Model predictive control method for bridge crane based on Koopman operator

Abstract

The present invention discloses a model predictive control method for a bridge crane based on the Koopman operator, belonging to the technical field of anti-sway control for bridge cranes. The method utilizes the Koopman operator's ability to identify nonlinear systems as high-dimensional linear models to obtain a high-dimensional linear representation of the bridge crane, a strongly nonlinear system. This method, combined with model predictive control technology, implements trajectory tracking control for the bridge crane and effectively suppresses cargo sway during transportation. This allows the bridge crane to achieve high positioning accuracy and anti-sway functionality without sacrificing transportation efficiency.

Description

Technical Field

[0001] The invention relates to a model predictive control method for a bridge crane based on a Koopman operator, and belongs to the technical field of anti-sway control of bridge cranes. Background Art

[0002] Bridge cranes are essential transportation tools in modern industrial production. They primarily consist of a bridge, a trolley, and flexible cables. An electric motor converts input voltage into force applied to the trolley, changing the trolley's position and thus moving the load to the target location. Due to their simple structure and high load capacity, bridge cranes are widely used in modern production, transportation, and construction processes.

[0003] With the continuous expansion of production and construction, the modern transportation industry is placing increasing demands on the efficiency and safety of bridge cranes. Considering that cargo is connected to the trolley via flexible steel ropes, any swaying of the cargo not only poses a significant safety hazard during transportation but also leads to reduced positioning accuracy and difficulty in securing the landing point during unloading. Therefore, designing a state feedback controller that minimizes swaying of the loaded cargo while ensuring fast and accurate transport, thereby ensuring safe transportation, is of great theoretical and practical significance.

[0004] Current control methods can be broadly divided into two categories. One involves controller design based on a known system model. This involves establishing a corresponding mathematical model based on the dynamic information in the actual application scenario to achieve control. However, accurate system models are often difficult to obtain in real applications, and the designed controllers are complex, computationally intensive, and difficult to determine parameters. Furthermore, the data generated by the system itself is not effectively utilized, and most current algorithms are unable to effectively handle state constraints and control input constraints, greatly limiting their application in real-world scenarios. The other category involves data-driven models, where output prediction models are established based on the system's input and output data to achieve control. This type of approach is suitable for applications where accurate physical models are unavailable, and has therefore gained widespread application.

[0005] However, control methods based on data-driven models require a large amount of input and output data to accurately reflect the actual system conditions when building output prediction models. However, in some practical applications, sufficient input and output data is not available. In some applications, the system dynamic characteristics are complex, and existing local linearization methods cannot effectively reflect these systems. However, the Koopman operator theory considered has the ability to globally linearize nonlinear systems and therefore has broad application prospects. In particular, "MO Williams, IG Kevrekidis, and C W Rowley, "A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition," Journal of Nonlinear Science, vol. 25, pp. 1307–1346, 2015," proposes an extended dynamic mode decomposition (EDMD) method based on Koopman theory. However, this method requires subjective experience to select certain basis functions when approximating the Koopman operator. Therefore, this method is significantly influenced by subjective experience. If the basis functions are not selected appropriately, the prediction accuracy of high-dimensional linear models will be affected.

[0006] In order to avoid the influence of the empirical selection of basis functions in the EDMD method on the prediction accuracy, and thus the inability to control the bridge crane well, that is, to ensure its transportation speed and accuracy while suppressing the swing of the goods as much as possible, ensuring its transportation efficiency and improving safety performance, this application proposes a new method to approximate the Koopman operator, and further uses the model predictive control method to realize the control of the bridge crane. Summary of the Invention

[0007] In order to ensure the transportation efficiency of the bridge crane while ensuring the accurate positioning of the cargo transportation destination and reducing the swing of the cargo during the lifting process, a model predictive control method for the bridge crane based on the Koopman operator is proposed. The method includes:

[0008] Step 1: Use the bridge crane system to collect two types of data. The first type of data includes the output data of the bridge crane system without control input, and the second type of data includes the output data of the system with control input. The control input of the bridge crane system refers to the input voltage. The system output data is defined as X = [x1, x2, x3, x4], where x1 represents the position of the trolley; x2 represents the running speed of the trolley; x3 represents the swing angle of the payload; and x4 represents the angular velocity of the swing angle. The system output data represents the system state.

[0009] Step 2 is based on the Koopman operator theory and the two types of data collected. The optimization method is used to solve the Koopman eigenvalue, boundary function and characteristic function, thereby constructing the Koopman high-dimensional linear model of the bridge crane.

[0010] In Step 3, given a reference trajectory, the Koopman high-dimensional linear model constructed in Step 2 is used to design a model predictive controller. The model predictive controller obtains the optimal control signal for each closed-loop control step through optimization, ensures that the optimal control signal obtained satisfies the control input constraints, and sends the control signal to the bridge crane system, thereby controlling the various states of the bridge crane to track the reference trajectory. The reference trajectory refers to the ideal trajectory of the cargo that meets the constraints during the operation of the bridge crane system.

[0011] Optionally, the two types of data collected in Step 1 are assumed to be M t There are different equally spaced sampling trajectories in the form of data, each trajectory corresponds to M s +1 sampling point data, sampling interval is T s ;

[0012] The first type of data is the system output data obtained without control input by changing the initial swing angle of the load end when no control input is included.

[0013] Optionally, Step 2 includes:

[0014] Step 2.1 Use the Koopman operator to construct a high-dimensional linear model of the unknown nonlinear system:

[0015]

[0016] z0=φ(x0)

[0017] y=Cz

[0018] Where z = φ(x) is the Koopman characteristic function, y is the output of the unknown nonlinear system, and A, B, and C are matrices to be determined.

[0019] Step 2.2 Determine matrices A, C and Koopman characteristic function φ based on the first type of data set collected in step 1, and determine matrix B based on the second type of data set collected in step 1, thereby obtaining a high-dimensional linear model of the bridge crane control system constructed using the Koopman operator.

[0020] Optionally, in Step 2.2, determining matrices A, C and Koopman characteristic function φ based on the first type of data set collected in Step 1 includes:

[0021] Determine the characteristic function of the bridge crane control system when there is no input voltage, that is, the Koopman characteristic function when u = 0:

[0022] Select the N eigenvalues ​​of the Koopman operator Λ=[λ1,λ2,…,λ N ] and N boundary functions [g1,g2,…,g N ], calculate the values ​​of each boundary function at all trajectory initial points in the first type of data set, and define:

[0023] matrix and i∈{1,…,N}; then, for k∈{0,…,M s} and j∈{1,…,M t}, calculate the Koopman characteristic function value Right now:

[0024]

[0025] in, Represents the sampling point data corresponding to the initial point of the j-th trajectory, Represents the sampling point data corresponding to the k-th sampling point of the j-th trajectory;

[0026] The calculated characteristic function value As labels, the first type of data collected is trained through a neural network to obtain a low-dimensional approximation of the Koopman operator and calculate the required initial Koopman eigenvalue Λ0; N characteristic functions correspond to N neural networks, each of which has a four-layer structure, with two hidden layers containing 80 neurons, and the input layer and output layer containing 4 neurons and 1 neuron respectively. The activation function selects the linear rectifier function;

[0027] definition in, Represents a set of natural numbers. After determining the parameter d, we can determine Further, we obtain a set lattice(Λ0) consisting of different linear combinations of initial eigenvalues;

[0028] Select Satisfaction d; according to The N initial eigenvalues ​​used to solve the eigenvalue optimization problem are obtained, and then the optimized eigenvalues ​​are obtained by minimizing the following objective function:

[0029]

[0030] Among them, h σ Represents sampling data of any dimension output by the system, that is, the length is M t (Ms +1), σ∈{1,…,p}, p=4, corresponding to the output dimension of the system; N is divided into p parts, that is, satisfy Introduction Definition Then i∈{1,…,N}={1,…,N1,N1+1,…,N2,…,N p-1 +1,…,N p};

[0031] For ease of expression, let N0 = 0; that is, when calculating the output of the σth dimension, i will get {N σ-1 +1,…,N σ}; Here, for all σ∈{1,…,p}, take That is, N = 40;

[0032]

[0033]

[0034] in The number is M t , at this time i takes {N σ-1 +1,…,N σ}, Indicated by Λ σ The dimensions of the construction are The column rank matrix is ​​full, so is Orthogonal projection operator on the column space of ;

[0035] According to the above definition, the row vectors of matrix G are given by Description; corresponding to any σ-dimensional system output, define the vector The boundary function is obtained by solving the following optimization problem:

[0036]

[0037] After the optimized eigenvalues ​​are determined, the value of the boundary function at the initial point is calculated directly according to the analytical solution of the above formula, that is:

[0038]

[0039] At this time, the matrix G is composed of describe;

[0040] Determination of the linear model coefficient matrix:

[0041] According to the construction process of the above characteristic function, the forms of matrices A and C are directly determined as follows:

[0042] A=diag(λ1,λ2,…,λ N ),

[0043] Optionally, determining the matrix B according to the second type of data set collected in step 1 in step 2.2 includes:

[0044] By minimizing the following multi-step forecast error:

[0045]

[0046] in,

[0047]

[0048] and are the coefficient matrices of the discretized linear system corresponding to A and B respectively; since A and C are known, they can be directly obtained according to the solution of the above problem:

[0049]

[0050] in, and

[0051]

[0052] in, stands for Kronecker product.

[0053] Optionally, Step 3 includes:

[0054] Determine the target position of the cargo during the operation of the bridge crane system and determine the reference point y r With z r , satisfying y k -y r =C(z k -z r );

[0055] The performance index function of model predictive control is defined as follows:

[0056]

[0057] Among them, N p is the number of prediction steps, Q, P, and R are the semi-weight matrices corresponding to the state, terminal state, and control input; the P matrix is ​​the solution of the following Lyapunov equation:

[0058]

[0059] in, K is such that F = A d+B d K is the Schur stable gain matrix;

[0060] The optimal control signal for each step of closed-loop control is obtained by solving the following quadratic programming problem:

[0061]

[0062] stz k+δ+1 =A d z k+δ +B d u k+δ ,δ=0,1,…,N k -1

[0063]

[0064] z k+δ ∈Z

[0065] u k+δ ∈U

[0066] Among them, Z and U represent the state constraint set and the control input constraint set respectively; for the bridge crane control system, the state constraint set Z refers to the state constraint set for the system The control input constraint set U represents the constraint conditions for the input voltage;

[0067] Solve to get the optimal control sequence u k+δ ,δ=0,1,…,N k -1, take u k+0 As the control quantity at time k, u k+δ ,δ=1,…,N k -1 will be discarded, u k+0 Input it into the crane system and get the state feedback at the next moment. After mapping it to the linear space using the Koopman characteristic function, the next step of control input solution calculation is performed. Thus, the entire closed-loop control is dynamically solved at each step to obtain the control signal of each step, and the control of each state of the bridge crane is realized according to the control signal of each step.

[0068] Optionally, the dynamic model of the bridge crane system is as follows:

[0069]

[0070]

[0071] Here, the parameter M represents the mass of the cart, m is the mass of the payload, x represents the position of the cart, l is the length of the cable, θ represents the swing angle of the payload, F represents the control force, and g is the acceleration due to gravity.

[0072] The beneficial effects of the present invention are:

[0073] The present application utilizes the characteristics of the Koopman operator to identify nonlinear systems as high-dimensional linear models, obtains a high-dimensional linear representation of the strong nonlinear system of the bridge crane, and combines it with the model predictive control technology to realize trajectory tracking control for the bridge crane and effectively suppress the swing of the goods during transportation, so that the bridge crane can obtain higher positioning accuracy and realize the anti-sway function without sacrificing its transportation efficiency; in approximating the Koopman operator, the present application is different from the existing EDMD method that requires the selection of basis functions based on the subjective experience of technicians. The present application first uses a neural network to calculate the initial Koopman eigenvalue, then obtains the optimized eigenvalue by solving the optimization problem, and further calculates the value of the boundary function at the initial point, thereby obtaining the Koopman characteristic function value according to the linear evolution characteristics of the Koopman operator. Then, an interpolation method or a neural network is used to complete supervised learning with the Koopman characteristic function value as a label to obtain an approximate representation of the Koopman characteristic function. Then, the input matrix of the linear predictor is calculated using data containing control input, so that a high-dimensional linear model corresponding to the crane model can be constructed, thereby achieving accurate control of the bridge crane. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0075] Figure 1 It is a schematic diagram of the bridge crane structure.

[0076] Figure 2 This is a closed-loop control block diagram for a bridge crane based on Koopman operator theory and model predictive control.

[0077] Figure 3 It is an overall flow chart of obtaining a high-dimensional controlled linear model of a bridge crane in a bridge crane model predictive control method based on the Koopman operator provided in one embodiment of the present invention.

[0078] Figure 4 It is a diagram showing the prediction effect of the Koopman linear model of the present invention on the actual model state without control input, including a comparison with the prediction of the EDMD method and the local linearization method.

[0079] Figure 5This is a simulation diagram of a square wave control input with a period of 0.3 seconds and an amplitude of 1.

[0080] Figure 6 yes Figure 5 The Koopman linear model of the present invention predicts the actual model state under the control input shown, including a comparison with the prediction of the EDMD method and the local linearization method.

[0081] Figure 7 It is a simulation diagram of the state change curve and control input change curve of the bridge crane trajectory tracking control experimental system. DETAILED DESCRIPTION

[0082] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0083] First, the theoretical basis of the Koopman operator involved in the technical solution of this application is introduced as follows:

[0084] For uncontrolled systems System flow S t (x) Satisfy:

[0085]

[0086] The so-called uncontrolled system refers to a system without control input. For example, a bridge crane is an uncontrolled system when the input voltage is 0.

[0087] According to the Koopman operator The definition of observation function

[0088]

[0089] Let λ represent the eigenvalue of the Koopman operator and φ be the corresponding characteristic function, then φ satisfies:

[0090]

[0091] Right now

[0092] φ(S t (x))=e λt φ(x)

[0093] Taking the derivative of both sides of the equation with respect to t, we can obtain the following linear model:

[0094]

[0095] Therefore, it is obvious to conclude that any characteristic function of the Koopman operator provides a coordinate along the linear evolution of the dynamical system, and the characteristic function of the Koopman operator can be obtained by solving the equation.

[0096] Furthermore, for the initial point x0 belonging to the acyclic set, define:

[0097] φ λ,g (S t (x0))=e λt g(x0)

[0098] Where g(x0) = φ λ,g (x0).

[0099] Therefore, according to the above formula, the characteristic function can be constructed forward in time using the initial points in the acyclic set and the boundary function values ​​thereon.

[0100] This application uses the Koopman operator theory to construct a Koopman high-dimensional linear model for a bridge crane.

[0101] Example 1:

[0102] This embodiment provides a model predictive control method for a bridge crane based on a Koopman operator, the method comprising:

[0103] Step 1: Use the bridge crane system to collect two types of data sets: one is a system state data set without control inputs, and the other is an input and output data set that uses control inputs that can stimulate the crane's dynamic characteristics. The two types of data sets can have different numbers and lengths of trajectories.

[0104] Assume that M t The available data is in the form of different equally spaced sampling trajectories, each with M s +1 sample, sampling interval is T s .

[0105] As we all know, the bridge crane converts the input voltage into a force applied to the trolley through the motor to change the trolley position, thereby driving the load to the target position. Therefore, for the bridge crane control system, its input data is the input voltage u, and the output data is defined as described above. Among them, x1, that is, x represents the position of the car; x2, that is, represents the running speed of the car; x3, also known as θ, represents the swing angle of the payload; and x4, also known as The angular velocity represents the swing angle. The output data is the system status data.

[0106] See also Figure 1 The schematic diagram of the bridge crane structure is shown in the figure. The two-dimensional bridge crane dynamic model is as follows:

[0107]

[0108]

[0109] Among them, the parameter M represents the mass of the car, m is the mass of the payload, x represents the position of the car, l represents the length of the flexible steel cable, F represents the control force on the car, and g is the acceleration due to gravity.

[0110] The parameter values ​​introduced in the subsequent simulation experiments are shown in Table 1.

[0111] Table 1. Dynamic parameters of the 2D bridge crane model.

[0112]

[0113] In the control system of the bridge crane, when no input voltage is added, the initial swing angle of the load end is changed to obtain a data set without control input, which is recorded as the first type of data set and subsequently used to calculate the optimized eigenvalues ​​and characteristic functions, thereby obtaining a high-dimensional linear model without control input; when the control input is added, the input voltage is given to the crane to obtain the trolley position speed data and load swing angle angular velocity data at the next moment, which is recorded as the second type of data set and subsequently used to solve the high-dimensional linear model input matrix, thereby obtaining the high-dimensional linear model corresponding to the original controlled nonlinear model.

[0114] Step 2: Based on the two data sets collected in step 1, and according to the Koopman operator theory, a Koopman high-dimensional linear model of the bridge crane is constructed:

[0115] First, the initial Koopman eigenvalue is calculated using a neural network (the neural network structure used here is consistent with the autoencoder proposed by B. Lusch, et al.) or the EDMD method;

[0116] Next, the optimized Koopman eigenvalue is obtained by solving the optimization problem;

[0117] The value of the boundary function at the initial point is further calculated, so that the value of the Koopman characteristic function can be obtained according to the linear evolution characteristics of the Koopman operator.

[0118] Then, an interpolation method, a regression problem method or a neural network is used to complete supervised learning with the Koopman characteristic function value as a label to obtain an approximate representation of the Koopman characteristic function;

[0119] Then, the input matrix of the linear predictor is calculated using the data containing the control input, so that the high-dimensional linear model corresponding to the crane model can be constructed.

[0120] Specifically, the high-dimensional linear model of the original unknown nonlinear system constructed using the Koopman operator is described by the following formula:

[0121]

[0122] z0=φ(x0)

[0123] y=Cz

[0124] Where z = φ(x) is the Koopman characteristic function, y is the system output, and A, B, and C are matrices to be determined. Subsequently, the matrices A, B, and C and the characteristic function φ are determined based on the two data sets collected in step 1, thereby obtaining a high-dimensional linear model of the bridge crane control system constructed using the Koopman operator, including:

[0125] Step 2.1, determine the characteristic function of the uncontrolled system (i.e., the bridge crane control system when there is no input voltage), that is, the characteristic function when u = 0. In the process of approximating the characteristic function, determine the forms of matrices A and C.

[0126] Step 2.2, calculate matrix B. Solve the optimization problem based on the second type of data set collected in step 1 to obtain matrix B.

[0127] For step 2.1, the Koopman characteristic function is constructed by solving the optimization problem;

[0128] Assume that N eigenvalues ​​Λ={λ1,λ2,…,λ N} and N boundary functions G={g1,g2,…,g N}, calculate the value of the boundary function at all initial points in the data set, and define and Then, for k∈{0,…,M s} and j∈{1,…,M t}, can be calculated Right now

[0129]

[0130] Use well-defined interpolation or approximation methods to learn the characteristic function.

[0131] Given a set of basis functions, By solving the following interpolation problem:

[0132]

[0133]

[0134] Or a regression problem:

[0135]

[0136] Among them, the regularization term δ1≥0,δ2≥0, ‖·‖1 represents the 1-norm, and ‖·‖2 represents the 2-norm, thus obtaining an approximate expression of the characteristic function. That is,

[0137] In addition, considering the fact that neural networks can accurately approximate any nonlinear function, when constructing the Koopman characteristic function, the characteristic function value that has been calculated can also be used. As labels, supervised learning is performed using a neural network. The trained network model can then be used as an approximation of the characteristic function. Here, N characteristic functions are approximated by N neural networks. Each network has four layers, including an input and output layer and two hidden layers. The number of neurons in each hidden layer is 80, the number of neurons in the input layer is 4, and the number of neurons in the output layer is 1. The activation function is a linear rectifier function.

[0138] The determination of eigenvalues ​​and boundary functions is accomplished with the help of neural network methods and optimization methods, specifically including:

[0139] First, the neural network is trained using the collected first-category data to obtain a low-dimensional approximation of the Koopman operator and calculate the required initial Koopman eigenvalue Λ0. The structure of the neural network here is consistent with the autoencoder proposed by B. Lusch et al., and can be referenced in "B. Lusch, J. N. Kutz, and S. L. Brunton, "Deep learning for universal linear embeddings of nonlinear dynamics," Nature Communications, vol. 9, no. 1, p. 4950, 2018." The input and output of the network are both system states. The settings of the hidden layer neurons and activation function can be adjusted based on the training results. Here, the encoding and decoding parts are set to two hidden layers with 64 neurons. The encoding and decoding part is connected by two linear layers with 4 neurons. The activation function is selected as the linear rectifier function.

[0140] Here, the dynamic mode decomposition method can also be used to directly calculate the initial eigenvalues.

[0141] Next, define in, Represents a set of natural numbers. After determining the parameter d, we can determine We can further obtain a set of lattice(Λ0) consisting of different linear combinations of initial eigenvalues, and select According to Get the N initial eigenvalues ​​used to solve the eigenvalue optimization problem, and then get the optimized eigenvalues ​​by minimizing the following optimization problem.

[0142]

[0143] Among them, h σ Represents sampling data of any dimension output by the system, that is, the length is M t (M s +1), σ∈{1,…,p}, where p=4, corresponding to the output dimension of the system. Divide N into p parts, that is, satisfy Introduction Definition Then i∈{1,…,N}={1,…,N1,N1+1,…,N2,…,N p-1 +1,…,N p For the sake of convenience, let N0 = 0. That is, when calculating the output of the σth dimension, i will get {N σ-1 +1,…,N σ Here for all σ∈{1,…,p}, we take That is, N=40. in The number is M t , at this time i takes {N σ-1 +1,…,N σ}, Indicated by Λ σ The dimensions of the construction are The column rank matrix is ​​full, so is The orthogonal projection operator on the column space of .

[0144] According to the above definition, the row vectors of matrix G are given by Description. For any σ-dimensional system output, define the vector The boundary function is obtained by solving the following optimization problem:

[0145]

[0146] After the optimized eigenvalues ​​are determined, the value of the boundary function at the initial point is calculated directly according to the analytical solution of the above formula, that is:

[0147]

[0148] At this time, the matrix G is composed of describe.

[0149] Determination of the linear model coefficient matrix:

[0150] According to the construction process of the above characteristic function, the forms of matrices A and C can be directly determined as follows:

[0151]

[0152] Next, we use the second dataset containing the control inputs to calculate the B matrix by minimizing the following multi-step forecast error:

[0153]

[0154] in,

[0155]

[0156] and are the coefficient matrices of the discretized linear system corresponding to A and B. Since A and C are known, the solution to the above problem can be directly obtained:

[0157]

[0158] in, and

[0159]

[0160] in, stands for Kronecker product.

[0161] Step 3: Given a reference trajectory, design a model predictive controller using the high-dimensional linear model obtained in step 2. First, given a reference point y r With z r , satisfying y k -y r =C(z k -z r ), the performance index function of model predictive control is defined as follows:

[0162]

[0163] Among them, N p is the number of prediction steps, Q, P, and R are the semi-weight matrices corresponding to the state, terminal state, and control input. In particular, the P matrix is ​​the solution of the following Lyapunov equation:

[0164]

[0165] in, K is such that F = A d +B d K is the Schur stable gain matrix.

[0166] Furthermore, the optimal control signal for each step of closed-loop control can be obtained by solving the following quadratic programming problem:

[0167]

[0168] stz k+δ+1 =A d z k+δ +B d u k+δ ,δ=0,1,…,N k -1

[0169]

[0170] z k+δ ∈Z

[0171] u k+δ ∈U, where Z and U represent the state constraint set and the control input constraint set respectively; for the bridge crane control system, the state constraint set Z refers to the system state The constraints include the trolley speed and acceleration, as well as the load swing angle and angular velocity restrictions during the operation of the bridge crane; and the control input constraint set U represents the constraints on the input voltage.

[0172] In this way, the solution of model predictive control is transformed into a linear quadratic programming problem, and the optimal control sequence u can be obtained by solving it. k+δ ,δ=0,1,…,N k -1 takes u k+0 As the control quantity at this time, u k+δ ,δ=1,…,N k -1 will be discarded, u k+0 Input it into the crane system and get the state feedback at the next moment. After mapping it to the linear space using the characteristic function, the next step of control input solution calculation is performed, so that the entire closed-loop control can perform dynamic solution at each step.

[0173] In summary, given a reference trajectory, a model predictive controller is designed with the help of a high-dimensional linear model. The optimal control signal for each step of closed-loop control is obtained through optimization. The solved control quantity is within the constraints of the actuator, and the control signal is sent to the crane system to control the mobile trolley to track the reference trajectory.

[0174] In order to verify the effectiveness of the control algorithm designed by the present invention, MATLAB and Arm Keil are used as experimental platforms, and a bridge crane, such as Figure 1 As shown, the system trajectory tracking control experiment is carried out to verify the control object. The following is a detailed description of the data-driven model predictive control method for bridge cranes based on Koopman operator proposed in the present invention, combined with the experimental results and the accompanying drawings. Figure 2 This is an illustration of the combination of Koopman operator and model predictive control. Figure 3 This is a schematic diagram of the steps for building a Koopman linear model.

[0175] After obtaining the Koopman high-dimensional linear model, we first need to test its degree of fit to the real model, such as Figure 4 As shown in , when there is no control input for both the Koopman high-dimensional linear model and the true model, the state output of the high-dimensional model is basically consistent with the state output of the true model, while the classic local linearization method and the EDMD method under the Koopman framework produce large prediction errors, which are far from the true state. Figure 5 The square wave voltage shown compares the predicted state with the actual state. Figure 6 It can be seen that under the same input, the state output of the high-dimensional model is well consistent with the state output of the real model. When the input is consistent, the prediction performance of the local linearization method and the EDMD method is still worse than the Koopman approximation method in this invention. Then, based on the Koopman linear model, a model predictive controller is designed to track the given trajectory. The control goal is to move the car from the initial position 0.1 to the target position 0.5, while ensuring that the swing of the load end is as small as possible, that is, the reference point is [0.5, 0, 0, 0]. T .

[0176] The parameters used in the model predictive controller based on the Koopman operator are as follows:

[0177] The number of characteristic functions N = 40, the prediction time domain length N p =150, optimize the target error gain The voltage optimization term gain R=0.0001, and the voltage output is constrained to be within the acceptable range of the actuator. In the present invention, the actuator output range is [-100, 100].

[0178] like Figure 7As shown in the figure, we achieved the control goal of stabilizing the trolley at the elevated target position by applying the feedback strategy generated by the model predictive control based on the Koopman operator to the two-dimensional bridge crane system. The trolley position measurement error is 0.003. Due to the existence of static friction, the control input eventually converges to a constant value. After the trolley reaches the target position, the trolley speed stabilizes to zero, the payload swings back and forth within a small angle range, and the angular velocity changes accordingly within a small value range. Therefore, the method proposed in this application can suppress the swing of goods as much as possible without relying on the experience of technical personnel, ensure its transportation efficiency, and improve safety performance.

[0179] Some steps in the embodiments of the present invention may be implemented using software, and the corresponding software program may be stored in a readable storage medium, such as a CD or a hard disk.

[0180] 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 in the scope of protection of the present invention.

Claims

1. A model predictive control method for a bridge crane based on Koopman operator, characterized in that: The method comprises: Step 1: Use the bridge crane system to collect two types of data. The first type of data is the output data of the bridge crane system without control input, and the second type of data is the system output data with control input. The control input of the bridge crane system refers to the input voltage. The system output data is defined as X = [x1, x2, x3, x4], where x1 represents the position of the trolley; x2 represents the running speed of the trolley; x3 represents the swing angle of the payload; and x4 represents the angular velocity of the swing angle. The system output data represents the system state. Step 2 is based on the Koopman operator theory and the two types of data collected. The optimization method is used to solve the Koopman eigenvalue, boundary function and characteristic function, thereby constructing the Koopman high-dimensional linear model of the bridge crane. In Step 3, given a reference trajectory, a model predictive controller is designed using the Koopman high-dimensional linear model constructed in Step 2. The model predictive controller obtains the optimal control signal for each closed-loop control step through optimization, ensures that the optimal control signal obtained satisfies the control input constraints, and sends the control signal to the bridge crane system, thereby controlling the various states of the bridge crane to track the reference trajectory. The reference trajectory refers to the ideal trajectory of the cargo that meets the constraints during the operation of the bridge crane system. Step 2 includes: Step 2.1 Use the Koopman operator to construct a high-dimensional linear model of the unknown nonlinear system: z0=φ(x0) y=Cz Where z = φ(x) is the Koopman characteristic function, y is the output of the unknown nonlinear system, and A, B, and C are matrices to be determined. Step 2.2: Determine the matrices A and C and the Koopman characteristic function φ based on the first type of data set collected in Step 1, and determine the matrix B based on the second type of data set collected in Step 1, thereby obtaining a high-dimensional linear model of the bridge crane control system constructed using the Koopman operator; In Step 2.2, the matrices A and C and the Koopman characteristic function φ are determined based on the first type of data set collected in Step 1, including: Determine the characteristic function of the bridge crane control system when there is no input voltage, that is, the Koopman characteristic function when u = 0: Select the N eigenvalues ​​of the Koopman operator Λ=[λ1,λ2,…,λ N ] and N boundary functions [g1,g2,…,g N ], calculate the values ​​of each boundary function at all trajectory initial points in the first type of data set, and define: matrix and i∈{1,…,N}; then, for k∈{0,…,M s } and j∈{1,…,M t }, calculate the Koopman characteristic function value Right now: in, Represents the sampling point data corresponding to the initial point of the j-th trajectory, Represents the sampling point data corresponding to the k-th sampling point of the j-th trajectory; The calculated characteristic function value As labels, the first type of data collected is trained through a neural network to obtain a low-dimensional approximation of the Koopman operator and calculate the required initial Koopman eigenvalue Λ0; N characteristic functions correspond to N neural networks, each of which has a four-layer structure, with two hidden layers containing 80 neurons, and the input layer and output layer containing 4 neurons and 1 neuron respectively. The activation function selects the linear rectifier function; definition in, Represents a set of natural numbers. After determining the parameter d, we can determine Further, we obtain a set lattice(Λ0) consisting of different linear combinations of initial eigenvalues; Select Satisfaction d; according to The N initial eigenvalues ​​used to solve the eigenvalue optimization problem are obtained, and then the optimized eigenvalues ​​are obtained by minimizing the following objective function: Among them, h σ Represents sampling data of any dimension output by the system, that is, the length is M t (M s +1), σ∈{1,…,p}, p=4, corresponding to the output dimension of the system; N is divided into p parts, that is, satisfy Introduction Definition Then i∈{1,…,N}={1,…,N1,N1+1,…,N2,…,N p-1 +1,…,N p }; For ease of expression, let N0 = 0; that is, when calculating the output of the σth dimension, i will get {N σ-1 +1,…,N σ }; Here, for all σ∈{1,…,p}, take That is, N = 40; in The number is M t , at this time i takes {N σ-1 +1,…,N σ }, Indicated by Λ σ The dimensions of the construction are The column rank matrix is ​​full, so is Orthogonal projection operator on the column space of ; According to the above definition, the row vectors of matrix G are given by Description; corresponding to any σ-dimensional system output, define the vector The boundary function is obtained by solving the following optimization problem: After the optimized eigenvalues ​​are determined, the value of the boundary function at the initial point is calculated directly according to the analytical solution of the above formula, that is: At this time, the matrix G is composed of describe; Determination of the linear model coefficient matrix: According to the construction process of the above characteristic function, the forms of matrices A and C are directly determined as follows: In Step 2.2, the matrix B is determined based on the second type of data set collected in Step 1, including: By minimizing the following multi-step forecast error: in, and are the coefficient matrices of the discretized linear system corresponding to A and B respectively; since A and C are known, they can be directly obtained according to the solution of the above problem: in, and in, stands for Kronecker product.

2. The method according to claim 1, characterized in that The two types of data collected in Step 1 are assumed to be M t There are different equally spaced sampling trajectories in the form of data, each trajectory corresponds to M s +1 sampling point data, sampling interval is T s ; The first type of data is the system output data obtained without control input by changing the initial swing angle of the load end when no control input is included.

3. The method according to claim 2, characterized in that Step 3 includes: Determine the target position of the cargo during the operation of the bridge crane system and determine the reference point y r With z r , satisfying y k -y r =C(z k -z r ); The performance index function of model predictive control is defined as follows: Among them, N k is the number of prediction steps, Q, P, and R are the semi-weight matrices corresponding to the state, terminal state, and control input; the P matrix is ​​the solution of the following Lyapunov equation: in, K is such that F = A d +B d K is the Schur stable gain matrix; The optimal control signal for each step of closed-loop control is obtained by solving the following quadratic programming problem: s.t.z k+δ+1 =A d z k+δ +B d u k+δ ,δ=0,1,…,N k -1 With k+δ ∈Z in k+δ ∈U Among them, Z and U represent the state constraint set and the control input constraint set respectively; for the bridge crane control system, the state constraint set Z refers to the state constraint set for the system The control input constraint set U represents the constraint conditions for the input voltage; Solve to get the optimal control sequence u k+δ ,δ=0,1,…,N k -1, take u k+0 As the control quantity at time k, u k+δ ,δ=1,…,N k -1 will be discarded, u k+0 Input it into the crane system and get the state feedback at the next moment. After mapping it to the linear space using the Koopman characteristic function, the next step of control input solution calculation is performed. Thus, the entire closed-loop control is dynamically solved at each step to obtain the control signal of each step, and the control of each state of the bridge crane is realized according to the control signal of each step.

4. The method according to claim 3, characterized in that The dynamic model of the bridge crane system is as follows: Here, the parameter M represents the mass of the cart, m is the mass of the payload, x represents the position of the cart, l is the length of the cable, θ represents the swing angle of the payload, F represents the control force, and g is the acceleration due to gravity.

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