CN113232822B - Control method and system for marine servo valve hydraulic steering engine - Google Patents

Control method and system for marine servo valve hydraulic steering engine Download PDF

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CN113232822B
CN113232822B CN202011031070.5A CN202011031070A CN113232822B CN 113232822 B CN113232822 B CN 113232822B CN 202011031070 A CN202011031070 A CN 202011031070A CN 113232822 B CN113232822 B CN 113232822B
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servo valve
rudder
opening
transition process
angle
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CN113232822A (en
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宋林章
李放
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Beijing Highlandr Digital Technology Co ltd
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Beijing Highlandr Digital Technology Co ltd
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    • BPERFORMING OPERATIONS; TRANSPORTING
    • B63SHIPS OR OTHER WATERBORNE VESSELS; RELATED EQUIPMENT
    • B63HMARINE PROPULSION OR STEERING
    • B63H25/00Steering; Slowing-down otherwise than by use of propulsive elements; Dynamic anchoring, i.e. positioning vessels by means of main or auxiliary propulsive elements
    • B63H25/06Steering by rudders
    • B63H25/08Steering gear
    • B63H25/14Steering gear power assisted; power driven, i.e. using steering engine
    • B63H25/26Steering engines
    • B63H25/28Steering engines of fluid type
    • B63H25/30Steering engines of fluid type hydraulic
    • GPHYSICS
    • G05CONTROLLING; REGULATING
    • G05BCONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B11/00Automatic controllers
    • G05B11/01Automatic controllers electric
    • G05B11/36Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential
    • G05B11/42Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential for obtaining a characteristic which is both proportional and time-dependent, e.g. P. I., P. I. D.

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  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
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  • Combustion & Propulsion (AREA)
  • Mechanical Engineering (AREA)
  • Ocean & Marine Engineering (AREA)
  • Feedback Control In General (AREA)

Abstract

The embodiment of the invention provides a control method and a control system of a marine servo valve hydraulic steering engine, wherein the method comprises the following steps: acquiring a rudder order generated by a marine servo valve hydraulic rudder machine; obtaining an output result of the arrangement transition process by arranging the transition process according to the helm; according to the output result, calculating and outputting the opening angle of the target servo valve by a PID controller which is controlled by proportion, integral and differential; and controlling the opening of the servo valve according to the opening angle of the target servo valve. According to the scheme, the steering engine can be smoothly controlled.

Description

Control method and system for marine servo valve hydraulic steering engine
Technical Field
The invention relates to the technical field of steering engine control processing, in particular to a control method and a control system of a marine servo valve hydraulic steering engine.
Background
Among marine equipment, steering engines are one of the most important equipment. It directly affects the safe sailing of the ship. The hydraulic system of the ship steering engine comprises a main oil way (power oil way), an auxiliary oil way (control oil way), an oil cylinder, a hydraulic oil tank and a power control cabinet. The main oil circuit and the auxiliary oil circuit are hydraulic elements such as a check valve, a throttle valve, an overflow valve, a reversing valve and the like. The oil cylinder is a mechanical structure responsible for pushing the rudder blade, and the hydraulic oil cylinder is used for storing hydraulic oil. The power control cabinet mainly comprises a booster pump set (providing stable hydraulic pressure for a power oil path), a servo valve (adjusting and controlling the flow direction and flow of hydraulic oil of the oil path), a filter, a piping system and the like. And a steering command of automatic steering or manual steering is sent to the servo controller, and the servo controller controls the opening and the direction of the servo valve through an actual steering angle acquired by the steering angle feedback module to perform steering control.
The steering engine is a core system for ship steering control, and a ship steering engine control system determines the main performance of the ship steering engine. When a ship navigates, a ship steering engine control system which requires course control must have good stability, robustness and flexibility.
The steering engine system is a nonlinear system, load change and interference caused by working under different working conditions and different water area environments are avoided, the classical PID controller is the current mainstream automatic control scheme, and steering control is carried out by setting a rudder angle and an actual rudder angle error fed back by a rudder angle feedback unit so as to achieve the purpose of course control; the PID scheme is based on error control, is a passive error correction, is small in correction force for small errors, is prone to causing inaccurate steering, is possibly too large in correction force for large errors, causes oil cylinder impact and rudder angle overshoot, and generally adopts a method of increasing a control dead zone on the basis of PID automatic control. The method often causes inaccurate rudder angle, so that the ship drifts, the steering engine can carry out rudder moving correction for many times, and the traditional PID control method cannot achieve perfect control.
And when the state of the ship booster pump group changes, if the number of the opened pump groups is different, the traditional PID controller can show different control states.
No matter current traditional PID servo controller is monocycle or dicyclo and is all can appear unsmooth start-up and stop process taking the rudder in-process, and the shake of corresponding servovalve itself has the interior oil pressure fluctuation of jar, strikes the sealing ring, aggravates steering wheel wearing and tearing. The increase of the dead zone of the rudder angle can be improved to a certain extent, but the course keeping effect is also deteriorated, the flight path goes forward in a large S-turn, the navigation distance is increased, the oil consumption of the main engine is increased, and the like.
Disclosure of Invention
The invention aims to provide a control method and a control system of a marine servo valve hydraulic steering engine. Thereby realizing the smooth control of the steering engine.
In order to solve the technical problems, the technical scheme of the invention is as follows:
a control method of a marine servo valve hydraulic steering engine comprises the following steps:
acquiring a helm command generated by a marine servo valve hydraulic helm machine;
obtaining an output result of the arrangement transition process by arranging the transition process according to the helm;
outputting a target servo valve opening angle through PID operation controlled by proportion, integral and differential according to the output result;
and controlling the opening degree of the servo valve according to the target opening degree of the servo valve.
Optionally, obtaining an output result of the transition process by arranging the transition process according to the helm order, where the output result includes:
obtaining an output result of the arrangement transition process through the following equation according to the steering order;
Figure GDA0003598819330000021
wherein x1 is an input variable, specifically the steering command, x2 is a differential of the input variable x1, h is a step length, i.e., a sampling time, fhan is a fast optimal control synthesis function, and v (t) is a differential signal of x 2.
Optionally, the outputting the target opening angle of the servo valve through proportional, integral and derivative PID operations according to the output result includes:
by the following formula:
Figure GDA0003598819330000022
outputting a target servo valve opening angle;
wherein e (i) is the ith rudder angle error, u (K) is the target servo valve opening angle, K p 、K I 、K D And e (k) is the k-th rudder angle error, and e (k-1) is the k-1 rudder angle error.
Optionally, controlling the opening of the servo valve according to the target opening of the servo valve includes:
feeding back a position signal of the valve as a voltage signal through the feedback of a linear variable differential transformer according to the opening angle of the target servo valve;
and controlling the opening of the servo valve through PID operation according to the voltage signal.
Optionally, the controlling the opening of the servo valve through PID operation includes:
by the following formula:
Figure GDA0003598819330000031
obtaining a duty ratio, and controlling the opening of the servo valve according to the duty ratio;
wherein e1 (i) is the ith valve opening error, u1 (K) is the output duty ratio, K p 、K I 、K D For three parameters of PID operation, e1 (k) is the kth valve opening error, and e1 (k-1) is the kth valve opening error.
The embodiment of the invention also provides a control system of the marine servo valve hydraulic steering engine, which comprises the following components:
the acquisition module is used for acquiring a rudder order generated by a marine servo valve hydraulic rudder machine;
the transition process arrangement module is used for obtaining an output result of the transition process arrangement through the transition process arrangement according to the rudder order;
the rudder angle control module is used for outputting a target servo valve opening angle through proportional, integral and differential PID operation for control according to the output result;
and the servo valve control module is used for controlling the opening of the servo valve according to the opening angle of the target servo valve.
Optionally, the schedule transition process module is specifically configured to:
obtaining an output result of the arrangement transition process through the following equation according to the steering order;
Figure GDA0003598819330000041
wherein x1 is an input variable, specifically the steering command, x2 is a differential of the input variable x1, h is a step length, i.e., a sampling time, fhan is a fast optimal control synthesis function, and v (t) is a differential signal of x 2.
Optionally, the rudder angle control module is specifically configured to:
by the following formula:
Figure GDA0003598819330000042
outputting a target servo valve opening angle;
wherein e (i) is the ith rudder angle error, u (K) is the opening angle of the target servo valve, K p 、K I 、K D And e (k) is the k-th rudder angle error, and e (k-1) is the k-1 rudder angle error.
Optionally, the servo valve control module is specifically configured to:
according to the opening angle of the target servo valve, feeding back a position signal of the valve as a voltage signal through the feedback of a linear variable differential transformer;
and controlling the opening of the servo valve through PID operation according to the voltage signal.
Optionally, the input end of the servo valve is connected with the servo valve controller, and the output end of the servo valve is connected with the linear variable differential transformer and the rudder engine oil cylinder; the rudder angle of the output of the rudder engine cylinder is fed back to the front end of the arrangement transition process.
The scheme of the invention at least comprises the following beneficial effects:
according to the scheme, the rudder order generated by the marine servo valve hydraulic rudder machine is obtained; obtaining an output result of the arrangement transition process by arranging the transition process according to the helm; according to the output result, calculating and outputting the opening angle of the target servo valve by a PID controller which is controlled by proportion, integral and differential; and controlling the opening of the servo valve according to the opening angle of the target servo valve. Therefore, the steering engine start-stop process is smoother, the working load of the hydraulic steering engine is reduced, the abrasion of the hydraulic oil cylinder is reduced, the impact on the sealing gasket is reduced, and the maintenance period and the service life of the steering engine are effectively prolonged. The accurate degree of helm steering is improved, the phenomenon that dead zones are too large is avoided, the course accuracy is improved to a certain extent, and the total oil consumption is reduced.
Drawings
FIG. 1 is a schematic flow chart of a control method of a marine servo valve hydraulic steering engine;
FIG. 2 is a specific flow chart of the control method of the marine servo valve hydraulic steering engine of the invention;
fig. 3 is a schematic diagram of a control system architecture of the marine servo valve hydraulic steering engine.
Detailed Description
Exemplary embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be embodied in various forms and should not be limited by the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.
As shown in fig. 1, a method for controlling a marine servo valve hydraulic steering engine includes:
step 11, acquiring a rudder order generated by a marine servo valve hydraulic rudder machine;
step 12, obtaining an output result of the transition process arrangement through the transition process arrangement according to the helm order;
step 13, calculating and outputting a target servo valve opening angle through a PID controller which performs proportional, integral and differential control according to the output result;
and step 14, controlling the opening of the servo valve according to the target opening angle of the servo valve.
In the embodiment of the invention, the control input quantity and the first-order differential component thereof can be continuously led by arranging the transition process, the large error can be slowly and smoothly input by arranging the transition process for the input state of the large error, and the controller can stably and continuously output under any error input.
In an alternative embodiment of the present invention, step 12 may include:
obtaining an output result of the arrangement transition process through the following equation according to the steering order;
Figure GDA0003598819330000051
wherein x1 is an input variable, specifically the steering order, x2 is a differential of the input variable x1, h is a step length, i.e., a sampling time, fhan is a fast optimal control synthesis function, and v (t) is a differential signal of x 2.
In an alternative embodiment of the present invention, step 13 may include:
by the following formula:
Figure GDA0003598819330000061
outputting a target servo valve opening angle;
wherein e (i) is the ith rudder angle error, u (K) is the opening angle of the target servo valve, K p 、K I 、K D Is three parameters of PID, e (k) is the numberk rudder angle errors, and e (k-1) is the k-1 st rudder angle error.
In an alternative embodiment of the present invention, step 14 may include:
step 141, feeding back a position signal of the valve as a voltage signal through feedback of a linear variable differential transformer according to the opening angle of the target servo valve;
and 142, controlling the opening of the servo valve through PID calculation according to the voltage signal.
In an optional embodiment of the present invention, the controlling the opening of the servo valve through PID calculation includes:
by the following formula:
Figure GDA0003598819330000062
obtaining a duty ratio, and controlling the opening of a servo valve according to the duty ratio;
wherein e1 (i) is the ith valve opening error, u1 (K) is the output duty ratio, K p 、K I 、K D For three parameters of PID operation, e1 (k) is the kth valve opening error, and e1 (k-1) is the kth valve opening error.
As shown in fig. 2, the rudder angle controller is an outer ring controlled by the steering engine, receives an output result of the transition process, and outputs a target servo valve opening angle through PID operation; the servo valve controller is an inner ring controlled by a steering engine, receives the output of the steering engine angle controller and the feedback of the LVDT, controls the opening of the servo valve through PID operation, is a linear variable differential transformer, and belongs to a linear displacement sensor. The position signal of the valve is fed back as a voltage signal. The rudder angle feedback is a measuring and transmitting device which converts a rotation signal of a rudder blade into a voltage signal. The high-precision diaphragm potentiometer is formed by driving a module and a digital circuit.
According to the embodiment of the invention, the starting and stopping processes of the steering engine can be smoother, the working load of the hydraulic steering engine is reduced, the abrasion of the hydraulic oil cylinder is reduced, the impact on the sealing gasket is reduced, and the maintenance period and the service life of the steering engine are effectively prolonged. The accurate degree of helm steering is improved, the phenomenon that dead zones are too large is avoided, the course accuracy is improved to a certain extent, and the total oil consumption is reduced.
As shown in fig. 3, an embodiment of the present invention further provides a control system 30 for a marine servo valve hydraulic steering engine, including:
the acquiring module 31 is used for acquiring a rudder order generated by a marine servo valve hydraulic rudder machine;
a transition process arrangement module 32, configured to obtain an output result of the transition process arrangement by arranging the transition process according to the rudder order;
the rudder angle control module 33 is used for outputting a target servo valve opening angle through proportional, integral and differential PID operation controlled according to the output result;
and the servo valve control module 34 is used for controlling the opening degree of the servo valve according to the target opening degree of the servo valve.
Optionally, the schedule transition process module 32 is specifically configured to:
obtaining an output result of the arrangement transition process through the following equation according to the helm;
Figure GDA0003598819330000071
wherein x1 is an input variable, specifically the steering command, x2 is a differential of the input variable x1, h is a step length, i.e., a sampling time, fhan is a fast optimal control synthesis function, and v (t) is a differential signal of x 2.
Optionally, the rudder angle control module is specifically configured to:
by the following formula:
Figure GDA0003598819330000072
outputting a target servo valve opening angle;
wherein e (i) is the ith rudder angle error, u (K) is the target servo valve opening angle, K p 、K I 、K D And e (k) is the k-th rudder angle error, and e (k-1) is the k-1 rudder angle error.
Optionally, the servo valve control module 34 is specifically configured to:
feeding back a position signal of the valve as a voltage signal through the feedback of a linear variable differential transformer according to the opening angle of the target servo valve;
and controlling the opening of the servo valve through PID operation according to the voltage signal.
Optionally, an input end of the servo valve is connected with a servo valve control module (specifically, a servo valve controller), and an output end of the servo valve is connected with the linear variable differential transformer and is connected with a rudder engine oil cylinder; the rudder angle of the output of the rudder engine cylinder is fed back to the front end of the arrangement transition process.
It should be noted that all the implementation manners in the method embodiment described above are applicable to the embodiment of the apparatus, and the same technical effects can be achieved.
Embodiments of the present invention also provide a computer-readable storage medium including instructions that, when executed on a computer, cause the computer to perform the method as described above. All the implementation manners in the method embodiment are applicable to the embodiment, and the same technical effect can be achieved.
Those of ordinary skill in the art will appreciate that the various illustrative elements and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, or combinations of computer software and electronic hardware. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the implementation. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention.
It can be clearly understood by those skilled in the art that, for convenience and simplicity of description, the specific working processes of the above-described systems, apparatuses and units may refer to the corresponding processes in the foregoing method embodiments, and are not described herein again.
In the embodiments provided in the present invention, it should be understood that the disclosed apparatus and method may be implemented in other manners. For example, the above-described apparatus embodiments are merely illustrative, and for example, the division of the units is only one type of logical functional division, and other divisions may be realized in practice, for example, multiple units or components may be combined or integrated into another system, or some features may be omitted, or not executed. In addition, the shown or discussed mutual coupling or direct coupling or communication connection may be an indirect coupling or communication connection through some interfaces, devices or units, and may be in an electrical, mechanical or other form.
The units described as separate parts may or may not be physically separate, and parts displayed as units may or may not be physical units, may be located in one position, or may be distributed on multiple network units. Some or all of the units can be selected according to actual needs to achieve the purpose of the solution of the embodiment.
In addition, functional units in the embodiments of the present invention may be integrated into one processing unit, or each unit may exist alone physically, or two or more units are integrated into one unit.
The functions may be stored in a computer-readable storage medium if they are implemented in the form of software functional units and sold or used as separate products. Based on such understanding, the technical solution of the present invention may be embodied in the form of a software product, which is stored in a storage medium and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device) to execute all or part of the steps of the method according to the embodiments of the present invention. And the aforementioned storage medium includes: a U disk, a removable hard disk, a ROM, a RAM, a magnetic disk or an optical disk, and various media capable of storing program codes.
Furthermore, it should be noted that in the apparatus and method of the present invention, it is obvious that each component or each step may be decomposed and/or recombined. These decompositions and/or recombinations are to be considered as equivalents of the present invention. Also, the steps of performing the series of processes described above may naturally be performed chronologically in the order described, but need not necessarily be performed chronologically, and some steps may be performed in parallel or independently of each other. It will be understood by those skilled in the art that all or any of the steps or elements of the method and apparatus of the present invention may be implemented in any computing device (including processors, storage media, etc.) or network of computing devices, in hardware, firmware, software, or any combination thereof, which can be implemented by those skilled in the art using their basic programming skills after reading the description of the present invention.
The object of the invention is thus also achieved by a program or a set of programs running on any computing device. The computing device may be a well-known general purpose device. The object of the invention is thus also achieved solely by providing a program product comprising program code for implementing the method or the apparatus. That is, such a program product also constitutes the present invention, and a storage medium storing such a program product also constitutes the present invention. It is to be understood that the storage medium may be any known storage medium or any storage medium developed in the future. It is also noted that in the apparatus and method of the present invention, it is apparent that each component or step can be decomposed and/or recombined. These decompositions and/or recombinations are to be considered as equivalents of the present invention. Also, the steps of executing the series of processes described above may naturally be executed chronologically in the order described, but need not necessarily be executed chronologically. Some steps may be performed in parallel or independently of each other.
While the foregoing is directed to the preferred embodiment of the present invention, it will be understood by those skilled in the art that various changes and modifications may be made without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (4)

1. A control method of a marine servo valve hydraulic steering engine is characterized by comprising the following steps:
acquiring a helm command generated by a marine servo valve hydraulic helm machine;
obtaining an output result of the arrangement transition process by arranging the transition process according to the helm;
outputting a target servo valve opening angle through PID operation controlled by proportion, integral and differential according to the output result;
controlling the opening of the servo valve according to the opening angle of the target servo valve; the input end of the servo valve is connected with a servo valve controller, and the output end of the servo valve is connected with the linear variable differential transformer and the rudder engine oil cylinder; the rudder angle output by the rudder engine oil cylinder is fed back to the front end of the arrangement transition process, the input of the arrangement transition process is a rudder order generated by an automatic rudder, and the output end of the arrangement transition process is connected with a rudder angle controller;
the rudder angle controller is an outer ring controlled by the steering engine, receives an output result of the arrangement transition process, and outputs a target servo valve opening angle through PID operation; the servo valve controller is an inner ring controlled by a steering engine, receives the output of a rudder angle controller and the feedback of a Linear Variable Differential Transformer (LVDT), and controls the opening of a servo valve through PID operation;
wherein, according to the helm, obtaining an output result of the transition process by the transition process, comprising: obtaining an output result of the arrangement transition process through the following equation according to the helm;
Figure FDA0003838085760000011
wherein x1 is an input variable, specifically the steering order, x2 is a differential of the input variable x1, h is a step length, i.e., a sampling time, fhan is a fast optimal control synthesis function, and v (t) is a differential signal of x 2; wherein, according to the output result, through proportional, integral and differential PID operation that the differentiation carries out control output target servovalve opening angle, include:
by the following formula:
Figure FDA0003838085760000012
outputting a target servo valve opening angle;
wherein e (i) is the ith rudder angle error, u (K) is the target servo valve opening angle, K p 、K I 、K D E (k) is the k-th rudder angle error, and e (k-1) is the k-1 rudder angle error; wherein, according to target servovalve opening angle, control the servovalve opening degree, include:
according to the opening angle of the target servo valve, feeding back a position signal of the valve as a voltage signal through the feedback of a linear variable differential transformer;
and controlling the opening of the servo valve through PID operation according to the voltage signal.
2. The control method of the marine servo valve hydraulic steering engine according to claim 1, wherein the controlling of the opening degree of the servo valve through PID operation comprises:
by the following formula:
Figure FDA0003838085760000021
obtaining a duty ratio, and controlling the opening of a servo valve according to the duty ratio;
wherein e1 (i) is the ith valve opening error, u1 (K) is the output duty cycle, K p 、K I 、K D For three parameters of PID operation, e1 (k) is the kth valve opening error, and e1 (k-1) is the kth valve opening error.
3. The utility model provides a control system of marine servovalve hydraulic pressure steering wheel which characterized in that includes:
the acquisition module is used for acquiring a rudder order generated by a marine servo valve hydraulic rudder machine;
the transition process arrangement module is used for obtaining an output result of the transition process arrangement through the transition process arrangement according to the rudder order;
the rudder angle control module is used for outputting a target servo valve opening angle through proportional, integral and differential PID operation controlled according to the output result;
the servo valve control module is used for controlling the opening of the servo valve according to the opening angle of the target servo valve;
the input end of the servo valve is connected with a servo valve controller, and the output end of the servo valve is connected with the linear variable differential transformer and the rudder engine oil cylinder; the rudder angle output by the rudder engine oil cylinder is fed back to the front end of the arrangement transition process, the input of the arrangement transition process is a rudder order generated by an automatic rudder, and the output end of the arrangement transition process is connected with a rudder angle controller;
the rudder angle controller is an outer ring controlled by the steering engine, receives an output result of the arrangement transition process, and outputs a target servo valve opening angle through PID operation;
the servo valve controller is an inner ring controlled by a steering engine, receives the output of a rudder angle controller and the feedback of a Linear Variable Differential Transformer (LVDT), and controls the opening of a servo valve through PID operation; wherein the schedule transition process module is specifically configured to:
obtaining an output result of the arrangement transition process through the following equation according to the steering order;
Figure FDA0003838085760000031
wherein x1 is an input variable, specifically the steering order, x2 is a differential of the input variable x1, h is a step length, i.e., a sampling time, fhan is a fast optimal control synthesis function, and v (t) is a differential signal of x 2; the rudder angle control module is specifically used for:
by the following formula:
Figure FDA0003838085760000032
outputting a target servo valve opening angle;
wherein e (i) is the ith rudder angle error, u (K) is the opening angle of the target servo valve, K p 、K I 、K D Three parameters of PID, e (k) is the k-th rudder angle errorE (k-1) is the error of the k-1 th rudder angle; wherein the servo valve control module is specifically configured to:
according to the opening angle of the target servo valve, feeding back a position signal of the valve as a voltage signal through the feedback of a linear variable differential transformer;
and controlling the opening of the servo valve through PID operation according to the voltage signal.
4. The control system of marine servo valve hydraulic steering engine according to claim 3, wherein the servo valve opening is controlled by PID calculation, comprising:
by the following formula:
Figure FDA0003838085760000033
obtaining a duty ratio, and controlling the opening of a servo valve according to the duty ratio;
wherein e1 (i) is the ith valve opening error, u1 (K) is the output duty ratio, K p 、K I 、K D And e1 (k) is the k-th valve opening error, and e1 (k-1) is the k-1 valve opening error.
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