Nonlinear system event triggering model-free adaptive control method based on iterative learning

Abstract

In order to reduce the data transmission and calculation frequency of iterative learning control, the invention provides an event-triggered model-free iterative learning control method for a repeated nonlinear system, which comprises the following steps: firstly, converting the nonlinear system into a virtual linear system through a pseudo-partial derivative, and then, on the basis of a virtual linear model, carrying out iterative learning control on the nonlinear system; an event triggering condition based on the direction of an iteration axis is designed. Adaptive parameters are introduced into the triggering condition, so that the system improves the real-time performance by dynamically adjusting the triggering condition. And then respectively constructing a pseudo-partial derivative estimated value and a target function of control input, minimizing the target function to obtain the pseudo-partial derivative estimated value and a control input updating rate, and expanding the algorithm to a multiple-input multiple-output nonlinear system. Traditional iterative learning control can effectively improve tracking precision of repeated tasks by extracting rules from historical execution data and optimizing control input by using task repetition characteristics, but depends on a global updating mechanism of a fixed period, and may cause redundant calculation or communication burden in non-uniform sampling or resource limited scenes. By designing an iterative updating strategy triggered by an event, the system can selectively utilize historical data under a dynamic triggering condition, and only executes learning and optimization when necessary, so that the resource consumption is reduced while the control precision is guaranteed.

Description

Technical Field

[0001] In order to reduce the data transmission and calculation frequency in traditional iterative learning control, the present invention proposes an event-triggered model-free adaptive control method for nonlinear systems based on iterative learning for repetitive nonlinear systems. Background Art

[0002] Iterative learning control effectively improves the tracking accuracy of repetitive tasks by leveraging the repetitive nature of tasks, extracting patterns from historical execution data, and optimizing control inputs. However, its reliance on a fixed-period global update mechanism can lead to redundant computation or communication overhead in non-uniform sampling or resource-constrained scenarios. By designing an event-triggered iterative update strategy, the system can selectively utilize historical data under dynamic triggering conditions, performing learning and optimization only when necessary, thereby ensuring control accuracy while reducing resource consumption. Summary of the Invention

[0003] To address the model-dependent and computationally intensive problems of iterative learning control in nonlinear systems, an event-triggered model-free adaptive iterative learning control method is proposed. This method effectively reduces the system's computational load by dynamically adjusting the update frequency of the control input.

[0004] The specific technical solution of the present invention is as follows: A nonlinear system event-triggered model-free adaptive control method based on iterative learning, comprising the following steps:

[0005] For the following discrete-time nonlinear system: y k (t+1)=f(y k (t),...,y k (tn y ),u k (t),...,u k (tn u )),

[0006] Where, Indicates the system output, represents the control input, the nonlinear function f(·) is continuously differentiable, t∈{0, 1, ..., N} represents the sampling time, It's the end time, is the number of iterations, n y and n u is a positive integer; represents a real number, represents a positive integer;

[0007] There is a time-varying parameter φ associated with the iteration k (t), is the pseudo partial derivative, the system can be written as the following dynamic linearization model: y k (t+1)=y k-1 (t+1)+φ k (t)Δu k (t),

[0008] Where, φ k (t) For any time k and number of iterations i, Δy is bounded. k (t) = y k (t)-y k-1 (t), Δu k (t) = u k (t)-u k-1 (t);

[0009] Design the adaptive trigger function as follows:

[0010] Where, event trigger error k∈[k l , k l+1 ), δ k (t) is the iteratively changing adaptive event triggering condition parameter, {k l}, l = 0, 1, ... represents the event-triggered iterative sequence generated by the event-triggered function, and its update mechanism is:

[0011] Where, e k (t) = y d (t)-y k (t) represents the tracking error, y d (t) is the expected trajectory, δ k The initial value δ0(t) of (t) is given and δ0∈[0,1).σ and M represent the step size and threshold signal, respectively, which are usually positive constants;

[0012] Next, design an event-triggered nonlinear controller. l When , the objective function is defined as follows:

[0013] Convert the above formula

[0013] to Taking the derivative and setting it equal to 0, we get:

[0014] Where, Can be abbreviated as ρ s As a step size factor, it makes the control update rate more general. When k∈(k l-1, k l), the actuator is not triggered, so Therefore, the designed event-triggered iterative learning control input update law is:

[0015] Where, Can be abbreviated as is φ k (t) estimate;

[0016] In order to obtain parameter estimates Design a and The relevant objective function is:

[0017] Similarly, optimizing the above formula

[0017] yields Update rate:

[0018] Where, 0<η s <2 is the step size factor; in order to make the parameter estimation law have a strong tracking ability for iterative changes and time-varying parameters, and to ensure Non-zero, the following reset algorithm is proposed: if or

[0019] Where is a small enough positive constant and smaller than φ k (t), sign(·) is the sign function, and the controller is obtained as: if or

[0020] Using the proposed event-triggered iterative learning control scheme, if 0<η s <2,μ s >0 and 0<(ρ s b s / (λ s ) 1 / 2 )<4, the parameter estimation can be guaranteed It is bounded, and the corrected tracking error e(k, i) gradually converges to 0 as the number of iterations increases. The proof process is as follows:

[0021] C001: First, prove Boundedness, at the triggering moment, (k, i) = (k, i l ),Will Update rate combined with linear model:

[0022] C002: Subtract φ from both sides of B001 k (t) obtain:

[0023] C003: Δφ k (t) = φ k (t)-φ k-1 (t);

[0024] C004: Select 0<η s <2 and μ s > 0, there must be a positive constant d for all k and t s So that:

[0025] C005:

[0026] C006: Based on C004 and C002, and We can get:

[0027] C007: If at the triggering moment k=k l , then Δu k-1 (t)≠0, so d s <1 is always true. According to formula B006, is bounded, since the parameter φ k (t) is bounded, so it is easy to get is also bounded; in another case, during the trigger interval k∈(k l-1， k l ), the estimated value of the pseudo partial derivative The same as the last trigger moment, so is bounded, so we can get Boundedness on k∈{0, 1, 2, ...};

[0028] C008: Next, we prove the convergence of the tracking error. According to the linearized data model and the definition of the tracking error, it is the expected trajectory, which can be restated as: e k (t+1)=y d (t+1)-y k-1 (t+1)-φ k (t)Δu k (t) =ek-1 (t+1)-φ k (t)Δu k (t).

[0029] C009: At the triggering moment k=k l In this case, the control input update rate is substituted into C008 to obtain:

[0030] C010: Take the norm of both sides of C009, then:

[0031] C011: According to the inequality ab≤(1 / 2)a 2 +(1 / 2)b 2 ,and It can be seen that:

[0032] C012: According to 0<(ρ s b s / (λ s ) 1 / 2 )<4, C011 can be written as the following inequality:

[0033] C013: where d1 is a constant;

[0034] C014: According to the formulas C012 and C010, we can get:

[0035] C015: Substitute the control input update rate into B014;

[0036] C016: In summary, the tracking error of the system can converge to zero at the triggering moment, that is: For other cases k∈(k l-1 , k l ), the control input is the same as the previous iteration, so the tracking error is still converged within the trigger interval; therefore, the tracking error e k (t) gradually converges to 0 on k∈{0, 1, 2, ...} as the number of iterations increases. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 is a flow chart of a method according to an embodiment of the present invention;

[0038] Figure 2is the total number of event triggering moments in each iteration under different triggering thresholds of the method proposed in the present invention;

[0039] Figure 3 is the total number of event triggering moments in each iteration under different triggering thresholds of the method proposed in the present invention; DETAILED DESCRIPTION

[0040] The present invention is further illustrated below with reference to the examples. It should be understood that these examples 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, modifications of various equivalent forms of the present invention made by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0041] like Figure 1 As shown in FIG, an event-triggered model-free adaptive control method for nonlinear systems based on iterative learning includes the following steps:

[0042] Step 1: Set the initial value of the parameter;

[0043] Step 2: Update algorithm parameters;

[0044] Step 3: Based on the updated algorithm parameters, if the trigger condition is met, the control input u is generated in real time. k (t). Otherwise, maintain the control input u of the previous iteration k (t) = u k-1 (t);

[0045] Step 4: According to the control input u k (t) The tracking error generated is used to update the algorithm parameters synchronously;

[0046] Step 5: Repeat steps 3 and 4 until the current iteration ends and the next iteration begins;

[0047] Step 6: An embodiment of the present invention is described below: Consider a control problem of a non-affine nonlinear system, the corresponding mathematical model is:

[0048]

[0049] The desired reference orbit is: y d (k) = 0.7sin(kπ / 30) + 0.3cos(kπ / 10); For any iteration i, set the initial condition to y k (1) = y d (1), the control signal for the first iteration is set to u1(t) = 0, t∈{0, 1, ..., T}. The other controller parameters in (4-13)-(4-15) are T = 200, ρ s =0.5,λ s=0.7, μ s =η s =1 and ε=10 -5 In addition, the step size is selected to be σ=1×10 -3 ,σ=5×10 -3 ,σ=1×10 -4 .

[0050] Figure 1 Flowchart of a method according to an embodiment of the present invention; applying the proposed method, the figure shows the total number of event triggering moments in each iteration under different triggering thresholds of the proposed method of the present invention; Figure 3 is the average error of the method proposed in this invention under different step lengths; it can be seen from these two figures that the proposed method has a good application effect in nonlinear systems, and after the 30th iteration, a satisfactory tracking performance can be obtained.

[0051] References

[0052] [1] Wang

[0053] [2]Hu Y, Yan H, Zhang H, et al. Adaptive Neural Network Output-FeedbackControl for Uncertain Nonlinear Systems via Event-Triggered Output[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2024.

Claims

1. An event-triggered model-free adaptive control method for nonlinear systems based on iterative learning, characterized in that: The following steps are involved: The nonlinear system is converted into a virtual linear system through pseudo partial derivatives, as follows: For the following discrete-time nonlinear system: y k (t+1)=f(y k (t),...,y k (t-n y ),u k (t),…,u k (t-n u )), Where, Indicates the system output, represents the control input, the nonlinear function f(·) is continuously differentiable, t∈{0, 1, ..., N} represents the sampling time, It's the end time, is the number of iterations, n y and n u is a positive integer; represents a real number, represents a positive integer; There is a time-varying parameter φ associated with the iteration k (t), is a pseudo partial derivative, the system can be written as the following dynamic linearization model: k (t+1)=y k-1 (t+1)+φ k (t)Δu k (t), Where, φ k (t) For any time k and number of iterations i, Δu is bounded. k (t) = u k (t)-u k-1 (t) represents the input deviation between the kth and k-1th times; Construct an adaptive trigger mechanism as follows: Design the adaptive trigger function as follows: Where, event trigger error k∈[k l , k l+1 ), δ k (t) is the iteratively changing adaptive event triggering condition parameter, {k l }, l = 0, 1, ... represents the event-triggered iterative sequence generated by the event-triggered function, and its update mechanism is: Improve real-time performance by dynamically adjusting trigger conditions and introducing adaptive trigger condition parameters δ k (t), which is designed to: Where, e k (t) = y d (t)-y k (t) represents the tracking error, y d (t) is the expected trajectory; δ k The initial value δ0(t) of (t) is given and δ0∈[0,1); σ and M represent the step size and threshold signal, respectively, which are usually positive constants; Next, design an event-triggered nonlinear controller. l When , the objective function is defined as follows: Put the above formula Taking the derivative and setting it equal to 0, we get: Where, Can be abbreviated as λ s is an adjustable parameter, ρ s As a step size factor, it makes the control update rate more general; when k∈(k l-1 , k l ), the actuator is not triggered, so Therefore, the designed event-triggered iterative learning control input update law is: Where, Can be abbreviated as is φ k (t) estimate; In order to obtain parameter estimates Design a and The relevant objective function is: Similarly, optimizing the above formula yields Update rate: Where, 0<η s <2 is the step size factor, μ s is an adjustable parameter. In order to make the parameter estimation law have a strong tracking ability for iterative changes and time-varying parameters, and to ensure Non-zero, the following reset algorithm is proposed: if or Where ε is a sufficiently small positive constant and is smaller than φ k (t), sign(·) is the sign function; Therefore, the proposed event-triggered iterative learning control method is summarized as follows: if or Using the proposed event-triggered iterative learning control scheme, if 0<η s <2,μ s >0 and 0<(ρ s b s / (λ s ) 1 / 2 )<4, the parameter estimation can be guaranteed Is bounded, the tracking error e k (t) gradually converges to 0 as the number of iterations increases.

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