A finite time position tracking control method for a magnetic levitation system with output constraint

Abstract

The application provides a finite time position tracking control method of a magnetic suspension system with output constraint, and the steps are as follows: a mathematical model of the magnetic suspension system is established, an error system is determined, upper and lower limits of a suspension air gap constraint and system state information are obtained; a fractional type barrier Lyapunov function is designed by considering the upper and lower limits of the suspension air gap constraint, and a constraint processing mechanism function is obtained by derivation; a continuous finite time state feedback constraint controller is designed by using the system state information based on the constraint processing mechanism function and the finite time control technology and homogeneous system theory; when the system speed state information is unmeasurable, a non-smooth filter is designed based on the constraint processing mechanism function, and a finite time output feedback controller is designed by using the filter output information and the system output information. The application realizes unified finite time control of the magnetic suspension system under symmetric / asymmetric suspension gap constraints, and has high tracking precision, strong anti-disturbance robustness and engineering practicability.

Description

Technical Field

[0001] This invention relates to the technical field of magnetic levitation systems, and more particularly to a position tracking control method for magnetic levitation systems. Background Technology

[0002] Magnetic levitation systems, with their advantages of low noise, non-contact operation, and low friction, have been widely used in transportation, medical devices, and industrial equipment, especially in control engineering. However, these control systems are open-loop unstable systems with highly nonlinear characteristics, posing significant challenges to the design of position tracking control. In recent years, many advanced control methods have been proposed for the position tracking control of magnetic levitation systems, such as adaptive control, model predictive control, proportional-integral-derivative control, and sliding mode control. These diverse and effective control techniques have significantly improved the position tracking control performance of magnetic levitation systems from different perspectives.

[0003] Despite the abundant research achievements in position tracking control methods for magnetic levitation systems, these methods are largely limited to asymptotically stable control schemes, which clearly hinders in-depth exploration of position tracking control research for magnetic levitation systems. Finite-time control technology, as a practical alternative, has been widely applied in engineering control systems in recent years due to its advantages such as fast convergence speed, high control accuracy, and strong robustness. Simultaneously, researchers have conducted extensive studies on the application of finite-time control technology to magnetic levitation systems. For example, to accelerate convergence speed and obtain better stability and anti-interference capabilities, a hyperlocal model-free adaptive supertorsional nonsingular terminal sliding mode control strategy, an integral terminal sliding mode controller control scheme, a finite-time control method based on fractional power integrals of a series of exponential functions and nested symbolic functions, and an event-triggered strategy composed of sharp triggering rules and time-varying thresholds, etc., have been applied to magnetic levitation systems to ensure the finite-time convergence of the magnetic levitation system state.

[0004] In practical engineering control systems, the system output is often constrained due to system safety considerations or inherent physical structural limitations. For example, due to the limitations of its mechanical structure, a robot arm's joints need to operate within a specific range to ensure safety. To maintain the good performance and stability of the electrical circuit system, it is necessary to limit the current within a certain range. Any behavior that violates the output constraints will cause varying degrees of damage to the controlled system. Similarly, the levitation block of a magnetic levitation system must be kept confined within a specified air gap to ensure stable levitation. However, it should be noted that the control strategies mentioned above do not take into account the air gap constraints of magnetic levitation systems.

[0005] Currently, for control problems of output-constrained systems, a common approach is to incorporate the constraints into the controller design process beforehand. By designing a suitable controller, the system output is theoretically guaranteed to strictly meet the pre-given constraints. Examples include finite-time control based on barrier Lyapunov functions, event-triggered adaptive finite-time control with predetermined performance, and control with predetermined performance assisted by barrier functions, all of which are widely used in constrained systems. To address the challenge of position tracking control in magnetic levitation systems with air gap constraints, researchers have designed a continuous finite-time state feedback controller based on a tannic barrier Lyapunov function, successfully achieving convergence of the system state to zero within a finite time. Unfortunately, this method is no longer applicable when only the system output is available for feedback design. Summary of the Invention

[0006] To address the technical problem of low tracking accuracy caused by existing control strategies that do not consider the air gap constraint of the magnetic levitation system, this invention proposes a finite-time position tracking control method for magnetic levitation systems with output constraints. By comprehensively utilizing finite-time stability control theory, homogeneous system theory, and Lyapunov function stability analysis, a position trajectory tracking control scheme for magnetic levitation systems with higher tracking accuracy, simpler structure, and better performance is achieved. This method can more accurately track the desired trajectory of the magnetic levitation system within a finite time while satisfying the system output safety constraints.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows: a finite-time position tracking control method for a magnetic levitation system with output constraints, comprising the following steps:

[0008] Step 1: Establish a mathematical model of the magnetic levitation system, determine the error system, and obtain the upper and lower bounds of the suspension air gap constraint and the system state information;

[0009] Step 2: Considering the upper and lower bounds of the air gap constraint of magnetic levitation, design the fractional barrier Lyapunov function, and take the partial derivative of the fractional barrier Lyapunov function to obtain the constraint handling mechanism function.

[0010] Step 3: Applying finite-time control technology and homogeneous system theory, and based on the constraint handling mechanism function, design a continuous finite-time state feedback constraint controller using system state information;

[0011] Step 4: Based on the constraint processing mechanism function, when the system velocity state information is not measurable, design a non-smooth filter, and use the output information of the non-smooth filter and the system output information to design a finite-time output feedback controller.

[0012] Preferably, the mathematical model of the magnetic levitation system is as follows:

[0013]

[0014] Where m represents the mass of the steel ball, g is the gravitational acceleration, δ(t) represents the position of the steel ball at time t, F is the electromagnetic force generated by the electromagnet, i(t) represents the current flowing through the electromagnet at time t, μ0 is the permeability of free space, N represents the number of turns of the electromagnet coil, S represents the interaction area between the steel ball and the magnetic pole, and i0 and δ0 are the equilibrium current and equilibrium position of the steel ball in a stable levitation state, respectively.

[0015] Preferably, the electromagnetic force F is expanded using Taylor series at the equilibrium point, and higher-order terms are ignored, resulting in:

[0016]

[0017] Define system control inputs Given the output y(t) = x1(t), the mathematical model of the error system is:

[0018]

[0019] Where (x1, x2) T =(x1(t),x2(t)) T u = u(t) and y = y(t) represent the state, control input, and output of the error system, respectively. and These represent the system's nominal parameters;

[0020] The error system satisfies the output safety constraints. h represents the upper and lower bounds of the system's suspended air gap constraint, respectively.

[0021] Preferably, the fractional barrier Lyapunov function is:

[0022]

[0023] Where x1 = x1(t) represents the output state of the error system. The parameter is the barrier Lyapunov function, and τ is the homogeneity.

[0024] Preferably, the fractional barrier Lyapunov function V b (x1) is positive definite. Taking the partial derivative with respect to the output state x1, we get...

[0025]

[0026] Among them, the constraint handling mechanism function

[0027] Preferably, the finite-time state feedback constraint controller is:

[0028]

[0029] Where, k p1 and k d1 It is a positive number; The fractional power term representing the velocity state of the error system.

[0030] Preferably, the finite-time state feedback controller u1 is substituted into the error system to obtain the closed-loop system:

[0031]

[0032] For an error system whose initial output state x1 satisfies the output safety constraints, there exists a positive constant k. p1 and k d1 This ensures that the closed-loop system described above is stable for a finite time and does not violate safety constraints.

[0033] Preferably, the non-smooth filter is:

[0034]

[0035] Where η represents the state information of the filter, the filter gain parameters l1 and l2 are both positive numbers, and x3 represents the filter output information.

[0036] Preferably, the finite-time output feedback constraint controller is:

[0037]

[0038] Among them, the control parameter k p2 and k d2 All are normal numbers.

[0039] Preferably, the finite-time output feedback constraint controller u2 is substituted into the error system to obtain the closed-loop system:

[0040]

[0041] For an error system whose initial output state x1 satisfies the output safety constraints, there exists a positive constant k. p2 k d2 l1 and l2 ensure that the closed-loop system consisting of the finite-time output feedback constraint controller and the error system is finite-time stable and will not violate the safety constraints.

[0042] The constraints include symmetric output safety constraints and asymmetric output safety constraints for the output state x1 of the error system;

[0043] When considering symmetric output security constraints, set k p1=20 and k d1 =18; When considering asymmetric output security constraints, set k p1 =6 and k d1 =5;

[0044] When considering the symmetrical output safety constraints, set the filter gain parameters l1 = 15 and l2 = 95; when considering the asymmetrical output safety constraints, set the filter gain parameters l1 = 9 and l2 = 90.

[0045] When considering symmetrical output safety constraints, the control parameter k is set... p2 =25 and k d2 =45; When considering asymmetric output safety constraints, set the control parameter k. p2 =8 and k d2 =25.

[0046] The beneficial effects of this invention are mainly reflected in the following four aspects: (1) Constructing a novel fractional barrier Lyapunov function (BLF). In view of the limitation that the traditional tan-type function is only applicable to symmetric constraints, this invention constructs a novel fractional barrier Lyapunov function. This function can uniformly handle the symmetric and asymmetric suspension air gap constraints of the magnetic levitation system. There is no need to adjust the controller structure for different suspension air gap constraint types, which significantly improves the versatility and simplicity of the controller. (2) Integrating finite-time control and output constraint control. In order to solve the shortcomings of traditional asymptotic stability control in terms of dynamic response speed and disturbance rejection capability of magnetic levitation system, this invention introduces finite-time control theory into magnetic levitation control and organically combines it with output constraint conditions. This integrated design effectively solves the dynamic tracking stability problem of magnetic levitation system with suspension air gap hard constraints in finite time. By introducing the constraint processing mechanism into the design of finite-time controller, the magnetic levitation system can achieve stable control in finite time and meet the output constraint conditions of the magnetic levitation system. (3) Proposing an observer-free continuous finite-time output feedback controller. To address the shortcomings of existing finite-time control schemes that often rely on state observers or complete state feedback, this invention proposes a continuous finite-time output feedback controller that does not require the design of an observer. By cleverly utilizing the output information of a non-smooth filter and the actual output information of the system, a finite-time output-constrained feedback controller is constructed, achieving for the first time precise finite-time control of a magnetic levitation system with suspension gap constraints under the condition that only the output is measurable. The controller designed in this invention can uniformly handle magnetic levitation systems with symmetrical or asymmetrical output constraints, ensuring strong robustness and high tracking accuracy of the magnetic levitation control system. Moreover, when the output constraint condition is not considered, the controller naturally degenerates into an unconstrained controller without changing the controller structure. (4) Ensuring strong robustness and high tracking accuracy: Under the conditions of suspension gap constraints and external disturbances, the proposed controller can still guarantee the strong robustness and high-precision trajectory tracking performance of the system. This invention, through the innovative application of a novel fractional barrier Lyapunov function, a simplified controller structure, and a non-smooth filter, achieves for the first time globally a unified finite-time control of a magnetic levitation system under symmetric / asymmetric levitation gap constraints. It combines high tracking accuracy, strong disturbance rejection robustness, and engineering practicality, representing a significant breakthrough in the field of magnetic levitation control technology.

[0047] This invention innovatively combines fractional obstacle Lyapunov functions with finite-time control technology to propose two types of finite-time constraint controllers, solving the finite-time position tracking problem of magnetic levitation systems under air gap constraints. Its core advantages lie in its unified handling of symmetric / asymmetric constraints, simplified controller structure, and improved robustness. The control performance of this invention has been verified in simulations. Attached Figure Description

[0048] 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.

[0049] Figure 1 This is a schematic diagram of the structure of the present invention.

[0050] Figure 2 The diagram shows the control simulation results of the continuous finite-time state feedback constraint controller designed in this invention, where (a1) represents the safety constraint conditions considering the magnetic levitation system, and (a2) represents the safety constraint conditions not considering the magnetic levitation system.

[0051] Figure 3 The figure shows the numerical simulation results of the continuous finite-time output feedback constraint controller designed in this invention, where (b1) represents the safety constraint conditions considering the magnetic levitation system and (b2) represents the safety constraint conditions not considering the magnetic levitation system.

[0052] Figure 4 The simulation results of the continuous finite-time state feedback constraint controller designed for this invention on a magnetic levitation system with asymmetric output constraints are shown in the figure.

[0053] Figure 5 The numerical simulation results of the continuous finite-time output feedback constraint controller designed for this invention on a magnetic levitation system with asymmetric output constraints are shown in the figure. Detailed Implementation

[0054] 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.

[0055] like Figure 1As shown, a finite-time position tracking control method for a magnetic levitation system with output constraints is proposed, and two finite-time controllers are explored and designed:

[0056] 1. Combining finite-time stability theory and homogeneous system theory, and utilizing system states, a continuous finite-time state feedback constraint controller based on the obstacle Lyapunov function is designed. This controller features a simple structure, higher tracking performance, and strong robustness.

[0057] 2. When the system outputs only measurable information, design a non-smooth filter and use the output information of the non-smooth filter and the system output information to design a continuous finite-time output feedback constraint controller.

[0058] The specific steps of this invention include:

[0059] Step 1: Establish a mathematical model of the magnetic levitation system and determine the mathematical model of the error system to obtain the output constraints and system state information of the magnetic levitation system.

[0060] A magnetic levitation system includes a levitation system that uses magnetic force to keep the object in a non-contact state with the support surface; a sensor system that monitors parameters such as levitation gap, speed, and position in real time; a power supply system that provides stable and efficient power support for the entire system; and a control system that adjusts the electromagnet current according to control signals to maintain stable levitation.

[0061] The mathematical model of the magnetic levitation system is:

[0062]

[0063] Where m represents the mass of the steel ball, g is the gravitational acceleration, δ(t) represents the position of the steel ball at time t, F is the electromagnetic force generated by the electromagnet, i(t) represents the current flowing through the electromagnet at time t, μ0 is the permeability of free space, N represents the number of turns of the electromagnet coil, S represents the interaction area between the steel ball and the magnetic pole, and i0 and δ0 are the equilibrium current and equilibrium position of the steel ball in a stable levitation state, respectively.

[0064] Expanding the electromagnetic force F at the equilibrium point using Taylor and neglecting higher-order terms simplifies the control law design, we obtain:

[0065]

[0066] Define system control inputs Given the output y(t) = x1(t), we can obtain the following mathematical model of the error system.

[0067]

[0068] in, u and y represent the state, control input, and output of the error system, respectively. and These represent the system's nominal parameters. Furthermore, this error system must meet output safety constraints. h represents the upper and lower bounds of the system's suspended air gap constraint, respectively. The specific values ​​are determined by the actual system requirements.

[0069] Step 2: Considering the output constraints of the magnetic levitation system, design a novel fractional obstacle Lyapunov function, and obtain the constraint handling mechanism function by taking the partial derivative of the novel fractional obstacle Lyapunov function.

[0070] Introducing fractional barrier Lyapunov functions:

[0071]

[0072] Where x1 = x1(t) represents the output state of the error system. Let τ be the parameter of the obstacle Lyapunov function, and τ be the homogeneity. In the specific implementation, constraints are set as symmetric output safety constraints (-0.01 < x1 < 0.01) and asymmetric output safety constraints (-0.005 < x1 < 0.01), with homogeneity set. Symmetric and asymmetric constraints are specifically manifested in whether the relative distances between the upper and lower bounds of the system output constraints and the system equilibrium point are symmetric or asymmetric. The values ​​of the output constraints directly affect whether the system meets the safety conditions.

[0073] V can be calculated b (0) = 0, and for any V b (x1)>0. Therefore, we can obtain the fractional barrier Lyapunov function V. b (x1) is positive definite, and taking the partial derivative with respect to x1, we get...

[0074]

[0075] Among them, the constraint handling mechanism function

[0076] Step 3: Applying finite-time control technology and homogeneous system theory, and based on the constraint handling mechanism function obtained in Step 2, design a continuous finite-time state feedback constraint controller using system state information.

[0077] Consider the following nonlinear system

[0078]

[0079] Where f(·): exist neighborhood of the origin x = 0 Upright continuous. If an open neighborhood of the origin exists. sum function T x : This allows the nonlinear system described above to start from the initial point Each initial solution trajectory x(t, x0) is well-defined and unique in forward time and for t ∈ [0, T] x (x0)), and the limit Then the equilibrium point x = 0 is locally finite-time stable. Here, T(x0) is called the convergence time (relative to the initial state x0). If the equilibrium of the system is Lyapunov stable and converges in finite time, then the equilibrium is finite-time stable. The origin is then in a globally stable equilibrium with a finite time.

[0080] Consider the following system

[0081]

[0082] Where f(x) is about A continuous homogeneous vector field with homogeneity τ < 0. Assume... The zero solution is asymptotically stable. If for i = 0, 1, 2, ..., n, ...

[0083]

[0084] Therefore, the zero solution above is locally finite-time stable.

[0085] Furthermore, the global asymptotic stability and local finite-time stability of the system at the equilibrium point x=0 indicate that the system possesses global finite-time stability.

[0086] The finite-time state feedback constraint controller is:

[0087]

[0088] Where, k p1 and k d1 It is a positive constant. When considering the safety constraints of symmetrical output, set k... p1 =20 and k d1 =18. When considering asymmetric output security constraints, set k... p1 =6 and k d1 =5. The symmetric output safety constraints and the asymmetric output safety constraints are -0.01 < x1 < 0.01 and -0.015 < x1 < 0.01, respectively. This represents the fractional power term of the velocity state of the error system. The finite-time state feedback controller of this invention has a simpler structure and is easier to implement in practice.

[0089] Substituting the finite-time state feedback controller u1 into the error system, we can obtain the following closed-loop system:

[0090]

[0091] For an error system whose initial state x1 satisfies the output safety constraints, there exists a positive constant k. p1 and k d1 This ensures that the above closed-loop system is stable in finite time and does not violate safety constraints. The following section will demonstrate that the above closed-loop system is stable in finite time; the specific stability analysis will be completed in three steps.

[0092] Step 1: For the closed-loop system consisting of the error system and the continuous finite-time state feedback controller, choose the following Lyapunov function.

[0093]

[0094] Differentiating the Lyapunov function V1 along the corresponding closed-loop system yields the derivative.

[0095]

[0096] Note the derivative Therefore when This means that the state x2 ≡ 0. According to the LaSalle invariant principle, all trajectories of the corresponding closed-loop system converge to the invariant set S = {(x1, x2) | x2 ≡ 0}, from which we can deduce the state x1 = 0. Therefore, on the invariant set S, (x1, x2) = (0, 0).

[0097] Step 2: Rewrite the corresponding closed-loop system as a nominal system plus a nonlinear term.

[0098]

[0099] In the formula, the nonlinear term

[0100] The corresponding nominal system is represented as:

[0101]

[0102] Choose the following Lyapunov functions:

[0103]

[0104] The derivative can be obtained by taking the derivative along the nominal system.

[0105]

[0106] Let r = (r1, r2), then according to the theory of homogeneous systems, it is easy to prove that the corresponding nominal system is about... A homogeneous system of order τ < 0.

[0107] Define the function g(x1) := 1-λ(x1), and perform a Taylor expansion of g(x1) to obtain

[0108]

[0109] In the formula, the coefficient function of the expansion term It is a higher-order term with respect to state x1.

[0110] Clearly, according to homogeneous system theory, for all states x1,

[0111]

[0112] Therefore, the closed-loop system is locally stable in finite time. Combining the proofs in steps one and two, we can further deduce that the closed-loop system is stable over the entire timescale. It is stable over a limited period of time.

[0113] Step 3: Confirm the output constraint boundaries. Let (x1(t), x2(t)) T The system starts from the initial state (x1(0), x2(0)). T Any initial solution.

[0114] Next, we will use a contradictory viewpoint to illustrate that the corresponding closed-loop system satisfies the output constraints.

[0115] Suppose there exists a time t′ > 0 such that According to the mean value theorem, there must exist another time t. * >0, making Based on the designed barrier Lyapunov function, we have

[0116] Combining the Lyapunov function V1, we have the following relationship:

[0117]

[0118] Obviously, with There is a contradiction. The position trajectory of the error system is well defined on [0, +∞) and satisfies the output constraint requirements. This completes the proof.

[0119] Step 4: Based on the constraint processing mechanism function obtained in Step 2, when the system velocity state information is not measurable (i.e., when only the system output information is measurable), design a non-smooth filter, and use the output information of the non-smooth filter and the system output information to design a finite-time output feedback controller.

[0120] In practical applications, sensor failures or other reasons may cause certain state information of the system to be unmeasurable. In this case, the error system only outputs measurable state information. Therefore, we first design a non-smooth filter, and then use the filter output information and the system output information to design a finite-time output feedback constraint controller.

[0121] The system speed status and output information are obtained by measuring the sensors and then fed back to the controller.

[0122] The non-smooth filter design is as follows:

[0123]

[0124] Where η represents the state information of the filter, the filter gain parameters l1 and l2 represent positive constants, and x3 represents the filter output information.

[0125] In practical implementation, when considering the symmetrical output safety constraints, the filter gain parameters l1 = 15 and l2 = 95 are set. When considering the asymmetrical output safety constraints, the filter gain parameters l1 = 9 and l2 = 90 are set.

[0126] The finite-time output feedback constraint controller is designed as follows:

[0127]

[0128] Among them, the control parameter k p2 and k d2 It is a positive number. The finite-time output feedback controller of this invention has a simpler structure, which is beneficial for practical implementation.

[0129] In the specific implementation process, when considering the symmetric output safety constraints, k is set. p2 =25 and k d2 =45. When considering asymmetric output security constraints, set k... p2 =8 and k d2 =25.

[0130] Substituting the finite-time output feedback constraint controller u2 into the error system, we can obtain the following closed-loop system:

[0131] Similarly, it can be proven that x1(t′) <- h The above scenario is also invalid.

[0132] Therefore, for a closed-loop system, when hour,

[0133]

[0134] For an error system whose initial state x1 satisfies the output safety constraints, there exists a positive constant k. p2 k d2 The values ​​l1 and l2 ensure that the closed-loop system consisting of the finite-time output feedback constraint controller and the error system is finite-time stable and does not violate the safety constraints. The following section demonstrates that the above closed-loop system is finite-time stable; the stability analysis is performed in three steps.

[0135] Step 1: First, prove that the corresponding closed-loop system is asymptotically stable. Choose the following Lyapunov function.

[0136]

[0137] The derivative can be obtained by differentiating the Lyapunov function V² along the closed-loop system.

[0138]

[0139] When the derivative This means x3 ≡ 3. According to LaSalle's invariance principle, all trajectories (x1, x2, x3) of the corresponding closed-loop system converge to the invariant set S = {(x1, x2, x3) | x3 ≡ 0}. Therefore, we can conclude that x2 = 0. Furthermore, we can conclude that x1 = 0. Thus, we can derive (x1, x2, x3) = (0, 0, 0) from the invariant set S.

[0140] Step 2: Rewrite the corresponding closed-loop system as a nominal system plus a nonlinear term.

[0141]

[0142] In the formula, the nonlinear term

[0143] The corresponding nominal system can be represented as follows:

[0144]

[0145] For the nominal system, choose the following Lyapunov function:

[0146]

[0147] The derivative can be obtained by taking the derivative along the nominal system.

[0148]

[0149] According to Lyapunov's stability theorem, it can be easily proven that the nominal system is asymptotically stable. Furthermore, letting r = {r1, r2, r1}, it can be proven that the closed-loop system is asymptotically stable with respect to... A continuous homogeneous vector field with homogeneity τ < 0.

[0150] Clearly, for all states x1, we have

[0151]

[0152] Based on the above proof, it has been demonstrated that the corresponding closed-loop system is locally stable in finite time.

[0153] Step 3: For the corresponding closed-loop system, let (x1(t), x2(t), x3(t)) T The system starts from the initial state (x1(0), x2(0), x3(0)). T Any initial solution.

[0154] Suppose there exists a time t′ > 0 such that According to the mean value theorem, there must exist another time t. * >0, making x1(t) * )=- h .

[0155] Based on the designed fractional barrier Lyapunov function V b (x1), has However, combined The following conclusions can be drawn.

[0156]

[0157] Obviously This contradicts the above equation. Similarly, it can be proven that… This situation is also not feasible.

[0158] Therefore, for the aforementioned closed-loop system, when hour, The position trajectory of the error system is well defined on [0, +∞) and satisfies the output constraints. This completes the proof.

[0159] In magnetic levitation systems, the goal of finite-time position tracking control is to quickly and accurately adjust the position of the suspended object to a predefined space. To achieve this goal, this invention designs two types of finite-time controllers: a finite-time state feedback controller and a finite-time output feedback controller. Both controllers are based on finite-time stability theory and homogeneous system theory, and can effectively complete the position tracking task even when the system has suspension air gap constraints.

[0160] First, the designed finite-time state feedback controller is implemented by introducing a novel fractional barrier Lyapunov function, which can uniformly handle symmetric and asymmetric suspended air gap constraints without changing the controller structure. Through this design, the controller can achieve stable control of the system state within a finite time and ensure that the system output always meets the given constraints.

[0161] Secondly, when only the system output is available, a continuous finite-time output feedback controller based on a non-smooth filter is designed. This filter avoids the reconstruction of unmeasured states, thus simplifying the controller design process. In this way, effective system control can be achieved even with incomplete state information.

[0162] The effectiveness of the two continuous finite-time controllers designed in this invention is verified through Matlab simulations. The simulation section will illustrate the universality of the designed control scheme through four case studies. Case 1 verifies the performance of the continuous finite-time state feedback constraint controller in handling symmetric constraints. Case 2 verifies the performance of the continuous finite-time output feedback constraint controller in handling symmetric constraints. Case 3 verifies the performance of the continuous finite-time state feedback constraint controller in handling asymmetric constraints, and adds external disturbances to verify the controller's anti-interference performance. Case 4 verifies the performance of the continuous finite-time output feedback constraint controller in handling asymmetric constraints, and adds external disturbances to verify the controller's anti-interference performance.

[0163] In Case 1, this invention simulates the control effect of the magnetic levitation system under a finite-time state feedback controller using Matlab numerical simulation. In this case, the position constraint boundary of the magnetic levitation system is set to 0.0325 < δ < 0.0525. Figure 2 The simulation results of the continuous finite-time state feedback constraint controller designed in this invention are shown in the figure. Figure 2 It can be clearly seen that when considering the safety constraints of the magnetic levitation system, the controller successfully constrains the system's position trajectory within a safe range, and the position trajectory converges to the equilibrium point within a finite time. Figure 2 As shown in (a1). When the safety constraints of the magnetic levitation system are not considered, it can be observed that the system's position trajectory will cross the safety constraint boundary at some point in time, as shown... Figure 2 As shown in (a2), this is not allowed in engineering. Therefore, the continuous finite-time state feedback constraint controller developed in this invention has great application value in handling magnetic levitation systems with output constraints. (Figure...) δ These represent the upper and lower bounds of the safety constraints for a magnetic levitation system.

[0164] In Case 2, the present invention simulates the control effect of the magnetic levitation system under a finite-time output feedback controller using Matlab numerical simulation. In this case, the position constraint boundary of the magnetic levitation system is set to 0.0325<δ<0.0525. Figure 3 The numerical simulation results of the continuous finite-time output feedback constraint controller designed for this invention are shown in the figure. It is clear from the figure that when considering the safety constraints of the magnetic levitation system, the controller achieves the goal of constraining the system's position trajectory within a symmetrical safety range, and the system's position trajectory converges to the equilibrium point within a finite time. Figure 3 As shown in (b1). When the safety constraints of the magnetic levitation system are not considered, it can be clearly seen that the system's position trajectory will cross the safety constraint boundary at a certain point in time, as shown... Figure 3 As shown in (b2), this scenario presents safety risks in industrial applications and should be avoided as much as possible. Therefore, the continuous finite-time output feedback constraint controller developed in this invention has great application potential in handling magnetic levitation systems with symmetrical output constraints.

[0165] In Case 3, the present invention simulates the control effect of the magnetic levitation system under the finite-time state feedback controller through Matlab numerical simulation. In this case, the position constraint boundary of the magnetic levitation system is set to 0.0275<δ<0.0525. In addition, in order to verify the anti-interference capability of the finite-time state feedback controller, an external disturbance d(t)=0.1sin(t) is introduced. Figure 4 The simulation results of the continuous finite-time state feedback constraint controller designed for this invention on a magnetic levitation system with asymmetric output constraints are shown in the figure. Figure 4 It can be clearly seen that despite external interference, the controller can still ensure the tracking performance of the system, which shows that the controller has strong anti-interference ability. Furthermore, under asymmetric constraint conditions, the controller can still achieve safe control of the magnetic levitation system.

[0166] In Case 4, the present invention simulates the control effect of the magnetic levitation system under the finite-time output feedback controller through Matlab numerical simulation. In this case, the position constraint boundary of the magnetic levitation system is set to 0.0325 < δ < 0.0575. In addition, in order to verify the anti-interference capability of the finite-time output feedback controller, an external disturbance d(t) = 0.2sin(5t) is introduced. Figure 5The figure shows the numerical simulation results of the continuous finite-time output feedback constraint controller designed for this invention on a magnetic levitation system with asymmetric output constraints. From the figure, we can draw the following conclusions: despite external disturbances, the controller can still ensure the system's tracking performance, demonstrating its strong anti-interference capability. Furthermore, even under asymmetric constraint conditions, this controller can still achieve safe control of the magnetic levitation system.

[0167] The contributions of this invention are as follows:

[0168] 1. Introducing the barrier Lyapunov function

[0169] This invention considers the output constraints of the mathematical modeling of magnetic levitation systems and designs a novel fractional barrier Lyapunov function, incorporating the system's output constraints into the function's design. Compared to the tan-type barrier Lyapunov function, the fractional barrier Lyapunov function designed in this invention can uniformly handle magnetic levitation systems with both symmetric and asymmetric output constraints.

[0170] 2. Design a continuous finite-time state feedback controller

[0171] This invention applies finite-time stability theory and homogeneous system theory to design a continuous finite-time state feedback constraint controller based on fractional barrier Lyapunov functions. This controller has a simpler structure and is easier to implement in practice compared to existing schemes.

[0172] 3. Design a continuous finite-time output feedback controller

[0173] When the magnetic levitation system only outputs information that can be used for controller design, this invention first designs a non-smooth filter, and then uses the system output information and the non-smooth filter output information to design for the first time a continuous finite-time output feedback controller based on a fractional barrier Lyapunov function.

[0174] This invention proposes a novel finite-time position tracking control method for magnetic levitation systems with output constraints, by uniformly handling symmetric / asymmetric air gap constraints (as demonstrated by simulation results). Figure 2 , 3 Figures 4 and 5 clearly show that the two controllers can uniformly handle symmetric / asymmetric air gap constraint problems in magnetic levitation systems, eliminate observer dependence (the designed finite-time output feedback controller does not introduce an observer to estimate the unmeasurable state x2), enhance anti-interference performance, and significantly improve the engineering applicability of position tracking control in magnetic levitation systems, thus having broad application prospects.

[0175] 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 finite-time position tracking control method for a magnetic levitation system with output constraints, characterized in that, The steps are as follows: Step 1: Establish a mathematical model of the magnetic levitation system, determine the error system, and obtain the upper and lower bounds of the suspension air gap constraint and the system state information; Step 2: Considering the upper and lower bounds of the air gap constraint of magnetic levitation, design the fractional barrier Lyapunov function, and take the partial derivative of the fractional barrier Lyapunov function to obtain the constraint handling mechanism function. Step 3: Applying finite-time control technology and homogeneous system theory, and based on the constraint handling mechanism function, design a continuous finite-time state feedback constraint controller using system state information; Step 4: Based on the constraint processing mechanism function, when the system velocity state information is not measurable, design a non-smooth filter, and use the output information of the non-smooth filter and the system output information to design a finite-time output feedback controller. The mathematical model of the magnetic levitation system is as follows: ； in, Indicates the mass of the steel ball. It is the acceleration due to gravity. Indicates the position of the steel ball at time t. The electromagnetic force generated by an electromagnet. This 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. and These represent the equilibrium current and equilibrium position of the steel ball in a stable suspended state, respectively. The fractional barrier Lyapunov function is: ； Where, x1= Indicates the output state of the error system. For the barrier Lyapunov function parameters, , These represent the upper and lower bounds of the system's suspended air gap constraint, respectively.

2. The finite-time position tracking control method for a magnetic levitation system with output constraints according to claim 1, characterized in that, Electromagnetic force Expanding at the equilibrium point using Taylor series and ignoring higher-order terms, we obtain: ； Define system control inputs and output The mathematical model of the error system is obtained as follows: ； in, , =u(t) and =y(t) represents the state, control input, and output of the error system, respectively. and These represent the system's nominal parameters; The error system satisfies the output safety constraints. .

3. The finite-time position tracking control method for a magnetic levitation system with output constraints according to claim 2, characterized in that, The fractional barrier Lyapunov function It is positive definite, for the output state. Taking the partial derivative, we get ； Among them, the constraint handling mechanism function , .

4. The finite-time position tracking control method for a magnetic levitation system with output constraints according to claim 3, characterized in that, The finite-time state feedback constraint controller is: ； in, and It is a positive number; The fractional power term representing the velocity state of the error system.

5. The finite-time position tracking control method for a magnetic levitation system with output constraints according to claim 4, characterized in that, Finite-time state feedback controller Substituting the error system into the closed-loop system, we obtain: ； For the initial output state An error system that satisfies the output safety constraints has normal numbers. and This ensures that the closed-loop system described above is stable for a finite time and does not violate safety constraints.

6. The finite-time position tracking control method for a magnetic levitation system with output constraints according to any one of claims 3-5, characterized in that, The non-smooth filter mentioned is: ； in, This indicates the filter's state information and filter gain parameters. All are positive numbers. This indicates the filter output information.

7. The finite-time position tracking control method for a magnetic levitation system with output constraints according to claim 6, characterized in that, The finite-time output feedback constraint controller is: ； Among them, control parameters and All are normal numbers.

8. The finite-time position tracking control method for a magnetic levitation system with output constraints according to claim 7, characterized in that, Finite-time output feedback constraint controller Substituting the error system into the closed-loop system, we obtain: ； For the initial output state An error system that satisfies the output safety constraints has normal numbers. , , and This ensures that the closed-loop system consisting of the finite-time output feedback constraint controller and the error system is finite-time stable and does not violate safety constraints.

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