Hybrid event triggered iterative learning control method of variable iteration length nonlinear system

Abstract

The invention discloses an event triggering ILC of a mixed triggering strategy for a nonlinear system of which the iteration length is dynamically changed due to external interference or internal parameter fluctuation. The event triggering mechanism comprises a relative threshold strategy and a zero error tracking strategy, and the tracking performance of the system is ensured while the communication and calculation frequency is reduced. A Bernoulli distribution function is introduced to describe the probability of the random variable iteration length, and a corrected input updating strategy is provided. 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] Aiming at nonlinear systems whose iteration length changes dynamically due to external interference or internal parameter fluctuations, the present invention proposes an event-triggered iterative learning control scheme with a hybrid triggering strategy. Background Art

[0002] Iterative learning control can effectively improve the tracking accuracy of repetitive tasks by leveraging the repetitive nature of tasks, extracting patterns from historical execution data, and optimizing control inputs. However, it relies on a fixed-period global update mechanism, which may lead to redundant computation or communication burdens 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 reducing resource consumption while ensuring control accuracy. However, existing event-triggered iterative learning control methods rarely dynamically adjust triggering conditions in response to environmental changes, making it difficult to balance real-time performance and control accuracy. Therefore, research on the collaborative design of dynamic adaptive event triggering mechanisms and iterative learning update laws, while also addressing practical issues such as variable iteration length, will have important theoretical value and industrial application potential. Summary of the Invention

[0003] This paper proposes an event-triggered iterative learning control scheme with a hybrid triggering strategy for nonlinear systems whose iteration lengths vary dynamically due to external interference or internal parameter fluctuations. This scheme effectively addresses the problem of non-uniform periods in applications such as industrial robots and production lines, where fixed iteration lengths are ineffective in addressing these dynamic variations due to external interference or internal parameter fluctuations. This approach achieves excellent control results.

[0004] The specific technical solution of the present invention is as follows: an event-triggered iterative learning control solution with a hybrid trigger strategy, comprising the following steps:

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

[0006] Where i is the iteration index, k∈[0, 1, ..., T d ] is the sampling time, T d is the expected iteration length; u(k, i)∈R 1 and y(k,i)∈R 1 are the control input and output, respectively, and f(·) is an unknown nonlinear function;

[0007] There is a time-varying parameter φ(k, i) related to the iteration, which is a pseudo partial derivative. The system can be written as the following dynamic linearization model:

[0008] Where φ(k, i) is bounded for any time k and iteration number i, Δy(k+1, i) = y(k+1, i) - y(k+1, i - 1), Δu(k, i) = u(k, i) - u(k, i - 1);

[0009] In order to describe the randomness of the iteration length, a random variable γ(k, i) is defined, k∈[0 T d ] and satisfies the Bernoulli distribution:

[0010] Where A m It is the event that the iterative process ends at time k = T1 + m, where m∈{1,..., T d -T1}, from which we can get:

[0011] Since γ(k, i) satisfies the Bernoulli distribution, the expectation of γ(k, i) is as follows: E{γ(k,i)}=1·p(k)+0·(1-p(k))=p(k)

[0012] A hybrid trigger strategy is adopted, and the trigger mechanism is expressed as:

[0013] In the formula is the event triggering error, i l represents an event-triggered iterative sequence, l = 0, 1, ..., tracking error e(k, i) = y d (k)-y(k, i), σ indicates that the threshold signal is an adjustable positive constant;

[0014] Design an event-triggered nonlinear controller. When the iteration length is fixed, design u(k, i) and parameter estimates as follows: The relevant objective function is: J(u(k,i l ))=|y d (k+1)-y(k+1,i l )| 2 +λ|u(k,i l )-u(k,k l-1 )| 2 ,

[0015] Taking the derivative of the above formula and setting it equal to 0, we get u(k, i) and parameter φ k Estimated update rate of (t): u(k,il )=u(k,i l-1 )+P(k,i l )e(k+1,i l-1 ).

[0016] Where, ρφ(k,i l ) / (λ+φ(k,i l ) 2 ) is abbreviated as P(k, i l ), The pseudo partial derivative φ is abbreviated as ξ(k+1, i-1) k (t), ρ is the step size factor, 0<η<2 is the step size factor, is φ k (t) is estimated; when i∈(i l-1 ,i l ), the actuator is not triggered, so u(k, i)=u(k, i l-1 ); Thus, the event-triggered iterative learning control input update law is designed as:

[0017] Where, Abbreviated as 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:

[0018] Where β is a sufficiently small positive constant and is smaller than φ k (t), sign(·) is the sign function, and the controller is obtained as: if or

[0019] Considering the variable iteration length problem, the corrected tracking error is defined as:

[0020] can be written as: ε(k,i)=γ(k,i)e(k,i),k∈[0,T d ].

[0021] Similarly, the corrected pseudo partial derivative update term is defined as:

[0022] It can be written as: ζ(k, i) = γ(k, i)ξ(k, i), k ∈ [0, T d .

[0023] Therefore, the event-triggered iterative learning control method

[0019] can be reconstructed as:

[0024] Adopting the proposed event-triggered scheme, if μ, ρ > 0 and 0 < η < 1, it can be guaranteed that the parameter estimation is bounded, and the modified tracking error e(k, i) gradually converges to 0 as the number of iterations increases. The proof process is as follows:

[0025] C001: First, prove the boundedness. At the triggering moment, (k, i) = (k, i l ):

[0026] C002: Combining C001 with the system dynamic linear model gives the following equation;

[0027] C003:

[0028] C004: Subtracting φ(k, i) from both sides of C003 gives:

[0029] C005:

[0030] C006: In the formula

[0031] C007: Select μ > 0 and 0 < η < 1. Therefore, η|Δu(k, i - 1)| 2 < |Δu(k, i - 1)| 2 < μ + |Δu(k, i - 1)| 2 , there exists a positive constant q1 satisfying:

[0032] C008:

[0033] C009: γ(k + 1, i - 1) satisfies a random variable with a Bernoulli distribution. According to formulas C008 and C005, we can obtain:

[0034] C010: According to |φ(k, i)| < b, we can obtain: |Δφ(k, i)| = |φ(k, i) - φ(k, i - 1)| < 2b,

[0035] C011: The random variable γ(k + 1, i - 1) and are independent of each other. Taking the expectation on both sides of B009 simultaneously:

[0036] C012: Since 0 < p(k + 1) ≤ 1 and 0 < q1 < 1, there exists a constant d1 such that 0 < |1 - q1p(k + 1)| ≤ d1 < 1 holds. We can obtain:

[0037] C013: Since 0 < d1 < 1, according to B012, it is bounded at the triggering moment; in addition, E{|φ(k, i)|} is known to be bounded, and therefore, we can get is also bounded; in another case, during the triggering interval i ∈ (i l-1 , i l ), the estimated value of the pseudo partial derivative remains unchanged from the previous triggering moment. Therefore, is also bounded, and thus we can obtain is always bounded for i ∈ {0, 1, 2,...};

[0038] C014: Next, prove the convergence of the tracking error. According to the linearized data model and the definition of the tracking error e(k, i) = y d (k) - y(k, i), e(k, i) can be reformulated as: e(k + 1, i l ) = y d (k + 1) - y(k + 1, i - 1) - φ(k, i)Δu(k, i l ) = e(k + 1, i - 1) - φ(k, i)Δu(k, i l ),

[0039] C015: Substitute the control input update rate into B014 to get:

[0040] C016: When i - 1 ∈ (i l-1 , i l ), the control input u(k + 1, i - 1) is not updated and remains the input at the previous triggering moment. Therefore, e(k + 1, i - 1) = e(k + 1, i l-1 ), then:

[0041] C017: Take the norm of both sides of B016 to obtain the expectation:

[0042] C018: According to the inequality ab ≤ (1 / 2)a 2 +(1 / 2)b 2 , and P(k, i l ) = ρφ(k, i l ) / (λ + φ(k, i l )), there must exist a bounded constant λ such that the following inequality holds: 2 )

[0043] C019: Where q2 is a constant. According to 0 < p(k + 1) ≤ 1 and 0 < q2 < 1, there exists a constant d2 such that 0 < |1 - q2p(k + 1)| ≤ d1 < 1 holds. Applying this inequality to B017 gives:

[0044] C020: In summary, the expectation of the tracking error converges to zero at the triggering moment, that is: For other cases where i ∈ (i l-1 , i [[ID=三十二]] l ), the control input is the same as the previous iteration, so the tracking error remains convergent within the triggering interval; therefore, the tracking error e(k, i) gradually converges to 0 as the number of iterations increases for i ∈ {0, 1, 2,...}. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 is the flowchart of the method according to an embodiment of the present invention;

[0046] Figure 2 is the iteration step size for each iteration;

[0047] 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;

[0048] Figure 4 is the average error under different triggering thresholds of the method proposed in the present invention; DETAILED DESCRIPTION OF THE EMBODIMENTS

[0049] The present invention will be further clarified below in conjunction with 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 modifications made by those skilled in the art fall within the scope defined by the appended claims of this application.

[0050] As Figure 1 As shown in FIG, an event-triggered iterative learning control method for a nonlinear system with variable iteration length includes the following steps:

[0051] Step 1: Set the initial value of the parameter;

[0052] Step 2: Update algorithm parameters;

[0053] 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);

[0054] Step 4: According to the control input u k (t) The tracking error generated is used to update the algorithm parameters synchronously;

[0055] Step 5: Repeat steps 3 and 4 until the current iteration ends and the next iteration begins;

[0056] 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:

[0057]

[0058] Among them, a(k)=1+rand(k / 300) is a time-varying parameter, and the maximum iteration length is T d =100.

[0059] The desired reference orbit is: y d (k) = 0.4sin(kπ / 30) + 0.3cos(kπ / 10). For any iteration i, set the initial condition to y(1, k) = y d (1), the control signal for the first iteration is set to u(t, 1) = 0, t∈{0, 1, ..., T}. The other controller parameters are selected as T = 100, ρ = 0.5, λ = 0.5, μ = η = 1 and ε = 10 -5 In addition, in order to show the influence of different threshold signals in the trigger mechanism on the system tracking control, two different threshold signals σ=1×10 -3 ,σ=1×10 -4 In order to describe the length of random iterative changes, let T d -T1=20. Since the maximum iteration length is T d =100, the iterative change operation length satisfies T i ∈[80 100]. Further assume that T iIn the above discrete set, it is a random variable that obeys discrete uniform distribution, and we can get P[T i =T1+l]=1 / 20, where l∈[1,…,20].

[0060] Figure 1 is a flow chart of a method according to an embodiment of the present invention; applying the proposed method, Figure 2 is the iteration step size for each iteration; 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; Figure 4 is the average error of the method proposed in the present invention under different trigger thresholds; it can be seen from these two figures that the proposed method has a good application effect in nonlinear systems, and after the 20th iteration, a satisfactory tracking performance can be obtained.

[0061] References

[0062] [1] Wang X, Qin W, Park JH, et al. Event-triggered data-driven control of discrete-time nonlinear systems with unknown disturbance[J]. ISA transactions, 2022, 128: 256-264.

[0063] [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. A hybrid event-triggered iterative learning control method for a nonlinear system with variable iteration length, 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+1,i)=f(y(k,i),u(k,i)), Where i is the iteration index, k∈[0, 1, ..., T d ] is the sampling time, T d is the expected iteration length; u(k, i)∈R 1 and y(k,i)∈R 1 are the control input and output, respectively, and f(·) is an unknown nonlinear function. There is a time-varying parameter φ(k, i) related to the iteration, which is a pseudo partial derivative. The system can be written as the following dynamic linearization model: Where φ(k, i) is bounded for any time k and iteration number i, Δy(k+1, i) = y(k+1, i) - y(k+1, i - 1), Δu(k, i) = u(k, i) - u(k, i - 1); The Bernoulli distribution function is introduced to describe the probability of randomly changing iteration length, as follows: T i represents the actual length at each iteration, which is random. In practical applications, the actual iteration length T i may be less than the expected length T d , and the output of the system is missing in the time period [T i+1 T d , so it cannot be applied to update the control input; assume that the minimum operation length is denoted by T1. Therefore, the actual length of each iteration process lies in the integer set {T1,..., T d}, then the system output is always available for all input updates in [0 T1], and in the i-th iteration, the system output in [T1 T i is still obtainable, while in [T i+1 T d it is missing; in order to be able to describe the randomness of the iteration length, a random variable γ(k, i) is defined, k ∈ [0 T d , and it satisfies the Bernoulli distribution; the event γ(k, i) = 1 means that there is a probability of p(k) to run to the time instant k in the i-th iteration process, where 0 < p(k) < 1 is a time-related function; the event γ(k, i) = 0 means that the i-th iteration process cannot run to the k-th sampling moment, and its probability is 1 - p(k), P[T i = T1 + m] = p l means that the probability of T i = T1 + m is p m , where m ∈ {1,..., T d - T1} and 0 ≤ p m < 1 is a known constant; Since each iteration can run to time T1, the event γ(k, i) = 1, k∈[0 T1] is valid for all iterations, so the probability p(k) = 1; and for k∈[T1+1 T i ] moment, let A m It is the event that the iterative process ends at time k = T1 + l, where m∈{1, ..., T d -T1}; thus we can get P[A m ]=p m ; Therefore, if T1+l≥k holds, for k∈[T1 T d ], the probability of γ(k, i) = 1 is given by event A m Decision: So we can get: In addition, since γ(k, i) satisfies the Bernoulli distribution, the expectation of γ(k, i) is as follows: E{γ(k,i)}=1×p(k)+0×(1-p(k))=p(k) Where, E{·} represents expectation; A model-free adaptive controller with a hybrid triggering strategy that includes a relative threshold strategy and a zero-error tracking strategy is constructed as follows: Using a hybrid triggering strategy, the triggering mechanism is expressed as: In the formula is the event triggering error, i l represents an event-triggered iterative sequence, l = 0, 1, ..., tracking error e(k, i) = y d (k)-y(k,i),y d (k) represents the expected trajectory, σ represents the threshold signal which is an adjustable positive constant; Next, the event-triggered nonlinear controller is designed. When the iteration length is fixed, the following u(k, i) and parameter estimates are designed: The relevant objective function is: J(u(k,i l ))=|y d (k+1)-y(k+1,i l )| 2 +λ|u(k,i l )-u(k,k l-1 )| 2 , In the formula, λ and μ are adjustable parameters. Taking the derivative of the above formula and setting it equal to 0, we can get u(k, i) and parameter φ k Estimated update rate of (t): u(k,i l )=u(k,i l-1 )+P(k,i l )e(k+1,i l-1 ), Among them, ρφ(k,i l ) / (λ+φ(k,i l ) 2 ) is abbreviated as P(k, i l ), Abbreviated as ξ(k+1, i-1), it is the pseudo partial derivative φ k (t), ρ is the step size factor, 0<η<2 is the step size factor, is φ k (t) is estimated; when i∈(i l-1 ,i l ), the actuator is not triggered, so u(k, i)=u(k, i l-1 ), and thus the event-triggered iterative learning control input update law is designed as: in, Abbreviated as 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 β is a sufficiently small positive constant and is smaller than φ k (t), sign(·) is the sign function; The event-triggered iterative learning control method is: if or Considering the variable iteration length problem, the corrected tracking error is defined as: can be written as: ε(k,i)=γ(k,i)e(k,i),k∈[0,T d ] Similarly, the corrected pseudo partial derivative update term is defined as: Rewritten as: ζ(k,i)=γ(k,i)ξ(k,i),k∈{0,T d ], Therefore, the event-triggered iterative learning control method can be restructured as: if or Using the proposed event-triggered iterative learning control scheme, if μ, ρ>0 and 0<η<1, 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.

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