An adaptive backstepping control method based on iterative learning

Abstract

The application discloses an adaptive backstepping control method based on iterative learning, which comprises the following steps: 1, establishing a nonstrict feedback system model; 2, estimating unknown functions in the nonstrict feedback system model by using RBFNN and the bounded characteristics of radial basis functions; 3, designing a neural network state observer; 4, introducing a desired trajectory correction function, so that the reference trajectory is equivalent to the desired trajectory; 5, designing a Lyapunov function based on a tracking error system of a backstepping framework; and 6, designing an actual control law and an adaptive law based on the Lyapunov function, so that the tracking error asymptotically converges to zero with iterations. The application estimates unknown functions in the nonstrict feedback system by using RBFNN, and avoids the algebraic loop problem in control design by using the bounded characteristics of radial basis functions. Meanwhile, the desired trajectory correction function is introduced, the reference trajectory is corrected again, and the limitation of ideal initial conditions on the control algorithm is effectively avoided.

Description

Technical Field

[0001] This invention belongs to the field of control system technology and relates to an adaptive backstepping control method based on iterative learning. Background Technology

[0002] With the booming development of industrial manufacturing, intelligent robotics, and other fields, control system design faces increasingly complex model objects and diverse engineering tasks. Conventional strict feedback systems, due to their structural characteristics, are insufficient to meet the control requirements of complex systems. Non-strict feedback systems, which better describe real-world physical systems, suffer from algebraic loop problems, making conventional backstepping methods difficult to apply directly. Furthermore, considering that many controlled objects in practical engineering need to repeatedly perform the same tasks, and that backstepping methods primarily study the system's control performance on a single time axis, they cannot effectively utilize historical operating data during repeated processes, making it difficult to achieve performance improvement through iterative experience accumulation. In contrast, while iterative learning control can utilize information from the iterative domain to optimize control effects, traditional design methods based on compression mapping struggle to effectively handle nonlinear characteristics, and system convergence is limited by the Lipschitz condition. Moreover, iterative learning control has strict requirements for initial conditions due to its convergence requirements. To ensure the feasibility of the control strategy, the system typically needs ideal initial conditions in each iteration, meaning the initial value of the system state matches the expected value. However, in reality, due to system characteristics or external disturbances, these ideal initial conditions are often difficult to meet. Therefore, for a class of nonlinear, non-strict feedback systems, how to combine the recursive design framework of backstepping control with the learning mechanism of iterative control to design a general and effective control strategy without the need for ideal initial conditions, and to ensure the convergence of the system along the time axis and the iteration axis, is an urgent problem to be solved. Summary of the Invention

[0003] In order to overcome at least one deficiency of the prior art, the present invention provides an adaptive backstepping control method based on iterative learning.

[0004] To achieve the above objectives, the present invention adopts the following technical solution: an adaptive backstepping control method based on iterative learning, comprising:

[0005] Step 1: Establish a non-strict feedback system model; Step 2: Estimate the unknown functions in the non-strict feedback system model using RBFNN and the bounded properties of radial basis functions; Step 3: Design the neural network state observer; Step 4: Introduce the desired trajectory correction function This makes the reference trajectory equivalent to the desired trajectory; Step 5: Design the Lyapunov function for the tracking error system based on the backstepping framework; Step 6: Design the actual control law and adaptive law based on the Lyapunov function so that the tracking error asymptotically converges to zero with iteration; Step 7: Perform stability analysis on the convergence of the tracking error within a finite time. Step 8: Simulation verification.

[0006] Furthermore, the mathematical expression of the non-strict feedback system model is as follows: (0-4) in and They represent the first time. k The system state and output at the next iteration. For vectors The estimated value, and These represent the system order and the number of iterations, respectively. The system control input is... , Refers to an unknown nonlinear function. The system state matrix Output injection matrix Nonlinear allocation matrix Input matrix Output matrix , , For the estimated value vector, Design the parameters for the observer and ensure the matrix Satisfying the Hurwitz condition, for any positive definite symmetric matrix There exists a matrix satisfy (0-5).

[0007] Furthermore, in step 2, the unknown function... Using RBFNN to approximate the estimate is ,in Ideal weights for a neural network The estimated value, and Defined as (0-6) Where arg min is the parameter that minimizes the error, and sup is the supremum. and They respectively represent the corresponding and Bounded compact set.

[0008] Furthermore, the neural network state observer in step 3 is... (0-8).

[0009] Furthermore, step 4 sets the system tracking reference trajectory. (0-13) (0-14) in , , It is the expected trajectory. When it is the kth iteration initial value, When it is the kth iteration initial value, When it is the kth iteration Real-time status variables, It is the corrected reference trajectory, making express l The time derivative of order 1.

[0010] Furthermore, the aforementioned It is a correction function that satisfies the following conditions: (1) yes n Differentiable; (2) In time interval The above is uniformly bounded; (3) , and when hour ; (4) For ,exist .

[0011] Furthermore, the correction function for: (0-15) Where ! represents factorial, that is .

[0012] Furthermore, the tracking error system in the backstepping framework in step 5 is designed as follows: (0-16) in It is the error variable at the k-th iteration. It is the state variable at the k-th iteration. The virtual control law representing the subsystem; Lyapunov functions are designed for (0-18).

[0013] Furthermore, step 6 is... Design virtual control laws and adaptive laws for (0-32) (0-33) in For design parameters, ; The actual control law and adaptive law are designed as follows: (0-39) (0-40) in For design parameters, .

[0014] In summary, the advantages of this invention are: This invention employs RBFNN to estimate unknown functions in non-strict feedback systems, utilizing the bounded property of radial basis functions to avoid algebraic loop problems in control design. Simultaneously, a desired trajectory correction function is introduced to effectively avoid the limitations imposed by ideal initial conditions on the control algorithm by re-correcting the reference trajectory. Attached Figure Description

[0015] Figure 1 The maximum error trajectory of this invention Schematic diagram.

[0016] Figure 2 The root mean square error trajectory of this invention Schematic diagram.

[0017] Figure 3 The output trajectory of this invention Schematic diagram.

[0018] Figure 4 Error trajectory of the present invention Schematic diagram.

[0019] Figure 5 This is a schematic diagram of the control input for the 20th iteration of the present invention. Detailed Implementation

[0020] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that, unless otherwise specified, the following embodiments and features described therein can be combined with each other.

[0021] Example: like Figures 1-5 As shown, an adaptive backstepping control method based on iterative learning includes... Step 1: Establish a non-strict feedback system model; Step 2: Estimate the unknown functions in the non-strict feedback system model using RBFNN and the bounded properties of radial basis functions; Step 3: Design the neural network state observer; Step 4: Introduce the desired trajectory correction function This makes the reference trajectory equivalent to the desired trajectory; Step 5: Design the Lyapunov function for the tracking error system based on the backstepping framework; Step 6: Design the actual control law and adaptive law based on the Lyapunov function so that the tracking error asymptotically converges to zero with iteration; Step 7: Perform stability analysis on the convergence of the tracking error within a finite time. Step 8: Simulation verification.

[0022] To facilitate control over the design, some necessary assumptions and lemmas are proposed as follows: Assumption 2.1: Expected Trajectory and i Derivative It is bounded and satisfies the conditions. as well as ,in , ,as well as It is a positive number.

[0023] Assumption 2.2: The initial conditions for each iteration satisfy... .

[0024] Assumption 2.3: For nonlinear functions There exists a positive constant. Satisfy the following inequalities (0-1) in For vectors The estimated value.

[0025] Lemma 2.1: For n dimensional vector and If in Timely satisfaction Then the following inequality holds. (0-2) Where the saturation function , , Represents a symbolic function. It is a scalar The upper boundary.

[0026] Lemma 2.2: For The following Young's inequality holds. (0-3) in , , , .

[0027] The mathematical expression for the non-strict feedback system model in step 1 is: (0-4) in and They represent the first time. k The system state and output at the next iteration. For vectors The estimated value. and These represent the system order and the number of iterations, respectively. The system control input is... , Refers to an unknown nonlinear function.

[0028] The system state matrix Output injection matrix Nonlinear allocation matrix Input matrix Output matrix , , This is the estimated value vector. Design the parameters for the observer and ensure the matrix It satisfies the Hurwitz condition. For any positive definite symmetric matrix... There exists a matrix satisfy (0-5) In step 2, for the unknown function Using RBFNN to approximate the estimate is ,in Ideal weights for a neural network The estimated value, and Defined as (0-6) Where arg min is the parameter that minimizes the error, and sup is the supremum. and They respectively represent the corresponding and The bounded compact set. Estimation error Defined as (0-7) in , It is a positive constant.

[0029] Step 3: Neural Network State Observer (0-8) For observation error According to (0-4) and (0-8), its time derivative is: (0-9) in To estimate the error, It is a basis function of neural networks.

[0030] For the observer described above, the corresponding Lyapunov function is designed as follows: (0-10) Based on assumption 2.3 and the boundedness of radial basis functions This can be obtained through Young's inequality in Lemma 2.2. (0-11) Combining (0-9) and (0-11), taking the derivative of (0-10) with respect to time yields... (0-12) in It is a matrix Q The smallest eigenvalue, intermediate variable intermediate variables n is a positive integer. This is the upper bound of the estimation error.

[0031] Step 4 introduces the desired trajectory correction function. This makes the reference trajectory equivalent to the desired trajectory; Set system tracking reference trajectory , (0-13) (0-14) in , , It is the expected trajectory. When it is the kth iteration initial value, When it is the kth iteration initial value, When it is the kth iteration Real-time status variables, It is the corrected reference trajectory, making express l The time derivative of order 1. It is a correction function that satisfies the following conditions: (1) yes n Differentiable; (2) In time interval The above is uniformly bounded; (3) , and when hour ; (4) For ,exist .

[0032] To satisfy the above conditions, the correction function... Optional (0-15) Where ! represents factorial, that is .

[0033] By correction function The effect can be obtained This satisfies the ideal initial conditions required by traditional iterative learning control methods. Furthermore, when At the same time, it can also be guaranteed that the reference trajectory is equivalent to the desired trajectory, that is... .if If it is small enough, then the time interval It can approximate the complete time domain .

[0034] The tracking error system in the backstepping framework in step 5 is designed as follows: (0-16) Among them It is the error variable at the k-th iteration. It is the state variable at the k-th iteration. This represents the virtual control law of the subsystem.

[0035] For (0-16) Differentiating by time, we get (0-17) Therefore, the following Lyapunov function is designed. (0-18) Then the time derivative of (0-18) is (0-19) Therefore, RBFNN is used to estimate the unknown nonlinear function. ,Right now (0-20) Among them, ideal network weights , and They are respectively at and The bounded compact set. Estimation error satisfy , It is a positive constant.

[0036] because It concerns all state variables. The function, when estimated using RBFNN, has its basis functions. The input signal also involves all variables, and the backstepping control method requires a virtual controller. Cannot directly contain higher-order states Therefore, by using Lemmas 2.2 and 2.3, we can obtain the following inequality. (0-21) (0-22) in It is a normal number.

[0037] Define ideal weight values , for The estimated value, estimation error Substituting (0-21) and (0-22) into (0-19), we get (0-23) Step 6: Design the actual control law and adaptive law based on the Lyapunov function so that the tracking error asymptotically converges to zero with iteration.

[0038] Design the initial virtual control law: (0-24) Design the initial adaptive law: (0-25) in For design parameters, saturation function Defined as (0-26) in and These are the maximum and minimum values ​​of the parameter estimates, used to prevent parameter estimation from diverging and to ensure the convergence of the control strategy. The estimation limits can usually be set based on prior knowledge and trial and error. If there is a lack of sufficient understanding of the characteristics of the system parameters, a larger conservative value can be selected as the upper and lower bounds of the estimation in the initial stage to ensure that the system parameters do not exceed the set estimation range. The parameter estimation limits can then be adaptively adjusted based on the system performance and control effect in subsequent iterations.

[0039] Substituting (0-24) and (0-25) into (0-23), we get (0-27) . The time derivative is (0-28) Similar to (0-21)-(0-22), the following inequality can be obtained. (0-29) (0-30) in , , yes Bounded compact set and .

[0040] For Lyapunov functions Substituting (0-29)-(0-30), its time derivative is... It can be represented as (0-31) Therefore, virtual control laws and adaptive laws are designed as follows: (0-32) (0-33) in For design parameters, .

[0041] Substituting (0-32)-(0-33) into (0-31), we get (0-34) against The time derivative is (0-35) Using RBFNN for unknown functions By estimation and combining (0-35), the Lyapunov function is... The time derivative is (0-36) Similarly, the following inequalities can be obtained. (0-37) (0-38) Therefore, the actual control law and adaptive law are designed as follows: (0-39) (0-40) in For design parameters, .

[0042] Substituting (0-37)-(0-40) into (0-36), we get (0-41) Based on (0-27), (0-34), and (0-41), we can finally obtain (0-42) The stability analysis method in step 7 is as follows: Constructing a composite energy function (0-43) in as well as .

[0043] Taking the derivative of (0-43) with respect to time, we get (0-44) For the first iteration as well as , can be obtained Therefore, (0-44) can be rewritten as (0-45) intermediate variables as well as It is bounded, as can be seen from Lemma 2.1. (0-46) We can obtain (0-46) By combining the selection of initial parameters, it can be guaranteed that It is bounded, at the same time as well as It is also bounded over a finite interval. Therefore, we can further conclude that... It is bounded.

[0044] Then, between two consecutive iterations The difference is as follows (0-47) According to the inequality (0-47) can be rewritten as (0-48) According to (0-42), the difference between two consecutive iterations It can be represented as (0-49) Based on the properties of the correction function, we can directly obtain .for ,according to as well as , can be obtained Next, according to... hour as well as , can be obtained Therefore, Similarly, we can eventually obtain... , .

[0045] Combining (0-48) and (0-49), the difference of CEF can be obtained. for (0-50) in for The minimum value of . From this, we can conclude that within a finite number of iterations, the tracking error can converge to a small region near zero.

[0046] pass , , can be obtained (0-51) Therefore, (0-51) can be rewritten as (0-52) Due to related items and Boundedness and The positive definiteness can be derived from (0-52). For any k The iterations are bounded. Furthermore, for any constant... There exists a finite number of iterations. ,exist Timely satisfaction That is, error term It can converge to a specific bound within a finite number of iterations. Therefore, we can conclude that the tracking error... It can asymptotically converge to a small region near zero after a finite number of iterations.

[0047] The simulation verification method for step 8 is as follows: In the field of chemical production, certain continuous stirred tank reactors and chemical reaction processes can actually be represented as non-strict feedback systems. The Brusselator model, representing chemical oscillatory reactions, is adopted as the simulation object, and its kinetic model is as follows: (0-53) in and It is the concentration of the reaction intermediate. Indicates system input, and Indicates the supply of chemical storage materials. and Represents unknown disturbances, among which With iteration The trajectory varies randomly between these parameters. The desired trajectory is set as follows: .

[0048] The parameters of the iterative learning controller are set as follows: , as well as , with Gaussian function As the activation function of RBFNN, its function center is uniformly distributed in the interval Its width is .

[0049] Next, the maximum error and root mean square error As a performance indicator, such as Figure 1 and Figure 2 As shown, the performance index function converges continuously with the increase of the number of iterations. Figure 5 The output trajectories for iterations 1, 10, and 20 are shown. and tracking error trajectory As can be seen, with each iteration, the control performance continuously improves, and the final output trajectory... It can track the expected trajectory very well. And tracking error It gets smaller.

[0050] Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort should fall within the scope of protection of the present invention.

Claims

1. An adaptive backstepping control method based on iterative learning, characterized in that: include Step 1: Establish a non-strict feedback system model; Step 2: Estimate the unknown functions in the non-strict feedback system model using RBFNN and the bounded properties of radial basis functions; Step 3: Design the neural network state observer; Step 4: Introduce the desired trajectory correction function This makes the reference trajectory equivalent to the desired trajectory; Step 5: Design the Lyapunov function for the tracking error system based on the backstepping framework; Step 6: Design the actual control law and adaptive law based on the Lyapunov function so that the tracking error asymptotically converges to zero with iteration; Step 7: Perform stability analysis on the convergence of the tracking error within a finite time. Step 8: Simulation verification.

2. The adaptive backstepping control method based on iterative learning according to claim 1, characterized in that: The mathematical expression for the non-strict feedback system model is: (0-4) in and They represent the first time. k The system state and output at the next iteration. For vectors The estimated value, and These represent the system order and the number of iterations, respectively. The system control input is... , Refers to an unknown nonlinear function. The system state matrix Output injection matrix Nonlinear allocation matrix Input matrix Output matrix , , For the estimated value vector, Design the parameters for the observer and ensure the matrix Satisfying the Hurwitz condition, for any positive definite symmetric matrix There exists a matrix satisfy (0-5)。 3. The adaptive backstepping control method based on iterative learning according to claim 1, characterized in that: In step 2, the unknown function Using RBFNN to approximate the estimate is ,in Ideal weights for a neural network The estimated value, and Defined as (0-6) Where arg min is the parameter that minimizes the error, and sup is the supremum. and They respectively represent the corresponding and Bounded compact set.

4. The adaptive backstepping control method based on iterative learning according to claim 1, characterized in that: The neural network state observer in step 3 is (0-8)。 5. The adaptive backstepping control method based on iterative learning according to claim 1, characterized in that: Step 4 sets the system tracking reference trajectory 6. (0-13) (0-14) in , , It is the expected trajectory. When it is the kth iteration initial value, When it is the kth iteration initial value, When it is the kth iteration Real-time status variables, It is the corrected reference trajectory, making express l The time derivative of order 1.

7. The adaptive backstepping control method based on iterative learning according to claim 5, characterized in that: The It is a correction function that satisfies the following conditions: (1) yes n Differentiable; (2) In time interval The above is uniformly bounded; (3) , and when hour ; (4) For ,exist .

8. The adaptive backstepping control method based on iterative learning according to claim 6, characterized in that: The correction function for: (0-15) Where ! represents factorial, that is .

9. The adaptive backstepping control method based on iterative learning according to claim 1, characterized in that: The tracking error system in the backstepping framework in step 5 is designed as follows: (0-16) in It is the error variable at the k-th iteration. It is the state variable at the k-th iteration. The virtual control law representing the subsystem; Lyapunov functions are designed for (0-18)。 10. The adaptive backstepping control method based on iterative learning according to claim 1, characterized in that: Step 6 is Design virtual control laws and adaptive laws for (0-32) (0-33) in For design parameters, ; The actual control law and adaptive law are designed as follows: (0-39) (0-40) in For design parameters, .

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