Sliding mode model-free adaptive control method and system for second-order nonlinear system

Abstract

The application discloses a sliding mode model-free adaptive control method and system for a second-order nonlinear system, the model is decomposed into two discrete subsystems in series, and an expression of the overall system PPD is given; then, expected states of the output of the second discrete subsystem are designed by using output data of the two subsystems, and then the tracking control of the overall target is realized by continuously tracking the expected states updated on line by using MFAC. The application designs a SM-MFAC control algorithm for the second-order nonlinear system which does not satisfy the generalized lipschitz condition and the PPD sign invariance, and the SM-MFAC control algorithm can overcome the influence of the nonlinear characteristics in the complex system by using only the measured input and output data and order information, and quickly converges to the expected trajectory without establishing the object model.

Description

Technical Field

[0001] This invention belongs to the field of system control technology and relates to a sliding mode model-free adaptive control method and system for second-order nonlinear systems. Background Technology

[0002] Currently, most complex systems are composed of subsystems interconnected in series, parallel, and feedback configurations. However, for a system consisting of two subsystems connected in series, the relationship between their direct control input and output may not necessarily conform to the original generalized Lipschitz condition. If the information of the constructed series model is ignored and it is assumed that it satisfies all the premises of MFAC, and the MFAC control scheme is applied to the whole system, the parameter values ​​in the controller are very likely to be misestimated, resulting in a significant reduction in control effect, instability, or even irreversible damage to the system.

[0003] Solutions for high-order systems mainly include distributed estimation centralized control MFAC, PID-MFAC, PFDL-MFAC, and FFDL-MFAC to achieve control of the entire system.

[0004] Distributed estimation and centralized control MFAC requires all subsystems to satisfy the generalized Lipschitz condition, and the estimated parameters in the control algorithm increase exponentially, leading to a significant increase in control algorithm complexity. For a class of unknown SISO nonlinear non-affine discrete-time systems, incremental PI and PID controllers are designed based on MFAC, and the convergence domain of the parameters is given through a compression mapping method. An improved compact-format model-free adaptive control (iCF-MFAC) adds a time-varying proportional control term compared to the original controller (CF-MFAC) which only has a time-varying integral term, providing better dynamic performance. These two control algorithms combine MFAC and PID, improving control performance, but do not provide controller design or stability proofs for second-order nonlinear systems. For stable control of high-order systems, PFDL-MFAC, FFDL-MFAC, and other linearization methods such as feedback linearization and Taylor linearization have significant advantages over other methods. The introduction of pseudo-order avoids the design of high-order controllers, reducing computational burden and difficulty. However, for high-order systems, using only input-output data and ignoring order information results in unsatisfactory control performance.

[0005] The above methods improve control performance by making full use of known structural information or other state data. However, these schemes are only applicable to systems that meet the generalized Lipschitz condition and PPD satisfies sign invariance. Moreover, many parameters to be estimated are introduced in the design process, which increases the complexity of the controller sharply. Some control schemes also lack rigorous stability and convergence proofs. Summary of the Invention

[0006] The purpose of this invention is to solve the problems in the prior art and provide a sliding mode model-free adaptive control method and system for second-order nonlinear systems. This invention realizes trajectory tracking for a class of second-order nonlinear systems that do not satisfy the generalized Lipschitz condition and the sign invariance of PPD. First, the model is decomposed into two discrete subsystems in series, and the expression for the PPD of the overall system is given. Then, the desired state of the output of the second discrete subsystem is designed using the output data of the two subsystems. Finally, the tracking control of the overall target is achieved by continuously tracking the online updated desired state using MFAC.

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

[0008] In a first aspect, the present invention provides a sliding mode model-free adaptive control method for second-order nonlinear systems, comprising the following steps:

[0009] The model is decomposed into two discrete subsystems in series based on its order;

[0010] Establish a dynamic linearized equation relating the input and output changes of two discrete subsystems;

[0011] The PPD expression for the second-order nonlinear system is obtained from the dynamic linearization equation;

[0012] Based on the output data of the two discrete subsystems, the desired state of the output of the second discrete subsystem is obtained;

[0013] MFAC is used to track the desired state and achieve adaptive control.

[0014] Secondly, the present invention provides a sliding mode model-free adaptive control system for second-order nonlinear systems, comprising:

[0015] The model decomposition module is used to decompose the model into two discrete subsystems in series according to the order of the model; the dynamic linearization equation construction module is used to establish the dynamic linearization equation between the input changes and output changes of the two discrete subsystems.

[0016] The first calculation module is used to obtain the PPD expression of the second-order nonlinear system based on the dynamic linearization equation;

[0017] The second calculation module is used to obtain the desired state of the output of the second discrete subsystem based on the output data of the two discrete subsystems.

[0018] The tracking control module is used to track the desired state using MFAC to achieve adaptive control.

[0019] Thirdly, the present invention provides a computer device including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the method described above.

[0020] Fourthly, the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the method described above.

[0021] Compared with the prior art, the present invention has the following beneficial effects:

[0022] For MFAC control of second-order nonlinear systems, existing control algorithms only utilize input-output information to continuously improve MFAC, but still cannot achieve stable control. This invention decomposes the second-order nonlinear system model and gives a PPD expression reflecting the overall characteristics of a class of second-order nonlinear systems. It proves that this class of systems does not meet the assumptions of general MFAC. For this class of second-order nonlinear systems that do not satisfy the generalized Lipschitz condition and PPD sign invariance, an SM-MFAC control algorithm is designed. This algorithm can overcome the influence of nonlinear characteristics in complex systems and quickly converge to the desired trajectory using only measured input-output data and order information, without establishing an object model. In addition, the stability and convergence of the SM-MFAC controller are proven. Simulation experiments on the joint control of a free-floating space robotic arm verify the analysis of the PPD characteristics of a class of second-order nonlinear systems, which is convincing. Figure 5 As shown, the experimental results of joint control under PID, PD-MFAC and SM-MFAC are compared, which shows that the control algorithm designed in this invention can converge to the desired trajectory faster and better. Attached Figure Description

[0023] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0024] Figure 1 This is a flowchart of the method of the present invention.

[0025] Figure 2 This is a schematic diagram of the system of the present invention.

[0026] Figure 3 This is a block diagram of two discrete subsystems connected in series in this invention.

[0027] Figure 4 This is a block diagram of the controller design for the present invention.

[0028] Figure 5 This is a simulation experiment trajectory tracking diagram of the present invention. Detailed Implementation

[0029] 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 embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0030] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0031] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0032] In the description of the embodiments of the present invention, it should be noted that if terms such as "upper," "lower," "horizontal," or "inner" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, or the orientation or positional relationship commonly used when the product of the invention is in use, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the present invention. Furthermore, terms such as "first" and "second" are only used to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0033] Furthermore, the use of the term "horizontal" does not imply that the component must be absolutely horizontal, but rather that it can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal than "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted.

[0034] In the description of the embodiments of the present invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in the present invention according to the specific circumstances.

[0035] The present invention will now be described in further detail with reference to the accompanying drawings:

[0036] See Figure 1 This invention discloses a sliding mode model-free adaptive control method for second-order nonlinear systems, comprising the following steps:

[0037] Step S1: Decompose the model into two discrete subsystems connected in series according to the order of the model;

[0038] Step S2: Establish the dynamic linearized equations between the input and output changes of the two discrete subsystems;

[0039] Step S3: Obtain the PPD expression for the second-order nonlinear system based on the dynamic linearization equation;

[0040] Step S4: Based on the output data of the two discrete subsystems, obtain the desired state of the output of the second discrete subsystem;

[0041] Step S5: Use MFAC to track the desired state and achieve adaptive control.

[0042] This embodiment also provides a sliding mode model-free adaptive control method for second-order nonlinear systems. Step S1 may include the following steps:

[0043] Based on the model order, the second-order non-affine nonlinear system is decomposed into two cascaded discrete subsystems: a first discrete subsystem and a second discrete subsystem. The second-order non-affine nonlinear system is as follows:

[0044]

[0045] Where x1 and x2 are the system output states, g is an unknown nonlinear function, u is the system control input, and t is time;

[0046] The first discrete subsystem is as follows:

[0047] x2(k+1) = x1(k) + Tx2(k)

[0048] Where x1(k) is the output of the first discrete subsystem, k is the time of the discrete system, T is the unit time step, and x2(k) is the output of the second discrete subsystem.

[0049] The second discrete subsystem is as follows:

[0050]

[0051] Where u(k) is the input of the second discrete subsystem. n is the length of the input sliding time window of the control system. u Let f() be the length of the output sliding time window of the control system, and let f() be an unknown nonlinear function.

[0052] This embodiment also provides a sliding mode model-free adaptive control method for second-order nonlinear systems, where step S2 may include the following steps:

[0053] Assumptions are made regarding the characteristics of the second discrete subsystem:

[0054] Assumption 1: Except for finite time points, f() with respect to the th The partial derivatives of the variables are continuous;

[0055] Assumption 2: Except at finite time points, the input and output of the second discrete subsystem satisfy the generalized Lipschitz condition;

[0056] Assuming the input change at the current moment is not zero, i.e., |Δu(k)|≠0; the second discrete subsystem is transformed into a partially oriented dynamically linearized PFDL data model, yielding the dynamic linearized equations between the input and output changes of the two discrete subsystems:

[0057]

[0058] in, The time-varying parameter of the pseudopartial derivative PPD. And the sign remains unchanged; Δu(k) is the change in control input at time k.

[0059] This embodiment also provides a sliding mode model-free adaptive control method for second-order nonlinear systems. Step S3 may include the following steps:

[0060] The partial pseudo-derivative of the first discrete subsystem can be easily obtained from the linear dynamic equation:

[0061]

[0062] Based on the data model of the overall system, the partial pseudo-derivative of the second discrete subsystem is obtained:

[0063]

[0064] Wherein, Δx2(k) is the output change of the second discrete subsystem at time k;

[0065] Based on the expressions for the partial pseudo-derivatives of the first and second discrete subsystems, the PPD expression for the second-order nonlinear system, which reflects the input-output characteristics of the overall system, is obtained:

[0066]

[0067] in, This is the PPD expression for a second-order nonlinear system.

[0068] This embodiment also provides a sliding mode model-free adaptive control method for second-order nonlinear systems. Step S4 may include the following steps:

[0069] The desired output state of the second discrete subsystem is as follows:

[0070] The expected curves for the first discrete subsystem and the second discrete subsystem are as follows:

[0071]

[0072] The current state error is calculated based on the measured state data:

[0073]

[0074] in:

[0075]

[0076] Select to switch manifolds:

[0077] s(k)=ce1(k)+e2(k)

[0078] Based on the output data of the first discrete subsystem and the second discrete subsystem at the current moment, the virtual desired input of the first discrete subsystem is designed using backstepping control. The desired state of the second discrete subsystem is obtained:

[0079]

[0080] Where ω is the arrival parameter, s(k) is the switching manifold, c(k) is the switching manifold parameter, and ρ is the control step size factor. λ is the estimated value of the pseudo-partial derivative of the second discrete subsystem, and λ is the control weight factor.

[0081] This embodiment also provides a sliding mode model-free adaptive control method for second-order nonlinear systems. Step S5 may include the following steps:

[0082] For the second discrete subsystem, according to the PPD estimation criterion function Seeking information about The extreme values ​​are used to obtain the PPD estimation algorithm;

[0083] The PPD estimation criterion function as follows:

[0084]

[0085] in, The pseudo-partial derivative of the second discrete subsystem. This is an estimate of the pseudo-partial derivative of the second discrete subsystem;

[0086] The PPD estimation algorithm is as follows:

[0087]

[0088] In the formula, η∈(0,1] is the estimated step size factor, and μ>0 is the estimated weight factor;

[0089] Based on the control criterion function J(u(k)), find the extreme value of Δu(k) to obtain the control input u(k):

[0090] The control criterion function J(u(k)) is as follows:

[0091]

[0092] The control input u(k) is as follows:

[0093]

[0094] In the formula, ρ∈(0,1] is the control step size factor, and λ>0 is the control weight factor, which is used to limit the change of control input.

[0095] See Figure 2 This embodiment provides a sliding mode model-free adaptive control system for second-order nonlinear systems, including:

[0096] The model decomposition module is used to decompose the model into two discrete subsystems in series according to the order of the model;

[0097] The dynamic linearization equation construction module is used to establish dynamic linearization equations between the input and output changes of two discrete subsystems.

[0098] The first calculation module is used to obtain the PPD expression of the second-order nonlinear system based on the dynamic linearization equation;

[0099] The second calculation module is used to obtain the desired state of the output of the second discrete subsystem based on the output data of the two discrete subsystems.

[0100] The tracking control module is used to track the desired state using MFAC to achieve adaptive control.

[0101] Example

[0102] For a typical SISO (Single-Input Single-Output) second-order nonlinear system, various nonlinear characteristics such as friction, dead zone, and input saturation often exist in reality. Although there have been fruitful research results on nonlinearity with different characteristics, the accuracy of models built for complex systems is still limited, and theoretical research is also difficult to advance. Understanding these nonlinear characteristics, efforts should be made to avoid modeling the aforementioned complex nonlinear features, such as... Figure 3 As shown, it is decomposed into two cascaded subsystems based solely on the model order, so its overall data model is as follows:

[0103]

[0104] Assumption 1: Except for finite time points, f() with respect to the th The partial derivatives of the variables are continuous.

[0105] Assumption 2: Except at finite time points, the input and output of the second discrete subsystem satisfy the generalized Lipschitz condition.

[0106] For the second discrete subsystem, assuming it satisfies Assumptions 1 and 2, a dynamic linearization equation between input and output changes can be obtained. For the first discrete subsystem, it clearly satisfies Assumption 1, and based on the established overall system model, a similar dynamic linearization equation between input and output changes can be established. Combining these two points, an expression for the PPD of the overall system can be derived. The results prove that it does not satisfy the generalized Lipschitz condition and the sign invariance assumption, verifying the conjecture about this class of second-order nonlinear systems.

[0107] For the second discrete subsystem, consider the following PPD estimation criterion function:

[0108]

[0109] Find the expression about the above. The extreme value of PPD can be obtained by finding the PPD estimation algorithm.

[0110] Consider the following criterion function:

[0111]

[0112] Finding the extreme value of u(k) in the above equation yields the estimation algorithm for PPD.

[0113] Define the expected curve x of the first discrete subsystem 1d And select the switching manifold s(k);

[0114] Based on the output data of the first discrete subsystem and the second discrete subsystem at the current moment, the desired input of the first discrete subsystem is defined. That is, the desired state of the second discrete subsystem:

[0115]

[0116] Finally, the design of the sliding mode model-free adaptive controller (SM-MFAC) was completed, and its detailed process is as follows: Figure 4 As shown, the following theorem holds:

[0117] Theorem 1: For the overall system (2), under the premise of satisfying assumptions 1, 2, 3 and 4, any given bounded expected state x 1d If (k+1) is used with the SM-MFAC scheme, then there exists a positive integer λ. min >0, such that when λ>λ min Sometimes:

[0118] (1) If the output tracking error of the first discrete subsystem is monotonically convergent, then:

[0119]

[0120] (2) If the output of the second discrete subsystem monotonically converges to zero, then:

[0121]

[0122] (3) The entire closed-loop system is BIBO stable, that is, the output sequences x1(k), x2(k) and the input sequence u(k) are all bounded.

[0123] To prove the above theorem, we first calculate the manifold switching error. The results show that its magnitude is positively correlated with the PPD estimation error. Referring to Hou Zhongsheng's definition of PPD, we know that PPD changes very slowly. Assuming that when k is sufficiently large, its change value is ignored and is a constant, then as time increases, the PPD estimation error approaches zero. At this time, the output error of the first discrete subsystem and the second discrete subsystem will reach the switching manifold in a finite time and slide along the switching manifold.

[0124] To prove that the input of the designed controller is convergent, let the flow switching increment be zero to obtain the output error of the second discrete subsystem. Further, the difference of the equivalent control can be calculated. By continuously expanding the equivalent control into the series sum of the equivalent control difference, it can be found that its magnitude is positively correlated with the cumulative sum of the output of the second discrete subsystem. As proved by the conclusion in Theorem 1 above, x2(k) is exponentially convergent, so its series sum must be absolutely convergent, thus the control input is bounded.

[0125] In conclusion, the SM-MFAC controller exhibits good convergence and stability, thus proving Theorem 1.

[0126] A single-DOF free-floating space robot was constructed using the joint simulation platform of MBDyn and Simulink, multibody kinematics and dynamics simulation software. The robot consists of a base and links connected by rotary joints. To more closely approximate real-world conditions, friction, actuator output dead zone, and saturation characteristics were introduced into the joint system. PID and PD-MFAC were used as comparative control methods, and SM-MFAC was applied to the joint control respectively. The simulation results are as follows: Figure 5 As shown.

[0127] Depend on Figure 5 It can be seen that the system response speed is the fastest under PID control, but the overshoot is large. Due to the influence of dead zone, the steady-state error increases. In addition, it is very sensitive to external disturbances. Although the output of the system under PD-MFAC control has chattering, its anti-interference ability is stronger than that of PID. The SM-MFAC proposed in this invention can overcome complex nonlinearity, has a smaller steady-state error, and can also have a certain anti-interference ability while ensuring fast response and high tracking accuracy. In summary, the method proposed in this invention has better dynamic response capability, tracking effect and robustness.

[0128] A computer device is provided according to an embodiment of the present invention. This computer device includes a processor, a memory, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps in the various method embodiments described above. Alternatively, when the processor executes the computer program, it implements the functions of each module / unit in the various device embodiments described above.

[0129] The computer program can be divided into one or more modules / units, which are stored in the memory and executed by the processor to complete the present invention.

[0130] The computer device may be a desktop computer, laptop, handheld computer, or cloud server, etc. The computer device may include, but is not limited to, a processor and memory.

[0131] The processor may be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc.

[0132] The memory can be used to store the computer program and / or module, and the processor implements various functions of the computer device by running or executing the computer program and / or module stored in the memory, and by calling the data stored in the memory.

[0133] If the modules / units integrated into the computer device are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, all or part of the processes in the methods of the above embodiments of the present invention can also be implemented by a computer program instructing related hardware. The computer program can be stored in a computer-readable storage medium, and when executed by a processor, it can implement the steps of the various method embodiments described above. The computer program includes computer program code, which can be in the form of source code, object code, executable files, or certain intermediate forms. The computer-readable medium can include: any entity or device capable of carrying the computer program code, recording media, USB flash drives, portable hard drives, magnetic disks, optical disks, computer memory, read-only memory (ROM), random access memory (RAM), electrical carrier signals, telecommunication signals, and software distribution media, etc. It should be noted that the content included in the computer-readable medium can be appropriately added or removed according to the requirements of legislation and patent practice in the jurisdiction. For example, in some jurisdictions, according to legislation and patent practice, computer-readable media do not include electrical carrier signals and telecommunication signals.

[0134] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A sliding mode model-free adaptive control method for second-order nonlinear systems, characterized in that, Includes the following steps: Based on the order of the corresponding model of the second-order nonlinear system applied to the joint system of a single-degree-of-freedom free-floating space robot, the model is decomposed into a first discrete subsystem and a second discrete subsystem in series. The single-degree-of-freedom free-floating space robot includes a base and a link, which are connected by a rotary joint. The joint system has friction, actuator output dead zone and saturated nonlinear characteristics. Based on the changes in joint control input and joint output state of the joint system, a dynamic linearization equation is established between the input and output changes of the first discrete subsystem and the second discrete subsystem. Based on the dynamic linearization equation, the PPD expression for a second-order nonlinear system is obtained, which characterizes the input-output relationship between the joint control input and the joint output state of the joint system. Based on the output data of the first discrete subsystem and the second discrete subsystem, the desired state of the output of the second discrete subsystem is obtained. The desired state is used to make the joint output state of the joint system track the desired trajectory. The desired state of the output of the second discrete subsystem is as follows: The expected curves for the first discrete subsystem and the second discrete subsystem are as follows: The current state error is calculated based on the measured state data of the joint system: in: Select to switch manifolds: Based on the output data of the first discrete subsystem and the second discrete subsystem at the current moment, the virtual desired input of the first discrete subsystem is designed using backstepping control. The desired state of the second discrete subsystem is obtained: in, To reach the parameters, To switch manifolds, To switch manifold parameters, To control the step size factor, This is an estimate of the pseudopartial derivative of the second discrete subsystem. To control the weighting factor; MFAC is used to track the desired state to obtain the joint control input, which is then used for the adaptive control of the joint system to make the joint output state of the joint system track the desired trajectory. The specific method is as follows: For the second discrete subsystem, according to the PPD estimation criterion function Seeking information about The extreme values ​​are used to obtain the PPD estimation algorithm; The PPD estimation criterion function as follows: in, The pseudo-partial derivative of the second discrete subsystem. This is an estimate of the pseudo-partial derivative of the second discrete subsystem; The PPD estimation algorithm is as follows: In the formula, To estimate the step size factor, To estimate the weighting factors; According to the control criterion function Seeking information about The extreme value is used to obtain the control input. : The control criterion function as follows: The control input as follows: In the formula, To control the step size factor, It is a control weighting factor used to limit changes in the control input.

2. The sliding mode model-free adaptive control method for second-order nonlinear systems according to claim 1, characterized in that, The decomposition of the model into a first discrete subsystem and a second discrete subsystem in series, based on the order of the second-order nonlinear system corresponding to the joint system of a single-degree-of-freedom free-floating space robot, includes: Based on the model order, the second-order non-affine nonlinear system is decomposed into two cascaded discrete subsystems: a first discrete subsystem and a second discrete subsystem. The second-order non-affine nonlinear system is as follows: in, This is the system output status. For an unknown nonlinear function, For the system's control input, For time; The first discrete subsystem is as follows: in, The output of the first discrete subsystem, For the time of the discrete system, For a unit time step, This is the output of the second discrete subsystem; The second discrete subsystem is as follows: in, It is the input of the second discrete subsystem. The length of the input sliding time window for the control system. The length of the output sliding time window of the control system. It is an unknown nonlinear function.

3. The sliding mode model-free adaptive control method for second-order nonlinear systems according to claim 2, characterized in that, The dynamic linearization equations between the input and output changes of the joint control input and the joint output state based on the joint system are established, including: Assumptions are made regarding the characteristics of the second discrete subsystem: Assumption 1: Except for finite points in time, Regarding the first The partial derivatives of the variables are continuous; Assumption 2: Except at finite time points, the input and output of the second discrete subsystem satisfy the generalized Lipschitz condition; Assuming the change in input at the current moment is not zero, that is The second discrete subsystem is transformed into a partially oriented dynamically linearized PFDL data model, yielding the dynamic linearization equations between the input and output changes of the two discrete subsystems: in, The time-varying parameter of the pseudopartial derivative PPD. And the sign remains unchanged; for The amount of change in the control input at any given time.

4. The sliding mode model-free adaptive control method for second-order nonlinear systems according to claim 3, characterized in that, The PPD expression for the second-order nonlinear system, which characterizes the input-output relationship between the joint control input and the joint output state, is obtained based on the dynamic linearization equation, including: The partial pseudo-derivative of the first discrete subsystem can be easily obtained from the linear dynamic equation: Based on the data model of the overall system, the partial pseudo-derivative of the second discrete subsystem is obtained: in, for The change in the output of the second discrete subsystem at time t; Based on the expressions for the partial pseudo-derivatives of the first and second discrete subsystems, the PPD expression for the second-order nonlinear system, which reflects the input-output characteristics of the overall system, is obtained: in, This is the PPD expression for a second-order nonlinear system.

5. A sliding mode model-free adaptive control system for a second-order nonlinear system for implementing the method of claim 1, characterized in that, include: The model decomposition module is used to decompose the model into a first discrete subsystem and a second discrete subsystem in series, based on the order of the model corresponding to the second-order nonlinear system applied to the joint system of a single-degree-of-freedom free-floating space robot. The single-degree-of-freedom free-floating space robot includes a base and a link, which are connected by a rotary joint. The joint system has friction, actuator output dead zone, and saturation nonlinearity characteristics. The dynamic linearization equation construction module is used to establish dynamic linearization equations between the input changes and output changes of the first discrete subsystem and the second discrete subsystem based on the changes in joint control input and joint output state of the joint system. The first calculation module is used to obtain, based on the dynamic linearization equation, the PPD expression of the second-order nonlinear system that characterizes the input-output characteristic relationship between the joint control input and the joint output state of the joint system. The second calculation module is used to obtain the desired state of the output of the second discrete subsystem based on the output data of the first discrete subsystem and the second discrete subsystem. The desired state is used to make the joint output state of the joint system track the desired trajectory. The tracking control module is used to track the desired state using MFAC, obtain joint control input, and use the joint control input for adaptive control of the joint system, so that the joint output state of the joint system tracks the desired trajectory.

6. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the method as described in any one of claims 1-4.

7. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method as described in any one of claims 1-4.

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