An underactuated control system and method based on a variable structure active disturbance rejection controller

Abstract

The application discloses an underactuated control system and method based on a variable structure active disturbance rejection controller, which comprises an underactuated inverted pendulum platform, a host computer, a variable structure active disturbance rejection controller, a photoelectric encoder, a digital signal processor, a servo driver and a direct current driving motor, the underactuated inverted pendulum platform is used for controlling a self-unstable swing rod, the host computer is used for recording swing rod and slider displacement data of the inverted pendulum platform in real time and transmitting required data of the inverted pendulum platform in real time, and the variable structure active disturbance rejection controller is used for controlling stability of the underactuated inverted pendulum platform; the underactuated control system has strong anti-interference ability when a motor of the underactuated control system is subjected to a sudden disturbance, and the speed of the motor can be recovered in time; the linear tracking differentiator, the extended state observer and the extended state observer are combined to weaken system chattering well, improve the response speed and anti-interference performance of the system, and are suitable for being widely promoted and used.

Description

Technical Field

[0001] This invention relates to the field of underactuated control technology, and specifically to an underactuated control system and method based on a variable structure active disturbance rejection controller. Background Technology

[0002] Underactuated inverted pendulums, characterized by complexity, instability, and nonlinearity, are suitable for practical engineering applications. They are a hot research topic, and studying underactuated inverted pendulum systems effectively reveals representative problems encountered in real-world applications, such as nonlinearity, robustness, stabilization, following, and tracking. Using underactuated inverted pendulums as the controlled object is also a current practice in control engineering to test the stability and nonlinearity handling capabilities of advanced control algorithms. Underactuated inverted pendulum systems also have wide applications in industrial fields, such as aerospace, military, robotics, rocket recovery and launch, and satellite attitude control.

[0003] Currently, when the motor is subjected to sudden disturbances, existing control methods generally cannot restore the motor speed in a timely manner, resulting in poor anti-interference performance of the underactuated control system and easy to cause poor stability or inability to maintain stability of the inverted pendulum system. Therefore, it is necessary to construct an underactuated control system and method based on a variable structure active disturbance rejection controller. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing technologies. To better and more effectively solve the problem that current underactuated control systems generally cannot restore the motor speed in a timely manner when the motor is subjected to sudden disturbances, resulting in poor anti-interference performance of the underactuated control system and easy occurrence of poor stability or inability to maintain stability in the inverted pendulum system, this invention provides an underactuated control system and method based on a variable structure active disturbance rejection controller. This invention achieves strong anti-interference capability when the motor of the underactuated control system is subjected to sudden disturbances, and the motor speed can be restored in a timely manner. The combined construction of a linear tracking differentiator, an extended state observer, and an extended state observer can effectively reduce system chattering and improve the system's response speed and anti-interference performance.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] An underactuated control system based on a variable structure active disturbance rejection controller includes an underactuated inverted pendulum platform, a host computer, a variable structure active disturbance rejection controller, a photoelectric encoder, a digital signal processor, a servo driver, and a DC drive motor. The underactuated inverted pendulum platform is used to control an unstable pendulum rod.

[0007] The host computer is used to record the displacement data of the pendulum rod and slider of the inverted pendulum platform in real time, and to transmit the data required by the inverted pendulum platform in real time.

[0008] The variable structure active disturbance rejection controller is used to control the stability of the underactuated inverted pendulum platform;

[0009] The photoelectric encoder is used to detect the pendulum angle and obtain pendulum angle information;

[0010] The digital signal processor is used to combine the lever angle information with the position information output by the grating to obtain the control quantity, and then output the control signal.

[0011] The servo driver is used to amplify the control signal output by the digital signal processor and drive the linear motor to generate a corresponding force to perform reverse equalization on the swing arm.

[0012] The DC drive motor is used to receive the control signal amplified by the servo driver and send the rotation speed back to the servo driver.

[0013] The aforementioned underactuated control system based on a variable structure active disturbance rejection controller includes a linear tracking differentiator, an error feedback controller, and an extended state observer.

[0014] An underactuated control method based on a variable structure active disturbance rejection controller includes the following steps:

[0015] Step (A): Construct an underactuated control system based on a variable structure active disturbance rejection controller;

[0016] Step (B): Based on the constructed underactuated control system, establish the motion equations of the inverted pendulum;

[0017] Step (C): Based on the equation of motion of the inverted pendulum, a variable structure active disturbance rejection controller is constructed, wherein the variable structure active disturbance rejection controller includes a linear tracking differentiator, an error feedback controller and an extended state observer;

[0018] Step (D): Use a linear tracking differentiator to obtain the actual displacement and velocity of the trolley and the reference displacement and velocity, and synthesize the control quantity;

[0019] Step (E) involves using an extended state observer to measure the output angle and angular velocity of the underactuated control system in real time.

[0020] Step (F) involves using an error feedback controller to provide real-time feedback on the measurement error, thereby completing the control operation of the underactuated control system.

[0021] The aforementioned underactuated control method based on a variable structure active disturbance rejection controller includes step (A) of constructing an underactuated control system based on a variable structure active disturbance rejection controller, wherein the underactuated control system includes an underactuated inverted pendulum platform, a host computer, a variable structure active disturbance rejection controller, a photoelectric encoder, a digital signal processor, a servo driver, and a DC drive motor.

[0022] The aforementioned underactuated control method based on a variable structure active disturbance rejection controller, step (B), involves establishing the motion equations of the inverted pendulum based on the constructed underactuated control system. The specific steps are as follows.

[0023] Step (B1) involves applying Newton's second law in the horizontal direction, as shown in formula (1).

[0024]

[0025] Where M represents the mass of the cart, m represents the mass of the pendulum, l represents the length of the pendulum, x represents the displacement of the cart, θ represents the angle of the pendulum, and u represents the output.

[0026] Step (B2) involves applying Newton's second law in the direction perpendicular to the pendulum, as shown in formula (2).

[0027]

[0028] Where g represents the acceleration due to gravity;

[0029] Step (B3) involves solving the differential equation and linearizing it to obtain the equation of motion for the inverted pendulum, as shown in formula (3).

[0030]

[0031] The aforementioned underactuated control method based on a variable structure active disturbance rejection controller includes step (C), which involves constructing a variable structure active disturbance rejection controller based on the inverted pendulum motion equation. The variable structure active disturbance rejection controller comprises a linear tracking differentiator, an error feedback controller, and an extended state observer. The linear tracking differentiator is used to acquire the actual displacement and velocity of the vehicle and the reference displacement and velocity, and synthesize the control quantity. The extended state observer is used to perform real-time calculation of the output angle and angular velocity of the underactuated control system. The error feedback controller is used to provide real-time feedback of the calculation error.

[0032] In the aforementioned underactuated control method based on a variable structure active disturbance rejection controller, step (D) involves using a linear tracking differentiator to obtain the actual displacement and velocity of the vehicle and the reference displacement and velocity, and then synthesizing the control quantity. The linear tracking differentiator is shown in formula (4).

[0033]

[0034] Where m0 represents the actual displacement of the vehicle, m1 represents the tracked value of the vehicle displacement, m2 represents the calculated value of the vehicle speed, and h1 represents the adjustable parameter; in order to obtain the tracked value V1 and the differential value V2 of the vehicle reference displacement, a linear tracking differentiator is used, and the adjustable parameter of the linear tracking differentiator is h2.

[0035] In the aforementioned underactuated control method based on a variable structure active disturbance rejection controller, step (E) involves using an extended state observer to calculate the output angle and angular velocity of the underactuated control system in real time. The extended state observer is shown in formula (5).

[0036]

[0037] Where e represents the observation error, b0 represents the compensation factor, u represents the control output, zal(·) is a nonlinear function, and β 01 β 02 and β 03 σ1 and σ2 represent the gain of the extended state observer, α1, α2, δ1, δ2, γ1 and γ2 are all adjustable parameters;

[0038] The expression for the nonlinear function zal(·) is shown in formula (6).

[0039]

[0040] Where δ and γ represent the adjustable parameters of the controller, and α represents the width of the linear interval of the zal function.

[0041] In the aforementioned underactuated control method based on a variable structure active disturbance rejection controller, step (F) involves using an error feedback controller to provide real-time feedback of the measured error, thereby completing the control operation of the underactuated control system. The error feedback controller is shown in formula (7).

[0042]

[0043] Where n1, n2, n3, and n4 represent the gain of the error feedback controller.

[0044] The beneficial effects of this invention are as follows: This invention provides an underactuated control system and method based on a variable structure active disturbance rejection controller. First, a novel ZAL function is constructed, solving the problem of chattering and large errors in traditional functions due to unevenness at inflection points, which leads to increased system gain. Next, to obtain the two outputs of the underactuated inverted pendulum system, two linear tracking differentiators are constructed to obtain the actual displacement and velocity, as well as the reference displacement and velocity. Subsequently, an extended state observer is constructed based on the novel nonlinear ZAL function to perform real-time calculations of the output angle and angular velocity of the underactuated system. Furthermore, the control law part combines the two output quantities into a control quantity, thereby achieving control over the two... With effective control of the output, a control embodiment for the underactuated inverted pendulum system was constructed. The output response curve of this control method was compared and analyzed with that of traditional active disturbance rejection control. The results verified the superiority and feasibility of the constructed variable structure active disturbance rejection control method, improving the stability and anti-interference capability of the underactuated inverted pendulum system. In summary, this invention effectively achieves strong anti-interference capability when the motor of the underactuated control system is subjected to sudden disturbances, and the motor speed can be restored in a timely manner. The constructed linear tracking differentiator, extended state observer, and extended state observer can effectively reduce system chattering, improving the system's response speed and anti-interference performance. Attached Figure Description

[0045] Figure 1 This is a schematic diagram of an underactuated control system based on a variable structure active disturbance rejection controller according to the present invention;

[0046] Figure 2 This is a schematic diagram of the motion equation modeling of the inverted pendulum in the underactuated control system of the present invention;

[0047] Figure 3 This is a schematic diagram of the control structure of the underactuated control system in an embodiment of the present invention;

[0048] Figure 4 This is a schematic diagram comparing the zal(e,α,δ,γ) and fal(x,a,δ) functions in an embodiment of the present invention;

[0049] Figure 5 This is a schematic diagram comparing the error gain of the zal(e,α,δ,γ) and fal(x,a,δ) functions in an embodiment of the present invention;

[0050] Figure 6 This is a waveform comparison diagram of the response of the pendulum angle between the variable structure active disturbance rejection controller (ADRC) and the traditional ADRC in an embodiment of the present invention.

[0051] Figure 7 This is a comparison diagram of the response waveforms of the trolley displacement signal between the variable structure active disturbance rejection controller (ADRC) and the traditional ADRC in an embodiment of the present invention.

[0052] Figure 8 This is a comparison diagram of the sinusoidal signal response waveforms of the variable structure active disturbance rejection controller (ADRC) and the traditional ADRC in an embodiment of the present invention.

[0053] Figure 9 This is a comparison diagram of the step signal response waveforms of the variable structure active disturbance rejection controller (ADRC) and the traditional ADRC in an embodiment of the present invention. Detailed Implementation

[0054] The present invention will now be further described with reference to the accompanying drawings.

[0055] like Figure 1 As shown, an underactuated control system based on a variable structure active disturbance rejection controller according to the present invention includes an underactuated inverted pendulum platform, a host computer, a variable structure active disturbance rejection controller, a photoelectric encoder, a digital signal processor, a servo driver and a DC drive motor. The underactuated inverted pendulum platform is used to control an unstable pendulum rod.

[0056] The host computer is used to record the displacement data of the pendulum rod and slider of the inverted pendulum platform in real time, and to transmit the data required by the inverted pendulum platform in real time.

[0057] The variable structure active disturbance rejection controller is used to control the stability of the underactuated inverted pendulum platform;

[0058] The photoelectric encoder is used to detect the pendulum angle and obtain pendulum angle information;

[0059] The digital signal processor is used to combine the lever angle information with the position information output by the grating to obtain the control quantity, and then output the control signal.

[0060] The servo driver is used to amplify the control signal output by the digital signal processor and drive the linear motor to generate a corresponding force to perform reverse equalization on the swing arm.

[0061] The DC drive motor is used to receive the control signal amplified by the servo driver and send the rotation speed back to the servo driver.

[0062] Specifically, the variable structure active disturbance rejection controller includes a linear tracking differentiator, an error feedback controller, and an extended state observer.

[0063] An underactuated control method based on a variable structure active disturbance rejection controller includes the following steps:

[0064] Step (A) constructs an underactuated control system based on a variable structure active disturbance rejection controller, wherein the underactuated control system includes an underactuated inverted pendulum platform, a host computer, a variable structure active disturbance rejection controller, a photoelectric encoder, a digital signal processor, a servo driver, and a DC drive motor.

[0065] like Figure 2 As shown, in step (B), based on the constructed underactuated control system, the motion equations of the inverted pendulum are established. The specific steps are as follows:

[0066] Step (B1) involves applying Newton's second law in the horizontal direction, as shown in formula (1).

[0067]

[0068] Where M represents the mass of the cart, m represents the mass of the pendulum, l represents the length of the pendulum, x represents the displacement of the cart, θ represents the angle of the pendulum, and u represents the output.

[0069] Step (B2) involves applying Newton's second law in the direction perpendicular to the pendulum, as shown in formula (2).

[0070]

[0071] Where g represents the acceleration due to gravity;

[0072] Step (B3) involves solving the differential equation and linearizing it to obtain the equation of motion for the inverted pendulum, as shown in formula (3).

[0073]

[0074] Step (C): Based on the inverted pendulum motion equation, a variable structure active disturbance rejection controller is constructed. The variable structure active disturbance rejection controller includes a linear tracking differentiator, an error feedback controller, and an extended state observer. The linear tracking differentiator is used to obtain the actual displacement and velocity of the trolley and the reference displacement and velocity and synthesize the control quantity. The extended state observer is used to calculate the output angle and angular velocity of the underactuated control system in real time. The error feedback controller is used to provide real-time feedback on the calculation error.

[0075] In step (D), the actual displacement and velocity of the vehicle and the reference displacement and velocity are obtained using a linear tracking differentiator, and the control quantity is synthesized. The linear tracking differentiator is shown in formula (4).

[0076]

[0077] Where m0 represents the actual displacement of the vehicle, m1 represents the tracked value of the vehicle displacement, m2 represents the calculated value of the vehicle speed, and h1 represents the adjustable parameter; in order to obtain the tracked value V1 and the differential value V2 of the vehicle reference displacement, a linear tracking differentiator is used, and the adjustable parameter of the linear tracking differentiator is h2.

[0078] In step (E), the output angle and angular velocity of the underactuated control system are calculated in real time using an extended state observer, as shown in formula (5).

[0079]

[0080] Where e represents the observation error, b0 represents the compensation factor, u represents the control output, zal(·) is a nonlinear function, and β 01 β 02 and β 03 σ1 and σ2 represent the gain of the extended state observer, α1, α2, δ1, δ2, γ1 and γ2 are all adjustable parameters;

[0081] The expression for the nonlinear function zal(·) is shown in formula (6).

[0082]

[0083] Where δ and γ represent the adjustable parameters of the controller, and α represents the width of the linear interval of the zal function.

[0084] Step (F) involves using an error feedback controller to provide real-time feedback on the measurement error, thereby completing the control operation of the underactuated control system. The error feedback controller is shown in formula (7).

[0085]

[0086] Where n1, n2, n3, and n4 represent the gain of the error feedback controller.

[0087] To better illustrate the effects of the present invention, a specific embodiment of the method of the present invention is described below; wherein ADRC is a variable structure active disturbance rejection controller, TD is a linear tracking differentiator, LSEF is an error feedback controller and ESO is an extended state observer.

[0088] (1) In this embodiment, to facilitate the comparison of the performance of the improved nonlinear function qal and the traditional nonlinear function fal, the characteristic curves of the two functions are compared in the MATLAB simulation software while keeping α = 0.2, δ = 0.25, and γ = 0.1. Figure 3 As shown.

[0089] (2) The error feedback gain comparison diagram of the ZAL function of this invention and the existing FAL function in this embodiment is shown in the figure below. Figure 4 As shown, when the error is greater than 1, the ZAL function outputs a lower constant value, significantly smaller than the FAL function's output value. It also exhibits better convergence and smoothness near the origin, overcoming the inflection point problem of traditional nonlinear functions near the origin. Figure 4 It can be seen that when the input error is small, the output gain of the ZAL function is higher than that of the FAL function; when the input error is large, the output gain of the ZAL function is lower than that of the FAL function; and when the input error approaches 0, the output gain of the ZAL function is small and is less likely to cause system oscillation, which reflects the necessary principle of selecting nonlinear functions with large gain for small errors and small gain for large errors. This ensures that the subsequent extended state observer (ESO) will not deviate significantly when tracking the system state, has better observation capabilities, and improves the system's anti-interference ability.

[0090] (3) This embodiment simulates the stable control of the underactuated system in conjunction with the variable structure active disturbance rejection controller of the present invention. To illustrate the advantages of the variable structure active disturbance rejection controller of the present invention, a simulation comparison is performed with the traditional ADRC. The simulation verifies that the underactuated system controlled by the variable structure active disturbance rejection controller of the present invention can achieve better stable control and stronger disturbance rejection capability. To verify the performance difference under the same conditions, the parameters of the traditional ADRC and the ADRC of the present invention are selected in the same way. The selection of the platform parameters of the underactuated control system is shown in Table 1.

[0091] Table 1 Parameters of the Inverted Pendulum Platform

[0092]

[0093] The following explanation, illustrated with simulation diagrams, illustrates the following: From Figure 5 , Figure 6 As can be seen, the variable structure active disturbance rejection controller of this invention achieves the shortest control time for the pendulum's arm deflection angle and the trolley displacement. Traditional ADRC control strategies require 3.2 seconds for the pendulum to stabilize, while the variable structure active disturbance rejection controller of this invention stabilizes the pendulum in just 2 seconds. Furthermore, the response curve of the variable structure active disturbance rejection controller of this invention is smooth, with low amplitude and frequency. The curve showing the trolley displacement controlled by the variable structure active disturbance rejection controller of this invention also demonstrates that the stabilized position returns to the reference position, verifying that this invention can achieve both high-precision control of the pendulum's deflection angle and precise control of the trolley displacement, solving the problem that traditional ADRC methods cannot simultaneously control the pendulum's displacement while controlling the pendulum's angle. Figure 7 and Figure 8 By comparing the variable structure active disturbance rejection controller of the present invention with the traditional ADRC control strategy under the addition of interference, it can be concluded that when the motor receives external interference, the variable structure active disturbance rejection controller of the present invention has a faster speed recovery time than the traditional ADRC control strategy. Furthermore, the variable structure active disturbance rejection controller of the present invention reacts with smaller fluctuations than the traditional ADRC control strategy, demonstrating better anti-interference performance. This indicates that the control effect of the variable structure active disturbance rejection controller of the present invention is superior to that of the traditional active disturbance rejection algorithm control.

[0094] In summary, the present invention provides an underactuated control system and method based on a variable structure active disturbance rejection controller. First, an underactuated control system based on the variable structure active disturbance rejection controller is constructed. Then, the motion equation of an inverted pendulum is established based on the constructed underactuated control system. Next, a variable structure active disturbance rejection controller is constructed based on the inverted pendulum motion equation. The variable structure active disturbance rejection controller includes a linear tracking differentiator, an error feedback controller, and an extended state observer. Subsequently, the actual displacement and velocity of the trolley and the reference displacement and velocity are obtained using the linear tracking differentiator and synthesized into control quantities. Then, the output angle and angular velocity of the underactuated control system are calculated in real time using the extended state observer. Finally, the error feedback controller provides real-time feedback on the calculation error, completing the control operation of the underactuated control system. This invention effectively achieves strong anti-interference capability when the motor of the underactuated control system is subjected to sudden disturbances, and the motor speed can be restored in a timely manner. The combined linear tracking differentiator, extended state observer, and extended state observer can effectively reduce system chattering, improving the system's response speed and anti-interference performance.

[0095] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. An underactuated control method based on a variable structure active disturbance rejection controller, characterized in that: Includes the following steps, Step (A): Construct an underactuated control system based on a variable structure active disturbance rejection controller; Step (B): Based on the constructed underactuated control system, establish the motion equations of the inverted pendulum; Step (C): Based on the equation of motion of the inverted pendulum, a variable structure active disturbance rejection controller is constructed, wherein the variable structure active disturbance rejection controller includes a linear tracking differentiator, an error feedback controller and an extended state observer; Step (D): Use a linear tracking differentiator to obtain the actual displacement and velocity of the trolley and the reference displacement and velocity, and synthesize the control quantity; In step (E), the output angle and angular velocity of the underactuated control system are calculated in real time using an extended state observer, as shown in formula (5). Where e represents the observation error, b0 represents the compensation factor, u represents the control output, zal(·) is a nonlinear function, and β 01 β 02 and β 03 σ1 and σ2 represent the gain of the extended state observer, α1, α2, δ1, δ2, γ1 and γ2 are all adjustable parameters; The expression for the nonlinear function zal(·) is shown in formula (6). Where δ and γ represent the adjustable parameters of the controller, and α represents the width of the linear interval of the ZAL function; Step (F) uses an error feedback controller to provide real-time feedback on the measurement error, thereby completing the control operation of the underactuated control system.

2. The underactuated control method based on a variable structure active disturbance rejection controller according to claim 1, characterized in that: Step (A) constructs an underactuated control system based on a variable structure active disturbance rejection controller, wherein the underactuated control system includes an underactuated inverted pendulum platform, a host computer, a variable structure active disturbance rejection controller, a photoelectric encoder, a digital signal processor, a servo driver, and a DC drive motor.

3. The underactuated control method based on a variable structure active disturbance rejection controller according to claim 2, characterized in that: Step (B): Based on the constructed underactuated control system, establish the motion equations of the inverted pendulum. The specific steps are as follows: Step (B1) involves applying Newton's second law in the horizontal direction, as shown in formula (1). Where M represents the mass of the cart, m represents the mass of the pendulum, l represents the length of the pendulum, x represents the displacement of the cart, θ represents the angle of the pendulum, and u represents the output. Step (B2) involves applying Newton's second law in the direction perpendicular to the pendulum, as shown in formula (2). Where g represents the acceleration due to gravity; Step (B3) involves solving the differential equation and linearizing it to obtain the equation of motion for the inverted pendulum, as shown in formula (3).

4. The underactuated control method based on a variable structure active disturbance rejection controller according to claim 3, characterized in that: In step (D), the actual displacement and velocity of the vehicle and the reference displacement and velocity are obtained using a linear tracking differentiator, and the control quantity is synthesized. The linear tracking differentiator is shown in formula (4). Where m0 represents the actual displacement of the vehicle, m1 represents the tracked value of the vehicle displacement, m2 represents the calculated value of the vehicle speed, and h1 represents the adjustable parameter; in order to obtain the tracked value V1 and the differential value V2 of the vehicle reference displacement, a linear tracking differentiator is used, and the adjustable parameter of the linear tracking differentiator is h2.

5. The underactuated control method based on a variable structure active disturbance rejection controller according to claim 4, characterized in that: Step (F) involves using an error feedback controller to provide real-time feedback on the measurement error, thereby completing the control operation of the underactuated control system. The error feedback controller is shown in formula (7). Where n1, n2, n3, and n4 represent the gain of the error feedback controller.

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