Longitudinal plane depth control method for hybrid drive underwater glider

Abstract

The application relates to underwater vehicle intelligent control, in particular to a longitudinal plane depth control method of a hybrid driving underwater glider, a longitudinal motion mathematical model of the hybrid driving underwater glider is established; for the kinematics of the hybrid driving underwater glider, an adaptive line-of-sight guidance is adopted to convert the depth control problem into a line-of-sight distance tracking problem, and a pitch angle parameter is calculated; for the dynamics of the hybrid driving underwater glider, an active disturbance rejection controller based on a tracking differentiator and an extended observer is constructed to form a nonlinear state error feedback; the pitch angle parameter calculated by using the adaptive line-of-sight guidance is input into the active disturbance rejection controller, and a pitch angle control signal is output by the active disturbance rejection controller to realize real-time adjustment of the depth of the hybrid driving underwater glider; the technical scheme provided by the application can effectively overcome the defects of low control precision and complex control process existing in the prior art.

Description

Technical Field

[0001] This invention relates to intelligent control of underwater vehicles, specifically to a longitudinal depth control method for a hybrid-driven underwater glider. Background Technology

[0002] The hybrid-drive underwater glider is a novel underwater vehicle developed by combining underwater glider technology with underwater autonomous robotics. Its main features are: conventional gliding motion control does not rely on a propulsion system, but rather achieves vertical buoyancy by adjusting the glider's net buoyancy; horizontal wings attached to the fuselage generate upward or downward forces to control forward gliding. In hybrid-drive mode, propulsion is generated by thrusters to quickly traverse a specific area or achieve constant depth navigation. The hybrid-drive underwater glider overcomes the shortcomings of high power and short flight time in traditional underwater vehicles, extending endurance while significantly reducing manufacturing and operating costs, making it highly valuable for military applications and marine exploration research.

[0003] The longitudinal motion of hybrid-driven underwater gliders is easily affected by ocean currents and waves. Furthermore, the complex airframe structure leads to highly nonlinear dynamic models, making it difficult to obtain accurate model parameters. Additionally, models built for different aquatic environments lack universality. While many traditional control methods can achieve longitudinal depth control to some extent, these methods have low control accuracy and complex control processes. Summary of the Invention

[0004] (a) Technical problems to be solved

[0005] In view of the above-mentioned shortcomings of the existing technology, the present invention provides a longitudinal plane depth control method for a hybrid-driven underwater glider, which can effectively overcome the defects of low control accuracy and complex control process of the existing technology.

[0006] (II) Technical Solution

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] A method for longitudinal depth control of a hybrid-driven underwater glider includes the following steps:

[0009] S1. Establish a mathematical model of the longitudinal motion of the hybrid-driven underwater glider;

[0010] S2. For the kinematics of the hybrid-driven underwater glider, adaptive line-of-sight guidance is used to transform the depth control problem into a line-of-sight distance tracking problem, and pitch angle parameters are calculated.

[0011] S3. Regarding the dynamics of the hybrid-driven underwater glider, an active disturbance rejection controller based on a tracking differentiator and an extended observer to form a nonlinear state error feedback is constructed.

[0012] S4. The pitch angle parameters calculated using adaptive line-of-sight guidance are input into the active disturbance rejection controller, which outputs a pitch angle control signal to adjust the depth of the hybrid-driven underwater glider in real time.

[0013] Preferably, the longitudinal motion mathematical model of the hybrid-driven underwater glider is established in S1, including:

[0014] The mathematical model for the longitudinal motion of the hybrid-driven underwater glider is as follows:

[0015] ;

[0016] ;

[0017] ;

[0018] ;

[0019] ;

[0020] ;

[0021] ;

[0022] ;

[0023] ;

[0024] ;

[0025] ;

[0026] Where x and z are the axial position components, and v1 and v3 are the vertical and horizontal velocities of the glider, respectively. These are the pitch angle parameters for the glider; a glider pitching up is positive, and a glider pitching down is negative.

[0027] q is the angular velocity of the pitch angle change, J2 is the moment of inertia of the glider system, and m1 and m3 are the components of the glider's total mass. Let g be the mass of the moving point, g be the standard gravitational acceleration, and r be the mass of the moving point. P1 and r P3 Let P be the position of the slider in the glider coordinate system. P1 and P P3 M represents the momentum of the glider system acting on the slider. DLU1 is the viscous torque, u3 is the control input for the sliding mass, and u3 is the control input for the buoyancy system mass.

[0028] m0 represents the net buoyancy mass of the glider, which is the difference between the glider's total mass and its displacement mass, with downward values ​​being positive. L represents the lift force experienced by the glider, and D represents the drag force experienced by the glider. Let θ be the angle of attack, and T be the pitching moment acting on the glider.

[0029] m b Let u be the mass of the slider, and u3 be the rate of change of mass of the buoyancy system.

[0030] Preferably, in S2, for the kinematics of the hybrid-driven underwater glider, adaptive line-of-sight guidance is used to transform the depth control problem into a line-of-sight distance tracking problem, and pitch angle parameters are calculated, including:

[0031] S21. Calculate the desired trajectory P in the geodetic coordinate system. k P k+1 The angle between the coordinate system and the vertical axis of the geodetic coordinate system :

[0032] ;

[0033] Among them, (x k ,y k ) is the desired trajectory P k P k+1 Starting point P k Position coordinates, (x k+1 ,y k+1 ) is the desired trajectory P k P k+1 Central line point P k+1 Location coordinates, This represents the ATAN2 function;

[0034] S22. Calculate the glider's real-time position P0 and desired trajectory P. k P k+1 The vertical distance y between e (0):

[0035] ;

[0036] Where (x0, y0) are the position coordinates of the glider's real-time position P0;

[0037] S23. Determine the glider's real-time position P0 on the desired trajectory P. k P k+1 The foot of the perpendicular and the line of sight P k+1 Forward sight distance between ;

[0038] Among them, forward sight distance It is n times the total length L' of the glider, that is n is an integer, and its value range is 1. ;

[0039] S24. Calculate the glider's actual pitch angle, i.e., the glider's pitch angle parameters. :

[0040] .

[0041] Preferably, when the glider is at a distance P from the desired trajectory k P k+1 When the distance is relatively far, it is necessary to quickly approach the desired trajectory P. k P k+1 To reduce lateral error, a smaller forward sight distance should be selected. When the glider is at a distance P from the desired trajectory k P k+1 When nearby, a larger forward sight distance should be selected. This allows the glider to slowly approach the desired trajectory P. k P k+1 To reduce overshoot;

[0042] Considering the above factors, at the forward sight distance The calculation formula introduces the vertical distance y e (0):

[0043] ;

[0044] in, , These represent the maximum and minimum forward sight distances, respectively. Design coefficient, .

[0045] Preferably, in S3, regarding the dynamics of the hybrid-driven underwater glider, an active disturbance rejection controller based on a tracking differentiator and an extended observer to form nonlinear state error feedback is constructed, including:

[0046] S31. For the tracking differentiator, construct a tracking differentiator based on tangent sigmoid function optimization, and use the glider pitch angle parameters calculated in S2. If we consider it as v(t), then the tracking differentiator derived from the second-order nonlinear system is expressed as:

[0047] ;

[0048] ;

[0049] ;

[0050] ;

[0051] Where v1(t) and v1(t+1) are the tracking signals obtained by the tracking differentiator after smoothing the input signal at times t and t+1, respectively; v2(t) and v2(t+1) are the differential signals of the tracking signal v1(t) at time t and the tracking signal v1(t+1) at time t+1, respectively, used to represent information about the changes in the input signal; h is the sampling period; and tansig represents the tangent sigmoid function. ;

[0052] k0, l1, l2, p and To adjust the parameters required for the tracking differentiator, increasing k0 and l1 increases the system bandwidth frequency, speeds up the tracking differentiator's response to the input signal, and enhances its ability to track the input signal, but weakens its ability to suppress high-frequency noise; increasing l2 enhances the filtering performance of the tracking differentiator, but reduces the system bandwidth; increasing p enhances the filtering performance of the tracking differentiator, but also increases the tracking time delay effect of the tracking differentiator. Increasing the size of the differential will improve the tracking accuracy of the tracking differential to a certain extent.

[0053] S32. For the extended observer, based on its online estimation of the internal nonlinear dynamics and the effect of external disturbances, a third-order extended observer is adopted:

[0054] ;

[0055] ;

[0056] ;

[0057] ;

[0058] Where e is the control error, z1 and z2 are the glider's state variables, y is the system output, z3 is the real-time action variable of the unknown disturbance and uncertain model, u(t) is the control input, and b is a predefined parameter. , and The damping coefficient of the third-order extended observer;

[0059] f1 and f2 represent the fal function:

[0060] ;

[0061] In the above formula, For the gain coefficient of the extended observer, To expand the bandwidth parameter of the observer;

[0062] In the extended observer, z3(t) can track the real-time action variable of acceleration in the open-loop system. If the system is observable and acceleration plays a role, the effect will be reflected in the system output. Therefore, the action variable is extracted from the system output. When the predefined parameter b is known, the control input u(t) is considered as:

[0063] ;

[0064] Where u0 is the initial variable, i.e., the nonlinear state error feedback law;

[0065] S33. Nonlinear state error feedback uses the output of the extended observer as the state feedback variable of the active disturbance rejection controller, comparing it with the output of the tracking differentiator. The nonlinear state error feedback law u0 is:

[0066] ;

[0067] Where e1 and e2 are the components of the control error e, and To control the gain.

[0068] (III) Beneficial Effects

[0069] Compared with the prior art, the longitudinal plane depth control method for a hybrid-driven underwater glider provided by the present invention has the following beneficial effects:

[0070] 1) The introduction of adaptive line-of-sight guidance technology effectively improves the depth tracking capability of the hybrid-driven underwater glider and reduces motion error;

[0071] 2) The active disturbance rejection controller is used to compensate for the nonlinear dynamics inside the system and external disturbances, so that the active disturbance rejection controller has good robustness and adaptability, thereby effectively improving the longitudinal plane depth control accuracy of the hybrid-driven underwater glider, and the control process is very simple. Attached Figure Description

[0072] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.

[0073] Figure 1 This is a schematic diagram of the process of the present invention;

[0074] Figure 2 This is a schematic diagram of the adaptive line-of-sight guidance principle in this invention;

[0075] Figure 3 This is a diagram illustrating the effect of longitudinal plane depth control on a conventional underwater glider without adaptive line-of-sight guidance.

[0076] Figure 4 This is a diagram illustrating the effect of longitudinal plane depth control on a conventional underwater glider combined with adaptive line-of-sight guidance.

[0077] Figure 5 A diagram illustrating the effect of longitudinal plane depth control on a hybrid-drive underwater glider without adaptive line-of-sight guidance.

[0078] Figure 6 This is a diagram illustrating the effect of longitudinal plane depth control combined with adaptive line-of-sight guidance for a hybrid-driven underwater glider. Detailed Implementation

[0079] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0080] A method for longitudinal depth control of a hybrid-driven underwater glider, such as... Figure 1 As shown, S1, establishing a mathematical model of the longitudinal motion of a hybrid-driven underwater glider, specifically including:

[0081] The mathematical model for the longitudinal motion of the hybrid-driven underwater glider is as follows:

[0082] ;

[0083] ;

[0084] ;

[0085] ;

[0086] ;

[0087] ;

[0088] ;

[0089] ;

[0090] ;

[0091] ;

[0092] ;

[0093] Where x and z are the axial position components, and v1 and v3 are the vertical and horizontal velocities of the glider, respectively. These are the pitch angle parameters for the glider; a glider pitching up is positive, and a glider pitching down is negative.

[0094] q is the angular velocity of the pitch angle change, J2 is the moment of inertia of the glider system, and m1 and m3 are the components of the glider's total mass. Let g be the mass of the moving point, g be the standard gravitational acceleration, and r be the mass of the moving point. P1 and r P3 Let P be the position of the slider in the glider coordinate system. P1 and P P3 M represents the momentum of the glider system acting on the slider. DL U1 is the viscous torque, u3 is the control input for the sliding mass, and u3 is the control input for the buoyancy system mass.

[0095] m0 represents the net buoyancy mass of the glider, which is the difference between the glider's total mass and its displacement mass, with downward values ​​being positive. L represents the lift force experienced by the glider, and D represents the drag force experienced by the glider. Let θ be the angle of attack, and T be the pitching moment acting on the glider.

[0096] m b Let u be the mass of the slider, and u3 be the rate of change of mass of the buoyancy system.

[0097] S2. Regarding the kinematics of the hybrid-drive underwater glider, adaptive line-of-sight guidance is used to transform the depth control problem into a line-of-sight distance tracking problem, and pitch angle parameters are calculated, such as... Figure 2 As shown, it specifically includes:

[0098] S21. Calculate the desired trajectory P in the geodetic coordinate system. k P k+1 The angle between the coordinate system and the vertical axis of the geodetic coordinate system :

[0099] ;

[0100] Among them, (x k ,y k ) is the desired trajectory P k P k+1 Starting point P k Position coordinates, (x k+1 ,y k+1 ) is the desired trajectory P k P k+1 Central line point P k+1 Location coordinates, This represents the ATAN2 function;

[0101] S22. Calculate the glider's real-time position P0 and desired trajectory P. k P k+1 The vertical distance y between e (0):

[0102] ;

[0103] Where (x0, y0) are the position coordinates of the glider's real-time position P0;

[0104] S23. Determine the glider's real-time position P0 on the desired trajectory P. k P k+1 The foot of the perpendicular and the line of sight P k+1 Forward sight distance between ;

[0105] Among them, forward sight distance It is n times the total length L' of the glider, that is n is an integer, and its value range is 1. ;

[0106] S24. Calculate the glider's actual pitch angle, i.e., the glider's pitch angle parameters. :

[0107] .

[0108] Specifically, when the glider is at a distance P from the desired trajectory k P k+1 When the distance is relatively far, it is necessary to quickly approach the desired trajectory P. k P k+1 To reduce lateral error, a smaller forward sight distance should be selected. When the glider is at a distance P from the desired trajectory k P k+1 When nearby, a larger forward sight distance should be selected. This allows the glider to slowly approach the desired trajectory P. k P k+1 To reduce overshoot;

[0109] Considering the above factors, at the forward sight distance The calculation formula introduces the vertical distance y e (0):

[0110] ;

[0111] in, , These represent the maximum and minimum forward sight distances, respectively. Design coefficient, .

[0112] like Figures 3 to 6 As shown in the figure, it can be seen that whether it is a conventional underwater glider or a hybrid-driven underwater glider, the use of adaptive line-of-sight guidance technology can significantly improve the accuracy of longitudinal depth control.

[0113] S3. Regarding the dynamics of the hybrid-driven underwater glider, an active disturbance rejection controller based on a tracking differentiator and an extended observer to form nonlinear state error feedback is constructed, specifically including:

[0114] S31. For the tracking differentiator, construct a tracking differentiator based on the tangent sigmoid function optimization (this tracking differentiator has the characteristics of high accuracy, fast response speed, and strong noise suppression capability) to improve the system's ability to resist and adjust to unknown observations and environmental disturbances, and to use the glider pitch angle parameters calculated in S2. If we consider it as v(t), then the tracking differentiator derived from the second-order nonlinear system is expressed as:

[0115] ;

[0116] ;

[0117] ;

[0118] ;

[0119] Where v1(t) and v1(t+1) are the tracking signals obtained by the tracking differentiator after smoothing the input signal at times t and t+1, respectively; v2(t) and v2(t+1) are the differential signals of the tracking signal v1(t) at time t and the tracking signal v1(t+1) at time t+1, respectively, used to represent information about the changes in the input signal; h is the sampling period; and tansig represents the tangent sigmoid function. ;

[0120] k0, l1, l2, p and To adjust the parameters required for the tracking differentiator, increasing k0 and l1 increases the system bandwidth frequency, speeds up the tracking differentiator's response to the input signal, and enhances its ability to track the input signal, but weakens its ability to suppress high-frequency noise; increasing l2 enhances the filtering performance of the tracking differentiator, but reduces the system bandwidth; increasing p enhances the filtering performance of the tracking differentiator, but also increases the tracking time delay effect of the tracking differentiator. Increasing the size of the differential will improve the tracking accuracy of the tracking differential to a certain extent.

[0121] S32. For the extended observer (the result of the extended observer determines the control quality), based on its online estimation of the internal nonlinear dynamics and the effect of external disturbances, a third-order extended observer is adopted:

[0122] ;

[0123] ;

[0124] ;

[0125] ;

[0126] Where e is the control error, z1 and z2 are the glider's state variables, y is the system output, z3 is the real-time action variable of the unknown disturbance and uncertain model, u(t) is the control input, and b is a predefined parameter. , and The damping coefficient of the third-order extended observer;

[0127] f1 and f2 represent the fal function:

[0128] ;

[0129] In the above formula, For the gain coefficient of the extended observer, To expand the bandwidth parameter of the observer;

[0130] In the extended observer, z3(t) can track the real-time action variable of acceleration in the open-loop system. If the system is observable and acceleration plays a role, the effect will be reflected in the system output. Therefore, the action variable is extracted from the system output. When the predefined parameter b is known, the control input u(t) is considered as:

[0131] ;

[0132] Where u0 is the initial variable, i.e., the nonlinear state error feedback law;

[0133] S33. Nonlinear state error feedback uses the output of the extended observer as the state feedback variable of the active disturbance rejection controller, comparing it with the output of the tracking differentiator. The nonlinear state error feedback law u0 is:

[0134] ;

[0135] Where e1 and e2 are the components of the control error e, and To control the gain.

[0136] S4. The pitch angle parameters calculated using adaptive line-of-sight guidance are input into the active disturbance rejection controller, which outputs a pitch angle control signal to adjust the depth of the hybrid-driven underwater glider in real time.

[0137] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions will not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for longitudinal plane depth control of a hybrid-driven underwater glider, characterized in that: Includes the following steps: S1. Establish a mathematical model of the longitudinal motion of the hybrid-driven underwater glider; S2. Regarding the kinematics of the hybrid-drive underwater glider, adaptive line-of-sight guidance is used to transform the depth control problem into a line-of-sight distance tracking problem, and pitch angle parameters are calculated, specifically including: S21. Calculate the desired trajectory P in the geodetic coordinate system. k P k+1 The angle between the coordinate system and the vertical axis of the geodetic coordinate system : ； Among them, (x k ,y k ) is the desired trajectory P k P k+1 Starting point P k Position coordinates, (x k+1 ,y k+1 ) is the desired trajectory P k P k+1 Central line point P k+1 Location coordinates, This represents the ATAN2 function; S22. Calculate the glider's real-time position P0 and desired trajectory P. k P k+1 The vertical distance y between e (0): ； Where (x0, y0) are the position coordinates of the glider's real-time position P0; S23. Determine the glider's real-time position P0 on the desired trajectory P. k P k+1 The foot of the perpendicular and the line of sight P k+1 Forward sight distance between ; Among them, forward sight distance It is n times the total length L' of the glider, that is n is an integer, and its value range is 1. ; S24. Calculate the glider's actual pitch angle, i.e., the glider's pitch angle parameters. : ； S3. Regarding the dynamics of the hybrid-driven underwater glider, an active disturbance rejection controller based on a tracking differentiator and an extended observer to form a nonlinear state error feedback is constructed. S4. The pitch angle parameters calculated using adaptive line-of-sight guidance are input into the active disturbance rejection controller, which outputs a pitch angle control signal to adjust the depth of the hybrid-driven underwater glider in real time.

2. The longitudinal plane depth control method for a hybrid-driven underwater glider according to claim 1, characterized in that: In S1, a mathematical model of the longitudinal motion of a hybrid-driven underwater glider is established, including: The mathematical model for the longitudinal motion of the hybrid-driven underwater glider is as follows: ； ； ； ； ； ； ； ； ； ； ； Where x and z are the axial position components, and v1 and v3 are the vertical and horizontal velocities of the glider, respectively. These are the pitch angle parameters for the glider; a glider pitching up is positive, and a glider pitching down is negative. q is the angular velocity of the pitch angle change, J2 is the moment of inertia of the glider system, and m1 and m3 are the components of the glider's total mass. Let g be the mass of the moving point, g be the standard gravitational acceleration, and r be the mass of the moving point. P1 and r P3 Let P be the position of the slider in the glider coordinate system. P1 and P P3 M represents the momentum of the glider system acting on the slider. DL U1 is the viscous torque, u3 is the control input for the sliding mass, and u3 is the control input for the buoyancy system mass. m0 represents the net buoyancy mass of the glider, which is the difference between the glider's total mass and its displacement mass, with downward values ​​being positive. L represents the lift force experienced by the glider, and D represents the drag force experienced by the glider. Let θ be the angle of attack, and T be the pitching moment acting on the glider. m b Let u be the mass of the slider, and u4 be the rate of change of mass of the buoyancy system.

3. The longitudinal plane depth control method for a hybrid-driven underwater glider according to claim 1, characterized in that: When the glider is at a distance P from the desired flight path k P k+1 When the distance is relatively far, it is necessary to quickly approach the desired trajectory P. k P k+1 To reduce lateral error, a smaller forward sight distance should be selected. When the glider is at a distance P from the desired trajectory k P k+1 When nearby, a larger forward sight distance should be selected. This allows the glider to slowly approach the desired trajectory P. k P k+1 To reduce overshoot; Considering the above factors, at the forward sight distance The calculation formula introduces the vertical distance y e (0): ； in, , These represent the maximum and minimum forward sight distances, respectively. Design coefficient, .

4. The longitudinal plane depth control method for a hybrid-driven underwater glider according to claim 1, characterized in that: In S3, regarding the dynamics of the hybrid-driven underwater glider, an active disturbance rejection controller based on a tracking differentiator and an extended observer to form nonlinear state error feedback is constructed, including: S31. For the tracking differentiator, construct a tracking differentiator based on tangent sigmoid function optimization, and use the glider pitch angle parameters calculated in S2. If we consider it as v(t), then the tracking differentiator derived from the second-order nonlinear system is expressed as: ； ； ； ； Where v1(t) and v1(t+1) are the tracking signals obtained by the tracking differentiator after smoothing the input signal at times t and t+1, respectively; v2(t) and v2(t+1) are the differential signals of the tracking signal v1(t) at time t and the tracking signal v1(t+1) at time t+1, respectively, used to represent information about the changes in the input signal; h is the sampling period; and tansig represents the tangent sigmoid function. ; k0, l1, l2, p and To adjust the parameters required for the tracking differentiator, increasing k0 and l1 increases the system bandwidth frequency, speeds up the tracking differentiator's response to the input signal, and enhances its ability to track the input signal, but weakens its ability to suppress high-frequency noise; increasing l2 enhances the filtering performance of the tracking differentiator, but reduces the system bandwidth; increasing p enhances the filtering performance of the tracking differentiator, but also increases the tracking time delay effect of the tracking differentiator. Increasing the size of the differential will improve the tracking accuracy of the tracking differential to a certain extent. S32. For the extended observer, based on its online estimation of the internal nonlinear dynamics and the effect of external perturbations, a third-order extended observer is adopted: ； ； ； ； Where e is the control error, z1 and z2 are the glider's state variables, y is the system output, z3 is the real-time action variable of the unknown disturbance and uncertain model, u(t) is the control input, and b is a predefined parameter. , and The damping coefficient of the third-order extended observer; f1 and f2 represent the fal function: ； In the above formula, For the gain coefficient of the extended observer, To expand the bandwidth parameter of the observer; In the extended observer, z3(t) can track the real-time action variable of acceleration in the open-loop system. If the system is observable and acceleration plays a role, the effect will be reflected in the system output. Therefore, the action variable is extracted from the system output. When the predefined parameter b is known, the control input u(t) is considered as: ； Where u0 is the initial variable, i.e., the nonlinear state error feedback law; S33. Nonlinear state error feedback uses the output of the extended observer as the state feedback variable of the active disturbance rejection controller, comparing it with the output of the tracking differentiator. The nonlinear state error feedback law u0 is: ； Where e1 and e2 are the components of the control error e, and To control the gain.

Images