Interval type-2 fuzzy sliding mode control method based on parameter uncertain inverted pendulum system

Abstract

The application discloses an interval type-2 fuzzy sliding mode control method based on a parameter uncertain inverted pendulum system and belongs to the technical field of nonlinear system control, and comprises the following steps: establishing an interval type-2 T-S fuzzy model of the parameter uncertain inverted pendulum system; constructing a sliding mode controller of the parameter uncertain inverted pendulum system based on the interval type-2 T-S fuzzy model; the sliding mode controller comprises a sliding surface, an equivalent controller, a switching controller and a comprehensive controller; the sliding surface is used as the only input variable of the fuzzy controller, the switching gain is dynamically controlled by using a fuzzy logic rule, and the sliding mode controller is optimized into an interval type-2 fuzzy sliding mode controller. The application solves the problem that the prior art cannot ensure that the parameter uncertain inverted pendulum system is stably operated on a sliding surface and is prone to chattering when parameters change and external disturbances occur.

Description

Technical Field

[0001] This invention relates to an interval type II fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters, belonging to the field of nonlinear system control technology. Background Technology

[0002] In recent years, modeling and controlling inverted pendulum systems with uncertain parameters has become a key research direction in control theory. As a typical nonlinear, multivariable, and strongly coupled unstable system, the inverted pendulum system exhibits complex dynamic characteristics, and parameter uncertainty further increases the difficulty of control. Traditional control methods often rely on precise model design to address this issue, but parameter uncertainty can easily lead to model failure. In recent years, with the development of control theory, advanced methods such as neural networks and fuzzy control have been introduced into the study of inverted pendulum systems. Among them, the interval-type II fuzzy logic controller, by introducing the core concept of "uncertainty footprint" and using intervals to represent membership, expands the degrees of freedom in system design and fuzzy reasoning capabilities, significantly improving the ability to handle nonlinearity and parameter uncertainty. Sliding mode control, as a nonlinear control method, can adapt to the nonlinear characteristics and dynamic changes of the system, exhibiting strong robustness to parameter variations and external disturbances, demonstrating unique advantages in inverted pendulum systems.

[0003] Existing technologies still have key shortcomings in handling parameter uncertainties in inverted pendulum systems: traditional control methods based on precise models rely on ideal assumptions and are prone to failure when parameter uncertainties are large; while sliding mode control is robust, its chattering phenomenon directly affects system stability and control accuracy; although interval type II fuzzy logic controllers can effectively handle uncertainties, how to deeply integrate them with sliding mode control to suppress chattering and further improve robustness and control accuracy remains a challenge for current research. Furthermore, the adaptability of existing methods under complex operating conditions still needs optimization, and integrating the flexibility of interval type II fuzzy logic with the strong robustness of sliding mode control remains an urgent technical problem to be solved. Summary of the Invention

[0004] The purpose of this invention is to provide an interval type II fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters. By using the sliding mode surface as the only input variable of the fuzzy controller, and after fuzzy inference, type reduction and defuzzification operations, a switching gain that matches the current state of the inverted pendulum system with uncertain parameters is output. This solves the problem that the existing technology cannot ensure the stable operation of the inverted pendulum system with uncertain parameters on the sliding mode surface, and that chattering is easy to occur when parameters change and external disturbances occur.

[0005] To solve the above-mentioned technical problems, the present invention is implemented using the following technical solution:

[0006] This invention provides an interval type II fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters, comprising:

[0007] Based on the physical dynamics of the inverted pendulum system, the dynamic equations of the inverted pendulum system with parameter uncertainty, including the mass of the pendulum and the mass of the car, are established.

[0008] Based on the dynamic equation of the inverted pendulum system with uncertain parameters, an interval type II TS fuzzy model is constructed using preset fuzzy logic rules to address the parameter uncertainty between the mass of the pendulum and the mass of the car.

[0009] A sliding mode controller for an inverted pendulum system with uncertain parameters is constructed based on a type II TS fuzzy model. The sliding mode controller includes a sliding surface characterizing the degree of deviation of the system state from the equilibrium point, an equivalent controller, a switching controller, and a comprehensive controller.

[0010] Using the sliding surface as the sole input variable of the interval type II TS fuzzy model, and employing preset fuzzy logic rules, the value of the switching gain is adaptively adjusted according to the real-time state of the sliding surface. Through the inference process of the interval type II TS fuzzy model, the fixed switching gain in the sliding controller is optimized into the adaptively adjusted switching gain, thereby optimizing the sliding controller into an interval type II fuzzy sliding controller.

[0011] Furthermore, the step of constructing an interval type II TS fuzzy model based on the dynamic equations of the inverted pendulum system with uncertain parameters, using preset fuzzy logic rules, for the parameter uncertainty between the mass of the pendulum and the mass of the car, includes:

[0012] Based on the dynamic equation of the inverted pendulum system with uncertain parameters, the mass of the pendulum and the mass of the car are taken as uncertain parameters. Based on the angular displacement of the pendulum, the length of the pendulum and the external force imposed on the car, the dynamic equation of the inverted pendulum system with uncertain parameters is established.

[0013] Based on the dynamic equations of the inverted pendulum system with uncertain parameters, an interval type II TS fuzzy model is established for the uncertain parameters based on preset fuzzy logic rules.

[0014] Furthermore, the dynamic equation of the inverted pendulum system with uncertain parameters is expressed as:

[0015] ;

[0016] In the formula, Indicates the current time of the pendulum. angular acceleration, Represents gravitational acceleration. Represents the sine function. Indicates the current time The angular velocity of the pendulum, Indicates the current time Angular displacement of the pendulum Indicates the length of the pendulum. Represents the cosine function. Indicates the current time External forces imposed on the car Indicates the quality coefficient. ,in, Indicates the mass of the pendulum. Indicates the mass of the vehicle;

[0017] No. The fuzzy logic rule is represented as follows:

[0018] if yes ,and yes , This represents the state vector under the first fuzzy logic rule level. Let the state vector at the second fuzzy logic rule level be... time derivative Represented as:

[0019] ;

[0020] In the formula, This indicates the first input variable corresponding to the first... A fuzzy logic rule, This indicates the second input variable corresponding to the first... A fuzzy logic rule, Indicates the first The state matrix of the fuzzy logic rules , Indicates transpose. Indicates the first The input matrix of the fuzzy logic rules;

[0021] The interval type II TS fuzzy model is represented as follows:

[0022] ;

[0023] ;

[0024] In the formula, This represents a type II TS fuzzy model for intervals. Indicates the first The trigger strength of the fuzzy logic rule, with a value range of [0, 1]. Indicates the first The product of the lower bound function of the trigger strength of a fuzzy logic rule and the corresponding weight function, where, Indicates the first The lower bound function of the trigger strength of a fuzzy logic rule. Indicates the first The weight function corresponding to the lower bound function of the trigger strength of the fuzzy logic rule. Indicates the first The product of the upper bound function of the trigger strength of a fuzzy logic rule and the corresponding weight function, where, Indicates the first An upper bound function for the trigger strength of a fuzzy logic rule. Indicates the first The weight function corresponding to the upper bound function of the trigger strength of the fuzzy logic rule;

[0025] No. The trigger strength range of a fuzzy logic rule is represented as follows:

[0026] ;

[0027] In the formula, Indicates the first The trigger strength range of a fuzzy logic rule. This represents the total number of fuzzy logic rules in the interval type II TS fuzzy model, with a value of 4.

[0028] Furthermore, the sliding mode controller for the inverted pendulum system with uncertain parameters constructed based on the interval type II TS fuzzy model includes:

[0029] Based on the angular displacement and angular velocity of the pendulum, the state vector of the inverted pendulum system with undefined parameters is determined. Determine the sliding surface ,in, Represents angular displacement parameters. Represents angular velocity parameters. The parameters of the sliding surface are given, and they satisfy the Hurwitz condition.

[0030] By designing the control input, This causes the state of the inverted pendulum system with uncertain parameters to converge to the sliding surface. To obtain the equivalent controller ,in, Represents the sliding surface Time derivative, This represents the convergence rate parameter of the sliding surface. It is a positive real number;

[0031] Based on saturation function Configure the switching controller ,in, , Indicates switching gain. ;

[0032] By combining the equivalent controller and the switching controller, the control input is obtained, which is the current time. External forces imposed on the car ;

[0033] By designing control inputs Make This causes the state of the inverted pendulum system with uncertain parameters to converge to the sliding surface. To obtain the equivalent controller ,in, Represents the sliding surface Time derivative, This represents the convergence rate parameter of the sliding surface. It is a positive real number. Indicates the current time External forces imposed on the vehicle;

[0034] Based on saturation function Configure the switching controller ,in, , Indicates switching gain. ;

[0035] By combining the equivalent controller and the switching controller, the control input is obtained, and the control input is: .

[0036] Furthermore, saturation function Represented as:

[0037] ;

[0038] In the formula, Indicates the boundary layer thickness, where, , This indicates a conditional statement "if". Represents the sliding surface The absolute value, Represents a symbolic function;

[0039] Current moment External forces imposed on the car Represented as:

[0040] .

[0041] Furthermore, the step of using the sliding surface as the sole input variable of the interval type II TS fuzzy model, adaptively adjusting the switching gain value according to the real-time state of the sliding surface using preset fuzzy logic rules, and optimizing the fixed switching gain in the sliding controller to the adaptively adjusted switching gain through the inference process of the interval type II TS fuzzy model, thereby optimizing the sliding controller into an interval type II fuzzy sliding controller, includes:

[0042] The precondition "if" yes The membership function of a fuzzy set is defined as an interval type II fuzzy set, which is composed of an upper membership function and a lower membership function, forming an uncertainty footprint used to describe the interval type II fuzzy set. Represents the nonlinearized sliding surface The conversion function, Indicates the first Interval type II fuzzy sets of interval type II fuzzy logic rules;

[0043] A fuzzy logic rule base is established based on the interval type II fuzzy set and interval type II fuzzy logic rules, and the interval type II fuzzy output is determined.

[0044] A type reduction method based on set center is used to convert the interval type II fuzzy output into type I fuzzy output;

[0045] By using fuzzy logic rules to dynamically control the switching gain, the type I fuzzy output is converted into a control signal, thereby optimizing the sliding mode controller into an interval type II fuzzy sliding mode controller.

[0046] Furthermore, the height of the lower membership function is expressed as: ,in, ,Will Defined as:

[0047] ;

[0048] In the formula, express The height of the subordinate members, express The height of the subordinate members, express The height of the subordinate members, This represents the only parameter that needs to be adjusted in the interval type II fuzzy sliding mode controller, which directly determines the degree of uncertainty of the interval type II fuzzy set.

[0049] Furthermore, the step involves establishing a fuzzy logic rule base based on interval type-2 fuzzy sets and interval type-2 fuzzy logic rules, and determining the interval type-2 fuzzy output, wherein the first... The interval type II fuzzy logic rule is represented as follows:

[0050] if yes ,but yes ;

[0051] In the formula, The output variable represents the fuzzy rule. Represents the nonlinearized sliding surface The conversion function, Indicates the first Interval type II fuzzy sets of interval type II fuzzy logic rules. Indicates the first The output of the interval type II rule, where, .

[0052] Furthermore, the type reduction method based on set center is used to convert the interval type II fuzzy output into type I fuzzy output, wherein the type I fuzzy output is represented as:

[0053] ;

[0054] In the formula, This indicates gain switching, i.e., type I fuzzy output. This represents the left endpoint of the type reduction output, where Indicates the left switch point identifier. This represents the right endpoint of the type reduction output, where Indicates the right switch point identifier. Indicates the left switch point index This represents the total number of fuzzy logic rules in the fuzzy sliding mode controller. Indicates the right switch point index. Represents the nonlinearized sliding surface conversion function The membership function of , Represents the nonlinearized sliding surface conversion function The subordinate membership function.

[0055] Furthermore, the control signal is represented as:

[0056] ;

[0057] In the formula, Indicates control signal, The output variable represents the fuzzy logic rule. Represents the nonlinearized sliding surface conversion function The gain of the nonlinear function will be used to transform the sliding surface into a nonlinear form. conversion function nonlinear function gain Defined as:

[0058]

[0059] In the formula, Represents the nonlinearized sliding surface conversion function The absolute value, This represents the only parameter that needs to be adjusted in the interval type II fuzzy sliding mode controller, which directly determines the degree of uncertainty of the interval type II fuzzy set.

[0060] Compared with the prior art, the beneficial effects achieved by the present invention are as follows:

[0061] 1. This invention effectively solves the chattering problem caused by fixed switching gain in traditional sliding mode control by using the sliding mode surface as the sole input variable of the interval type II fuzzy controller and dynamically adjusting the switching gain in combination with fuzzy logic rules. At the same time, it uses the interval type II TS fuzzy model to accurately quantify and characterize the uncertainty in the inverted pendulum system with uncertain parameters. By expanding the fuzzy inference capability through its "uncertainty footprint", the controller can adapt to the dynamic changes of the system and external disturbances, and finally achieve rapid convergence and high-precision stable control of the pendulum system's swing angle and swing angular velocity, which significantly improves the robustness and anti-interference capability of the system.

[0062] 2. This invention uses the mass of the pendulum and the mass of the vehicle as key uncertainty parameters, and quantifies the parameter fluctuations based on the interval type II TS fuzzy model. It leverages the "uncertainty footprint" characteristic to expand fuzzy inference capabilities, enabling the controller to dynamically adapt to parameter changes and external disturbances such as changes in the force imposed on the vehicle. Compared to traditional fixed-parameter models, this invention automatically adjusts the control strategy through a fuzzy logic rule base, significantly improving the system's robustness to multi-source uncertainties and ensuring that the inverted pendulum system can quickly converge to a stable state even under complex operating conditions.

[0063] 3. This invention effectively solves the chattering problem caused by fixed switching gain in traditional sliding mode control by using the sliding surface as the sole input variable of the interval type-2 fuzzy controller and dynamically adjusting the switching gain in conjunction with fuzzy logic rules. Specifically, the interval type-2 fuzzy controller automatically optimizes the switching gain based on the real-time state of the sliding surface, such as angular displacement and angular velocity deviation, ensuring rapid convergence of the sliding surface while suppressing high-frequency oscillations, thus making the control signal smoother. Attached Figure Description

[0064] Figure 1 This is a flowchart illustrating the interval type II fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters provided in this embodiment of the invention.

[0065] Figure 2 This is a schematic diagram of the inverted pendulum system with uncertain parameters provided in an embodiment of the present invention;

[0066] Figure 3 This is a schematic diagram of the structure of the interval type II fuzzy sliding mode controller provided in an embodiment of the present invention;

[0067] Figure 4 This is a schematic diagram illustrating the state changes of the angular displacement and angular velocity of the pendulum provided in an embodiment of the present invention.

[0068] Figure 5 This is a partially enlarged schematic diagram of the angular displacement state change of a pendulum provided in an embodiment of the present invention;

[0069] Figure 6 This is a partially enlarged schematic diagram of the state change of the pendulum's angular velocity provided in an embodiment of the present invention;

[0070] Figure 7 This is a schematic diagram of the state response under the interval type II fuzzy sliding mode controller provided in the embodiment of the present invention;

[0071] Figure 8 This is a schematic diagram of the state response under the sliding mode controller provided in an embodiment of the present invention. Detailed Implementation

[0072] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the embodiments of the present invention and the specific features in the embodiments are detailed descriptions of the technical solution of the present invention, rather than limitations thereof. In the absence of conflict, the embodiments of the present invention and the technical features in the embodiments can be combined with each other.

[0073] Example 1

[0074] like Figure 1 As shown in the figure, this embodiment introduces an interval type II fuzzy sliding mode control method based on a parameter-uncertain inverted pendulum system, including:

[0075] Step 1: Based on the physical dynamics of the inverted pendulum system, establish the dynamic equations of the inverted pendulum system with uncertain parameters, including the mass of the pendulum and the mass of the car; based on the dynamic equations of the inverted pendulum system with uncertain parameters, construct an interval type II TS fuzzy model for the parameter uncertainty between the mass of the pendulum and the mass of the car using preset fuzzy logic rules.

[0076] This invention constructs a type-II TS fuzzy model for key parameter uncertainties in an inverted pendulum system, such as the mass of the pendulum and the mass of the cart. The core value of the type-II TS fuzzy model lies in using the "uncertainty footprint" formed by the upper and lower membership functions to quantitatively represent the uncertain parameters in the nonlinear dynamic equations of the system in an interval-based manner. Compared with traditional type-I fuzzy models or fixed parameter models, it significantly expands the degrees of freedom and capabilities of fuzzy inference, enabling it to accurately cover the actual fluctuation range of parameters and providing a robust model foundation for controller design that can accommodate parameter perturbations.

[0077] Step 2: Construct a sliding mode controller for the inverted pendulum system with uncertain parameters based on the interval type II TS fuzzy model.

[0078] This invention designs a sliding mode controller based on the interval type II TS fuzzy model established in step one. The key to step two is defining a sliding surface that satisfies the Hurwitz condition. Ensuring that the system state converges to the sliding surface means that the angular displacement and angular velocity are stable. The equivalent controller is derived through this process. The system is made to move along the sliding surface, and a switching controller with a saturation function is introduced. To provide robustness to unmodeled dynamics and external disturbances. However, fixed switching gain The design contains an inherent contradiction: too large While it enhances robustness, it exacerbates chattering; too small This may reduce the ability to resist interference.

[0079] In this invention, the sliding mode controller includes a sliding surface that characterizes the degree to which the system state deviates from the equilibrium point, an equivalent controller, a switching controller, and a comprehensive controller.

[0080] Step 3: Using the sliding surface as the sole input variable of the interval type II TS fuzzy model, and using the preset fuzzy logic rules, the value of the switching gain is adaptively adjusted according to the real-time state of the sliding surface. Through the inference process of the interval type II TS fuzzy model, the fixed switching gain in the sliding controller is optimized into the adaptively adjusted switching gain, and the sliding controller is optimized into an interval type II fuzzy sliding controller.

[0081] The interval type II fuzzy sliding mode control method based on the inverted pendulum system with uncertain parameters provided by this invention can perform high-precision anti-interference control on nonlinear, strongly coupled, and parameter-fluctuating dynamic systems such as inverted pendulums, so that complex devices such as inverted pendulums, which are unstable and easily affected by external factors, can maintain a highly stable and accurate working state.

[0082] This invention uses a sliding surface As the sole input variable of the type-2 fuzzy controller, it overcomes the drawbacks of fixed switching gain. Through preset fuzzy logic rules and utilizing a unique nonlinearized sliding surface... conversion function nonlinear function gain ,in Represents the nonlinearized sliding surface The conversion function dynamically and adaptively adjusts the switching gain. The size of the interval type-II fuzzy output is transformed into type-I fuzzy output using a type reduction method based on set center of ensemble (COS), i.e., dynamically switching the gain. Ultimately, a smooth control signal is generated. This optimization allows the controller to increase gain to accelerate convergence when it is far from the sliding surface, and decrease gain to effectively suppress chattering when it is close to the sliding surface, achieving the best balance between control performance and robustness.

[0083] Example 2

[0084] Based on the same inventive concept as Embodiment 1, this embodiment introduces the implementation steps of an interval type II fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters, including:

[0085] Step 1: Based on the physical dynamics of the inverted pendulum system, establish the dynamic equation of the inverted pendulum system with uncertain parameters, including the mass of the pendulum and the mass of the car; based on the dynamic equation of the inverted pendulum system with uncertain parameters, construct an interval type II TS fuzzy model for the parameter uncertainty between the mass of the pendulum and the mass of the car using preset fuzzy logic rules.

[0086] In this embodiment, based on the dynamic equation of the inverted pendulum system with uncertain parameters, a type II TS fuzzy model is constructed using preset fuzzy logic rules to address the parameter uncertainty between the pendulum's mass and the car's mass, including:

[0087] Step 1.1: Based on the dynamic equation of the inverted pendulum system with uncertain parameters, the mass of the pendulum and the mass of the car are taken as uncertain parameters. Based on the angular displacement of the pendulum, the length of the pendulum and the external force imposed on the car, the dynamic equation of the inverted pendulum system with uncertain parameters is established.

[0088] This embodiment uses Matlab for simulation to verify that the method provided enables stable control of the inverted pendulum system with uncertain parameters, and that the pendulum angle and angular velocity converge to the desired values. A schematic diagram of the inverted pendulum system with uncertain parameters is provided below. Figure 2 As shown, the system parameters are set as follows:

[0089] In this embodiment, the dynamic equation of the inverted pendulum system with uncertain parameters is expressed as:

[0090] ;

[0091] In the formula, Indicates the current time of the pendulum. angular acceleration, Represents gravitational acceleration. Represents the sine function. Indicates the current time The angular velocity of the pendulum, Indicates the current time Angular displacement of the pendulum Indicates the length of the pendulum. Represents the cosine function. Indicates the current time External forces imposed on the car Indicates the quality coefficient. ,in, Indicates the mass of the pendulum. This indicates the mass of the vehicle.

[0092] In this embodiment, the gravitational acceleration g = 9.8, and the mass of the pendulum is... The quality of the car , Length of the pendulum .

[0093] Step 1.2: Based on the dynamic equation of the inverted pendulum system with uncertain parameters, establish an interval type II TS fuzzy model for the uncertain parameters based on the preset fuzzy logic rules.

[0094] In this embodiment, when and When taking constant values, the inverted pendulum system with uncertain parameters is a type TS fuzzy model, represented as:

[0095] ;

[0096] In the formula, This indicates the first input variable corresponding to the first... A fuzzy logic rule, This indicates the second input variable corresponding to the first... A fuzzy logic rule, Represents the state vector at the fuzzy logic rule level. Time derivative, Indicates the first The state matrix of the fuzzy logic rules, where, , Indicates transpose. Indicates the first The input matrix of the fuzzy logic rules, This represents the state vector under the first fuzzy logic rule level. This represents the state vector under the second fuzzy logic rule level. This represents the input variable, i.e., the first state vector.

[0097] In this embodiment, ,in, , They are represented as follows:

[0098] , , , ;

[0099] In the formula, Indicates the first The state matrix of the fuzzy logic rules Indicates the first The state matrix of the fuzzy logic rules Represents state variables The lower limit of the first type of nonlinear transformation, Indicates the first The state matrix of the fuzzy logic rules Indicates the first The state matrix of the fuzzy logic rules Represents state variables The upper limit of the first type of nonlinear transformation, Indicates the first The input matrix of the fuzzy logic rules, Indicates the first The input matrix of the fuzzy logic rules, Represents state variables The lower limit of the second type of nonlinear transformation, Indicates the first The input matrix of the fuzzy logic rules, Indicates the first The input matrix of the fuzzy logic rules, Represents state variables The upper limit of the second type of nonlinear transformation;

[0100] In this embodiment, , , , .

[0101] In this embodiment, Represented as:

[0102] ;

[0103] In the formula, This indicates the first nonlinear transformation of the input variable corresponding to the... The incentive strength of a fuzzy logic rule Represents state variables The first type of nonlinear transformation, This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The lower bound of the excitation intensity of a fuzzy logic rule. Represents state variables The second type of nonlinear transformation, This indicates the first nonlinear transformation of the input variable corresponding to the... A fuzzy logic rule, This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... A fuzzy logic rule, Indicates the first The incentive intensity of a fuzzy logic rule.

[0104] ;

[0105] ;

[0106] ;

[0107] ;

[0108] ;

[0109] ;

[0110] In the formula, This indicates the first nonlinear transformation of the input variable corresponding to the... The incentive strength of a fuzzy logic rule This indicates the first nonlinear transformation of the input variable corresponding to the... The incentive strength of a fuzzy logic rule This indicates the first nonlinear transformation of the input variable corresponding to the... The incentive strength of a fuzzy logic rule This represents the excitation intensity of the fourth fuzzy logic rule corresponding to the first nonlinear transformation of the input variable. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The incentive strength of a fuzzy logic rule This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The incentive strength of a fuzzy logic rule This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The incentive strength of a fuzzy logic rule This represents the excitation intensity of the fourth fuzzy logic rule corresponding to the second nonlinear transformation of the input variable.

[0111] In summary, when the mass of the pendulum The quality of the car When there is uncertainty, the membership value is also uncertain. For an inverted pendulum system with uncertain parameters, this embodiment establishes an interval type II TS fuzzy model with four fuzzy logic rules, where the first... The fuzzy logic rule is represented as follows:

[0112] if yes ,and yes , This represents the state vector under the first fuzzy logic rule level. Let the state vector at the second fuzzy logic rule level be... time derivative Represented as:

[0113] ;

[0114] In the formula, This indicates the first input variable corresponding to the first... A fuzzy logic rule, This indicates the second input variable corresponding to the first... A fuzzy logic rule, Indicates the first The state matrix of the fuzzy logic rules , Indicates transpose. Indicates the first The input matrix of the fuzzy logic rules.

[0115] In this embodiment, the interval type II TS fuzzy model and the first The trigger strengths of the fuzzy logic rules are expressed as follows:

[0116] ;

[0117] ;

[0118] In the formula, This represents the interval type II TS fuzzy model. In this embodiment, the interval type II TS fuzzy model is defined as the state vector at the interval type II TS fuzzy model level. Time derivative, Indicates the first The trigger strength of the fuzzy logic rule, with a value range of [0, 1]. Indicates the first The product of the lower bound function of the trigger strength of a fuzzy logic rule and the corresponding weight function, where, Indicates the first The lower bound function of the trigger strength of a fuzzy logic rule. Indicates the first The weight function corresponding to the lower bound function of the trigger strength of the fuzzy logic rule. Indicates the first The product of the upper bound function of the trigger strength of a fuzzy logic rule and the corresponding weight function, where, Indicates the first An upper bound function for the trigger strength of a fuzzy logic rule. Indicates the first The weight function corresponding to the upper bound function of the trigger strength of the fuzzy logic rule;

[0119] In summary, when the mass of the pendulum The quality of the car When the problem is uncertain but bounded, this embodiment treats the interval type II TS fuzzy model as a set of multiple type I TS fuzzy models to cover the mass of the pendulum. The quality of the car The entire range.

[0120] This embodiment aims to determine the excitation intensity. and The lower and upper membership functions must satisfy the following conditions:

[0121] ;

[0122] ;

[0123] In the formula, This indicates the first nonlinear transformation of the input variable corresponding to the... The incentive strength of a fuzzy logic rule This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. This indicates the first nonlinear transformation of the input variable corresponding to the... The upper bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The incentive strength of a fuzzy logic rule This represents the second type of nonlinear transformation of the input variable. The lower bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the incentive intensity of a fuzzy logic rule.

[0124] This embodiment will use the current time No. The second function corresponding to the fuzzy logic rule The mass of the pendulum The quality of the car Within its scope of work and Let the inner values ​​be constants, and consider all their combinations to determine... and The lower and upper bounds are used as the lower and upper membership functions of the interval type II TS fuzzy model.

[0125] Regarding the quality of the pendulum The quality of the car Due to the uncertainty, this embodiment defines the lower membership function and the upper membership function as follows:

[0126] Rule 1: When the input variable is a type 1 nonlinear transformation, hour:

[0127] ;

[0128] ;

[0129] In the formula, This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. Represents state variables The first type of nonlinear transformation, Represents state variables The upper limit of the first type of nonlinear transformation, State variables The lower limit of the first type of nonlinear transformation.

[0130] Rule 2, when the input variable is the second type of nonlinear transformation, when hour:

[0131] ;

[0132] ;

[0133] In the formula, This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. Represents state variables The second type of nonlinear transformation, Represents state variables The upper limit of the second type of nonlinear transformation, State variables The lower limit of the second type of nonlinear transformation.

[0134] Rule 3, when the input variable is the first type of nonlinear transformation, hour:

[0135] ;

[0136] ;

[0137] In the formula, This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. Represents state variables The first type of nonlinear transformation, Represents state variables The upper limit of the first type of nonlinear transformation, State variables The lower limit of the first type of nonlinear transformation.

[0138] Rule 4, when the input variable is the second type of nonlinear transformation, when hour:

[0139] ;

[0140] ;

[0141] In the formula, This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. Represents state variables The second type of nonlinear transformation, Represents state variables The upper limit of the second type of nonlinear transformation, State variables The lower limit of the second type of nonlinear transformation.

[0142] In this embodiment, the first The trigger strength range of a fuzzy logic rule is represented as follows:

[0143] ;

[0144] In the formula, Indicates the first The trigger strength range of a fuzzy logic rule. This represents the total number of fuzzy logic rules in the interval type II TS fuzzy model, with a value of 4.

[0145] In this embodiment, , Let represent the upper and lower bounds of the membership degree controlled by the lower and upper membership functions, respectively. That is, the upper bound of the membership degree is always greater than or equal to the lower bound of the membership degree. In other words, for any fuzzy logic rule, there is always an upper bound of the trigger strength of the fuzzy logic rule that is greater than or equal to the lower bound of the trigger strength. .

[0146] In this embodiment, the first The lower bound and upper bound of the trigger strength of a fuzzy logic rule are expressed as follows:

[0147] ;

[0148] ;

[0149] In the formula, Indicates the first The lower bound function of the trigger strength of a fuzzy logic rule. This indicates the first nonlinear transformation of the input variable corresponding to the... The lower bound of the excitation intensity of a fuzzy logic rule. This represents the first state variable of the system. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The lower bound of the excitation intensity of a fuzzy logic rule. Indicates the first The lower bound of the excitation intensity of the fuzzy logic rule. This represents the sum of the lower bounds of the excitation strengths of all four fuzzy logic rules. This indicates the first nonlinear transformation of the input variable corresponding to the... The upper bound of the excitation intensity of a fuzzy logic rule. This indicates the second type of nonlinear transformation of the input variable, corresponding to the first... The upper bound of the excitation intensity of a fuzzy logic rule. Indicates the first The upper bound of the excitation intensity of a fuzzy logic rule. It represents the sum of the upper bounds of the excitation strengths of all fuzzy logic rules.

[0150] Step 2: Construct a sliding mode controller for an inverted pendulum system with uncertain parameters based on a type II TS fuzzy model of the interval.

[0151] In this embodiment, the sliding mode controller includes a sliding surface that characterizes the degree to which the system state deviates from the equilibrium point, an equivalent controller, a switching controller, and a comprehensive controller.

[0152] Step 2.1: Based on the angular displacement and angular velocity of the pendulum, define the state vector of the parametrically uncertain inverted pendulum system. Determine the sliding surface ,in, Represents angular displacement parameters. Represents the angular velocity parameter. This represents the parameters of the sliding surface, and it satisfies the Hurwitz condition.

[0153] By designing the control input, This causes the state of the inverted pendulum system with uncertain parameters to converge to the sliding surface. To obtain the equivalent controller ,in, Represents the sliding surface Time derivative, This represents the convergence rate parameter of the sliding surface. It is a positive real number;

[0154] In this embodiment, the state variable is set to The sliding surface is designed as .

[0155] Step 2.2: Design the control input to make This causes the state of the inverted pendulum system with uncertain parameters to converge to the sliding surface. ,therefore:

[0156] ;

[0157] The equivalent controller is obtained by solving. ,in, Represents the sliding surface Time derivative, This represents the convergence rate parameter of the sliding surface. It is a positive real number. Indicates the current time External forces imposed on the car.

[0158] The equivalent controller is obtained by solving. = .

[0159] Step 2.3: Based on the saturation function Configure the switching controller ,in, , Indicates switching gain. .

[0160] In this embodiment, the saturation function Represented as:

[0161] ;

[0162] In the formula, Indicates the boundary layer thickness, where, , This indicates a conditional statement "if". Represents the sliding surface The absolute value, Represents a symbolic function.

[0163] In this embodiment, the following settings are provided: , , , .

[0164] Step 2.4: Combine the equivalent controller and the switching controller to obtain the control input, which is the current time. External forces imposed on the car .

[0165] In this embodiment, the current time External force applied to the car Represented as:

[0166]

[0167] ;

[0168] In the formula, This indicates the trigger strength of the nth fuzzy logic rule. , Indicates the first The state matrix of the fuzzy logic rules Indicates the first The input matrix of the fuzzy logic rules, State vector

[0169] Step 3: Using the sliding surface as the sole input variable of the interval type II TS fuzzy model, and using the preset fuzzy logic rules, the value of the switching gain is adaptively adjusted according to the real-time state of the sliding surface. Through the inference process of the interval type II TS fuzzy model, the fixed switching gain in the sliding controller is optimized into the adaptively adjusted switching gain, and the sliding controller is optimized into an interval type II fuzzy sliding controller.

[0170] In this embodiment, to reduce chattering in sliding mode control, fuzzy logic rules are used to dynamically adjust the switching gain. The fuzzy logic controller adjusts the switching gain based on the current sliding surface. To adjust the switching gain The value of makes the control input smoother.

[0171] like Figure 3 As shown, the SI-IT2-FP controller employs an interval type-II fuzzy mapping (FM), i.e., proportional control, where the input proportional factor... The purpose is to normalize the input sliding surface s to the universe of discourse [−1, 1] so as to meet the preconditions of the interval type II fuzzy sliding mode controller (IT2-FLC), i.e., if yes Consistent domain, sliding mode surface after nonlinearization The conversion function is denoted as ,Right now .

[0172] Step 3.1: Define the membership function of the precondition as an interval type II fuzzy set. The uncertainty footprint is composed of the upper membership function and the lower membership function and is used to describe the interval type II fuzzy set.

[0173] In this embodiment, the precondition "if" is... yes The membership function of a fuzzy set is defined as an interval type II fuzzy set, which is composed of the upper membership function and the lower membership function, forming an uncertainty footprint used to describe the interval type II fuzzy set. Represents the nonlinearized sliding surface The conversion function, Indicates the first The height of the membership function of the interval type II fuzzy set of interval type II fuzzy logic rules is expressed as follows: ,in, ,Will Defined as:

[0174] ;

[0175] In the formula, express The height of the subordinate members, express The height of the subordinate members, express The height of the subordinate members, It is the only parameter that needs to be adjusted in the interval type II fuzzy sliding mode controller, and it is used to directly determine the degree of uncertainty of the interval type II fuzzy set.

[0176] Step 3.2: Establish a fuzzy logic rule base based on the interval type II fuzzy set and interval type II fuzzy logic rules, and determine the interval type II fuzzy output.

[0177] In this embodiment, the first The interval type II fuzzy logic rule is represented as follows:

[0178] if yes ,but yes ;

[0179] In the formula, The output variable represents the fuzzy rule. Indicates the first The precise output of the interval type II fuzzy logic rule, where, .

[0180] Step 3.3: Use a type reduction method based on set center to convert the interval type II fuzzy output into type I fuzzy output.

[0181] Type reduction is a key step in interval type-II fuzzy sliding mode controllers. Its core objective is to reduce the interval type-II fuzzy output to type-I fuzzy output, thereby simplifying the calculation process and improving the practicality and efficiency of inverted pendulum systems with uncertain parameters. This embodiment uses a center-of-sets-based type reduction method to calculate the geometric center of the fuzzy set for type reduction.

[0182] Therefore, in this embodiment, type I fuzzy output is represented as:

[0183] ;

[0184] In the formula, This indicates gain switching, i.e., type I fuzzy output. This represents the left endpoint of the type reduction output, where Indicates the left switch point identifier. This represents the right endpoint of the type reduction output, where Indicates the right switch point identifier. Indicates the left switch point index This represents the total number of fuzzy logic rules in the fuzzy sliding mode controller. Indicates the right switch point index. Represents the nonlinearized sliding surface conversion function The membership function of , Represents the nonlinearized sliding surface conversion function The membership function is given by L = R = 1, where, since the interval type II fuzzy sets completely overlap, the input σ always activates two adjacent rules.

[0185] Step 3.4: Use fuzzy logic rules to dynamically control the switching gain to convert the type I fuzzy output into a control signal, thereby optimizing the sliding mode controller into an interval type II fuzzy sliding mode controller.

[0186] In this embodiment, the control signal is represented as:

[0187] ;

[0188] In the formula, Indicates control signal, The output variable represents the fuzzy logic rule. Represents the nonlinearized sliding surface conversion function The gain of the nonlinear function will be used to transform the sliding surface into a nonlinear form. conversion function nonlinear function gain Defined as:

[0189]

[0190] In the formula, Represents the nonlinearized sliding surface conversion function The absolute value, This represents the only parameter that needs to be adjusted in the interval type II fuzzy sliding mode controller, which directly determines the degree of uncertainty of the interval type II fuzzy set.

[0191] Step 4: Perform Lyapunov stability analysis on the inverted pendulum system with uncertain parameters.

[0192] This embodiment uses the sliding surface as the sole input variable of the interval type II TS fuzzy model. By using preset fuzzy logic rules, the value of the switching gain is adaptively adjusted according to the real-time state of the sliding surface. Through the inference process of the interval type II TS fuzzy model, the fixed switching gain in the sliding controller is optimized into the adaptively adjusted switching gain, and the sliding controller is optimized into an interval type II fuzzy sliding controller, thereby obtaining the optimized parameter uncertain inverted pendulum system.

[0193] This embodiment performs Lyapunov stability analysis on the optimized parameter-uncertain inverted pendulum system. By constructing two Lyapunov functions based on sliding mode variables and state space, it is proved that the parameter-uncertain inverted pendulum system has sliding surface reachability and global asymptotic stability under interval type II fuzzy sliding mode control, thereby ensuring the robust control performance of the parameter-uncertain inverted pendulum system under parameter uncertainty.

[0194] This embodiment selects the Lyapunov function. Prove the stability of the sliding surface, where, Indicates based on sliding surface The Lyapunov energy function is constructed.

[0195] Its derivative, as derived, satisfies:

[0196] ;

[0197] In the formula, Indicates based on sliding surface Constructed Lyapunov energy function Time derivative, Denotes the convergence rate parameter, which, according to Lyapunov's stability theorem, when and Only when At that time, the system state will asymptotically converge to the sliding surface. .

[0198] This embodiment further selects a state-space Lyapunov function. The global asymptotic stability of the optimized inverted pendulum system with uncertain parameters is proved, where, This represents the Lyapunov energy function constructed based on the system state. This represents the transpose of the state vector. This represents the positive definite matrix to be designed.

[0199] Where P is a positive definite matrix and Lyapunov function The derivative can be derived as follows:

[0200] ;

[0201] In the formula, Represents Lyapunov functions The time derivative.

[0202] This proves that the optimized inverted pendulum system with uncertain parameters has global asymptotic stability under a sliding mode controller.

[0203] The final simulation result obtained in this embodiment is as follows: Figures 4 to 8 As shown, Figure 4 The changes in angular displacement and angular velocity over time were shown. The angular displacement rapidly converged to near 0 rad within about 1.5 seconds and remained stable. The angular velocity also converged to 0 rad / s in sync, indicating that the controller effectively suppressed chattering and the steady-state error was extremely small. Figure 5 The image shows a magnified view of the angular displacement, revealing minute fluctuations (±0.05 rad) within the 0.9-1.1 second range, which decay rapidly, demonstrating the controller's ability to quickly suppress high-frequency interference. Figure 6 This is a magnified view of the angular velocity, showing smooth switching of control inputs and avoiding the "sawtooth" velocity curve of traditional sliding mode control. Figure 7 and Figure 8 The state response diagrams under the interval type II fuzzy sliding mode controller and the sliding mode controller are shown respectively. Figure 7 The steady-state accuracy is higher, which reflects the adaptive compensation capability of fuzzy logic rules for parameter uncertainty.

[0204] Among them, the interval type II fuzzy sliding mode controller exhibits faster convergence speed and smaller oscillation amplitude in both angular displacement and angular velocity control, indicating that the controller has superior performance in suppressing system oscillation and enhancing system stability.

[0205] In summary, this embodiment demonstrates that the optimized parameter-uncertain inverted pendulum system possesses sliding surface reachability and global asymptotic stability under interval type II fuzzy sliding mode control by constructing two Lyapunov functions based on sliding mode variables and state space, thereby ensuring the robust control performance of the optimized parameter-uncertain inverted pendulum system under parameter uncertainty.

[0206] Example 3

[0207] Based on the same inventive concept as other embodiments, this embodiment describes a computer-readable storage medium having computer instructions stored thereon, which, when executed by a processor, implement the steps of the methods of Embodiment 1 or 2 described above.

[0208] Example 4

[0209] Based on the same inventive concept as other embodiments, this embodiment introduces a computer program product, including computer instructions that, when executed by a processor, implement the steps of the methods described in Embodiment 1 or 2 above.

[0210] In summary, this invention effectively solves the chattering problem caused by fixed switching gain in traditional sliding mode control by using the sliding mode surface as the sole input variable of the interval type II fuzzy controller and dynamically adjusting the switching gain in combination with fuzzy logic rules. At the same time, it uses the interval type II TS fuzzy model to accurately quantify and characterize the uncertainty in the inverted pendulum system with uncertain parameters, and expands the fuzzy inference capability through its "uncertainty footprint", enabling the controller to adapt to dynamic changes in the system and external disturbances. Ultimately, it achieves rapid convergence and high-precision stable control of the pendulum system's swing angle and angular velocity, significantly improving the system's robustness and anti-interference capability.

[0211] This invention uses the pendulum mass and the car mass as key uncertainty parameters, and quantifies parameter fluctuations based on a type-II TS fuzzy model. It leverages the "uncertainty footprint" characteristic of this model to expand fuzzy inference capabilities, enabling the controller to dynamically adapt to parameter changes and external disturbances such as changes in the force imposed on the car. Compared to traditional fixed-parameter models, this invention automatically adjusts the control strategy through a fuzzy logic rule base, significantly improving the system's robustness to multi-source uncertainties and ensuring that the inverted pendulum system can quickly converge to a stable state even under complex operating conditions.

[0212] This invention effectively solves the chattering problem caused by fixed switching gain in traditional sliding mode control by using the sliding surface as the sole input variable of an interval type-two fuzzy controller and dynamically adjusting the switching gain according to fuzzy logic rules. Specifically, the interval type-two fuzzy controller automatically optimizes the switching gain based on the real-time state of the sliding surface, such as angular displacement and angular velocity deviation, ensuring rapid convergence of the sliding surface while suppressing high-frequency oscillations, thus making the control signal smoother.

[0213] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0214] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0215] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0216] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0217] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. An interval-based type-II fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters, characterized in that, include: Based on the physical dynamics of the inverted pendulum system, the dynamic equations of the inverted pendulum system with parameter uncertainty, including the mass of the pendulum and the mass of the car, are established. Based on the dynamic equations of the inverted pendulum system with uncertain parameters, an interval type II TS fuzzy model is constructed using pre-defined fuzzy logic rules to address the parameter uncertainties between the pendulum's mass and the car's mass, including: Based on the dynamic equation of the inverted pendulum system with uncertain parameters, the mass of the pendulum and the mass of the car are taken as uncertain parameters. Based on the angular displacement of the pendulum, the length of the pendulum and the external force imposed on the car, the dynamic equation of the inverted pendulum system with uncertain parameters is established. Based on the dynamic equations of the inverted pendulum system with uncertain parameters, an interval type II TS fuzzy model is established for the uncertain parameters based on the preset fuzzy logic rules. A sliding mode controller for an inverted pendulum system with uncertain parameters is constructed based on a type II TS fuzzy model. The sliding mode controller includes a sliding surface characterizing the degree of deviation of the system state from the equilibrium point, an equivalent controller, a switching controller, and a comprehensive controller. The sliding mode controller for the inverted pendulum system with uncertain parameters constructed based on the interval type II TS fuzzy model includes: Based on the angular displacement and angular velocity of the pendulum, the state vector of the inverted pendulum system with undefined parameters is determined. Determine the sliding surface ,in, Represents angular displacement parameters. Represents angular velocity parameters. The parameters of the sliding surface are given, and they satisfy the Hurwitz condition. By designing the control input, This causes the state of the inverted pendulum system with uncertain parameters to converge to the sliding surface. To obtain the equivalent controller ,in, Represents the sliding surface Time derivative, This represents the convergence rate parameter of the sliding surface. It is a positive real number; Based on saturation function Configure the switching controller ,in, , Indicates switching gain. ; By combining the equivalent controller and the switching controller, the control input is obtained, which is the current time. External forces imposed on the car ; Using the sliding surface as the sole input variable of the interval type II TS fuzzy model, and employing preset fuzzy logic rules, the switching gain is adaptively adjusted based on the real-time state of the sliding surface. Through the inference process of the interval type II TS fuzzy model, the fixed switching gain in the sliding controller is optimized to an adaptively adjusted switching gain, thus optimizing the sliding controller into an interval type II fuzzy sliding controller. This includes: If the preconditions are yes The membership function is defined as an interval type II fuzzy set, and the uncertainty footprint is formed by the upper membership function and the lower membership function, which is used to describe the interval type II fuzzy set. Represents the nonlinearized sliding surface The conversion function, Indicates the first Interval type II fuzzy sets of interval type II fuzzy logic rules; A fuzzy logic rule base is established based on the interval type II fuzzy set and interval type II fuzzy logic rules, and the interval type II fuzzy output is determined. A type-order reduction method based on set center is used to convert the interval type-II fuzzy output into type-I fuzzy output as a switching gain. By using fuzzy logic rules to dynamically control the switching gain, the type I fuzzy output is converted into a control signal, thereby optimizing the sliding mode controller into an interval type II fuzzy sliding mode controller.

2. The interval type-two fuzzy sliding mode control method based on an inverted pendulum system with uncertain parameters according to claim 1, characterized in that, The dynamic equation of the inverted pendulum system with uncertain parameters is expressed as follows: ； In the formula, Indicates the current time of the pendulum. angular acceleration, Represents gravitational acceleration. Represents the sine function. Indicates the current time The angular velocity of the pendulum, Indicates the current time Angular displacement of the pendulum Indicates the length of the pendulum. Represents the cosine function. Indicates the current time External forces imposed on the car Indicates the quality coefficient. ,in, Indicates the mass of the pendulum. Indicates the mass of the vehicle; No. The fuzzy logic rule is represented as follows: if yes ,and yes , This represents the state vector under the first fuzzy logic rule level. Let the state vector at the second fuzzy logic rule level be... time derivative Represented as: ； In the formula, This indicates the first input variable corresponding to the first... A fuzzy logic rule, This indicates the second input variable corresponding to the first... A fuzzy logic rule, Indicates the first The state matrix of the fuzzy logic rules , Indicates transpose. Indicates the first The input matrix of the fuzzy logic rules; The interval type II TS fuzzy model is represented as follows: ； ； In the formula, This represents a type II TS fuzzy model for intervals. Indicates the first The trigger strength of the fuzzy logic rule, with a value range of [0, 1]. Indicates the first The product of the lower bound function of the trigger strength of a fuzzy logic rule and the corresponding weight function, where, Indicates the first The lower bound function of the trigger strength of a fuzzy logic rule. Indicates the first The weight function corresponding to the lower bound function of the trigger strength of the fuzzy logic rule. Indicates the first The product of the upper bound function of the trigger strength of a fuzzy logic rule and the corresponding weight function, where, No. An upper bound function for the trigger strength of a fuzzy logic rule. Indicates the first The weight function corresponding to the upper bound function of the trigger strength of the fuzzy logic rule; No. The trigger strength range of a fuzzy logic rule is represented as follows: ； In the formula, Indicates the first The trigger strength range of a fuzzy logic rule. This represents the total number of fuzzy logic rules in the interval type II TS fuzzy model, with a value of 4.

3. The interval type-two fuzzy sliding mode control method based on an inverted pendulum system with uncertain parameters according to claim 1, characterized in that, Saturation function Represented as: ； In the formula, Indicates the boundary layer thickness, where, , This indicates a conditional statement "if". Represents the sliding surface The absolute value, Represents a symbolic function; Current moment External forces imposed on the car Represented as: .

4. The interval type-two fuzzy sliding mode control method based on an inverted pendulum system with uncertain parameters according to claim 1, characterized in that, The height of the lower membership function is expressed as ,in, ,Will Defined as: ； In the formula, express The height of the subordinate members, express The height of the subordinate members, express The height of the subordinate members, This represents the only parameter that needs to be adjusted in the interval type II fuzzy sliding mode controller, which directly determines the degree of uncertainty of the interval type II fuzzy set.

5. The interval type-two fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters according to claim 4, characterized in that, The step involves establishing a fuzzy logic rule base based on interval type-two fuzzy sets and interval type-two fuzzy logic rules, and determining the interval type-two fuzzy output, wherein the first... The interval type II fuzzy logic rule is represented as follows: if yes ,but yes ; In the formula, The output variable represents the fuzzy rule. Indicates the first The output of the interval type II fuzzy logic rule, where, .

6. The interval type-two fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters according to claim 5, characterized in that, The method of type reduction based on set center is used to convert the interval type II fuzzy output into type I fuzzy output, wherein the type I fuzzy output is represented as: ； In the formula, This indicates gain switching, i.e., type I fuzzy output. This represents the left endpoint of the type reduction output, where Indicates the left switch point identifier. This represents the right endpoint of the type reduction output, where Indicates the right switch point identifier. Indicates the left switch point index. This represents the total number of fuzzy logic rules in the fuzzy sliding mode controller. Indicates the right switch point index. Represents the nonlinearized sliding surface Conversion function The membership function of , Represents the nonlinearized sliding surface Conversion function The subordinate membership function.

7. The interval type-two fuzzy sliding mode control method for an inverted pendulum system with uncertain parameters according to claim 6, characterized in that, The control signal is represented as follows: ； In the formula, Indicates control signal, The output variable represents the fuzzy logic rule. Represents the nonlinearized sliding surface Conversion function The gain of the nonlinear function will be used to transform the sliding surface into a nonlinear form. Conversion function nonlinear function gain Defined as: ； In the formula, Represents the nonlinearized sliding surface Conversion function The absolute value, This represents the only parameter that needs to be adjusted in the interval type II fuzzy sliding mode controller, which directly determines the degree of uncertainty of the interval type II fuzzy set.

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