A data-driven adaptive backstepping control method for fractional order uncertain nonlinear systems under external time-varying disturbance

Abstract

The application discloses a data-driven adaptive backstepping control method for a fractional-order uncertain nonlinear system under external time-varying interference and relates to the technical field of automatic control algorithms.The method innovatively introduces a data-driven control strategy.First, a fractional-order uncertain system model in strict feedback form is established;on the premise of not depending on an accurate prior model, an identification matrix is constructed by collecting system offline data to realize accurate data-driven identification of unknown nonlinear terms;secondly, according to tracking error, a virtual control signal of each order is generated by using a simulation chain rule to perform recursive calculation of the backstepping method;finally, a hyperbolic tangent function is introduced to calculate an actual control input signal, and adaptive estimated values are updated through fractional-order operation in cooperation with real-time online data.The application effectively solves the derivative explosion problem, realizes robust suppression of external time-varying interference, and significantly improves the control precision and engineering application value of the uncertain system.

Description

Technical Field

[0001] This invention belongs to the field of automatic control algorithm technology, specifically relating to a data-driven adaptive backstepping control method for fractional-order uncertain nonlinear systems under external time-varying disturbances. Background Technology

[0002] In real physical systems, unidentified system nonlinearities caused by system uncertainties or unmodeled dynamics are common, which can seriously impair control accuracy and jeopardize the stability of closed-loop systems.

[0003] In recent years, fractional calculus has been increasingly widely used in system dynamics modeling. Extensive research and engineering practice have demonstrated that fractional dynamics provides an extremely powerful mathematical framework for modeling complex physical systems with genetic characteristics or memory-related behaviors. Compared to traditional integer-order systems, fractional models not only significantly broaden the descriptive dimensions of system dynamics but also exhibit superior control performance, providing significantly improved modeling accuracy with simpler mathematical expressions. However, for fractional nonlinear systems, the differentiation of composite functions in fractional calculus is extremely complex, posing significant challenges to traditional nonlinear control methods (such as fuzzy logic approximation based on precise mathematical models or relying on expert experience). Therefore, there is an urgent need for a data-driven control method that does not rely on precise system models, has strong anti-interference capabilities, and possesses a clearly defined and executable algorithm flow. Summary of the Invention

[0004] To achieve the above-mentioned technical objectives and effects, the present invention provides the following technical solution:

[0005] A data-driven adaptive backstepping control method for a fractional-order uncertain nonlinear system under external time-varying disturbances includes the following steps:

[0006] S1: Establish a fractional-order uncertain nonlinear system model with strict feedback, and sample at a preset time interval. Collect offline datasets of system inputs, status, and corresponding fractional derivatives;

[0007] S2: Based on the offline dataset, identify the unknown nonlinear terms in the nonlinear system model by constructing a data identification matrix;

[0008] S3: Construct tracking error variables based on system reference signals and actual outputs. Based on the tracking error variables and the online estimate of the fractional derivative of the composite function, perform backstepping recursive calculations using the simulated chain rule to obtain virtual control signals of each order.

[0009] S4: Based on the tracking error variable of the final order of the system, the upper bound estimate of the unknown external disturbance and the compensation parameter estimate, the actual control input signal acting on the nonlinear system is calculated;

[0010] S5: Using the current state and error data, update the adaptive estimate online through fractional derivative calculation, and feed the updated adaptive estimate back to the control signal calculation in steps S3 and S4 in real time to form closed-loop control.

[0011] Specifically, in step S1, the offline dataset includes:

[0012] System control input matrix System state variable matrix Fractional derivative matrix of state variables and nonlinear basis function matrix Furthermore, the number of samples meets the preset data scale requirements.

[0013] Specifically, step S2 is as follows:

[0014] For each order of the fractional-order uncertain nonlinear system model, the unknown nonlinear terms in the system are transformed into linear parameter vectors, and linear equations are established using the nonlinear basis function matrix and the corresponding estimates of the unknown constant vectors. The data identification matrix is ​​obtained by solving this equation. This ensures that the result of the operation between the basis function matrix and the data identification matrix satisfies the condition of a negative identity matrix.

[0015] Specifically, in step S3, the recursive calculation of the virtual control signal using the backstepping method is as follows:

[0016] Construct the first Tracking error variable ,in For system status, This is the next higher-order virtual control signal;

[0017] For the intermediate level, a simulated chain rule is used to apply the virtual control signal of the previous level. Perform fractional differentiation, decomposing it into a feedforward compensation term with known derivative and an unknown bounded nonlinear function. ;

[0018] Based on the backstepping control law and the introduced upper bound estimate of the unknown bounded nonlinear function Calculate the first Virtual control signal .

[0019] Specifically, in step S4, the method for calculating the actual control input signal includes:

[0020] Introducing the hyperbolic tangent function Construct a robust compensation term, where For the final-order tracking error variable, For smoothing parameters;

[0021] Combined with the upper bound estimate of external time-varying disturbances Based on the dynamic state of the system, generate actual control input signals. This is to counteract the effects of external time-varying disturbances and eliminate control chattering.

[0022] Specifically, in step S5, the adaptive estimate includes:

[0023] Upper bound estimate of unknown nonlinear function obtained by taking the derivative of a composite function Upper bound estimate of external time-varying disturbance and estimated values ​​of system dynamic compensation parameters ;

[0024] Each estimate is updated online in a collaborative manner through mutually independent parallel adaptive laws, and the update process is implemented using a fractional numerical integration algorithm.

[0025] On the other hand, the present invention also provides a data-driven adaptive backstepping control system for a fractional-order uncertain nonlinear system under external time-varying disturbances, for performing the method described in any of the preceding claims, comprising:

[0026] The data acquisition module is used to collect offline data of system input, status, and fractional derivatives.

[0027] The offline identification module is used to construct a data identification matrix based on offline data and identify nonlinear terms in the system.

[0028] The virtual control signal calculation module is used to construct tracking error variables based on the system reference signal and the actual output, and to perform backstepping recursive calculations using the simulated chain rule based on the online estimation values ​​of the tracking error variables and the fractional derivatives of the composite function to obtain virtual control signals of each order.

[0029] The actual control signal generation module is used to calculate the actual control input signal based on the tracking error variable of the final order of the system, the upper bound estimate of the unknown external disturbance, and the compensation parameter estimate.

[0030] The adaptive update module is used to update the adaptive estimate online and feed the adaptive estimate back to the control signal calculation of the virtual control signal calculation module and the actual control signal generation module in real time.

[0031] Compared with the prior art, the beneficial effects of the present invention are:

[0032] By constructing a data-driven adaptive backstepping control architecture, high-precision output tracking of a disturbed fractional-order nonlinear system was achieved. The offline data identification method transforms the unknown nonlinear dynamics into a data matrix solution, completely eliminating the reliance on expert experience in traditional neural network or fuzzy logic methods.

[0033] The online backstepping controller effectively solves the problem of non-closed derivatives of composite functions in fractional calculus by introducing the analog chain rule and online adaptive law. Based on rigorous Lyapunov stability analysis, it ensures that all signals in the closed-loop system are uniformly bounded. Ultimately, while guaranteeing that the preset trajectory tracking error converges to the minimum neighborhood of the origin, it significantly reduces the computational energy consumption related to control and effectively optimizes the resource allocation for engineering implementation.

[0034] Furthermore, by constructing a control input mechanism based on the hyperbolic tangent function in the final-order control law, robust suppression of time-varying external disturbances is achieved. Combined with multiple sets of adaptive estimates updated in parallel and dynamically, the system can smoothly handle various unmodeled dynamics and unknown sudden disturbances, avoiding the system chattering problem caused by traditional high-gain feedback, and greatly improving the safety and reliability of fractional-order uncertain systems under complex operating conditions. Attached Figure Description

[0035] To more clearly illustrate the technical solution of the present invention, the accompanying drawings used in the description of the specific embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0036] Figure 1 This invention provides a data-driven adaptive backstepping control method for fractional-order uncertain nonlinear systems under external time-varying disturbances.

[0037] Figure 2 This is a flowchart of the offline data-driven identification process in an embodiment of the present invention;

[0038] Figure 3 This is a flowchart of the backstepping control recursion in an embodiment of the present invention;

[0039] Figure 4 This is a flowchart illustrating the online control and adaptive update process in an embodiment of the present invention. Detailed Implementation

[0040] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0041] This invention provides a data-driven adaptive backstepping control method for a fractional-order uncertain nonlinear system under external time-varying disturbances. The control method includes:

[0042] Step S1: Establish a fractional-order uncertain nonlinear system model with strict feedback, and collect offline data of system input, state and its fractional derivative at a preset sampling time interval;

[0043] Step S2: Based on the offline data, identify the unknown system nonlinear terms in the nonlinear system model by constructing a data identification matrix;

[0044] Step S3: Construct tracking error variables based on system reference signals and actual outputs, and based on the tracking error variables and the online estimation of the fractional derivative of the composite function, perform backstepping recursive calculations using the simulated chain rule to obtain virtual control signals of each order;

[0045] Step S4: Based on the tracking error variable of the final order of the system, the upper bound estimate of the unknown external disturbance, and the compensation parameter estimate, calculate the actual control input signal acting on the nonlinear system;

[0046] Step S5: Using the current state and error data, update the adaptive estimate online through fractional derivative calculation, and feed the adaptive estimate back to the control signal calculation in steps S3 and S4 in real time.

[0047] Furthermore, the offline data collected in step S1 specifically includes: data collected during the sampling time interval. The number of samples obtained is as follows. System control input matrix System state variable matrix Fractional derivative matrix of state variables and nonlinear basis function matrix .

[0048] Further, step S2 includes: for each order of the nonlinear system model, transforming the unknown nonlinear terms in the system into a linear parameter vector form, establishing a linear equation using the nonlinear basis function matrix and the corresponding unknown constant vector estimates, and solving the equation... Obtain the data identification matrix such that the product of the basis function matrix and the data identification matrix satisfies the identity negative matrix condition, thereby eliminating the dependence on the exact mathematical model in the controller design.

[0049] Furthermore, the specific process of recursively calculating the virtual control signal using the backstepping method in step S3 is as follows: constructing the first-order tracking error variable. ,in For system status, For the previous level virtual control signal, an analog chain rule is used to apply the previous level virtual control signal to the intermediate level. Perform fractional differentiation, decomposing it into a feedforward compensation term with known derivative and an unknown bounded nonlinear function. Based on the backstep control law and the introduced upper bound estimate of the unknown bounded nonlinear function. The first-order virtual control signal is calculated. .

[0050] Furthermore, the method for calculating the actual control input signal in step S4 includes: introducing a hyperbolic tangent function. Construct robust compensation terms; combine with the tracking error variable of the final order of the system. Upper bound estimate of external time-varying disturbance Based on the dynamic status of the system, generate actual control input signals. This is to counteract the effects of external time-varying disturbances and eliminate control chattering.

[0051] Furthermore, the adaptive estimate in step S5 includes: the upper bound estimate of the unknown nonlinear function generated by the derivative of the composite function. Upper bound estimate of external time-varying disturbance and the estimated value of compensation parameters Each estimate is updated online in a parallel adaptive manner using mutually independent laws.

[0052] The core idea of ​​this invention lies in proposing a data-driven control strategy that does not rely on a precise mathematical model of the controlled object. This strategy combines backstepping with an online adaptive mechanism to solve the stable tracking problem of fractional-order nonlinear systems with external time-varying disturbances. The invention will be further described below with reference to the accompanying drawings.

[0053] This embodiment provides a data-driven adaptive backstepping control method for a fractional-order uncertain nonlinear system under external time-varying disturbances. In practical engineering deployments, this system mainly consists of a data acquisition module (sensor network), a core microprocessor (such as a DSP, FPGA, or industrial control computer), and actuators (such as inverters, motor drivers, etc.).

[0054] This embodiment uses a fractional single-machine infinite bus (SMIB) power system as the controlled object for detailed explanation. The dynamic model of this system is described by the following non-homogeneous fractional differential equations in strict feedback form:

[0055]

[0056] The fractional order of the system is set as follows: System state variables and These represent the rotor angle and angular frequency of the generator, respectively. The actual control input signal is calculated by the microprocessor and sent to the actuator; This indicates the external time-varying disturbances faced by the power system (such as combined disturbances caused by load changes and grid faults), which manifest as time-varying signals with abrupt step terms and high-frequency fluctuations.

[0057] The complete execution cycle of the control method in this embodiment includes an "offline identification stage" and an "online control stage," and the specific implementation steps are as follows:

[0058] Phase 1: Offline Data Acquisition and Matrix Processing. This phase aims to utilize offline data to extract the unknown nonlinear characteristics of the system, replacing traditional precise mechanism modeling.

[0059] Specifically, the system's data sampling interval is set to... Seconds. Before the system is officially put into closed-loop control, in the open-loop or preliminary closed-loop state (e.g., setting test detection signals). During the 30-second run, the microprocessor continuously collects system data via sensors, forming a sample count. Offline dataset.

[0060] Microprocessor extracts the second-order nonlinear basis function vector of the system And based on the state data collected at each sampling time, a dimension is constructed as follows. basis function data matrix and the state fractional derivative matrix .

[0061] Furthermore, the microprocessor solves algebraic matrix equations. (in Given a 2D identity matrix (the pseudo-inverse matrix can be obtained using the least squares method in practical engineering), the data identification matrix can be calculated. Subsequently, offline estimates of the unknown constant vector are calculated. In this embodiment, the calculation yields... This completes the identification of data-driven parameters for unknown nonlinear dynamics within the system.

[0062] Phase Two: Online Algorithm Execution (Control Loop). The system enters real-time closed-loop control mode, and the microprocessor executes the algorithm in each sampling cycle. The internal operation strictly follows the following backstep recursive logic:

[0063] Step 2.1, Control Parameter and State Initialization: Preset the controller's gain parameters and compensation parameters. To ensure system tracking accuracy and convergence speed, this embodiment sets the backstepping error gain parameter. , Hyperbolic tangent function smoothing parameter Adaptive gain update , , Adaptive damping parameters Simultaneously, the online adaptive estimation states within the microprocessor are initialized, and settings are configured. , , .

[0064] Step 2.2, First-order virtual control law calculation: The microprocessor reads the target reference signal sent by the power grid dispatching layer. Calculate the first-order tracking error of the generator angle at the current moment. Based on the data-driven identification parameters obtained in the first stage, the first-level virtual control quantity used to stabilize the first-order subsystem is calculated. :

[0065]

[0066] (Note: Since the first order in this embodiment is a linear integral, there are no unknown nonlinear terms.) (Item 0).

[0067] Step 2.3, Processing the fractional derivative of the composite function: In the backstepping recursion, the virtual control signal needs to be processed. Find the fractional derivative. This invention uses the "simulated chain rule" to expand it into a linear combination of partial derivative terms and an unknown bounded nonlinear error term. :

[0068]

[0069] because Since analytical calculation is difficult, this invention introduces an online estimate for dynamic compensation in the next step, thereby avoiding the "derivative explosion" problem caused by differentiating complex fractional composite functions.

[0070] Step 2.4, Final Calculation and Issuance of Actual Control Signal: Calculate the second-order intermediate tracking error. The microprocessor is based on the backstepping Lyapunov stabilizing function, integrating error terms, derivative feedforward compensation terms, and nonlinear estimation terms, and innovatively introduces a hyperbolic tangent function-based approach. Robust control terms are used to calculate the final actual control command. :

[0071]

[0072] The microprocessor converts digital signals into analog signals or PWM signals and sends them to the actuators of the power system (such as generator excitation controllers) to smoothly cancel out external time-varying disturbances. .

[0073] Step 2.5, Online Adaptive Law Numerical Update Calculation: At the end of the current sampling period, the microprocessor calculates the numerical update based on the current error. Given the state data, run a fractional-order numerical integration algorithm (such as the Oustaloup filter approximation algorithm or the Adams-Bashforth-Moulton predictor-corrector algorithm) to solve for and update the adaptive term for the next period in real time.

[0074] Update of upper bound estimate for unknown bounded nonlinear terms:

[0075]

[0076] Updated upper bound estimate of external time-varying disturbances:

[0077]

[0078] System dynamic compensation parameter estimation update:

[0079]

[0080] Step 2.6: The microprocessor saves the updated estimate. , , The system waits for a timer interrupt to trigger the next sampling cycle, then returns to step 2.2, repeating this process to form a data-driven adaptive disturbance rejection closed-loop control.

[0081] Simulation and experimental results show that by implementing the above-mentioned hardware and software collaborative technical solution, even under conditions where an accurate nonlinear analytical model of the generator is lacking and there are high-frequency fluctuations in a complex power grid, the generator angle can be improved. It can still accurately and quickly track the reference instructions. Compared to existing fuzzy logic control methods that rely on multiple sets of Gaussian membership functions, this invention achieves a significant reduction in both the integral absolute error (IAE) and integral square control (ISC) performance indicators. This effectively demonstrates that the method in this embodiment not only improves the robustness and tracking accuracy of the system but also greatly saves the computing power of the microprocessor and the control energy consumption of the actuator.

[0082] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0083] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A data-driven adaptive backstepping control method for fractional order uncertain nonlinear systems under external time-varying disturbance, characterized in that, include: S1: Establish a fractional-order uncertain nonlinear system model with strict feedback, and sample at a preset time interval. Collect offline datasets of system inputs, status, and corresponding fractional derivatives; S2: Based on the offline dataset, identify the unknown nonlinear terms in the nonlinear system model by constructing a data identification matrix; S3: Construct tracking error variables based on system reference signals and actual outputs. Based on the tracking error variables and the online estimate of the fractional derivative of the composite function, perform backstepping recursive calculations using the simulated chain rule to obtain virtual control signals of each order. S4: Based on the tracking error variable of the final order of the system, the upper bound estimate of the unknown external disturbance and the compensation parameter estimate, the actual control input signal acting on the nonlinear system is calculated; S5: Using the current state and error data, update the adaptive estimate online through fractional derivative calculation, and feed the updated adaptive estimate back to the control signal calculation in steps S3 and S4 in real time to form closed-loop control.

2. The method according to claim 1, characterized in that, In step S1, the offline dataset includes: System control input matrix System state variable matrix Fractional derivative matrix of state variables and nonlinear basis function matrix Furthermore, the number of samples meets the preset data scale requirements.

3. The method according to claim 1, characterized in that, Step S2 specifically involves: For each order of the fractional-order uncertain nonlinear system model, the unknown nonlinear terms in the system are transformed into linear parameter vectors, and linear equations are established using the nonlinear basis function matrix and the corresponding estimates of the unknown constant vectors. The data identification matrix is ​​obtained by solving this equation. This ensures that the result of the operation between the basis function matrix and the data identification matrix satisfies the condition of a negative identity matrix.

4. The method according to claim 1, characterized in that, In step S3, the recursive calculation of the virtual control signal using the backstepping method specifically involves: Construct the first Tracking error variable ,in For system status, This is the next higher-order virtual control signal; For the intermediate level, a simulated chain rule is used to apply the virtual control signal of the previous level. Perform fractional differentiation, decomposing it into a feedforward compensation term with known derivative and an unknown bounded nonlinear function. ; Based on the backstepping control law and the introduced upper bound estimate of the unknown bounded nonlinear function Calculate the first Virtual control signal .

5. The method according to claim 1, characterized in that, In step S4, the method for calculating the actual control input signal includes: Introducing the hyperbolic tangent function Construct a robust compensation term, where For the final-order tracking error variable, For smoothing parameters; Combined with the upper bound estimate of external time-varying disturbances Based on the dynamic state of the system, generate actual control input signals. This is to counteract the effects of external time-varying disturbances and eliminate control chattering.

6. The method according to claim 1, characterized in that, In step S5, the adaptive estimate includes: Upper bound estimate of unknown nonlinear function obtained by taking the derivative of a composite function Upper bound estimate of external time-varying disturbance and estimated values ​​of system dynamic compensation parameters ; Each estimate is updated online in a collaborative manner through mutually independent parallel adaptive laws, and the update process is implemented using a fractional numerical integration algorithm.

7. A data-driven adaptive backstepping control system for a fractional-order uncertain nonlinear system under external time-varying disturbances, characterized in that, For performing the method as described in any one of claims 1-6, comprising: The data acquisition module is used to collect offline data of system input, status, and fractional derivatives. The offline identification module is used to construct a data identification matrix based on offline data and identify nonlinear terms in the system. The virtual control signal calculation module is used to construct tracking error variables based on the system reference signal and the actual output, and to perform backstepping recursive calculations using the simulated chain rule based on the online estimation values ​​of the tracking error variables and the fractional derivatives of the composite function to obtain virtual control signals of each order. The actual control signal generation module is used to calculate the actual control input signal based on the tracking error variable of the final order of the system, the upper bound estimate of the unknown external disturbance, and the compensation parameter estimate. The adaptive update module is used to update the adaptive estimate online and feed the adaptive estimate back to the control signal calculation of the virtual control signal calculation module and the actual control signal generation module in real time.

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