Event-triggered robust model prediction control method for suspension system of magnetically levitated train

Abstract

The event triggering robust model prediction control method for the maglev train suspension system comprises the following steps: S1, considering the influence of track irregularity interference, and establishing a mathematical model of the maglev train suspension system interfered by the track irregularity; s2, according to the established mathematical model of the suspension system and the state of the balance point of the suspension system, obtaining a tracking error model of the suspension system; s3, establishing an equivalent linear model, and calculating an equivalent state constraint and a control input constraint; s4, designing an event triggering robust model predictive control strategy by considering the state constraint and the control input constraint; s5, verifying recursive feasibility and system stability of the event-triggered robust model predictive control strategy; according to the method, the influence of track irregularity on the suspension system of the electromagnetic suspension train is eliminated, the suspension precision of the suspension system is improved, and the problems of calculation pressure and communication resource shortage caused by high-frequency control in the suspension control process of the magnetic suspension train are relieved on the basis of ensuring the suspension stability of the magnetic suspension train.

Description

Technical Field

[0001] The present invention relates to the technical field of magnetic levitation system control, and more particularly to an event-triggered robust model predictive control method based on a magnetic levitation train suspension system. Background Art

[0002] With rapid economic development and increasing technological, economic, and information exchange between cities, people's demand for transportation has gradually increased. Travel speed has become an increasingly important consideration, second only to safety, when choosing a mode of transportation. This is where the development of magnetic levitation trains comes in. Compared to traditional wheel-rail trains, magnetic levitation trains offer numerous advantages. They are electrically powered, and their contactless suspension design reduces friction with the track, enabling higher speeds, further reducing energy consumption and noise emissions, and offering enhanced earthquake resistance during operation.

[0003] As one of the core systems of a maglev train, the suspension system is closely linked to its safety and comfort. Therefore, the design of its control system has attracted considerable attention from researchers. First, the maglev system exhibits strong nonlinearities and open-loop instabilities, necessitating a controller to ensure stable suspension. Furthermore, given the strict suspension height restrictions, smoothness and comfort requirements, and input saturation constraints inherent in maglev trains, the multi-constraint nature of the maglev system must be considered in the control process.

[0004] However, research on magnetic levitation system controllers has rarely considered the computational complexity and limited communication resources. A more comprehensive approach to addressing the multi-constraint characteristics of magnetic levitation trains, ensuring their levitation control performance, improving their anti-interference capabilities, and reducing the consumption of computational and communication resources remains a challenging task.

[0005] According to the above analysis, at present, only electromagnetic levitation medium-low speed and high-speed maglev trains are available.

[0006] Currently, domestic and international research on the suspension control of electromagnetically suspended high-speed maglev trains only considers the anti-interference performance of the suspension system, and there is little research on the multi-constraint problem of the suspension system and the computational burden of the control method.

[0007] Therefore, the present invention proposes an event-triggered robust model predictive control strategy for this situation, which effectively eliminates the impact of track unevenness on the suspension accuracy of magnetic levitation trains and alleviates the computational pressure and communication resource constraints caused by high-frequency control during the suspension control process. It has theoretical significance, strong feasibility, and is conducive to improving economic benefits. Summary of the Invention

[0008] The object of the present invention is to provide an event-triggered robust model predictive control method based on a maglev train suspension system, which eliminates the influence of track irregularity interference on the suspension accuracy of the maglev train.

[0009] In order to achieve the above object, the present invention adopts the following technical solutions:

[0010] An event-triggered robust model predictive control method for a maglev train suspension system is characterized by comprising the following steps:

[0011] S1. Based on the working principle of the electromagnetic levitation train suspension system and considering the influence of track irregularity interference, a mathematical model of the electromagnetic levitation train suspension system affected by track irregularity interference is established;

[0012] S2. Obtaining a suspension system tracking error model based on the established suspension system mathematical model and the state of the suspension system at the equilibrium point;

[0013] S3. Aiming at the strong nonlinearity of the suspension system, the input state of the suspension system tracking error model is linearized, an equivalent linear model is established, and the equivalent state constraints and control input constraints are calculated;

[0014] S4. Based on the equivalent linear model, the event-triggered robust model predictive control strategy is designed considering the state constraints and control input constraints;

[0015] S5. Verify the recursive feasibility and system stability of the event-triggered robust model predictive control strategy.

[0016] Furthermore, in step S1, a mathematical model of the electromagnetic levitation train suspension system affected by track irregularities is established, specifically as follows:

[0017] According to Newton's second law and Maxwell's stress tensor method, the equations for the vertical dynamics, suspension force and voltage of the electromagnet can be obtained:

[0018]

[0019] Where F(t) is the suspension force generated by the electromagnet at time t; m is the mass of the electromagnet; g is the acceleration due to gravity; δ(t) is the suspension gap value at time t; is the vertical velocity of the suspension system at time t; is the vertical acceleration of the suspension system at time t; B(t) is the magnetic flux density of the suspension gap at time t; u(t) is the voltage value applied to the two ends of the electromagnet conductive coil at time t; R, μ0, A m and N are magnetic resistance, air permeability, effective electromagnet pole area and number of coil turns respectively; when the suspension system reaches equilibrium, the suspension force is equal to the gravity of the suspension system, that is, F(0) = A m B(0)2 / μ0=mg;

[0020] Select the suspension gap value δ(t) between the track and the electromagnet, the vertical speed The suspension gap magnetic flux density B(t) is used as the state variable z(t), and the additional voltage u(t) across the electromagnet is used as the control input. The mathematical model of the suspension system is established:

[0021]

[0022] in, is the state variable, is the control input; to ensure the safe and smooth operation of the train, the state variable z(t) needs to be constrained Design of the maximum allowable input voltage value of the electromagnetic levitation train suspension system The state and control input parameters of the equilibrium position of the suspension system are expressed as (z eq ,u eq ),in Track irregularities There is an upper bound, that is, the condition is satisfied Among them, f d (t) represents the vertical velocity disturbance caused by track irregularities.

[0023] Furthermore, the tracking error model of the suspension system in step S2 is as follows:

[0024] The error between the suspension system and the expected value at the equilibrium point of the suspension system is expressed as That is, the state variable and the control input variable minus the expected value to obtain the suspension system tracking error model:

[0025]

[0026] At the same time, the equivalent constraints of the suspension system tracking error model are obtained:

[0027]

[0028] Furthermore, in step S3, the equivalent linear model is established and the equivalent state constraints and control input constraints are calculated, as follows:

[0029] Use the input-state linearization method to handle the nonlinear terms of the system and calculate and The zero-order, first-order and second-order Lie brackets are and Get the vector field It is linearly independent and closed, satisfying the input-state linearization condition; next, calculate the new state variable, assuming that the first state satisfy and Condition; therefore define a new state variable New control inputs:

[0030]

[0031] Bringing the new control input v into the nonlinear suspension system, we get the equivalent linear model:

[0032]

[0033]

[0034] in, and are the state variables and control input variables of the linearized system respectively; the constraints of the equivalent linear model are calculated as follows:

[0035]

[0036] The new control input is simplified as From this, the upper and lower bounds of the equivalent control input of the linearized system can be calculated:

[0037]

[0038] Furthermore, step S4 includes the following sub-steps:

[0039] S4.1. Description of the optimization control problem:

[0040] Based on the equivalent linear model established in S3, the prediction time domain and trigger time series are defined as T and t respectively. k : The execution time is expressed as t k The optimal control problem at

[0041]

[0042] where s∈[t k :t k +T], is the initial state, and t k The feasible control input and corresponding state at time, and Respectively represent t k The optimal control input predicted at each moment and the corresponding optimal predicted state; and are the sets of robust tightening state constraints and control input constraints, respectively; is the terminal invariant set, where κ is determined by the terminal invariant set and the terminal control rate property; express The optimal value of

[0043] Input by feasible control The obtained cost function Recorded as Its expression is:

[0044]

[0045] in, represents the stage cost function, is the terminal cost function, Q, R, P>0 are weighted matrices with appropriate dimensions; by solving the optimization control problem Get the trajectory of the optimal control input and the corresponding optimal predicted state trajectory Therefore, according to the rolling horizon control principle of nominal model predictive control, the control input is defined as follows:

[0046]

[0047] Then, the optimization control problem is determined The specific expression of , including the robust tightening constraint and the trigger time sequence t k ;

[0048] S4.2 Robust Tightening Constraint Design:

[0049] Assume that the actual system state x(s; t k ) and nominal system state At the triggering time t k With the same initial state x(t k ), solve the state difference between the actual system and the nominal system, and use the triangle inequality, the Lipschitz condition L = 1, and the Gronwall-Bellman inequality to obtain:

[0050]

[0051] in, Finally, the tightening state constraint is estimated using Minkowski set subtraction and expressed as

[0052] S4.3. Terminal invariant set and terminal control rate design:

[0053] Since the equivalent linear model at the equilibrium point (0,0) is stable, there exists a local linear control law v = Kx, so that the control system is asymptotically stable; therefore, for the nominal system And given the weight matrices Q>0 and R>0, there exists a unique matrix P>0, a constant κ>0, and a state feedback gain K that can ensure that A+BK satisfies the Hurwitz stability condition, such that:

[0054] 1) Lyapunov equation Has a unique positive definite matrix P;

[0055] 2) For any condition that satisfies The control input is and satisfy It is a closed-loop system A positive invariant set of ; that is, if Then for any time t>t0,

[0056] 3) For any condition that satisfies Lyapunov equation Established, of which

[0057] S4.4. Trigger condition design:

[0058] Due to track irregularities The existence of the suspension system's true state trajectory x(t k ) and predicted state trajectory There are deviations between them; accordingly, the triggering conditions of the event-triggered robust model predictive control strategy are designed as follows:

[0059]

[0060] in is the trigger threshold, is the minimum trigger time interval that needs to be designed to meet Therefore, the next triggering moment is:

[0061]

[0062] in,

[0063]

[0064] Furthermore, step S5 includes the following sub-steps:

[0065] S5.1. Recursive feasibility analysis:

[0066] First, based on the trigger time t kThe optimal control input sequence Construct t k+1 The feasible control input sequence at time Expressed as:

[0067]

[0068] With the above-mentioned feasible control input The corresponding feasible state sequence is expressed as in

[0069] The equivalent linear model under the event-triggered robust model predictive control strategy is given an initial value x(0), and it is assumed that there is a solution at the initial moment; the control input In s∈[t k+1 ,t k +T], it is clear that the control input constraint is satisfied In s∈[t k +T,t k+1 +T], the control input is designed according to the terminal control rate The control input constraint is also satisfied, so the control input Satisfy constraints Right now s∈[t k+1 ,t k+1 +T] established;

[0070] When s∈[t k +T,t k+1 +T], we get Right now Enter the terminal domain; define Taking the derivative we get:

[0071]

[0072] where λ min and λ max Represent the minimum and maximum eigenvalues ​​of the matrix respectively; using the comparison principle and the minimum trigger time interval, we can get:

[0073]

[0074] At this point, it has been verified Able to use local feedback control law Enter the terminal domain Ω(ακ), where Due to the interference factor, for s∈[t k+1 ,t k+1 +T], Not necessarily equal to Set s=t k+1+T substitution to obtain:

[0075]

[0076] According to the triangle inequality and The above formula is simplified to:

[0077]

[0078] in The design trigger threshold is:

[0079]

[0080] get illustrate Satisfy the terminal constraints, that is

[0081] Since the optimization control problem is k It is feasible at the moment s∈[t k+1 ,t k +T] can be obtained calculate According to the event trigger threshold, triangle inequality and Gronwall-Bellman inequality, we can get:

[0082]

[0083] Substitute s=t k +T and get:

[0084]

[0085] Given and s∈[t k+1 :t k +T], if and If established, we can get therefore s∈[t k+1 ,t k +T] holds; when s∈[t k +T,t k+1 +T], design as well as get Where r>κ, so the state sequence Satisfy the state tightening constraint, that is, s∈[t k+1 ,t k+1 +T]; In summary, it can be verified that event-triggered robust model predictive control has recursive feasibility;

[0086] S5.2. Stability Theory Analysis:

[0087] Under the conditions of the existence of terminal invariant set and terminal control rate, the event-triggered robust model predictive control strategy is recursively feasible; when When , the optimal cost function is selected as the Lyapunov function, that is, Can get:

[0088]

[0089] Where, ΔV η ,η=1,2,3; for ΔV1, we can get:

[0090]

[0091] For ΔV2 we get:

[0092]

[0093] Simplifying ΔV3 yields:

[0094]

[0095] Design the controller parameters to satisfy the following inequalities:

[0096]

[0097] in And concluded It is then concluded that the state converges to the terminal domain within a finite time;

[0098] In the case of x∈Ω(κ), the control input is v(t)=Kx(t), and the choice As a Lyapunov function, the derivative can be obtained:

[0099]

[0100] When the perturbation upper bound satisfies the inequality hour:

[0101]

[0102] The above can verify that the equivalent linear model is stable, and further conclude that the mathematical model of the suspension system is stable.

[0103] The beneficial effects of the present invention are as follows:

[0104] 1. Effectively suppress the impact of track unevenness on the suspension accuracy and stability of maglev trains.

[0105] 2. Effectively ensure the safety constraints and input saturation constraints of the maglev train suspension system.

[0106] 3. It alleviates the computing pressure and communication resource shortage caused by high-frequency control during the suspension control of magnetic levitation trains. BRIEF DESCRIPTION OF THE DRAWINGS

[0107] The specific embodiments of the present invention are further described in detail below with reference to the accompanying drawings:

[0108] Figure 1 A flow chart showing an event-triggered robust model predictive control method for an electromagnetic levitation train suspension system according to the present invention;

[0109] Figure 2 A schematic diagram showing the suspension system of an electromagnetic levitation train affected by track irregularities according to the present invention;

[0110] Figure 3 A schematic diagram showing a comparison of the levitation gap, vertical velocity, and levitation gap magnetic flux density trajectory of the electromagnetic levitation system according to an embodiment of the present invention;

[0111] Figure 4 A schematic diagram showing a comparison of the trajectory of the input voltage on the electromagnet coil in an embodiment of the present invention is shown;

[0112] Figure 5 A schematic diagram showing a comparison of triggering conditions of an event-triggered robust model predictive control strategy controller in an embodiment of the present invention is shown;

[0113] Figure 6 A schematic diagram showing a comparison of single-step calculation time of the event-triggered robust model predictive control strategy in an embodiment of the present invention is shown. DETAILED DESCRIPTION

[0114] In order to more clearly illustrate the present invention, the present invention is further described below in conjunction with preferred embodiments and drawings. Those skilled in the art should understand that the following specific description is illustrative rather than restrictive and should not be used to limit the scope of protection of the present invention.

[0115] The event-triggered robust model predictive control method for a maglev train suspension system provided in this embodiment is used in an maglev train suspension system affected by track irregularities, such as Figure 1 As shown, the following steps are included:

[0116] S1. Based on the analysis of the working principle of the electromagnetic levitation train suspension system and considering the influence of track unevenness interference, a mathematical model of the electromagnetic levitation train suspension system affected by track unevenness interference is established.

[0117] Specifically, in step S1, a mathematical model of the electromagnetic levitation train suspension system affected by track irregularities is established, as follows:

[0118] According to Newton's second law and Maxwell's stress tensor method, the equations for the vertical dynamics, suspension force, and voltage of the electromagnet can be obtained:

[0119]

[0120] It is assumed that the magnetic permeability within the pole is infinite, the magnetic flux is uniformly distributed in the gap, and the leakage flux of the winding, the electromagnet, and the magnetic resistance of the track and the core are negligible. Where F(t) is the levitation force generated by the electromagnet at time t; m is the mass of the electromagnet; g is the acceleration due to gravity; and δ(t) is the value of the levitation gap at time t. is the vertical velocity of the suspension system at time t; is the vertical acceleration of the suspension system at time t; B(t) is the magnetic flux density of the suspension gap at time t; u(t) is the voltage value applied to the two ends of the electromagnet conductive coil at time t; R, μ0, A m and N are magnetic resistance, air permeability, effective electromagnet pole area and number of coil turns respectively; when the suspension system reaches equilibrium, the suspension force is equal to the gravity of the suspension system, that is, F(0) = A m B(0) 2 / μ0=mg;

[0121] Select the suspension gap value δ(t) between the track and the electromagnet, the vertical speed The suspension gap magnetic flux density B(t) is used as the state variable z(t), and the additional voltage u(t) across the electromagnet is used as the control input to establish a nonlinear model of the suspension system:

[0122]

[0123] in, is the state variable, is the control input; to ensure the safe and smooth operation of the train, the state variable z(t) needs to be constrained Design of the maximum allowable input voltage value of the electromagnetic levitation train suspension system The state and control input parameters of the equilibrium position of the suspension system are expressed as (z eq ,u eq ),in Track irregularities There is an upper bound, that is, the condition is satisfied Among them, f d (t) represents the vertical velocity disturbance caused by track irregularities.

[0124] S2. Obtaining a suspension system tracking error model based on the established suspension system mathematical model and the state of the suspension system at the equilibrium point;

[0125] Specifically, in step S2, the suspension system tracking error model is as follows:

[0126] The control goal of the suspension system is to control the suspension gap of the suspension system to track the expected value. Therefore, the tracking error model of the suspension system is established first. The state of the suspension equilibrium point position and the control input parameters are expressed as (z eq ,u eq ),in The error between the suspension system and the expected value at the equilibrium point of the suspension system is expressed as That is, the state variable and the control input variable minus the expected value to obtain the suspension system tracking error model:

[0127]

[0128] At the same time, the equivalent constraints of the suspension system tracking error model are obtained:

[0129]

[0130] S3. Aiming at the strong nonlinearity of the suspension system, the input state of the suspension system tracking error model is linearized, an equivalent linear model is established, and the equivalent state constraints and control input constraints are calculated;

[0131] Specifically, in step S3, the equivalent linear model is established and the equivalent state constraints and control input constraints are calculated as follows:

[0132] Lemma 3.1: (Input-state linearization condition) and is a smooth vector field function, then the nonlinear system The necessary and sufficient condition for input-state linearization is that there exists a region Ω that satisfies: 1) the vector field are linearly independent; 2) in the same region Ω or its subregion, the vector field It is closed, that is, these vector fields satisfy certain linear combination relationships;

[0133] Note 3.1: (Input-State Linearization Process) 1) Calculate the vector field 2) Determine its controllability and closure conditions; 3) If the above two conditions are met, calculate the first state x1 of the new state variable, and solve Get; 4) Get all other state quantities x1=x1,x2=L h x1,...,x n =L h x n-1 ; 5) Calculate the corresponding input transformation in,

[0134] Use the input-state linearization method to handle the nonlinear terms of the suspension system and calculate and The zero-order, first-order and second-order Lie brackets are and Get the vector field is linearly independent and closed, satisfying the input-state linearization condition, Lemma 3.1, so the established nonlinear suspension system can be linearized; next calculate the new state variables, assuming that the first state satisfy and Condition, the first state variable Established; According to the above calculation and input-state linearization law, we redefine the new state variable as New control input in and Get a new control input:

[0135]

[0136] Bringing the new control input v into the nonlinear suspension system, we get the equivalent linear model:

[0137]

[0138]

[0139] in, and are the state variables and control input variables of the equivalent linear model respectively; the constraints of the equivalent linear model are calculated as follows:

[0140]

[0141] The new control input is simplified as From this, the upper and lower bounds of the equivalent control input of the linearized system can be calculated:

[0142]

[0143] Next, a controller is designed based on the linear tracking error system while ensuring that the above constraints are met.

[0144] S4. Based on the equivalent linear model, an event-triggered robust model predictive control (ERMPC) strategy is designed considering state constraints and control input constraints, specifically including:

[0145] S4.1. Description of the optimization control problem:

[0146] Based on the equivalent linear model established in S3, the prediction time domain and trigger time series are defined as T and t respectively. k : The execution time is expressed as t k The optimal control problem at

[0147]

[0148]

[0149] where s∈[t k :t k +T], is the initial state, and t k The feasible control input and corresponding state at time, and Respectively represent t k The optimal control input predicted at each moment and the corresponding optimal predicted state; and are the sets of robust tightening state constraints and control input constraints, respectively; is the terminal invariant set, where κ is determined by the terminal invariant set and the terminal control rate property; express The optimal value of

[0150] For simplicity, the feasible control input The obtained cost function Recorded as Its expression is:

[0151]

[0152] in, represents the stage cost function, is the terminal cost function, Q, R, P>0 are weighted matrices with appropriate dimensions; by solving the optimization control problem Get the trajectory of the optimal control input and the corresponding optimal predicted state trajectory Therefore, according to the rolling horizon control principle of nominal model predictive control, the control input is defined as follows:

[0153]

[0154] Then, the optimization control problem is determined The specific expression of , including the robust tightening constraint and the trigger time sequence t k;

[0155] S4.2 Robust Tightening Constraint Design:

[0156] In each prediction time domain T, the solution process of the optimization control problem uses the nominal system; due to the influence of external interference, the actual state trajectory x(s; t k ) will deviate from the predicted optimal state trajectory Therefore, we need to calculate the upper bound of the state prediction deviation; in the process of solving the optimization problem, ensuring that the predicted state is within the tightening constraints can ensure that the actual state is within the set constraints;

[0157] Assume that the actual system state x(s; t k ) and nominal system state At the triggering time t k With the same initial state x(t k ), solve the state difference between the actual system and the nominal system, and use the triangle inequality and the Lipschitz condition L = 1 to obtain:

[0158]

[0159] Then use the Gronwall-Bellman inequality to simplify and get:

[0160]

[0161] in, Finally, the tightening state constraint is estimated using Minkowski set subtraction and expressed as Used to ensure that state constraints can be satisfied in the presence of disturbances;

[0162] S4.3. Terminal invariant set and terminal control rate design:

[0163] Since the equivalent linear model at the equilibrium point (0,0) is stable, there exists a local linear control law v = Kx, so that the control system It is asymptotically stable; without loss of generality, assume has at least one non-zero eigenvalue;

[0164] For nominal system Given the weight matrices Q > 0 and R > 0, there exists a unique matrix P > 0, a constant κ > 0, and a state feedback gain K that ensures that A + BK satisfies the Hurwitz stability condition, so that the following conclusion holds:

[0165] 1) Lyapunov equation Has a unique positive definite matrix P;

[0166] 2) For any condition that satisfies The control input is and satisfy It is a closed-loop system A positive invariant set of . That is, if Then for any time t>t0,

[0167]

[0168] 3) For any condition that satisfies Lyapunov equation Established, of which

[0169] S4.4. Trigger condition design:

[0170] Due to track irregularities The existence of the suspension system's true state trajectory x(t k ) and predicted state trajectory There is a deviation between the two. When the deviation exceeds a certain value, the suspension gap of the maglev train will deviate from the safe area, which will bring safety hazards to the operation of the maglev train. Based on the above discussion, the triggering conditions of the event-triggered robust model predictive control (ERMPC) strategy are designed as follows:

[0171]

[0172] in is the trigger threshold, is the minimum trigger time interval that needs to be designed to meet Therefore, the next triggering moment is:

[0173]

[0174] in,

[0175]

[0176] According to the above formula, it is easy to see that the upper and lower limits of the controller update time are Therefore, the designed event trigger mechanism will not have the Zeno effect.

[0177] S5. Verify the recursive feasibility and system stability of the event-triggered robust model predictive control (ERMPC) strategy, specifically including the following steps:

[0178] S5.1. Recursive feasibility analysis:

[0179] First, based on the trigger time t k The optimal control input Construct t k+1 The feasible control input sequence at time Expressed as:

[0180]

[0181] With the above-mentioned feasible control input The corresponding feasible state sequence is expressed as in The following lemma provides a theoretical basis for proving the feasibility of recursion:

[0182] Lemma 5.1: Given and s∈[t k+1 :t k +T], if and If established, we can get

[0183] Theorem 5.1: Given an initial value x(0) and assuming that there is a solution at the initial moment, the equivalent linear model under the ERMPC strategy is recursively feasible if the following conditions are met:

[0184]

[0185] where λ min and λ max denote the minimum and maximum eigenvalues ​​of the matrix respectively;

[0186] Proof: For a clearer description, the proof is divided into three parts. The first part explains the control input sequence Satisfy the constraints; the second part explains Satisfy the terminal constraints; finally, state sequence Satisfy the state tightening constraint:

[0187] 1) Verification s∈[t k+1 ,t k+1 +T]:

[0188] First control input In s∈[t k+1 ,t k +T], it is clear that the control input constraint is satisfied In s∈[t k +T,t k+1 +T], the control input is designed according to the terminal control rate The control input constraint is also satisfied, so the control input Satisfy constraints Right now s∈[tk+1 ,t k+1 +T] is established and the verification is completed;

[0189] 2) Verification

[0190] First, consider the state trajectory In the time interval s∈[t k +T,t k+1 +T], due to the optimization control problem at t k The moment is feasible, so Right now Entering the terminal domain, according to the dual-mode MPC principle, the local feedback control law is applied definition Therefore, according to the terminal control rate design, we can get:

[0191]

[0192] Using the comparison principle and the minimum trigger time interval, we can get:

[0193]

[0194] At this point, it has been verified The local feedback control law can be used Enter the terminal domain Ω(ακ), where Due to the interference factor, for s∈[t k+1 ,t k+1 +T], Not necessarily equal to Set s=t k+1 +T substitution can be obtained:

[0195]

[0196] According to the triangle inequality and The above formula is simplified to:

[0197]

[0198] in get illustrate Satisfy the terminal constraints, that is

[0199] 3) Proof s∈[t k+1 ,t k+1 +T]:

[0200] Since the optimization control problem is k It is feasible at the moment s∈[tk+1 ,t k +T] can be obtained calculate It can be concluded that:

[0201]

[0202] According to the event trigger conditions, we can get:

[0203]

[0204] Therefore, applying the triangle inequality yields:

[0205]

[0206] Where L = 1 is the Lipschitz constant; according to the Gronwall-Bellman inequality, for s∈[t k+1 ,t k+1 +T] can be obtained:

[0207]

[0208] Considering that when s=t k +T time and get:

[0209]

[0210] According to Lemma 5.1, s∈[t k+1 ,t k +T] holds; in s∈[t k +T,t k+1 +T] time interval, we get where r>κ. Therefore, s∈[t k+1 ,t k+1 +T];

[0211] In summary, it can be verified that the ERMPC strategy is recursively feasible;

[0212] S5.2. Stability Theory Analysis:

[0213] Theorem 5.2: Under the conditions of the existence of terminal invariant sets and terminal control rates, and the recursive feasibility of the ERMPC strategy, if the controller parameters satisfy the following inequality, it can be concluded that the closed-loop system enters the terminal region Ω(κ) in a finite time:

[0214]

[0215] in In addition, the upper bound of the interference satisfies the following conditions, and the closed-loop system remains stable in the terminal domain:

[0216]

[0217] Verification: This theorem is verified by discussing two cases of initial state. If The proof state will converge to the terminal domain Ω(κ) in finite time; if x∈Ω(κ), the proof state will always remain in the terminal domain Ω(κ), that is, Ω(κ) is a positive invariant set;

[0218] First, when When , the optimal cost function is selected as the Lyapunov function, that is, It can be seen that V(t k+1 )-V(t k ) meets the following conditions:

[0219]

[0220] Will Substitute it into the above formula and it can be transformed into:

[0221]

[0222] Where, ΔV η ,η=1,2,3 is divided into three parts for discussion:

[0223]

[0224] For the first part ΔV1, we can observe using the triangle inequality that:

[0225]

[0226] To quantify We know:

[0227]

[0228] By using the triangle inequality, we get:

[0229]

[0230] in, Similarly, we get:

[0231]

[0232] Theorem 5.1 is verified to be So we get:

[0233]

[0234] For the second part ΔV2, we get:

[0235]

[0236] The inequality and Substituting into the third part ΔV3 we get:

[0237]

[0238] According to the conditions of Theorem 5.1, we can get It can be concluded that the state variables of the equivalent linear model converge to the terminal domain in a finite time;

[0239] In the second step, we will verify that Ω(κ) is a positive invariant set of closed-loop systems: Assume x(t0)∈Ω(κ), for any t>t0, it does not mean x(t)∈Ω(κ), which means that there exists t>t0 and So that for x(t)∈Ω(κ) Then we define Right now

[0240] choose As a Lyapunov function, the derivative can be obtained:

[0241]

[0242] When x∈Ω(κ), the control input is converted to v(t)=Kx(t), substituting into the above formula:

[0243]

[0244] According to the terminal domain design, we get:

[0245]

[0246] because When the perturbation upper bound satisfies the inequality At this time, it is easy to know:

[0247]

[0248] Therefore, there exists t″∈(t0,t′) such that This contradicts the fact that t′ is a minimum value, and it can be concluded that Ω(κ) is an invariant set. Based on the discussion of the above two cases, the stability verification is completed.

[0249] In order to verify the effectiveness of the event-triggered robust model predictive control method for the electromagnetic levitation train suspension system provided in this embodiment, a simulation experiment is conducted using MATLAB and a detailed description is given below:

[0250] The mathematical model of the electromagnetic levitation train suspension system provided in this embodiment comprehensively considers track unevenness disturbances, safety constraints and physical constraints of train operation, designs an event-triggered robust model predictive control strategy, suppresses the impact of interference on the system control performance, and enables the electromagnetic levitation system to have good suspension performance. It alleviates the computational pressure and communication resource constraints caused by high-frequency control during the suspension control process of the magnetic levitation train, and ensures the stability of the suspension of the magnetic levitation train.

[0251] The parameters of the train are set as follows: m = 311.84 kg is the mass of the electromagnet, g = 9.8 m / s 2 is the acceleration due to gravity, N=324 is the number of turns of the magnetic coil, A m =0.0235m 2 is the effective area of ​​the magnetic pole, μ0=4π×10 -7 H / m is the atmospheric magnetic permeability, R=1Ω is the magnetic resistance; the suspension gap, vertical velocity and suspension gap magnetic flux density when the suspension system is in equilibrium are δ eq =0.01m, and B eq =0.40425T, and the voltage input at the equilibrium moment is u eq =19.863V; The goal of suspension control is to track the distance δ between the vehicle body and the track from the initial value δ(0) to the desired suspension gap δ eq , and ensure that the suspension system remains within the upper and lower limits of the safety gap, that is, 0.008≤δ≤0.02m; at the same time, considering the stability of the suspension, it is expected that the vertical speed of the maglev train will be within a certain range, that is, The magnetic flux density constraint of the suspension gap can be calculated as 0.402≤B≤0.406T. In addition, due to the characteristics and limitations of hardware devices such as actuators, the actual suspension system needs to meet the saturation constraint, that is, the voltage across the conductive coil has an upper bound of 0≤u≤100V. At the initial moment, the suspension system is set to δ(0)=0.02m, and B(0)=0.40425T, that is, the vertical initial velocity is 0, which is consistent with the train being in a suspended state. The initial voltage is calculated to be u(0)=39.74V; the vertical velocity disturbance caused by track irregularity is f d (t) = 0.002sin(5πt);

[0252] The ERMPC strategy parameters are designed as follows: the sampling time is designed to be τ = 0.02s; the prediction time domain is T = 50τ = 1s; the running time is t p=15s; In addition, given the weight matrix Q = [10, 0, 0; 0, 10, 0; 0, 0, 10] and R = 0.1, the state feedback gain K = [10, 18.2674, 11.6848] and the matrix

[0253]

[0254] And the terminal domain constant κ = 0.014; the RMPC strategy parameters are the same as the ERMPC strategy parameters, and the minimum trigger interval of the ERMPC strategy is designed to be 0.025s, 0.04s and 0.15s.

[0255] Based on the above parameters, the event-triggered robust model predictive control strategy proposed in this invention is simulated and verified, and it is obtained Figure 3 、 Figure 4 、 Figure 5 and Figure 6 ;in, Figure 3 The figure shows the comparison of the suspension gap, vertical velocity and suspension gap magnetic flux density trajectory under different minimum execution time control strategies of ERMPC and RMPC. Figure 4 A schematic diagram showing the comparison of control input voltage values ​​under different minimum execution time control strategies of ERMPC and RMPC strategies is shown. Figure 5 The diagram shows the triggering comparison of ERMPC controllers with different minimum execution times and RMPC controllers. Figure 6 A schematic diagram comparing the single-step calculation time of the controller under different minimum execution time control strategies of ERMPC and RMPC strategies is shown; it can be seen from the simulation diagram that the proposed ERMPC strategy has good control performance. Through simulation experiments, the optimal minimum execution time of 0.025s is selected by comprehensively considering the dynamic performance and the number of triggering times. It can enable the suspension system to quickly reach the set state, reduce the number of triggering times of the controller, save a lot of computing and communication resources, ensure the safety constraints and input saturation constraints of the magnetic levitation train suspension system, and ensure the suspension performance of the electromagnetic levitation train under the influence of track unevenness.

[0256] The above analysis verifies the effectiveness of the event-triggered robust model predictive control method for the electromagnetic levitation train suspension system provided in this embodiment, alleviates the computational pressure and communication resource constraints caused by high-frequency control during the suspension control of the maglev train, and ensures the stability of the maglev train suspension.

[0257] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not limitations on the implementation methods of the present invention. For ordinary technicians in the relevant field, other different forms of changes or modifications can be made based on the above description. It is impossible to list all the implementation methods here. All obvious changes or modifications derived from the technical solution of the present invention are still within the scope of protection of the present invention.

Claims

1. An event-triggered robust model predictive control method for a maglev train suspension system, characterized in that: The steps include: S1. Based on the working principle of the electromagnetic levitation train suspension system and considering the influence of track irregularity interference, a mathematical model of the electromagnetic levitation train suspension system affected by track irregularity interference is established; S2. Obtaining a suspension system tracking error model based on the established suspension system mathematical model and the state of the suspension system at the equilibrium point; S3. Aiming at the strong nonlinearity of the suspension system, the input state of the suspension system tracking error model is linearized, an equivalent linear model is established, and the equivalent state constraints and control input constraints are calculated; S4. Based on the equivalent linear model, the event-triggered robust model predictive control strategy is designed considering the state constraints and control input constraints; S5. Verify the recursive feasibility and system stability of the event-triggered robust model predictive control strategy.

2. The event-triggered robust model predictive control method for a maglev train suspension system according to claim 1, characterized in that: In step S1, a mathematical model of the electromagnetic levitation train suspension system affected by track irregularities is established, specifically as follows: According to Newton's second law and Maxwell's stress tensor method, the equations for the vertical dynamics, suspension force and voltage of the electromagnet can be obtained: Where F(t) is the suspension force generated by the electromagnet at time t; m is the mass of the electromagnet; g is the acceleration due to gravity; δ(t) is the suspension gap value at time t; is the vertical velocity of the suspension system at time t; is the vertical acceleration of the suspension system at time t; B(t) is the magnetic flux density of the suspension gap at time t; u(t) is the voltage value applied to the two ends of the electromagnet conductive coil at time t; R, μ0, A m and N are magnetic resistance, air permeability, effective electromagnet pole area and number of coil turns respectively; when the suspension system reaches equilibrium, the suspension force is equal to the gravity of the suspension system, that is, F(0) = A m B(0) 2 / μ0=mg; Select the suspension gap value δ(t) between the track and the electromagnet, the vertical speed The suspension gap magnetic flux density B(t) is used as the state variable z(t), and the additional voltage u(t) across the electromagnet is used as the control input. The mathematical model of the suspension system is established: in, is the state variable, is the control input; to ensure the safe and smooth operation of the train, the state variable z(t) needs to be constrained Design of the maximum allowable input voltage value of the electromagnetic levitation train suspension system The state and control input parameters of the equilibrium point position of the suspension system are expressed as (z eq ,u eq ),in Track irregularities There is an upper bound, that is, the condition is satisfied Among them, f d (t) represents the vertical velocity disturbance caused by track irregularities.

3. The event-triggered robust model predictive control method for a maglev train suspension system according to claim 2, characterized in that: The tracking error model of the suspension system in step S2 is as follows: The error between the suspension system and the expected value at the equilibrium point of the suspension system is expressed as That is, the state variable and the control input variable minus the expected value to obtain the suspension system tracking error model: At the same time, the equivalent constraints of the suspension system tracking error model are obtained:

4. The event-triggered robust model predictive control method for a maglev train suspension system according to claim 3, characterized in that: In step S3, the equivalent linear model is established and the equivalent state constraints and control input constraints are calculated, as follows: Use the input-state linearization method to handle the nonlinear terms of the system and calculate and The zero-order, first-order and second-order Lie brackets are and Get the vector field It is linearly independent and closed, satisfying the input-state linearization condition; next, calculate the new state variable, assuming that the first state satisfy and Condition; therefore define a new state variable New control inputs: Bringing the new control input v into the nonlinear suspension system, we get the equivalent linear model: in, and are the state variables and control input variables of the linearized system respectively; the constraints of the equivalent linear model are calculated as follows: The new control input is simplified as From this, the upper and lower bounds of the equivalent control input of the linearized system can be calculated:

5. The event-triggered robust model predictive control method for a maglev train suspension system according to claim 4, characterized in that: Step S4 includes the following sub-steps: S4.

1. Description of the optimization control problem: Based on the equivalent linear model established in S3, the prediction time domain and trigger time series are defined as T and The execution time is expressed as t k The optimization control problem at where s∈[t k :t k +T], is the initial state, and t k The feasible control input and corresponding state at time, and Respectively represent t k The optimal control input predicted at each moment and the corresponding optimal predicted state; and are the sets of robust tightening state constraints and control input constraints, respectively; is the terminal invariant set, where κ is determined by the terminal invariant set and the terminal control rate property; express The optimal value of Input by feasible control The obtained cost function Recorded as Its expression is: in, represents the stage cost function, is the terminal cost function, Q, R, P>0 are weighted matrices with appropriate dimensions; by solving the optimization control problem Get the trajectory of the optimal control input and the corresponding optimal predicted state trajectory Therefore, according to the rolling horizon control principle of nominal model predictive control, the control input at time t is defined as follows: Then, the optimization control problem is determined The specific expression of , including the robust tightening constraint and the trigger time sequence t k ; S4.2 Robust Tightening Constraint Design: Assume that the actual system state x(s; t k ) and nominal system state At the triggering time t k With the same initial state x(t k ), solve the state difference between the actual system and the nominal system, and use the triangle inequality, the Lipschitz condition L = 1, and the Gronwall-Bellman inequality to obtain: in, Finally, the tightening state constraint is estimated using Minkowski set subtraction and expressed as S4.

3. Terminal invariant set and terminal control rate design: Since the equivalent linear model at the equilibrium point (0,0) Is stable, there exists a local linear control law v = Kx, so that the control system has asymptotic stability; therefore, for the nominal system And given the weight matrices Q>0 and R>0, there exists a unique matrix P>0, a constant κ>0, and a state feedback gain K that can ensure that A+BK satisfies the Hurwitz stability condition, such that: 1) Lyapunov equation Has a unique positive definite matrix P; 2) For any condition that satisfies The control input is and satisfy It is a closed-loop system A positive invariant set of ; that is, if Then for any time t>t0, 3) For any condition that satisfies Lyapunov equation Established, of which S4.

4. Trigger condition design: Due to track irregularities The existence of the suspension system's true state trajectory x(t k ) and predicted state trajectory There are deviations between them; accordingly, the triggering conditions of the event-triggered robust model predictive control strategy are designed as follows: in is the trigger threshold, is the minimum trigger time interval that needs to be designed to meet Therefore, the next trigger moment is: in, 6. The event-triggered robust model predictive control method for a maglev train suspension system according to claim 5, characterized in that: Step S5 includes the following sub-steps: S5.

1. Recursive feasibility analysis: First, based on the trigger time t k The optimal control input sequence Construct t k+1 The feasible control input sequence at time Expressed as: With the above-mentioned feasible control input The corresponding feasible state sequence is expressed as in The equivalent linear model under the event-triggered robust model predictive control strategy is given an initial value x(0), and it is assumed that there is a solution at the initial moment; the control input In s∈[t k+1 ,t k +T], it is clear that the control input constraint is satisfied In s∈[t k +T,t k+1 +T], the control input is designed according to the terminal control rate The control input constraint is also satisfied, so the control input Satisfy constraints Right now Established; When s∈[t k +T,t k+1 +T], we get Right now Enter the terminal domain; define Taking the derivative we get: where λ min and λ max denote the minimum and maximum eigenvalues ​​of the matrix respectively; Using the comparison principle and the minimum trigger time interval, we can get: At this point, it has been verified A local feedback control law Enter the terminal domain Ω(ακ), where Due to the interference factor, for s∈[t k+1 ,t k+1 +T], Not necessarily equal to Set s=t k+1 +T substitution to obtain: According to the triangle inequality and The above formula is simplified to: in The design trigger threshold is: get illustrate Satisfy the terminal constraints, that is Since the optimization control problem is k It is feasible at the moment s∈[t k+1 ,t k +T] can be obtained calculate According to the event trigger threshold, triangle inequality and Gronwall-Bellman inequality, we can get: When s=t k +T time and get: Given and s∈[t k+1 :t k +T], if and If established, we can get therefore Established; when s∈[t k +T,t k+1 +T], design as well as get Where r>κ, so the state sequence Satisfy the state tightening constraint, that is, In summary, it can be verified that event-triggered robust model predictive control has recursive feasibility; S5.

2. Stability Theory Analysis: Under the conditions of the existence of terminal invariant set and terminal control rate, the event-triggered robust model predictive control strategy is recursively feasible; when When , the optimal cost function is selected as the Lyapunov function, that is, Can get: Where, ΔV η ,η=1,2,3; for ΔV1, we can get: For ΔV2 we get: Simplifying ΔV3 yields: Design the controller parameters to satisfy the following inequalities: in And concluded It is then concluded that the state converges to the terminal domain within a finite time; In the case of x∈Ω(κ), the control input is v(t)=Kx(t), and the choice As a Lyapunov function, the derivative can be obtained: When the perturbation upper bound satisfies the inequality hour: The above can verify that the equivalent linear model is stable, and further conclude that the mathematical model of the suspension system is stable.

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