Novel safety control method of magnetic suspension system based on control barrier function

Abstract

The invention provides a novel safety control method for a magnetic suspension system based on a control barrier function, and the method comprises the following steps: obtaining the real-time state information of the magnetic suspension system, and constructing a mathematical model of the magnetic suspension system; calculating a tracking error, and recursively constructing a non-singular terminal sliding mode surface; designing an adaptive finite time disturbance observer, and performing finite time estimation on lumped disturbance including unmodeled dynamics and external disturbance to obtain a disturbance estimation value; designing a continuous nonsingular terminal sliding mode composite controller as a nominal control law; constructing a control screen barrier function according to a preset output security constraint boundary of the magnetic suspension system; and constructing a quadratic programming problem by taking the nominal control law as an expected input and the control barrier function as a safety constraint, and carrying out online solving to obtain a final safety control input. According to the invention, rapid and high-precision position tracking control can be realized on the premise of strictly guaranteeing that the system output is always in the preset safety interval.

Description

A novel safety control method for magnetic levitation systems based on control obstacle functions Technical Field

[0001] This invention relates to the field of automatic control technology, and is a safety control method for magnetic levitation systems, particularly a novel safety control method for magnetic levitation systems based on a control obstacle function. Background Technology

[0002] Magnetic levitation systems (MLS) achieve contactless levitation of objects using electromagnetic force, offering significant advantages such as frictionless operation, low noise, and high precision, showing broad application prospects in high-speed transportation, precision manufacturing, and medical devices. However, its dynamic model is inherently open-loop unstable and highly nonlinear, posing a significant challenge to high-precision control. Furthermore, in actual operation, the position of the levitated object (i.e., the output) must be strictly limited within safe physical boundaries (e.g., preventing the steel ball from colliding with the electromagnet), imposing stringent safety requirements on the control system.

[0003] Existing magnetic levitation system control methods, such as PID control and linear feedback control, are usually effective near the equilibrium point, but they are difficult to handle large-scale dynamic processes and nonlinear characteristics. Although advanced methods such as sliding mode control and adaptive control have been introduced to improve robustness, the following key problems still exist: (1) Insufficient convergence performance: Many methods can only guarantee asymptotic convergence of tracking error, and cannot achieve better finite-time convergence, which limits the dynamic response speed of the system. (2) Contradiction between chattering and disturbance rejection: Traditional sliding mode control requires high-gain switching to suppress disturbances, which leads to harmful chattering of the control signal and damages the actuator. Although observers can be used for disturbance compensation, existing methods such as generalized proportional-integral observers can usually only achieve asymptotic estimation, and the real-time performance and accuracy of compensation are limited. (3) Lack of safety constraint guarantee: Most existing studies focus on tracking performance optimization and fail to systematically and provably guarantee output safety constraints as the core element of the design. A few studies have attempted to combine barrier Lyapunov functions, but their construction is complex and difficult to integrate naturally with high-performance finite-time controllers.

[0004] Therefore, there is an urgent need for a comprehensive control scheme that can simultaneously ensure high-precision tracking within a limited time, strong anti-disturbance and low jitter, and strict output safety constraints. Summary of the Invention

[0005] To address the technical challenge of simultaneously ensuring high-precision tracking and absolute safety in existing magnetic levitation system control methods under disturbance and output constraints, this invention proposes a novel safety control method for magnetic levitation systems based on a Control Barrier Function (CBF). This method aims to solve the technical difficulty of achieving fast and accurate position tracking while absolutely ensuring that the output does not exceed safety boundaries in disturbed environments. This invention achieves fast and high-precision position tracking control while strictly ensuring that the system output always remains within a preset safety range. It significantly improves the robustness, safety, and overall control performance of magnetic levitation systems under complex disturbance environments, and is particularly suitable for magnetic levitation systems with strict output constraints and external disturbances.

[0006] The specific technical solutions covered by this invention include: (a) for high-precision, fast tracking control of the controlled object's position in electromagnetic levitation devices, focusing on solving the control challenges caused by the inherent nonlinearity, open-loop instability, and parameter uncertainty of its dynamic model. (b) a control law design method that ensures the system tracking error converges to zero within a finite time. It particularly focuses on terminal sliding mode control and its improved forms, aiming to improve the system's transient response speed and steady-state accuracy, exceeding the performance limits of traditional asymptotic convergence control. (c) for lumped disturbances encountered during system operation, designing an observer with finite-time convergence characteristics for real-time accurate estimation, and performing feedforward compensation based on the estimated value to enhance system robustness and effectively suppress control signal chattering caused by high-gain switching. (d) to ensure that the system operation never violates physical safety boundaries, involving a formal description and enforcement method of safety constraints based on control barrier functions. It focuses on the processing of output constraints with a relative order of 2 by high-order control barrier functions, and how to seamlessly integrate safety conditions into high-performance control laws. (e) This invention investigates how to reconcile hard constraints that ensure safety with nominal control laws that pursue high performance through real-time optimization problem solving. The invention aims to achieve optimal or suboptimal control decisions under complex constraints by trading absolute safety for minimal performance cost, and is particularly suitable for magnetic levitation systems with strict output constraints and external disturbances.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows: a novel safety control method for a magnetic levitation system based on a control obstacle function, comprising the following steps:

[0008] Step 1: Obtain the real-time status information of the magnetic levitation system and construct a mathematical model of the magnetic levitation system;

[0009] Step 2: Based on real-time status information and desired position, calculate tracking error, and recursively construct a non-singular terminal sliding surface based on tracking error;

[0010] Step 3: Design an adaptive finite-time disturbance observer based on the non-singular terminal sliding surface, and perform finite-time estimation on the lumped disturbance that includes unmodeled dynamics and external disturbances to obtain the disturbance estimate.

[0011] Step 4: Based on the non-singular terminal sliding surface and disturbance estimates, design a continuous non-singular terminal sliding mode composite controller as the nominal control law;

[0012] Step 5: Construct the control panel obstacle function based on the preset output safety constraint boundary of the magnetic levitation system;

[0013] Step 6: Using the nominal control law as the desired input and the control barrier function as the safety constraint, construct a quadratic programming problem and solve it online to obtain the final safe control input;

[0014] Step 7: Apply the final safety control input to the electromagnetic coil of the magnetic levitation system.

[0015] Preferably, the real-time status information includes the position and velocity of the steel ball;

[0016] The mathematical model of the magnetic levitation system is as follows:

[0017]

[0018] The position and velocity of the steel ball are respectively , This represents the acceleration of the steel ball; For output, Indicates control input, intermediate coefficient Intermediate coefficient Centralized interference ; Indicates the mass of the steel ball. It is the acceleration due to gravity. (t) represents the position of the steel ball at time t. (t) represents the current flowing through the electromagnet at time t. The permeability of free space, Indicates the number of turns in the electromagnet coil. This represents the interaction area between the steel ball and the magnetic pole. For inductance, and These represent the equilibrium current and equilibrium position of the steel ball in a stable suspended state, respectively. This refers to the electromagnetic force generated by an electromagnet. Taylor expansion regarding current (t) and the position of the steel ball Higher-order terms of (t); External interference; This indicates the coil resistance.

[0019] Preferably, the method for recursively constructing a novel non-singular terminal sliding surface based on tracking error is as follows:

[0020] Tracking error ;

[0021] Design a non-singular terminal sliding surface: ;

[0022] in, For design constants, and It is a positive odd number and satisfies ; denoted by e, sgn(.) represents the derivative of the tracking error e;

[0023] Recursive construction of non-singular terminal sliding surfaces for: ;

[0024] in, The control gain of the PI controller, the control gain coefficient. .

[0025] Preferably, the adaptive positive function Adjust the observer gain in real time;

[0026] By dynamically injecting a nonlinear feedback term based on the estimation error into an adaptive finite-time perturbation observer, an error dynamic system that converges faster than the original system is constructed, and the error dynamic system is guaranteed to stabilize to zero within a finite time, thus obtaining the perturbation estimate. .

[0027] Preferably, let the derivative of the non-singular terminal sliding surface be... By combining the mathematical model of the magnetic levitation system, a non-singular terminal sliding mode controller is obtained:

[0028] Among them, equivalent control and the reaching law are respectively

[0029]

[0030] intermediate matrix , For the intermediate matrix The false rebellion, To control the gain;

[0031] In non-singular terminal sliding mode controller Based on the design of (t), the following continuous non-singular terminal sliding mode composite controller is designed as the nominal control law using disturbance compensation technology: Among them, equivalent control and the reaching law are respectively

[0032]

[0033] Among them, the reaching law control gain , This is the disturbance estimate output by the adaptive finite-time disturbance observer.

[0034] Preferably, consider the Lyapunov function: ;

[0035] Lyapunov function along the magnetic levitation system model Find the derivative and substitute it into the nominal control law. (t); the convergence time of the adaptive finite-time perturbation observer system is defined as . ,when When the derivative of the Lyapunov function is bounded, we can obtain the Lyapunov function. exist Time has boundaries; when hour, This holds true, and we can conclude that the derivative of the Lyapunov function is also bounded.

[0036] If the derivative of the tracking error Then the Lyapunov stability condition is satisfied; if the derivative of the tracking error is... , It is not an attractor;

[0037] According to the finite-time stability theorem, there exists a finite-time... This makes the non-singular terminal sliding surface In a limited time Converging to zero; on non-singular terminal sliding surfaces Above, the derivative of the non-singular terminal sliding surface simplifies to .

[0038] Preferably, the magnetic levitation system is subjected to the form of The output constraints, where, and The boundary constant is; the control barrier function is Where B1(y) and B2(y) both represent intermediate functions;

[0039] Output constraint safety set is ;in, It is the transpose of the matrix. Represents the set of real numbers;

[0040] To ensure the output constraint safety set The forward invariance means that the control input u(t) must satisfy the condition from the intermediate function. and The conditions for the exported control barrier function.

[0041] Preferably, the intermediate function and It is twice differentiable, and output safety constraints are implemented using a control barrier function, expressed as:

[0042]

[0043] in, The intermediate variables are the control barrier function to be designed. and These are design parameters that are positive constants;

[0044] intermediate variables Represented as the set of control inputs u(t):

[0045] ;

[0046] There exists a control input u(t) such that set K CBF It is not an empty set, which ensures that the system output always stays within the output safety constraints.

[0047] Preferably, the mathematical model of the magnetic levitation system is substituted into the set. get:

[0048]

[0049] Let intermediate variables

[0050] ;

[0051] ;

[0052] Then we obtain the following relationship

[0053]

[0054] Then there was

[0055]

[0056] gather It is a non-empty set.

[0057] Preferably, the optimal safety control input is obtained by minimizing the modification of the nominal control law through online optimization of the following quadratic programming problem:

[0058] .

[0059] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0060] (1) A non-singular terminal sliding surface with recursive construction is proposed. Combined with finite time control technology, it ensures finite time convergence in both the approach and sliding phases, overcoming the limitations of traditional methods that only achieve asymptotic convergence or two-stage convergence.

[0061] (2) An Adaptive Fnite-Time Disturbance Observer (AFTDO) is proposed to work in conjunction with a Continuous Non-singular Terminal Sliding Mode Composite Controller (CNTSMCC). While ensuring finite-time convergence, it significantly reduces chattering in traditional sliding mode control and improves system robustness and steady-state accuracy.

[0062] (3) For the first time, control barrier functions (CBFs) are combined with a non-singular terminal sliding mode controller to construct a safety control framework based on quadratic programming (QP). The safety controller is solved online to achieve a unified control that ensures strict output constraints of the magnetic levitation system and stable tracking within finite time.

[0063] (4) To address the second-order characteristics of the output constraints of the magnetic levitation system, a high-order control barrier function (CBF) is innovatively introduced to achieve mathematical modeling and control integration of the asymmetric output constraints, ensuring that the system operates within a safe region without affecting tracking performance. This invention combines high tracking accuracy, strong anti-disturbance robustness, and engineering practicality, representing a significant breakthrough in the field of magnetic levitation control technology. Attached Figure Description

[0064] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0065] Figure 1 is an overall flowchart of the present invention, illustrating the specific implementation steps of the present invention.

[0066] Figure 2 is a diagram of the overall technical roadmap of the present invention, showing the complete steps and logical relationships from system modeling, controller design to safety constraint processing and final control execution.

[0067] Figure 3 shows the simulation results of Case 1, illustrating the comparison between the system position tracking trajectory and the coil current response curves when using three control methods—PID-NTSMC (PID-type non-singular terminal sliding mode controller), GPIO-CITSMCC (continuous integral terminal composite controller), and the AFTDO-CNTSMCC (continuous non-singular terminal sliding mode composite controller) proposed in this invention—under the condition of no output constraints but with external sinusoidal disturbances. In Figure 3, (a1) represents the position tracking trajectory, and (a2) represents the coil current response curve. The results show that the method of this invention (AFTDO-CNTSMCC) has a faster convergence speed, higher tracking accuracy, and a smoother current response.

[0068] Figure 4 shows the simulation results of Case 1 in Figure 3, comparing the effects of the AFTDO method of this invention and the GPIO method on lumped perturbations. The estimation performance of the present invention, AFTDO, is demonstrated. The results show that AFTDO achieves faster and more accurate finite-time perturbation estimation.

[0069] Figure 5 shows the simulation results of Case 2, illustrating the system position tracking trajectory under four control methods—PID-NTSMC, GPIO-CITSMCC, AFTDO-CNTSMCC, and CBF-AFTDO-CNTSMCC (based on a control barrier function controller)—when output safety constraints (0.0325m ≤ y(t) ≤ 0.0525m) and external disturbances are present. (b1) shows the system output tracking effect under the three control methods PID-NTSMC, GPIO-CITSMCC, and CBF-AFTDO-CNTSMCC, and (b2) shows the system output tracking trajectory under the three control methods PID-NTSMC, GPIO-CITSMCC, and AFTDO-CNTSMCC. The results show that only the CBF-AFTDO-CNTSMCC of this invention can strictly guarantee that the output never violates the safety constraints while achieving finite-time tracking.

[0070] Figure 6 shows the current response curves of Case 2, illustrating the coil current response curves under four control methods: PID-NTSMC, GPIO-CITSMCC, AFTDO-CNTSMCC, and CBF-AFTDO-CNTSMCC. (c1) represents the tracking effect of the system drive current under the three control methods PID-NTSMC, GPIO-CITSMCC, and CBF-AFTDO-CNTSMCC, while (c2) represents the system current response curves under the three control methods PID-NTSMC, GPIO-CITSMCC, and AFTDO-CNTSMCC. The results show that all four methods drive the system current to the balanced current.

[0071] Figure 7 shows the simulation results for Case 2 in Figure 5, comparing the effects of the CBF-AFTDO method of this invention with the comparative method GPIO on lumped perturbations. The estimation performance of CBF-AFTDO is demonstrated. The results show that CBF-AFTDO can achieve faster and more accurate finite-time perturbation estimation. Detailed Implementation

[0072] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0073] As shown in Figure 1, this invention provides a novel safety control method for a magnetic levitation system (MLS) based on a control barrier function (CBF). This method includes the design of a non-singular terminal sliding surface, the design of an adaptive finite-time disturbance observer (AFTDO), the construction of a higher-order control barrier function, and an online optimization problem. First, this invention recursively constructs a novel non-singular terminal sliding surface to ensure finite-time convergence of the system state during both the approach and sliding phases. Then, it designs an adaptive finite-time disturbance observer (AFTDO) to accurately estimate and compensate for lumped disturbances, effectively suppressing the chattering phenomenon inherent in traditional sliding mode control. Finally, by constructing safety constraints based on a higher-order control barrier function and solving them online using quadratic programming (QP), the initial control law is minimized and modified to generate the final safety control command. This invention enables fast and high-precision position tracking control while strictly ensuring that the system output always stays within a preset safety range. It significantly improves the robustness, safety, and overall control performance of the magnetic levitation system under complex disturbance environments, representing a major breakthrough in the field of magnetic levitation control technology.

[0074] The specific embodiments of the present invention include the following:

[0075] Step 1: Obtain the real-time status information of the magnetic levitation system and construct a mathematical model of the magnetic levitation system.

[0076] System status information, including the position of the steel ball, is obtained through sensor measurements. and speed .

[0077] Based on the principles of electromagnetism and Newtonian mechanics, a mathematical model of the system is first constructed:

[0078]

[0079] in, Indicates the mass of the steel ball. It is the acceleration due to gravity. (t) represents the position of the steel ball at time t. The electromagnetic force generated by an electromagnet. (t) represents the current flowing through the electromagnet at time t. The permeability of free space, Indicates the number of turns in the electromagnet coil. This represents the interaction area between the steel ball and the magnetic pole. For inductance, and These represent the equilibrium current and equilibrium position of the steel ball in a stable suspended state, respectively. This refers to the electromagnetic force when the system is in equilibrium. Indicates control input, This indicates the coil resistance.

[0080] Electromagnetic force At the equilibrium point, a Taylor expansion is performed, and higher-order terms are ignored to simplify the control law design, resulting in:

[0081]

[0082] in, Represents the Taylor expansion of current. (t) and the position of the steel ball The higher-order term of (t).

[0083] Then, consider external interference to the system. Define the position and velocity of the steel ball as follows: The output is Therefore, the mathematical model of the magnetic levitation system can be obtained as follows.

[0084]

[0085] in, Let represent the state information and lumped disturbance of the magnetic levitation system at time t, respectively. This represents the acceleration of the steel ball; specifically, the intermediate coefficient. Intermediate coefficient Centralized interference .

[0086] The acquired real-time state information serves as a feedback signal, constituting the tracking error; it also acts as input to drive the state observer; and as a variable, it is substituted into the control law for calculation. Essentially, real-time state information is the bridge connecting physical objects and abstract mathematical models, transforming static equations into dynamic intelligent algorithms capable of sensing, making decisions, and controlling physical systems in real time.

[0087] Step 2: Based on the real-time status information and the desired position, calculate the tracking error, and recursively construct a novel non-singular terminal sliding surface based on the tracking error.

[0088] Define tracking error The design of the sliding surface is carried out in two steps: First, a non-singular terminal sliding surface is designed:

[0089]

[0090] in, For design constants, and It is a positive odd number and satisfies ; Let denot be the derivative of the tracking error at time t, and sgn(.) denote the sign function.

[0091] Figure 2 Indicates the desired position (i.e., equilibrium position). Non-singular terminal sliding surface. Designed based on system tracking error, the structure of this sliding surface is fixed.

[0092] Then, non-singular terminal sliding surfaces are recursively constructed. for:

[0093]

[0094] in, This is a sliding surface designed based on a PI controller, which has a fixed control gain. (The gain in this invention is 1) and The controller design parameters are also the control gain coefficient. .

[0095] Step 3: Design an Adaptive Finite-Time Perturbation Observer (AFTDO) based on the non-singular terminal sliding surface to observe lumped disturbances that include unmodeled dynamics and external disturbances. A finite-time estimation is performed to obtain the disturbance estimate.

[0096] Design a non-singular terminal sliding mode controller, so that Therefore, combining the mathematical model of the magnetic levitation system, we can obtain the following controller:

[0097]

[0098] in,

[0099]

[0100] here, (t) and (t) represents the equivalent control and the reaching law, respectively, and the intermediate matrix. , For the intermediate matrix The pseudo-inverse is to prevent the generation of strange phenomena. To control the gain. Controller (t) The derivative through the non-singular terminal sliding surface It was derived from this.

[0101] Assuming lumped disturbance It is twice differentiable and satisfies ,in, It is a positive continuous function (here we take...) Therefore, based on the assumptions, we define the lumped disturbance. , Therefore, the following adaptive finite-time perturbation observer can be designed:

[0102]

[0103] in, , and These are the estimated values ​​for the ball velocity, lumped perturbation, and derivative of the lumped perturbation, respectively. , and These are the estimated values. , and The derivative of The observer gain coefficient is positive, and here it is taken as a value of , An adaptive positive function that satisfies specific conditions.

[0104] Adaptive positive function The observer gain can be adjusted in real time. In some ordinary observers, the function here is a constant, which is an advantage of this invention. The core principle of the finite-time observer for estimation is: dynamically injecting a nonlinear feedback term based on the estimation error into the adaptive finite-time perturbation observer (specifically...) , and We construct an "error dynamic system" that converges faster than the original system, and ensure that this system stabilizes to zero within a finite time (rather than an infinite time). The perturbation estimate is obtained through finite-time estimation. .

[0105] Step 4: Based on the non-singular terminal sliding surface and disturbance estimates, design a continuous non-singular terminal sliding composite controller (CNTSMCC) as the nominal control law to ensure system stability, and perform secondary programming with the subsequent control obstacle function.

[0106] Based on the adaptive finite-time disturbance observer designed in step three, and building upon step two, the following continuous non-singular terminal sliding mode composite controller is designed as the nominal control law:

[0107]

[0108] in

[0109]

[0110] In particular, the intermediate matrix , For the intermediate matrix The false rebellion, To control the gain, the reaching law controls the gain. Used to suppress disturbance estimation errors. This is the disturbance estimate output by AFTDO. eq (t) and u r (t) represent the equivalent control and the reaching law, respectively. The parameters are selected as follows: , , , , , , The nominal control law in the controller This is obtained by using disturbance compensation technology based on the design of (t).

[0111] The perturbation observer converges, meaning the perturbation estimate is approximately equal to the true perturbation value. Once the convergence of the adaptive finite-time perturbation observer is established, consider the following Lyapunov function:

[0112]

[0113] Lyapunov function along the magnetic levitation system model Find the derivative and substitute it into the nominal control law. (t), we can obtain

[0114]

[0115] Among them, the nominal control law is substituted. (t) will give us the second equal sign.

[0116] Define the convergence time of the observer system as , then when At that time, we have the derivative of the Lyapunov function.

[0117]

[0118] in, This represents the disturbance estimation error. The control gain k is used to suppress the disturbance estimation error, even though this error is infinitely close to zero.

[0119] Lyapunov function exist Since time is bounded, the system state will not diverge to infinity in a finite amount of time.

[0120] when hour, This holds true. Therefore, the derivative of the Lyapunov function is obtained.

[0121]

[0122] Therefore, if the derivative of the tracking error Then the Lyapunov stability condition is satisfied (because The derivative of the Lyapunov function mentioned above holds and satisfies the Lyapunov stability condition. If the derivative of the tracking error... , It is not an attractor. This will not affect the Lyapunov function V's derivative satisfying the Lyapunov stability condition. According to the finite-time stability theorem, there exists a finite-time... This makes the sliding surface In a limited time It converges to zero. On the sliding flow surface... Up means The derivative of the first sliding surface is dynamically simplified to

[0123]

[0124] The above equation is a typical finite-time stability equation, which proves that the system state is finite-time stable.

[0125] Step 5: Based on the preset output safety constraint boundary of the magnetic levitation system, construct the corresponding high-order control screen obstacle function (HOCBF).

[0126] The magnetic levitation system is subject to the form of The output constraints, where and For constant boundary conditions (i.e., take symmetric constraints, i.e.) =0.0325, =0.0525, which can also be visually reflected in the simulation graph. Accordingly, the control barrier function (CBF) is defined as follows:

[0127]

[0128] Here, B1(y) and B2(y) both represent intermediate functions, which can be differentiated twice.

[0129] Define the output constraint safety set as

[0130]

[0131] in, It is the transpose of the matrix. It represents the set of real numbers.

[0132] To ensure the output constraint safety set Forward invariance (i.e., ensuring that all output constraint sets are safe) The trajectory originating from within always remains within the output constraint safety set. (Internal), the control input must meet the requirements from and Exported CBF conditions.

[0133] Given the intermediate function and It is twice differentiable, and high-order CBFs are used to strictly enforce these safety constraints. The specific formulation is as follows:

[0134]

[0135] in, These are intermediate variables for the higher-order control barrier function to be designed. and These are design parameters that are positive constants. The above formula can be expressed in another way (because...). All of them must be greater than or equal to 0, which means they are represented as a set:

[0136]

[0137] Taking the second derivative with respect to B1(y) and B2(y) yields a functional inequality with respect to the control input u(t), indicating that there exists a control input u(t) such that the set K CBF It is not an empty set, which ensures that the system output remains within the constraints. Only the set needs to be specified. If it is a non-empty set, it means that there exists a control input. Simply ensure that the set is non-empty.

[0138] Substituting the mathematical model of the magnetic levitation system into the set We can obtain:

[0139]

[0140] Let intermediate variables

[0141] ;

[0142] ;

[0143] Then we can easily obtain the following relationship.

[0144]

[0145] Then there was

[0146]

[0147] This proves the set It is a non-empty set.

[0148] Step 6: Using the initial control law as the desired input and the higher-order control barrier function as the safety constraint, construct a quadratic programming (QP) problem and solve it online to obtain the final safety control input.

[0149] To design a safety-critical controller for a magnetic levitation system subject to output constraints, the following QP online optimization is used to minimize modifications to the nominal control law, yielding the optimal safety control input.

[0150] .

[0151] The above minimization problem is optimized offline to obtain the safety control input. The design parameters here .

[0152] Step 7: Apply the final safety control input to the electromagnetic coil of the magnetic levitation system to achieve finite-time high-precision position tracking control that meets the output safety constraint boundary.

[0153] Due to limitations, this invention is implemented through simulation, and the simulation results are sufficient to demonstrate its effectiveness. Matlab simulations verify the effectiveness of the two continuous finite-time controllers designed in this invention. This embodiment uses the magnetic levitation system described in this invention as the controlled object to implement the safety control method described herein. System parameters are shown in Table 1.

[0154] Table 1 Parameters of the Magnetic Levitation System

[0155]

[0156] Based on the above parameters, the model parameters a = 0.3808 and b = 461.1765 were calculated. Two examples illustrate the universality of the designed control scheme. Example 1 verifies the performance of a continuous non-singular terminal sliding mode composite controller; the controller parameter values ​​are given, and the conclusions are explained in the illustrations. Example 2 verifies the performance of a non-singular terminal sliding mode composite controller based on a control obstacle function in handling a constrained magnetic levitation system; all figures are generated from MATLAB simulations. In particular, both examples are affected by external disturbances.

[0157] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A novel safety control method for a magnetic levitation system based on a control obstacle function, characterized in that, The steps are as follows: Step 1: Obtain the real-time state information of the magnetic levitation system and construct a mathematical model of the magnetic levitation system; Step 2: Calculate the tracking error based on the real-time state information and the desired position, and recursively construct a non-singular terminal sliding surface based on the tracking error. Step 3: Design an adaptive finite-time disturbance observer based on the non-singular terminal sliding surface to perform finite-time estimation of the lumped disturbance, which includes unmodeled dynamics and external disturbances, and obtain the disturbance estimate. Step 4: Based on the non-singular terminal sliding surface and the disturbance estimate, design a continuous non-singular terminal sliding composite controller as the nominal control law. Step 5: Construct the control barrier function according to the preset output safety constraint boundary of the magnetic levitation system. Step 6: Using the nominal control law as the desired input and the control barrier function as the safety constraint, construct a quadratic programming problem and solve it online to obtain the final safety control input. Step 7: Apply the final safety control input to the electromagnetic coils of the magnetic levitation system.

2. The novel safety control method for a magnetic levitation system based on a control obstacle function according to claim 1, characterized in that, The real-time status information includes the position and velocity of the steel ball; the mathematical model of the magnetic levitation system is: The position and velocity of the steel ball are respectively ， This represents the acceleration of the steel ball; For output, Indicates control input, intermediate coefficient Intermediate coefficient Centralized interference ； Indicates the mass of the steel ball. It is the acceleration due to gravity. (t) represents the position of the steel ball at time t. (t) represents the current flowing through the electromagnet at time t. The permeability of free space, Indicates the number of turns in the electromagnet coil. This represents the interaction area between the steel ball and the magnetic pole. For inductance, and These represent the equilibrium current and equilibrium position of the steel ball in a stable suspended state, respectively. This refers to the electromagnetic force generated by an electromagnet. Taylor expansion regarding current (t) and the position of the steel ball Higher-order terms of (t); External interference; This indicates the coil resistance.

3. The novel safety control method for a magnetic levitation system based on a control obstacle function according to claim 2, characterized in that, The method for recursively constructing a novel non-singular terminal sliding surface based on tracking error is as follows: Tracking error Design a non-singular terminal sliding surface: ;in, For design constants, and It is a positive odd number and satisfies ； The derivative of the tracking error e is represented by sgn(.), and the sign function is represented by sgn(.); non-singular terminal sliding surfaces are recursively constructed. for: ;in, The control gain of the PI controller, the control gain coefficient. 。 4. The novel safety control method for a magnetic levitation system based on a control obstacle function according to claim 3, characterized in that, The adaptive positive function The observer gain is adjusted in real time. By dynamically injecting a nonlinear feedback term based on the estimation error into the adaptive finite-time perturbation observer, an error dynamic system that converges faster than the original system is constructed, and the error dynamic system is guaranteed to stabilize to zero in a finite time, thus obtaining the perturbation estimate. 。 5. The novel safety control method for a magnetic levitation system based on a control obstacle function according to claim 3 or 4, characterized in that, Let the derivative of the non-singular terminal sliding surface By combining the mathematical model of the magnetic levitation system, a non-singular terminal sliding mode controller is obtained: Among them, equivalent control and the reaching law are respectively intermediate matrix ， For the intermediate matrix The false rebellion, To control gain; in a non-singular terminal sliding mode controller Based on the design of (t), the following continuous non-singular terminal sliding mode composite controller is designed as the nominal control law using disturbance compensation technology: Among them, equivalent control and the reaching law are respectively Among them, the reaching law control gain ， This is the disturbance estimate output by the adaptive finite-time disturbance observer.

6. The novel safety control method for a magnetic levitation system based on a control obstacle function according to claim 5, characterized in that, Consider the Lyapunov function: ; along the magnetic levitation system model, the Lyapunov function Find the derivative and substitute it into the nominal control law. (t); the convergence time of the adaptive finite-time perturbation observer system is defined as . ,when When the derivative of the Lyapunov function is bounded, we can obtain the Lyapunov function. exist Time has boundaries; when hour, If this holds true, then the derivative of the Lyapunov function is also bounded; if the derivative of the tracking error... Then the Lyapunov stability condition is satisfied; if the derivative of the tracking error is... ， It is not an attractor; according to the finite-time stability theorem, there exists a finite-time... This makes the non-singular terminal sliding surface In a limited time Converging to zero; on non-singular terminal sliding surfaces Above, the derivative of the non-singular terminal sliding surface simplifies to 。 7. The novel safety control method for a magnetic levitation system based on a control obstacle function according to any one of claims 2-4 and 6, characterized in that, The magnetic levitation system is subject to the form of The output constraints, where, and The boundary constant is; the control barrier function is Where B1(y) and B2(y) both represent intermediate functions; the output constraint safety set is ;in, It is the transpose of the matrix. Represents the set of real numbers; to ensure the output constraint safe set. The forward invariance means that the control input u(t) must satisfy the condition from the intermediate function. and The conditions for the exported control barrier function.

8. The novel safety control method for a magnetic levitation system based on a control obstacle function according to claim 7, characterized in that, intermediate function and It is twice differentiable, and output safety constraints are implemented using a control barrier function, expressed as: in, The intermediate variables are the control barrier function to be designed. and These are design parameters with positive constants; intermediate variables. Represented as the set of control inputs u(t): There exists a control input u(t) such that set K CBF It is not an empty set, which ensures that the system output always stays within the output safety constraints.

9. The novel safety control method for a magnetic levitation system based on a control obstacle function according to claim 8, characterized in that, Substituting the mathematical model of the magnetic levitation system into the set get: Let intermediate variables ； Then we obtain the following relationship: Furthermore, there are gather It is a non-empty set.

10. The novel safety control method for a magnetic levitation system based on a control obstacle function according to claim 8 or 9, characterized in that, The optimal safety control input is obtained by minimizing the modification of the nominal control law through online optimization of the following quadratic programming problem: 。

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