An iterative control method based on state constraints

Abstract

The application discloses an iterative control method based on state constraint, comprising the following steps: 1, establishing a non-strict feedback system model and a neural network state observer; 2, based on the backstepping framework, designing the tracking error of the system state; 3, introducing an auxiliary variable to update the tracking error of the system state; 4, constructing a tangent Lyapunov function to constrain the system state within a time-varying boundary; the application introduces a tangent BLF to constrain the system state within a time-varying boundary, and the function characteristics make the controller flexible and applicable to the application scenarios with or without constraints. Through the neural network observer, the unmeasurable system state is estimated, an adaptive hybrid update mechanism based on time and iteration domain is designed, the projection saturation limitation of the adaptive law is avoided, and the convergence of the tracking error and the estimation error along the iteration axis is ensured.

Description

Technical Field

[0001] This invention belongs to the field of control system technology and relates to an iterative control method based on state constraints. 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 struggling to meet the control requirements of complex systems. Meanwhile, non-strict feedback systems, which can better describe real-world physical systems, suffer from non-lower triangular structure characteristics, making conventional backstepping methods difficult to apply directly due to algebraic ring problems. Conventional non-strict feedback systems also exhibit unmeasurable states and constraint problems under state constraints. Summary of the Invention

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

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

[0005] Step 1: Establish a non-strict feedback system model and a neural network state observer; Step 2: Based on the backstepping framework, design the tracking error of the system state; Step 3: Introduce auxiliary variables to update the tracking error of the system state; Step 4: Construct a tangent-type Lyapunov function to constrain the system state within the time-varying boundary; Step 5: Adjust the parameters to set the adaptive law to a pure time domain, a pure iterative domain, and a hybrid time domain / iterative domain update mechanism; Step 6: Perform stability analysis on the convergence of the tracking error within a finite time. Step 7: Simulation verification.

[0006] Furthermore, the mathematical expression for the non-strict feedback system model in step 1 is: (2-9) 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 (2-10) Design a neural network state observer: (2-13).

[0007] Furthermore, in step 2, based on the backstepping framework, the system state tracking error is: (2-59) Among them, For the desired trajectory, The virtual control law representing the subsystem, and the system state. Constrained within a predefined set R represents the set of all real numbers. Combining (2-9) and (2-13), tracking error The time derivative is (2-60) in This is an estimated value. This is for estimating the error.

[0008] Furthermore, auxiliary variables are introduced in step 3. and ,error The time derivative is updated as follows: (2-61) in, For neural network weights, This is an estimated value. To estimate the error, To estimate the error, , It is a positive constant.

[0009] Furthermore, the tangent Lyapunov function in step 4 is designed as follows: (2-62) The time derivative of the tangent Lyapunov function is: (2-68) Among them, intermediate variables and constraint parameters To constrain the boundary, constraint parameters It is a very small positive constant.

[0010] Furthermore, the method in step 5 is as follows: Designing virtual control laws: (2-69) Design Adaptive Law: (2-70) in , , ; The virtual control law is updated as follows: (2-80) The adaptive law is updated as follows: (2-81) in ; The actual control law and adaptive law are designed as follows: (2-86) (2-87) in , For weight parameters, and For learning parameters, This is an estimated value.

[0011] Furthermore, the time derivative of the tangent Lyapunov function is updated as follows: (2-92) In summary, the advantages of this invention are: This invention introduces a tangent-type BLF to constrain the system state within a time-varying boundary, while its functional characteristics allow the controller to be flexibly applied to both constrained and unconstrained application scenarios. By estimating the unmeasurable system state through a neural network observer, an adaptive hybrid update mechanism based on time and iterative domains is designed. This avoids the projection saturation limitation of the adaptive law while ensuring the convergence of tracking and estimation errors along the iterative axis. Attached Figure Description

[0012] Figure 1 The maximum error trajectory of this invention Schematic diagram.

[0013] Figure 2 The mean square error trajectory of the present invention Schematic diagram.

[0014] Figure 3 The output trajectory of this invention Schematic diagram.

[0015] Figure 4 Error trajectory of the present invention Schematic diagram.

[0016] Figure 5 The observation trajectory of this invention and Schematic diagram.

[0017] Figure 6 Output trajectories of different methods of the present invention Schematic diagram.

[0018] Figure 7 Output trajectories of different methods of the present invention Schematic diagram.

[0019] Figure 8 Error trajectories of different methods of the present invention Schematic diagram.

[0020] Figure 9 Control inputs for different methods of the present invention Schematic diagram. Detailed Implementation

[0021] 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.

[0022] Example: like Figures 1-9 As shown, an iterative control method based on state constraints includes...

[0023] Step 1: Establish a non-strict feedback system model and a neural network state observer; Step 2: Based on the backstepping framework, design the tracking error of the system state; Step 3: Introduce auxiliary variables to update the tracking error of the system state; Step 4: Construct a tangent-type Lyapunov function to constrain the system state within the time-varying boundary; Step 5: Adjust the parameters to set the adaptive law to a pure time domain, a pure iterative domain, and a hybrid time domain / iterative domain update mechanism; Step 6: Perform stability analysis on the convergence of the tracking error within a finite time. Step 7: Simulation verification.

[0024] To facilitate control over the design, some necessary assumptions and lemmas are proposed as follows: Assumption 2.1: Expected Trajectory and i order derivative It is bounded and satisfies the conditions. as well as ,in , ,as well as It is a positive number.

[0025] Assumption 2.2: The initial conditions for each iteration satisfy... .

[0026] Assumption 2.3: For nonlinear functions There exists a positive constant. Satisfy the following inequalities in For vectors The estimated value.

[0027] Lemma 2.1: For n dimensional vector and If in Timely satisfaction Then the following inequality holds. Where the saturation function , , Represents a symbolic function. It is a scalar The upper boundary.

[0028] Lemma 2.2: For The following Young's inequality holds. in , , , .

[0029] The mathematical expression for the non-strict feedback system model in step 1 is: (2-9) 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 (2-10) Design a neural network state observer: (2-13) In step 2, based on the backstepping framework, the tracking error of the system state is... (2-59) Among them, For the desired trajectory, The virtual control law representing the subsystem, and the system state. Constrained within a predefined set R represents the set of all real numbers.

[0030] Combining (2-9) and (2-13), tracking error The time derivative is (2-60) in This is an estimated value. This is for estimating the error.

[0031] Introducing auxiliary variables in step 3 and ,error The time derivative is updated as follows: (2-61) in, For neural network weights, This is an estimated value. To estimate the error, To estimate the error, , It is a positive constant.

[0032] In step 4, the tangent Lyapunov function is designed as follows: (2-62) in for Constraint boundaries, Weight parameters Learning parameters ...

[0033] Taking the derivative of the tangent Lyapunov function with respect to time, we get: (2-63) Define intermediate variables and constraint parameters To constrain the boundary, constraint parameters It is a very small positive constant.

[0034] Combining (2-61) and (2-63), we can obtain (2-64) Using Young's inequality, Lemma 2.3, and Assumption 2.3, we can obtain (2-65) (2-66) (2-67) in It is a positive constant. It is the number of neurons in the RBFNN. It is a positive constant.

[0035] By using inequalities (2-65)-(2-67) and , can be obtained (2-68) The method in step 5 is as follows Designing virtual control laws: (2-69) Design Adaptive Law: (2-70) in , , Leaked items It helps solve parameter drift problems caused by noise or interference in practical applications. Substituting (2-69) and (2-70) into (2-68), we get (2-71) Note 2.1: When At that time, it can be obtained ,Right now (2-72) Therefore, no singular problems will occur in the control design.

[0036] Note 2.2: When there are no constraints, i.e. It can be obtained through L'Hôpital's rule. (2-73) This is equivalent to the tangent-type BLF function becoming a standard quadratic Lyapunov function without state constraints. Therefore, this control scheme is a universally applicable method suitable for both constrained and unconstrained scenarios. Furthermore, this method can restrict the system state to a time-varying constraint function. Within, and not limited to constant values. .

[0037] If the constraint range is time-invariant, we can obtain Control Law Simplified to (2-74) in , These are control parameters.

[0038] i . Combining (2-13) and (2-59), we can design corresponding auxiliary variables. and ,get (2-75) in .

[0039] Similarly, for Lyapunov functions Substituting (2-75) into its time derivative, we can obtain (2-76) in for Constraint boundaries, , It is a bounded set. , Through the following inequalities (2-77) (2-78) Substituting (2-77)-(2-78) into (2-76), we get (2-79) Therefore, the virtual control law is updated as follows: (2-80) The adaptive law is updated as follows: (2-81) in .

[0040] Substituting (2-80)-(2-81) into (2-79), we get (2-82) right Differentiating by time, we get (2-83) For Lyapunov functions Substituting (2-83) into its time derivative, we can obtain (2-84) Similarly, the following inequalities can be obtained. (2-85) Therefore, the actual control law and adaptive law are designed as follows: (2-86) (2-87) in , For weight parameters, and For learning parameters, This is an estimated value.

[0041] Substituting (2-85)-(2-87) into (2-84), we get (2-88) Combining (2-71), (2-82), and (2-88), we can obtain (2-89) in The derivative of the function. For weight parameters, and For learning parameters, For control parameters, This is the upper bound of the estimation error.

[0042] Using Young's inequality, we can obtain the following inequality. (2-90) Simultaneously utilize relational expressions , can be obtained (2-91) Based on (2-90) and (2-91), (2-89) can be rewritten as (2-92) The stability analysis method in step 6 is as follows: Theorem 2.2: For non-strict feedback systems, satisfying Assumptions 2.1-2.3 and initial conditions... By using the state observer (2-13), virtual control laws (2-69) and (2-80), and actual control law (2-86), combined with adaptive laws (2-70), (2-81), and (2-87), the closed-loop system in the interval... It is internally bounded and can guarantee the convergence of tracking error and observation error along the iteration axis without violating state constraints.

[0043] Proof: Based on the above analysis, in order to ensure In the interval Boundedness, and Based on the convergence and boundedness of the system state, the corresponding CEF is designed as follows: (2-93) in as well as .

[0044] Taking the derivative of (2-93) with respect to time, we get (2-94) Expanding (2-94) further, we can obtain the following inequality. (2-95) The design parameters of the control strategy must satisfy the following conditions (2-96) Therefore, (2-95) can be rewritten as (2-97) intermediate variables It is bounded.

[0045] During the initial iteration as well as ,therefore It is bounded, and at the same time, according to hour The expression can be obtained as follows: (2-98) Perform on both sides of (2-98) The points can be obtained. (2-99) Therefore, we can conclude In the interval The upper limit is bounded, that is... (2-100) Next, based on (2-93). The difference between two consecutive iterations is (2-101) Substituting (2-97) and (2-92) into (2-101), we get (2-102) intermediate variables , Control parameters The minimum value, the minimum value .

[0046] Based on the properties of the correction function, we can directly obtain .for ,according to as well as , can be obtained And according to the correction function exist The properties of time can be obtained through The conclusion is Therefore, we can ultimately obtain... , .

[0047] Based on assumption 2.2, the initial conditions can be obtained. , ,pass and combined and The positive definiteness can ultimately lead to... (2-103) pass , , can be obtained (2-104) Therefore, tracking error and observation error After a finite number of iterations, it can asymptotically converge to a small region near the origin.

[0048] Due to the positive definiteness and finiteness of CEF, we can obtain This can be further written as (2-105) Therefore, it can be in the interval Get it tracking error It can be guaranteed within the constraint limits Within. Based on assumptions 2.1 and (2-59), we can obtain ,in The upper bound of the trajectory, for The target value. Finally, through... You can get That is, system state Constrained by preset limits Within. Within the interval. Inside, for Due to initial conditions and correction function If all are bounded, then the system state In the interval The interior is also bounded. Similarly, it can be achieved through... Further obtain the system state The boundedness of, that is .

[0049] Note 2.3: The proposed iterative learning control strategy combines information from the time domain and the iteration domain, whereby... This represents adaptive learning in the pure time domain. This represents adaptive learning in a purely iterative domain, while This indicates that iterative learning algorithms can comprehensively utilize information from both the time domain and the iteration domain for updates, thereby effectively ensuring the estimation accuracy and convergence speed of the neural network. The performance of the control algorithm is mainly affected by the design parameters. , , and The influence of this factor means that appropriately increasing the parameter value can reduce the tracking error, but a trade-off needs to be struck between system stability and tracking accuracy. Furthermore, a leakage term is introduced into the adaptive law. This helps to eliminate parameter drift problems caused by noise and disturbances.

[0050] The simulation verification method in step 7 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: (2-106) 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: .

[0051] Control parameters are set to , , , , The constraint boundary is set as follows: and The observer parameters are set to , .

[0052] Maximum error and root mean square error As a performance indicator, such as Figure 1 and Figure 5 As shown, although the weights are updated at different times / iterations... It will affect the convergence speed of tracking error, but and It can converge to zero within a finite number of iterations. Meanwhile, it is comparable to pure time-domain updates. and pure iterative domain update In comparison, the adaptive hybrid update law It enables better control performance.

[0053] Meanwhile, to verify the effectiveness of this scheme, it was compared with several other control methods. The first method is the model-free PID method, which is widely used in practice. The second and third methods are backstepping control methods, namely, observer-based adaptive fuzzy control (O-AC) and observer-based finite-time adaptive control (O-FTAC), respectively. The main control parameters of these three methods are as follows: The control parameters for the PID method are set as follows: , as well as .

[0054] The control parameters for the O-AC method are set to... , as well as .

[0055] The control parameters for the O-FTAC method are set to... , as well as .

[0056] Comparison of simulation results as follows Figures 6 to 9 As shown. Figure 6 The figure shows the output trajectories of different control methods, although each trajectory They can almost track the expected trajectory. However, this control algorithm has better tracking performance. Figure 8 Error trajectory displayed Among them, O-AIBC has a faster convergence speed and a smaller steady-state error. Figure 9 The control inputs of different methods are shown. Overall, the proposed O-AIBC method achieves satisfactory control performance, demonstrating superior tracking accuracy and robustness compared to other control methods.

[0057] 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 iterative control method based on state constraints, characterized in that: include Step 1: Establish a non-strict feedback system model and a neural network state observer; Step 2: Based on the backstepping framework, design the tracking error of the system state; Step 3: Introduce auxiliary variables to update the tracking error of the system state; Step 4: Construct a tangent-type Lyapunov function to constrain the system state within the time-varying boundary; Step 5: Adjust the parameters to set the adaptive law to a pure time domain, a pure iterative domain, and a hybrid time domain / iterative domain update mechanism; Step 6: Perform stability analysis on the convergence of the tracking error within a finite time. Step 7: Simulation verification.

2. The iterative control method based on state constraints according to claim 1, characterized in that: The mathematical expression for the non-strict feedback system model in step 1 is: （2-9） 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 （2-10） Design a neural network state observer: （2-13）。 3. The iterative control method based on state constraints according to claim 1, characterized in that: In step 2, based on the backstepping framework, the tracking error of the system state is... （2-59） Among them, For the desired trajectory, The virtual control law representing the subsystem, and the system state. Constrained within a predefined set R represents the set of all real numbers. Combining (2-9) and (2-13), tracking error The time derivative is （2-60） in This is an estimated value. This is for estimating the error.

4. The iterative control method based on state constraints according to claim 1, characterized in that: In step 3, auxiliary variables are introduced. and ,error The time derivative is updated as follows: （2-61） in, For neural network weights, This is an estimated value. To estimate the error, To estimate the error, , It is a positive constant.

5. The iterative control method based on state constraints according to claim 1, characterized in that: In step 4, the tangent Lyapunov function is designed as follows: （2-62） The time derivative of the tangent Lyapunov function is: （2-68） Among them, intermediate variables and constraint parameters To constrain the boundary, constraint parameters It is a very small positive constant.

6. The iterative control method based on state constraints according to claim 1, characterized in that: The method in step 5 is as follows: Designing virtual control laws: （2-69） Design Adaptive Law: （2-70） in , , ; The virtual control law is updated as follows: （2-80） The adaptive law is updated as follows: （2-81） in ; The actual control law and adaptive law are designed as follows （2-86） （2-87） in , For weight parameters, and For learning parameters, This is an estimated value.

7. The iterative control method based on state constraints according to claim 6, characterized in that: The time derivative of the tangent Lyapunov function is updated as follows: （2-92）。

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