t)-y r,k (t) The constructed controller is shown below: In the formula, Let ξ(x) be an estimated value of θ(t) and be bounded. r,k (t) represents the local Lipschitz nonlinear function related to the current estimate; The current estimates have the following relationship: In the formula, ψ represents the transformation matrix, and from the above, we can derive x r,k (t) is bounded, and therefore we can deduce u k (t) is bounded; The loop current and the controller have the following relationship: In the formula, α1 and α2 represent k types of functions, because u k (t) is bounded, so x k (t) is bounded; The local Lipschitz nonlinearity has the following relationship: z k (t)=h(t)x k (t)+(1-h(t))x r,k (t) In the formula, Let α denote the local Lipschitz function, α3 denote the k-type function, and h(t) denote the function greater than zero and less than one. It is a function greater than or equal to zero, because x k (t),x r,k (t) is bounded, therefore z k (t) is bounded, thus leading to... It is bounded and satisfies Where β z It is an unknown constant greater than zero; The design parameter update law is in the following form: In the formula, proj θ (.),proj L (.) indicates parameter update projection. This indicates a gain greater than zero. (.) T Represents the transpose of a vector or matrix; To ensure continuity in the analysis, a virtual estimation error ω is defined in the following form. x,k (t): The convergence of the tracking error is analyzed by constructing a composite energy function, and the proof is as follows: B001: Select the following form of composite energy function: B002: In the formula, Where β θ ≥||θ(t)|| is an unknown constant greater than zero, λ1 is an eigenvalue greater than zero, and tr(.) represents the trace of the matrix; B003: Consider t≤T k In case E k (t) and E k-1 The difference ΔE between (t) k (t), assume PB = C, where P is a positive definite matrix, and: B004: Based on B003, the following formula can be obtained: B005: Consider T k E < t ≤ T k (t) and E k-1 The difference ΔE between (t) k (t), where: B006: Based on B005, the following formula can be obtained: B007: Based on B004 and B006, we can obtain ΔE for the case 0 < t ≤ T. k (t): B008: In the formula, B009: Consider t≤T k Boundedness of E1(t) under the following conditions: B010: According to B009, E1(t) is valid for t ≤ T. k Boundedness under certain conditions; B011: Consider T k Boundedness of E1(t) when t < T: B012: According to B011, E1(t) at T k Boundedness in the case of <t≤T; B013: According to B009 and B011, the boundedness of E1(t) in the case of 0 < t ≤ T can be obtained; B014: Provide a proof of the convergence of the tracking error: B015: By E k Given the positive definiteness of E(t) and the boundedness of E1(t), the convergent form of the virtual estimation error can be obtained as follows: B016: From B015, we can obtain... Therefore, we can obtain e y,k (t) in t∈[0,T) k It converges on [0,T], and on t∈[0,T] k [Up x] r,k (t)=x r (t),y k (t)=y r,k (t)=y r (t), which indicates that in T k Moment y k (t),y r,k (t) did not touch the boundary, which can be explained by contradiction. That is, the tracking error converges along the iteration axis.

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