CN107902555A - The shore container crane optimal control system that a kind of grid becomes more meticulous - Google Patents

Abstract

The invention discloses the shore container crane optimal control system that a kind of grid becomes more meticulous, by actuating motor, position sensor, fieldbus networks, DCS, control room is shown, electric machine controller is formed.Control room engineer specifies the start-stop position of container, DCS must be sent as an envoy to the optimal strategy of speed control of cargo handling process performance indicator by the grid method of becoming more meticulous, and be converted to the control instruction of motor, electric machine controller is sent to by fieldbus networks, actuating motor is set to perform corresponding actions, position sensor gathers the positional information of container and is passed back to DCS in real time, control room engineer is grasped cargo handling process at any time.The present invention can optimize cargo handling process performance indicator, improve the efficiency of loading and unloading of container.

Description

Bank container crane optimal control system with refined grids

Technical Field

The invention relates to the field of crane control, in particular to an optimal control system of a shore container crane with refined grids. The moving speed of the container can be automatically controlled to optimize the performance index of the loading and unloading process, thereby improving the throughput capacity of the port.

Background

A shore container crane (shore bridge for short) is a main device for loading and unloading containers between a container ship and the front edge of a wharf. The loading and unloading capacity and speed of the shore bridge directly determine the productivity of the quay operations. With the development of large-scale container transport vessels, particularly ultra-panama vessels, the demand for the production rate of shore bridges is increasing.

For a shore bridge with defined technical parameters, a key factor affecting its productivity is the control strategy of the container loading and unloading speed. Because the technical parameters and the operation requirements of different shore bridges of different ports are different, the automatic speed control of the shore container crane according to the specific parameters and the operation requirements has important significance.

At present, the optimal control theory and the corresponding method are rarely adopted in the domestic shore bridge control method, the parameters in the controller are often set according to the existing experience, and the productivity and the safety are required to be further improved. The safety of the shore bridge adopting the optimal control method can be guaranteed, and the production efficiency can be further improved.

Disclosure of Invention

In order to improve the productivity of the shore bridge, the invention provides an optimal control system of a shore container crane with a fine grid.

The purpose of the invention is realized by the following technical scheme: a grid refined optimal control system for a shore container crane can automatically control the moving speed of a container so as to optimize the performance index of the loading and unloading process. The system consists of an execution motor, a position sensor, a field bus network, a DCS, a control room display and a motor controller. The operation process of the system comprises the following steps:

step A1: a control room engineer appointing a starting and stopping position of the container, performance indexes of a loading and unloading process and speed control constraints;

step A2: the DCS executes an internal grid refinement method to obtain a speed control strategy for optimizing the performance index in the loading and unloading process;

step A3: the DCS converts the speed control strategy obtained by calculation into a control command of the motor, and sends the control command to the motor controller through the field bus network, so that the execution motor executes corresponding action according to the received control command;

step A4: the position sensor collects the position information of the container in real time, the position information is sent back to the DCS through the field bus network and displayed in the main control room, and an engineer in the control room can master the loading and unloading process at any time.

The DCS comprises an information acquisition module, an initialization module, a grid refinement module, an ODE solving module, a gradient calculation module, a Non-linear Programming (NLP) problem solving module, a refinement convergence judgment module and a control instruction output module. The information acquisition module comprises three submodules of container start-stop position acquisition, performance index acquisition and speed control constraint acquisition, and the NLP problem solving module comprises three submodules of optimizing direction calculation, optimizing step length calculation and NLP convergence judgment.

The working process of the shore bridge can be described as follows:

wherein t represents time, and u (t) represents a velocity vector consisting of velocity components in each direction; x (t) status information indicating handling procedures; f (-) is a system of differential equations that is built based on the principles of crane physics. From this description, it can be seen that the handling of a container can be represented mathematically by a set of differential equations.

To optimize the performance index of the container handling process, the final expression of the problem is:

wherein, t₀Indicating the start time, t, of the loading and unloading process_fRepresents the end time of the handling process and J represents the objective function to be minimized. This problem is essentially an optimal control problem. However, the conventional method has the defects of low efficiency and poor precision in solving the problems, and is difficult to meet the requirement of high efficiency in actual operation.

The technical scheme adopted by the invention for solving the technical problems is as follows: a grid refinement method is integrated in the DCS, and a set of control system is constructed on the basis of the grid refinement method. The complete structure of the system comprises an execution motor 11, a position sensor 12, a field bus network 13, a DCS14, a control room display 15 and a motor controller 16.

The operation process of the system comprises the following steps:

step A1: a control room engineer appointing a starting and stopping position of the container, performance indexes of a loading and unloading process and speed control constraints;

step A2: the DCS executes an internal grid refinement method to obtain a speed control strategy for optimizing the performance index in the loading and unloading process;

step A3: the DCS converts the speed control strategy obtained by calculation into a control command of the motor, and sends the control command to the motor controller through the field bus network, so that the execution motor executes corresponding action according to the received control command;

step A4: the position sensor collects the position information of the container in real time, the position information is sent back to the DCS through the field bus network and displayed in the main control room, and an engineer in the control room can master the loading and unloading process at any time.

The DCS comprises an information acquisition module, an initialization module, a grid refinement module, an ODE solving module, a gradient calculation module, a Non-linear Programming (NLP) problem solving module, a refinement convergence judgment module and a control instruction output module. The information acquisition module comprises three submodules of container start-stop position acquisition, performance index acquisition and speed control constraint acquisition, and the NLP problem solving module comprises three submodules of optimizing direction calculation, optimizing step length calculation and NLP convergence judgment.

In order to obtain a speed control strategy for optimizing the performance index of the container loading and unloading process, the grid refinement method executed by the DCS comprises the following operation steps:

step B1: the information acquisition module 21 acquires the starting and stopping positions of the container, the performance indexes of the loading and unloading process and the speed control constraint specified by an engineer;

step B2:the initialization module 22 starts to operate, parameterizes by using a piecewise constant, and sets the number of segments N of the loading and unloading process and the corresponding control grid toInitial guess value of parameterized vector of speed control strategySetting the computational accuracy tol of NLP problem₁Convergence accuracy tol of sum-adaptive approximation₂The number of iterations k₁And approximation degree k₂Setting zero;

step B3: when k is₂When 0, execute step B4; otherwise, the control mesh is refined by the mesh refinement module 23Refining to obtain new control gridAnd corresponding parameterized vector

Step B4: obtaining the state information of the iteration through the ODE solving module 24And an objective function value

Step B5: obtaining the gradient information of the iteration through the gradient calculation module 25When k is₁When 0, directly executing the step B7 by skipping the step B6;

step B6: the NLP problem solving module 26 operatesPerforming convergence judgment through the NLP convergence judgment module, ifObjective function value from last iterationThe absolute value of the difference is less than the accuracy tol₁If yes, determining that the convergence is satisfied, and executing step B9; if convergence is not satisfied, continuing to step B7;

step B7: by usingValue of (1) is coveredAnd will iterate the number of times k₁Increasing by 1;

step B8: the NLP problem solving module 26 obtains the ratio by calculating the optimizing direction and the optimizing step size using the objective function values and the gradient information obtained in steps B4 and B5More optimal new speed control strategyAfter the step is executed, jumping to the step B4 again;

step B9: the fine convergence determination module 27 operates to recordWhen k is₂When the value is 0, the step B10 is executed, otherwise, the judgment is madeAnd the last refined objective function valueWhether the absolute value of the difference is less than the accuracy tol₂If yes, the convergence is judged to be satisfied, the speed control strategy of the iteration is converted into a motor control command to be output, otherwise, the convergence is not satisfied, and the number of refinement times k is set₂:＝k₂+1, continue to execute step B3 until the refinement convergence determination module is satisfied.

The grid refinement module is realized by adopting the following steps:

step C1: calculating the left slope at a grid node by the following equationAnd right slope(k＝1,…,N-1)：

Wherein u is_kRepresenting the parametric representation of the velocity control strategy on the k-th parametric segment, t_kRepresents u_kAnd u_k+1A mesh node in between.

Step C2: if the mesh node t_kIf the left and right slopes meet the following requirements, the node is removed from the grid:

wherein epsilon_eIs a small positive real number. Grid node t_kAfter removal of u_kAnd u_k+1The corresponding grids are merged into a new grid, whichUpdate the parameter of (c) to (u)_k+u_k+1)/2。

Step C3: if the mesh node t_kThe left slope at (b) satisfies:

wherein epsilon_iIs one greater than epsilon_ePositive real number of (1), then at [ t_k-1,t_k]Inserting grid nodes upwards; if the mesh node t_kThe right slope at (b) satisfies:

then at [ t_k-1,t_k]And inserting grid nodes. In practical application, the number of the added nodes can be freely set according to the absolute value of the left slope and the right slope.

Step C4: and generating a new control grid and a corresponding parameterization vector according to the nodes removed and inserted in the steps C2 and C3.

The ODE solving module adopts a four-step Runge-Kutta method, and the calculation formula is as follows:

wherein t represents time, t_iDenotes the integration time, t, selected by the Runge-Kutta method_i+1Indicating that it is at time t_iThe integration step h is the difference between any two adjacent integration moments, x (t)_i) Indicating that the container is at t_iThe state information at the time, F () is a function describing a state differential equation, and K1, K2, K3, and K4 respectively represent function values of 4 nodes in the integration process of the Runge-Kutta method.

The gradient calculation module adopts an accompanying method:

step D1: let λ (t) be the co-modal vector, whose value is determined by the adjoint equation:

wherein, t_fDenotes the end time of the handling process, H denotes the Hamiltonian, and H ═ L + λ (t)^TF, L is the integral term of the objective function, Φ x (t)_f)]Is the steady state term of the objective function.

Step D2: for the adjoint equation, a four-step Runge-Kutta method is adopted to obtain the value of the co-modal vector lambda (t) at each integration moment, and the calculation formula is as follows:

wherein t represents time, t_iSolving the selected integration time, t, in the module for ODE_i+1Indicating that it is at time t_iThe latter integration time, and t_i+1＝t_i+ h, h is the integration step, and Q1, Q2, Q3 and Q4 respectively represent the function values of 4 nodes in the integration process of the Runge-Kutta method.

Step D3: based on the obtained value of the co-modal vector λ (t), gradient information is obtained by the following formula

Wherein,andto representThe first and second components, and so on.

The NLP problem solving module is realized by adopting the following steps:

step E1: if it is notObjective function value from last iterationIs less than the accuracy tol₁If yes, judging that the convergence is satisfied, and converting the speed control strategy of the iteration into a control instruction of the motor for output; if convergence is not satisfied, continuing to step E2;

step E2: by usingValue of (1) is coveredAnd will iterate the number of times k₁Increasing by 1;

step E3: velocity control strategyAs a point in vector space, denoted as P₁，P₁The corresponding objective function value is

Step E4: from point P₁Starting from the selected NLP algorithm and point P₁Information of the gradient ofConstructing a direction of optimization in vector spaceAnd step size

Step E5, by formulaConstructing correspondences in vector spaceAnother point P of₂So that P is₂Corresponding objective function valueRatio ofPreferably, wherein I isA vector of the same dimension.

The invention has the following beneficial effects: the optimal control system of the quayside container crane refined based on the self-adaptive grid can calculate the optimal control strategy of the quayside container crane, can adapt to the optimal control curve of the problem, particularly find the discontinuous point of the problem and can obtain higher precision; after the self-adaptive strategy is adopted, the initial estimation value of the next optimal control curve is the optimal curve of the current iteration, so that the higher convergence speed can be obtained, and the calculation time of the optimal strategy of the shore container crane is reduced. The invention can optimize the performance index of the loading and unloading process and improve the safety and efficiency of the loading and unloading of the container.

Drawings

FIG. 1 is a schematic structural view of the present invention;

FIG. 2 is a view showing the structure of the internal modules of the DCS according to the present invention.

Detailed Description

The working process of the shore bridge can be described as follows:

wherein t represents time, and u (t) represents a velocity vector consisting of velocity components in each direction; x (t) status information indicating handling procedures; f (-) is a system of differential equations that is built based on the principles of crane physics. From this description, it can be seen that the handling of a container can be represented mathematically by a set of differential equations.

To optimize the performance index of the container handling process, the final expression of the problem is:

wherein, t₀Indicating the start time, t, of the loading and unloading process_fRepresents the end time of the handling process and J represents the objective function to be minimized. This problem is essentially an optimal control problem. However, the conventional method has the defects of low efficiency and poor precision in solving the problems, and is difficult to meet the requirement of high efficiency in actual operation.

The technical scheme adopted by the invention for solving the technical problems is as follows: a grid refinement method is integrated in the DCS, and a set of control system is constructed on the basis of the grid refinement method. The complete structure of the system is shown in fig. 1, and comprises an execution motor 11, a position sensor 12, a field bus network 13, a DCS14, a control room display 15 and a motor controller 16.

The operation process of the system comprises the following steps:

step A1: a control room engineer appointing a starting and stopping position of the container, performance indexes of a loading and unloading process and speed control constraints;

step A2: the DCS executes an internal grid refinement method to obtain a speed control strategy for optimizing the performance index in the loading and unloading process;

step A3: the DCS converts the speed control strategy obtained by calculation into a control command of the motor, and sends the control command to the motor controller through the field bus network, so that the execution motor executes corresponding action according to the received control command;

step A4: the position sensor collects the position information of the container in real time, the position information is sent back to the DCS through the field bus network and displayed in the main control room, and an engineer in the control room can master the loading and unloading process at any time.

The DCS comprises an information acquisition module, an initialization module, a grid refinement module, an ODE solving module, a gradient calculation module, a Non-linear Programming (NLP) problem solving module, a refinement convergence judgment module and a control instruction output module. The information acquisition module comprises three submodules of container start-stop position acquisition, performance index acquisition and speed control constraint acquisition, and the NLP problem solving module comprises three submodules of optimizing direction calculation, optimizing step length calculation and NLP convergence judgment.

In order to obtain a speed control strategy for optimizing the performance index of the container loading and unloading process, the grid refinement method executed by the DCS comprises the following operation steps:

step B1: the information acquisition module 21 acquires the starting and stopping positions of the container, the performance indexes of the loading and unloading process and the speed control constraint specified by an engineer;

step B2: the initialization module 22 starts to operate, parameterizes by adopting a segment constant, and sets the number N of segments of the loading and unloading process and the corresponding numberThe control grid isInitial guess value of parameterized vector of speed control strategySetting the computational accuracy tol of NLP problem₁Convergence accuracy tol of sum-adaptive approximation₂The number of iterations k₁And approximation degree k₂Setting zero;

step B3: when k is₂When 0, execute step B4; otherwise, the control mesh is refined by the mesh refinement module 23Refining to obtain new control gridAnd corresponding parameterized vector

Step B4: obtaining the state information of the iteration through the ODE solving module 24And an objective function value

Step B5: obtaining the gradient information of the iteration through the gradient calculation module 25When k is₁When 0, directly executing the step B7 by skipping the step B6;

step B6: the NLP problem solving module 26 is operated, and the convergence judgment is carried out by the NLP convergence judgment module, if the convergence judgment module is operated, the NLP convergence judgment module carries out the convergence judgmentObjective function value from last iterationThe absolute value of the difference is less than the accuracy tol₁If yes, determining that the convergence is satisfied, and executing step B9; if convergence is not satisfied, continuing to step B7;

step B7: by usingValue of (1) is coveredAnd will iterate the number of times k₁Increasing by 1;

step B8: the NLP problem solving module 26 obtains the ratio by calculating the optimizing direction and the optimizing step size using the objective function values and the gradient information obtained in steps B4 and B5More optimal new speed control strategyAfter the step is executed, jumping to the step B4 again;

step B9: the fine convergence determination module 27 operates to recordWhen k is₂When the value is 0, the step B10 is executed, otherwise, the judgment is madeAnd the last refined objective function valueWhether the absolute value of the difference is less than the accuracy tol₂If yes, the convergence is judged to be fullIf not, the convergence is not satisfied, and the refined times k are set₂:＝k₂+1, continue to execute step B3 until the refinement convergence determination module is satisfied.

The grid refinement module is realized by adopting the following steps:

step C1: calculating the left slope at a grid node by the following equationAnd right slope(k＝1,…,N-1)：

Wherein u is_kRepresenting the parametric representation of the velocity control strategy on the k-th parametric segment, t_kRepresents u_kAnd u_k+1A mesh node in between.

Step C2: if the mesh node t_kIf the left and right slopes meet the following requirements, the node is removed from the grid:

wherein epsilon_eIs a small positive real number. Grid node t_kAfter removal of u_kAnd u_k+1The corresponding grids are merged into a new grid, and the parameters on the new grid are updated to (u)_k+u_k+1)/2。

Step C3: if the mesh node t_kThe left slope at (b) satisfies:

wherein epsilon_iIs one greater than epsilon_ePositive real number of (1), then at [ t_k-1,t_k]Inserting grid nodes upwards; if the mesh node t_kThe right slope at (b) satisfies:

then at [ t_k-1,t_k]And inserting grid nodes. In practical application, the number of the added nodes can be freely set according to the absolute value of the left slope and the right slope.

Step C4: and generating a new control grid and a corresponding parameterization vector according to the nodes removed and inserted in the steps C2 and C3.

The ODE solving module adopts a four-step Runge-Kutta method, and the calculation formula is as follows:

wherein t represents time, t_iDenotes the integration time, t, selected by the Runge-Kutta method_i+1Indicating that it is at time t_iThe integration step h is the difference between any two adjacent integration moments, x (t)_i) Indicating that the container is at t_iThe state information of the time, F (-) is a function describing a state differential equation, and K1, K2, K3 and K4 respectively represent function values of 4 nodes in the Runge-Kutta method integration process.

The gradient calculation module adopts an accompanying method:

step D1: let λ (t) be the co-modal vector, whose value is determined by the adjoint equation:

wherein, t_fDenotes the end time of the handling process, H denotes the Hamiltonian, and H ═ L + λ (t)^TF, L is the integral term of the objective function, Φ x (t)_f)]Is the steady state term of the objective function.

Step D2: for the adjoint equation, a four-step Runge-Kutta method is adopted to obtain the value of the co-modal vector lambda (t) at each integration moment, and the calculation formula is as follows:

wherein t represents time, t_iSolving the selected integration time, t, in the module for ODE_i+1Indicating that it is at time t_iThe latter integration time, and t_i+1＝t_i+ h, h is the integration step, and Q1, Q2, Q3 and Q4 respectively represent the function values of 4 nodes in the integration process of the Runge-Kutta method.

Step D3: based on the obtained value of the co-modal vector λ (t), gradient information is obtained by the following formula

Wherein,andto representThe first and second components, and so on.

The NLP problem solving module is realized by adopting the following steps:

step E1: if it is notObjective function value from last iterationIs less than the accuracy tol₁If yes, judging that the convergence is satisfied, and converting the speed control strategy of the iteration into a control instruction of the motor for output; if convergence is not satisfied, continuing to step E2;

step E2: by usingValue of (1) is coveredAnd will iterate the number of times k₁Increasing by 1;

step E3: velocity control strategyAs a point in vector space, denoted as P₁，P₁The corresponding objective function value is

Step E4: from point P₁Starting from the selected NLP algorithm and point P₁Information of the gradient ofConstructing a direction of optimization in vector spaceAnd step size

Step E5, by formulaConstructing correspondences in vector spaceAnother point P of₂So that P is₂Corresponding objective function valueRatio ofPreferably, wherein I isA vector of the same dimension.

Example 1

The crane is used to transfer the container from the vessel to the truck and how to operate optimizes the performance criteria. The mathematical model of the problem is:

where J denotes the objective function to be minimized. In order to obtain a speed control strategy for minimizing an objective function, the DCS runs a grid refinement method, the running process of which is shown in fig. 2, and the implementation steps are as follows:

step F1: the information acquisition module 21 obtains the starting and ending position x (0) of the container designated by the engineer as [0,22,0,0, -1, 0-]^TAnd x (9) ═ 10,14,0,2.5,0]^TPerformance index of loading and unloading processAnd velocity control constraint | u₁U is less than or equal to 2.83374 and-0.80865 is less than or equal to u₂≤0.71265；

Step F2: the initialization module 22 starts to operate, parameterizes by adopting a piecewise constant, sets the number of segments of the loading and unloading process to be 8, and corresponds to the control gridInitial guess value of parameterized vector of uniform division and speed control strategyTo 0.5, set the computation accuracy tol of the NLP problem₁Convergence accuracy tol of sum-adaptive approximation₂Are respectively 10^-6And 10^-4The number of iterations k₁And approximation degree k₂Setting zero;

step F3: when k is₂When 0, step F4 is executed; otherwise, the control mesh is refined by the mesh refinement module 23Refining to obtain new control gridAnd corresponding parameterized vector

Step F4: obtaining the state information of the iteration through the ODE solving module 24And an objective function value

Step F5: obtaining the gradient information of the iteration through the gradient calculation module 25When k is₁Step F7 is directly performed skipping step F6 when 0;

step F6: the NLP problem solving module 26 is operated, and the convergence judgment is carried out by the NLP convergence judgment module, if the convergence judgment module is operated, the NLP convergence judgment module carries out the convergence judgmentObjective function value from last iterationIs less than the accuracy tol₁If yes, determining that the convergence is satisfied, and executing step F9; if the convergence is not satisfied, continuing to perform step F7;

step F7: by usingValue of (1) is coveredAnd will iterate the number of times k₁Increasing by 1;

step F8: the NLP problem solving module 26 obtains the ratio by calculating the optimizing direction and the optimizing step size using the objective function values and the gradient information obtained in steps F4 and F5More optimal new speed control strategyAfter the step is completed, the step goes to step F4 again;

step F9: the fine convergence determination module 27 operates to recordWhen k is₂When 0, the process proceeds to step F10, otherwise, it is judgedAnd the last refined objective function valueWhether the absolute value of the difference is less than the accuracy tol₂If yes, the convergence is judged to be satisfied, the speed control strategy of the iteration is converted into a motor control command to be output, otherwise, the convergence is not satisfied, and the approximation times k are set₂:＝k₂+1, continue to execute step F3 until the refinement convergence determination module is satisfied.

And finally, the DCS converts the speed control strategy obtained by the grid refinement method into a control command of the motor, the control command is sent to the motor controller through the field bus network, the motor is executed to perform corresponding action, and meanwhile, the position sensor is used for collecting the position information of the container in real time and sending the position information back to the DCS, so that a control room engineer can master the loading and unloading process at any time.

The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and is not intended to limit the practice of the invention to these embodiments. For those skilled in the art to which the invention pertains, several simple deductions or substitutions may be made without departing from the inventive concept, which should be construed as falling within the scope of the present invention.

Claims (1)

1. A grid refined optimal control system for a shore container crane can automatically control the moving speed of a container so as to optimize the performance index of the loading and unloading process. The method is characterized in that: the system consists of an execution motor, a position sensor, a field bus network, a DCS, a control room display and a motor controller. The operation process of the system comprises the following steps:

step A1: a control room engineer appointing a starting and stopping position of the container, performance indexes of a loading and unloading process and speed control constraints;

step A2: the DCS executes an internal grid refinement method to obtain a speed control strategy for optimizing the performance index in the loading and unloading process;

step A3: the DCS converts the speed control strategy obtained by calculation into a control command of the motor, and sends the control command to the motor controller through the field bus network, so that the execution motor executes corresponding action according to the received control command;

step A4: the position sensor collects the position information of the container in real time, the position information is sent back to the DCS through the field bus network and displayed in the main control room, and an engineer in the control room can master the loading and unloading process at any time.

The DCS comprises an information acquisition module, an initialization module, a grid refinement module, an ODE solving module, a gradient calculation module, a Non-linear Programming (NLP) problem solving module, a refinement convergence judgment module and a control instruction output module. The information acquisition module comprises three submodules of container start-stop position acquisition, performance index acquisition and speed control constraint acquisition, and the NLP problem solving module comprises three submodules of optimizing direction calculation, optimizing step length calculation and NLP convergence judgment.

In order to obtain a speed control strategy for optimizing the performance index of the container loading and unloading process, the grid refinement method executed by the DCS comprises the following operation steps:

step B1: the information acquisition module 21 acquires the starting and stopping positions of the container, the performance indexes of the loading and unloading process and the speed control constraint specified by an engineer;

step B2: the initialization module 22 starts to operate, parameterizes by using a piecewise constant, and sets the number of segments N of the loading and unloading process and the corresponding control grid toInitial guess value of parameterized vector of speed control strategySetting the computational accuracy tol of NLP problem₁Convergence accuracy tol of sum-adaptive approximation₂The number of iterations k₁And approximation degree k₂Setting zero;

step B3: when k is₂When 0, execute step B4; otherwise, the control mesh is refined by the mesh refinement module 23Refining to obtain new control gridAnd corresponding parameterized vector

Step B4: obtaining the state information of the iteration through the ODE solving module 24And an objective function value

Step B5: obtaining the gradient information of the iteration through the gradient calculation module 25When k is₁When 0, directly executing the step B7 by skipping the step B6;

step B6: the NLP problem solving module 26 is operated, and the convergence judgment is carried out by the NLP convergence judgment module, if the convergence judgment module is operated, the NLP convergence judgment module carries out the convergence judgmentObjective function value from last iterationThe absolute value of the difference is less than the accuracy tol₁If yes, determining that the convergence is satisfied, and executing step B9; if convergence is not satisfied, continuing to step B7;

step B7: by usingValue of (1) is coveredAnd will iterate the number of times k₁Increasing by 1;

step B8: the NLP problem solving module 26 obtains the ratio by calculating the optimizing direction and the optimizing step size using the objective function values and the gradient information obtained in steps B4 and B5More optimal new speed control strategyAfter the step is executed, jumping to the step B4 again;

step B9: the fine convergence determination module 27 operates to recordWhen k is₂When the value is 0, the step B10 is executed, otherwise, the judgment is madeAnd the last refined objective function valueWhether the absolute value of the difference is less than the accuracy tol₂If yes, the convergence is judged to be satisfied, the speed control strategy of the iteration is converted into a motor control command to be output, otherwise, the convergence is not satisfied, and the number of refinement times k is set₂:＝k₂+1, continue to execute step B3 until the refinement convergence determination module is satisfied.

The grid refinement module is realized by adopting the following steps:

step C1: calculating the left slope at a grid node by the following equationAnd right slope

<mrow> <msubsup> <mi>s</mi> <mi>k</mi> <mo>-</mo> </msubsup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> </mrow> <mrow> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msubsup> <mi>s</mi> <mi>k</mi> <mo>+</mo> </msubsup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> </mrow> <mrow> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>

Wherein u is_kRepresenting the parametric representation of the velocity control strategy on the k-th parametric segment, t_kRepresents u_kAnd u_k+1A mesh node in between.

Step C2: if the mesh node t_kIf the left and right slopes meet the following requirements, the node is removed from the grid:

<mrow> <mo>|</mo> <msubsup> <mi>s</mi> <mi>k</mi> <mo>-</mo> </msubsup> <mo>|</mo> <mo>+</mo> <mo>|</mo> <msubsup> <mi>s</mi> <mi>k</mi> <mo>+</mo> </msubsup> <mo>|</mo> <mo>&amp;le;</mo> <msub> <mi>&amp;epsiv;</mi> <mi>e</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

wherein epsilon_eIs a small positive real number. Grid node t_kAfter removal of u_kAnd u_k+1The corresponding grids are merged into a new grid, and the parameters on the new grid are updated to (u)_k+u_k+1)/2。

Step C3: if the mesh node t_kThe left slope at (b) satisfies:

<mrow> <mo>|</mo> <msubsup> <mi>s</mi> <mi>k</mi> <mo>-</mo> </msubsup> <mo>|</mo> <mo>&amp;GreaterEqual;</mo> <msub> <mi>&amp;epsiv;</mi> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

wherein epsilon_iIs one greater than epsilon_ePositive real number of (1), then at [ t_k-1,t_k]Inserting grid nodes upwards; if the mesh node t_kThe right slope at (b) satisfies:

<mrow> <mo>|</mo> <msubsup> <mi>s</mi> <mi>k</mi> <mo>+</mo> </msubsup> <mo>|</mo> <mo>&amp;GreaterEqual;</mo> <msub> <mi>&amp;epsiv;</mi> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

then at [ t_k-1,t_k]And inserting grid nodes. In practical application, the number of the added nodes can be freely set according to the absolute value of the left slope and the right slope.

Step C4: and generating a new control grid and a corresponding parameterization vector according to the nodes removed and inserted in the steps C2 and C3.

The ODE solving module adopts a four-step Runge-Kutta method, and the calculation formula is as follows:

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mi>h</mi> <mn>6</mn> </mfrac> <mo>&amp;lsqb;</mo> <mi>K</mi> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>K</mi> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>K</mi> <mn>3</mn> <mo>+</mo> <mi>K</mi> <mn>4</mn> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>K</mi> <mn>1</mn> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>x</mi> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>K</mi> <mn>2</mn> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>+</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>x</mi> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>+</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mi>K</mi> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>K</mi> <mn>3</mn> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>+</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>x</mi> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>+</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mi>K</mi> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>K</mi> <mn>4</mn> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mi>x</mi> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>+</mo> <mi>h</mi> <mi>K</mi> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

wherein t represents time, t_iDenotes the integration time, t, selected by the Runge-Kutta method_i+1Indicating that it is at time t_iThe integration step h is the difference between any two adjacent integration moments, x (t)_i) Indicating that the container is at t_iThe state information of the time, F (-) is a function describing a state differential equation, and K1, K2, K3 and K4 respectively represent function values of 4 nodes in the Runge-Kutta method integration process.

The gradient calculation module adopts an accompanying method:

step D1: let λ (t) be the co-modal vector, whose value is determined by the adjoint equation:

<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mover> <mi>&amp;lambda;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mo>-</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>H</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>&amp;Phi;</mi> <mo>&amp;lsqb;</mo> <mi>x</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>&amp;part;</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

wherein, t_fDenotes the end time of the handling process, H denotes the Hamiltonian, and H ═ L + λ (t)^TF, L is the integral term of the objective function, Φ x (t)_f)]Is the steady state term of the objective function.

Step D2: for the adjoint equation, a four-step Runge-Kutta method is adopted to obtain the value of the co-modal vector lambda (t) at each integration moment, and the calculation formula is as follows:

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mi>h</mi> <mn>6</mn> </mfrac> <mo>&amp;lsqb;</mo> <mi>Q</mi> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>Q</mi> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>Q</mi> <mn>3</mn> <mo>+</mo> <mi>Q</mi> <mn>4</mn> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>Q</mi> <mn>1</mn> <mo>=</mo> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>x</mi> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>Q</mi> <mn>2</mn> <mo>=</mo> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>x</mi> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>-</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mi>Q</mi> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>Q</mi> <mn>3</mn> <mo>=</mo> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mi>x</mi> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>-</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mi>Q</mi> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>Q</mi> <mn>4</mn> <mo>=</mo> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mi>h</mi> <mo>,</mo> <mi>x</mi> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>-</mo> <mi>h</mi> <mi>Q</mi> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

wherein t represents time, t_iSolving the selected integration time, t, in the module for ODE_i+1Indicating that it is at time t_iThe latter integration time, and t_i+1＝t_i+h，h is an integration step, and Q1, Q2, Q3 and Q4 respectively represent function values of 4 nodes in the integration process of the Runge-Kutta method.

Step D3: based on the obtained value of the co-modal vector λ (t), gradient information is obtained by the following formula

<mrow> <msup> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </msup> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <msub> <mi>t</mi> <mi>f</mi> </msub> </msubsup> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>H</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>u</mi> <mn>1</mn> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </msubsup> </mrow> </mfrac> <mi>d</mi> <mi>t</mi> </mrow> </mtd> <mtd> <mrow> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <msub> <mi>t</mi> <mi>f</mi> </msub> </msubsup> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>H</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>u</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </msubsup> </mrow> </mfrac> <mi>d</mi> <mi>t</mi> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

Wherein,andto representThe first and second components, and so on.

The NLP problem solving module is realized by adopting the following steps:

step E1: if it is notObjective function value from last iterationIs less than the accuracy tol₁If yes, judging that the convergence is satisfied, and converting the speed control strategy of the iteration into a control instruction of the motor for output; if convergence is not satisfied, continuing to step E2;

step E2: by usingValue of (1) is coveredAnd will iterate the number of times k₁Increasing by 1;

step E3: velocity control strategyAs a point in vector space, denoted as P₁，P₁The corresponding objective function value is

Step E4: from point P₁Starting from the selected NLP algorithm and point P₁Information of the gradient ofConstructing a direction of optimization in vector spaceAnd step size

Step E5, by formulaConstructing correspondences in vector spaceAnother point P of₂So that P is₂Corresponding objective function valueRatio ofPreferably, wherein I isA vector of the same dimension.