A train adaptive polynomial approximation control method and system

Abstract

The present invention discloses a train adaptive polynomial approximation control method and system, which belongs to the field of train control technology. The method comprises the following steps: performing force analysis on the longitudinal motion of the train and establishing the dynamic equation of the longitudinal motion of the train; using polynomial approximation of unknown additional resistance to establish the dynamic equation of the longitudinal motion of the train under polynomial approximation; constructing an adaptive polynomial approximation controller, designing an adaptive law of unknown parameters, and establishing a closed-loop control system of the train under polynomial approximation; selecting a suitable Lyapunov function to prove the stability of the closed-loop control system of the train, and further using the adaptive law and the controller equation to control the actual running trajectory of the train to approach the expected running trajectory. The beneficial effect of the present invention is that the method of the present invention effectively copes with the uncertainty of the additional resistance during the operation of the train, and can ensure that the actual speed and displacement of the train can accurately track the preset target.

Description

Technical Field

[0001] The present invention relates to the technical field of train control, and more specifically, to a train adaptive polynomial approximation control method and system. Background Art

[0002] The operation control of trains plays a vital role in modern transportation systems, and the accuracy and stability of their operation directly affect the efficiency and safety of train operations. At present, researchers in this field have conducted a lot of research on the basic operating resistance, additional resistance, actuator failure and other factors that trains are subject to in actual operation. For example, the applicant has also conducted research on the safety of train operation under faults in the early stage, using neural networks to approximate its additional resistance, and applied for a patent application entitled A neural adaptive fault-tolerant control method for unknown disturbances of trains (publication number CN106249591 A). This invention is used to solve the impact of unknown disturbances and actuator failures on trains. The driving algorithm it provides is dedicated to ensuring the safe operation of trains in the event of a fault, but it is not suitable for normal operation of trains; and when it uses neural networks for approximation, its compensation effect depends on the number of neural network layers selected. Summary of the invention

[0003] In order to cope with the uncertainty of additional resistance during train operation, enable the actual speed and displacement of the train to accurately track preset targets, and ensure that the system remains stable during dynamic operation, the present invention provides a train adaptive polynomial approximation control method and system.

[0004] In order to achieve the above-mentioned object of the invention, the present invention provides a train adaptive polynomial approximation control method, the method comprising the following steps:

[0005] S1. Based on Newton's second law, the force analysis of the longitudinal motion of the train is carried out and the dynamic equation of the longitudinal motion of the train is established;

[0006] S2. Based on the unknown additional resistance of the train during operation and the dynamic equation of the longitudinal motion of the train, the unknown additional resistance is approximated by a polynomial, and the dynamic equation of the longitudinal motion of the train under the polynomial approximation is established;

[0007] S3. According to the train motion dynamics equation under polynomial approximation, an adaptive polynomial approximation controller is constructed, an adaptive law for unknown parameters is designed, and a closed-loop train control system under polynomial approximation is established;

[0008] S4. Select a suitable Lyapunov function to prove the stability of the train closed-loop control system, and further use the adaptive law and controller equation to control the actual running trajectory of the train to approach the expected running trajectory.

[0009] Wherein, in step S1, the dynamic equation of the longitudinal motion of the train is established as:

[0010] (1)

[0011] in, , is the running time of the train, and They represent the displacement, velocity and acceleration of the train during real-time operation respectively; Represents the total weight of the train during travel, including the train body, passengers and their luggage, etc.; Represents control input (traction / braking force); in actual operation, the resistance of the train is divided into two parts, namely basic operating resistance and additional resistance. represents the basic running resistance related to the train speed, , and is the unknown Davis equation coefficient; It represents the additional resistance encountered by trains running on fixed lines. It is difficult to obtain accurately in actual operation. It includes other resistances caused by curve resistance, slope resistance, tunnel resistance and unmodeled factors.

[0012] Wherein, the step S2 is specifically as follows:

[0013] S201. The formula for approximating the unknown additional resistance using a polynomial function is:

[0014] (2)

[0015] in, is the least squares polynomial, which is about of A polynomial of degree, of the form: ; is any non-negative real number, , is the polynomial dimension, is the independent variable in the polynomial function; is the reconstruction error, which is given by:

[0016] (3)

[0017] S202. Combining formulas (1) and (2), the train longitudinal motion dynamics equation under polynomial function approximation is obtained:

[0018] (4)

[0019] in, for The lateral quantity of the dimension, represented by the unknown coefficient An estimated value of for dimensional column vector that satisfies , Represents the transpose of a phasor.

[0020] Wherein, the step S3 is specifically as follows:

[0021] S301, design real-time displacement tracking error of train operation and velocity tracking error ;

[0022] (5)

[0023] in, and Tracking trajectory for expected displacement and expected speed of train operation;

[0024] S302, the constructed adaptive polynomial approximation controller is:

[0025] (6)

[0026] in, Its representative is The estimated value of ; sgn represents the sign function; is the normal number to be selected; for The estimated value of Reconstruction error The upper bound of Tracking trajectory for train operation expected acceleration;

[0027] S303, design the unknown parameter adaptive law as follows:

[0028] (7)

[0029] in, is the normal number to be selected; is the positive definite matrix to be selected.

[0030] The step S4 is specifically as follows:

[0031] S401. Constructing Lyapunov function under train adaptive polynomial approximation control as follows:

[0032] ;

[0033] S402, combining step S401 to derive the Lyapunov function, and get ,Through Lyapunov stability theory, the stability of the closed-loop control system under the ,adaptive polynomial approximation of the train can be verified, that ,under the adaptive polynomial approximation control algorithm, the system has good ,displacement and velocity tracking characteristics, and the actual displacement and velocity ,asymptotically track the expected displacement and velocity.

[0034] In order to better achieve the above-mentioned invention object, according to another aspect of the present application, a train adaptive polynomial approximation control system is also provided, the system comprising:

[0035] The power system building module analyzes the longitudinal motion of the train based on Newton's second law and establishes the dynamic equation of the longitudinal motion of the train;

[0036] The train power system establishment module uses polynomial approximation to approximate the unknown additional resistance during the train operation and the train longitudinal motion dynamic equation, and establishes the train longitudinal motion dynamic equation under polynomial approximation;

[0037] Controller design module: used to construct an adaptive polynomial approximation controller based on the train motion dynamics equation under polynomial approximation, design the adaptive law of unknown parameters, and establish a closed-loop train control system under polynomial approximation;

[0038] Stability analysis module: used to select appropriate Lyapunov functions to prove the stability of the train closed-loop control system, and further use adaptive laws and controller equations to control the actual running trajectory of the train to approach the expected running trajectory.

[0039] According to another aspect of an embodiment of the present application, a storage medium is further provided, which includes a stored program, and the above steps are executed when the program is run.

[0040] According to another aspect of the embodiment of the present application, there is also provided an electronic device, including an input unit, a memory, a processor, a feedback device and an output unit;

[0041] Memory, used to store computer programs;

[0042] The processor is used to implement the steps of the above-mentioned multiple embodiments when executing the computer program stored in the memory.

[0043] The embodiment of the present application also provides a computer program product including instructions, which, when executed on a computer, enables the computer to execute the steps in the above method.

[0044] The beneficial effects of the present invention are as follows: the present invention uses a polynomial algorithm to approximate the additional resistance and directly designs a feedback controller, which can effectively deal with the uncertainty of the additional resistance during train operation and ensure that the actual speed and displacement of the train can accurately track the preset target. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 Flowcharts of Embodiments 1 and 2 of the present invention.

[0046] Figure 2 This is the additional resistance response curve of the actual train operation in Example 2 of the present invention.

[0047] Figure 3 The figure is a schematic diagram of the expected displacement, actual displacement and displacement error curves under the train adaptive polynomial approximation control method in Example 2 of the present invention.

[0048] Figure 4 The figure is a schematic diagram of the expected speed, actual speed and speed error curves under the train adaptive polynomial approximation control method in Example 2 of the present invention.

[0049] Figure 5 Schematic diagram of the structure of the train adaptive control system in Example 2 of the present invention.

[0050] Figure 6 It is a structural block diagram of an electronic device under the train adaptive polynomial approximation control method in Example 4 of the present invention. DETAILED DESCRIPTION

[0051] In order to clearly illustrate the technical features of this solution, this solution is described below through a specific implementation method.

[0052] Example 1: See Figure 1 The embodiment of the present invention provides a train adaptive polynomial approximation control method, the method comprising the following steps:

[0053] S1. Based on Newton's second law, the force analysis of the longitudinal motion of the train is carried out and the dynamic equation of the longitudinal motion of the train is established;

[0054] S2. Based on the unknown additional resistance of the train during operation and the dynamic equation of the longitudinal motion of the train, the unknown additional resistance is approximated by a polynomial, and the dynamic equation of the longitudinal motion of the train under the polynomial approximation is established;

[0055] S3. According to the train motion dynamics equation under polynomial approximation, an adaptive polynomial approximation controller is constructed, an adaptive law for unknown parameters is designed, and a closed-loop train control system under polynomial approximation is established;

[0056] S4. Select a suitable Lyapunov function to prove the stability of the train closed-loop control system, and further use the adaptive law and controller equation to control the actual running trajectory of the train to approach the expected running trajectory.

[0057] Wherein, in step S1, the dynamic equation of the longitudinal motion of the train is established as:

[0058] (1)

[0059] in, , is the running time of the train, , and They represent the displacement, velocity and acceleration of the train during real-time operation respectively; Represents the total weight of the train during travel, including the train body, passengers and their luggage, etc.; Represents control input (traction / braking force); in actual operation, the resistance of the train is divided into two parts, namely basic operating resistance and additional resistance. represents the basic running resistance related to the train speed, , and is the unknown Davis equation coefficient; It represents the additional resistance encountered by trains running on fixed lines. It is difficult to obtain accurately in actual operation. It includes other resistances caused by curve resistance, slope resistance, tunnel resistance and unmodeled factors.

[0060] Wherein, the step S2 is specifically as follows:

[0061] S201. The formula for approximating the unknown additional resistance using a polynomial function is:

[0062] (2)

[0063] in, is the least squares polynomial, which is about of A polynomial of degree, of the form: ; is any non-negative real number, , is the polynomial dimension, is the independent variable in the polynomial function; is the reconstruction error, which is given by:

[0064] (3)

[0065] S202. Combining formulas (1) and (2), the train longitudinal motion dynamics equation under polynomial function approximation is obtained:

[0066] (4)

[0067] in, for The lateral quantity of the dimension, represented by the unknown coefficient An estimated value of for dimensional column vector that satisfies , Represents the transpose of a phasor.

[0068] Wherein, the step S3 is specifically as follows:

[0069] S301, design real-time displacement tracking error of train operation and velocity tracking error :

[0070] (5)

[0071] in, and Tracking trajectory for expected displacement and expected speed of train operation;

[0072] S302, the constructed adaptive polynomial approximation controller is:

[0073] (6)

[0074] in, Its representative is The estimated value of ; sgn represents the sign function; is the normal number to be selected; for The estimated value of Reconstruction error The upper bound of Tracking trajectory for train operation expected acceleration;

[0075] S303, design the unknown parameter adaptive law as follows:

[0076] (7)

[0077] in, is the normal number to be selected; is the positive definite matrix to be selected.

[0078] The step S4 is specifically as follows:

[0079] S401. Constructing Lyapunov function under train adaptive polynomial approximation control as follows:

[0080] ;

[0081] S402, combining step S401 to derive the Lyapunov function, and get ,Through Lyapunov stability theory, the stability of the closed-loop control system under the ,adaptive polynomial approximation of the train can be verified, that ,under the adaptive polynomial approximation control algorithm, the system has good ,displacement and velocity tracking characteristics, and the actual displacement and velocity ,asymptotically track the expected displacement and velocity.

[0082] Embodiment 2: This embodiment provides an adaptive polynomial approximation control method for train additional resistance, which is used in train position and speed tracking control technology; Figure 1 As shown, the method comprises the following steps:

[0083] S1. Based on Newton's second law, the longitudinal motion of the train is subjected to force analysis and the longitudinal motion dynamics equation of the train is established;

[0084] In a specific implementation, as a preference, step S1 may include the following sub-steps:

[0085] The dynamic equation of the longitudinal motion of the train is:

[0086] (1)

[0087] in, , is the running time of the train, , and They represent the displacement, velocity and acceleration of the train during real-time operation respectively; Represents the total weight of the train during travel, including the train body, passengers and their luggage, etc.; Represents control input (traction / braking force); In actual operation, the resistance of the train is divided into two parts, namely basic running resistance and additional resistance. represents the basic running resistance related to the train speed, , and is the unknown Davis equation coefficient; It represents the additional resistance encountered by trains running on fixed lines. It is difficult to obtain accurately in actual operation. It includes other resistances caused by curve resistance, slope resistance, tunnel resistance and unmodeled factors.

[0088] S2. According to the additional resistance encountered by the train during operation and the dynamic equation of the longitudinal motion of the train, the unknown additional resistance is approximated by polynomials, and the dynamic equation of the longitudinal motion of the train under polynomial approximation is established.

[0089] In a specific implementation, preferably, step S2 may include the following sub-steps:

[0090] S2.1. Use polynomial function to approximate the unknown additional resistance:

[0091] (2)

[0092] in, is the additional resistance after approach; is the least squares polynomial, which is about of A polynomial of the form: ; is any non-negative real number, , is the polynomial dimension, is the independent variable in the polynomial function; is the reconstruction error, which is given by:

[0093] (3)

[0094] S2.2. Combining formulas (1) and (2), the train longitudinal motion dynamics equation under polynomial function approximation is obtained:

[0095] (4)

[0096] in, for The lateral quantity of the dimension, represented by the unknown coefficient An estimated value of for dimensional column vector that satisfies , Represents the transpose of a phasor.

[0097] S3. According to the train motion dynamics equation under polynomial approximation, design the adaptive law of unknown parameters, construct an adaptive polynomial approximation controller, and establish a train closed-loop control system under polynomial approximation.

[0098] In a specific implementation, as a preference, step S3 may include the following sub-steps:

[0099] S3.1、Design of real-time displacement tracking error of train operation and velocity tracking error :

[0100] (5)

[0101] in, and Track the expected displacement and speed of the train.

[0102] Preferably, step S3 includes the following sub-steps:

[0103] S3.2, design controller:

[0104] (6)

[0105] in, Its representative is The estimated value of ; sgn represents the sign function; is the normal number to be selected; for The estimated value of Reconstruction error The upper bound of Track the trajectory for the desired acceleration of the train.

[0106] S3.3, construct the unknown parameter adaptive law as follows:

[0107] (7)

[0108] in, is the normal number to be selected; is the positive definite matrix to be selected.

[0109] S4. Select a suitable Lyapunov function to prove the stability of the closed-loop system, and further use the adaptive law and controller equation to control the actual running trajectory of the train to approach the expected running trajectory.

[0110] In a specific implementation, preferably, step S4 may include the following sub-steps:

[0111] S4.1. Construction of Lyapunov function under train adaptive polynomial approximation control as follows:

[0112]

[0113] S4.2. Combined with step S4.1, the Lyapunov function is derived and the result is , proving the stability of the closed-loop control system under the adaptive polynomial approximation of the train, that is, under the adaptive polynomial approximation control algorithm, the system has good displacement and velocity tracking characteristics, and the actual displacement and velocity asymptotically track the expected displacement and velocity.

[0114] In order to verify the feasibility and effectiveness of the train adaptive polynomial approximation control method provided in Example 2, this experimental example uses MATLAB to conduct simulation experiments and makes detailed explanations.

[0115] The train single-particle model provided in this embodiment 2 takes into account the influence of complex additional resistance on the train position and speed tracking performance during train operation, and analyzes the longitudinal motion of the train. Figure 2As shown, the polynomial approximation method is used to adaptively adjust the polynomial variables, and the unknown additional resistance is modeled and compensated in real time, which can effectively deal with the unknown additional resistance in complex operating environments and improve the control accuracy and system stability.

[0116] Considering the resistance generated by curve resistance, slope resistance, tunnel resistance and unmodeled factors during the operation of HST, the effectiveness of the control method is verified through simulation experiments. In the simulation experiment, the train running time is 2000s, the total mass is 500t, and the basic running resistance coefficient of the train is , and They are , and During the train operation, the center angle of the curve is 120° and the slope angle is The tunnel length is 1500m. The additional resistance during the actual operation of the train like Figure 2 shown.

[0117] Based on the above simulation conditions, MATLAB is used to carry out simulation experiments and the response curves of train displacement and speed under the proposed adaptive polynomial approximation control method are obtained as follows: Figure 3 and Figure 4 shown.

[0118] Combined with the above simulation results, it can be concluded that during the train operation, the train can obtain good displacement tracking performance and speed tracking performance under the adaptive polynomial approximation control method. The displacement tracking error is in the range of [-0.03, 0.04]m, and the speed tracking error is close to 0. Only in the train startup stage does the speed tracking error value fluctuate significantly, achieving the expected effect, proving the effectiveness of the control method adopted in this embodiment.

[0119] Example 3: See Figure 5 This embodiment provides a train adaptive polynomial approximation control system A0 capable of implementing embodiment 1 or embodiment 2, the system comprising:

[0120] Power system building module A1, based on Newton's second law, performs force analysis on the longitudinal motion of the train and establishes the dynamic equation of the longitudinal motion of the train;

[0121] The train power system establishment module A2 uses a polynomial to approximate the unknown additional resistance during the train operation and the train longitudinal motion dynamic equation to establish the train longitudinal motion dynamic equation under the polynomial approximation;

[0122] Controller design module A3: used to construct an adaptive polynomial approximation controller based on the train motion dynamics equation under polynomial approximation, design the adaptive law of unknown parameters, and establish a train closed-loop control system under polynomial approximation;

[0123] Stability analysis module A4: used to select the appropriate Lyapunov function to prove the stability of the train closed-loop control system, and further use the adaptive law and controller equation to control the actual running trajectory of the train to approach the expected running trajectory.

[0124] Embodiment 4: This embodiment provides an electronic device, such as Figure 6 As shown, the electronic device B0 may include: an input unit B1, a memory B2, a processor B3, a feedback device B4 and an output unit B5.

[0125] Memory B2, for storing computer programs;

[0126] The processor B3 is used to implement the steps of the above-mentioned multiple embodiments when executing the computer program stored in the memory B2.

[0127] The communication bus mentioned in the above terminal can be a Peripheral Component Interconnect (PCI) bus or an Extended Industry Standard Architecture (EISA) bus, etc. The communication bus can be divided into an address bus, a data bus, a control bus, etc. For ease of representation, only one thick line is used in the figure, but it does not mean that there is only one bus or one type of bus.

[0128] The communication interface is used for communication between the above terminal and other devices.

[0129] The memory may include a random access memory (RAM) or a non-volatile memory, such as at least one disk memory. Optionally, the memory may also be at least one storage device located away from the aforementioned processor.

[0130] The above-mentioned processor can be a general-purpose processor, including a central processing unit (CPU), a network processor (NP), etc.; it can also be a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components.

[0131] In another embodiment provided in the present application, a computer-readable storage medium is provided. The computer-readable storage medium stores instructions. When the instructions are executed on a computer, the computer executes any of the methods described in the above embodiments.

[0132] In another embodiment provided by the present application, a computer program product including instructions is also provided, which, when executed on a computer, enables the computer to execute any of the methods described in the above embodiments.

[0133] In the above embodiments, it can be implemented in whole or in part by software, hardware, firmware or any combination thereof. When implemented using software, it can be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, the process or function described in the embodiment of the present application is generated in whole or in part. The computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions may be stored in a computer-readable storage medium, or transmitted from one computer-readable storage medium to another computer-readable storage medium. For example, the computer instructions may be transmitted from a website site, computer, server or data center to another website site, computer, server or data center by wired (e.g., coaxial cable, optical fiber, digital subscriber line) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium may be any available medium that a computer can access or a data storage device such as a server or data center that includes one or more available media integrated. The available medium may be a magnetic medium (e.g., a floppy disk, a hard disk, a tape), an optical medium (e.g., a DVD), or a semiconductor medium (e.g., a solid-state hard disk SolidState Disk), etc.

[0134] The above description is only a preferred embodiment of the present application and is not intended to limit the protection scope of the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application are included in the protection scope of the present application.

[0135] The above description is only a specific implementation of the present application, so that those skilled in the art can understand or implement the present application. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application will not be limited to the embodiments shown herein, but will conform to the widest range consistent with the principles and novel features applied for herein.

Claims

1. A train adaptive polynomial approximation control method, characterized in that: The method comprises the following steps: S1. Based on Newton's second law, the force analysis of the longitudinal motion of the train is carried out and the dynamic equation of the longitudinal motion of the train is established; S2. Based on the unknown additional resistance of the train during operation and the dynamic equation of the longitudinal motion of the train, the unknown additional resistance is approximated by a polynomial, and the dynamic equation of the longitudinal motion of the train under the polynomial approximation is established; S3. According to the train motion dynamics equation under polynomial approximation, an adaptive polynomial approximation controller is constructed, an adaptive law for unknown parameters is designed, and a closed-loop train control system under polynomial approximation is established; S4. Select Lyapunov function to prove the stability of the train closed-loop control system, and further use the adaptive law and controller equation to control the actual running trajectory of the train to approach the expected running trajectory; Wherein, the step S3 is specifically as follows: S301, design real-time displacement tracking error of train operation and velocity tracking error : (5) in, and Tracking trajectory for expected displacement and expected speed of train operation; Represents the displacement of the train during real-time operation; S302, the constructed adaptive polynomial approximation controller is: (6) in, Its representative is The estimated value of , is 1×( n +1)-dimensional lateral quantity, represented by the unknown coefficient An estimated value of for( n +1)×1-dimensional column vector that satisfies , T represents the transpose of the vector; sgn represents the sign function; , is the normal number to be selected; for The estimated value of Reconstruction error The upper bound of Tracking trajectory for train operation expected acceleration; represents the basic running resistance related to the train speed, , and is the unknown Davis equation coefficient; Represents the total weight of the train during travel; Among them, S303, the design of the unknown parameter adaptive law is as follows: (7) in, is the normal number to be selected; is the positive definite matrix to be selected.

2. The method according to claim 1, characterized in that In step S1, the dynamic equation of the longitudinal motion of the train is established as: (1) in, t , is the running time of the train, , and They represent the displacement, velocity and acceleration of the train during real-time operation respectively; Represents the total weight of the train during travel; Represents control input; in actual operation, the resistance of the train is divided into two parts, namely basic operation resistance and additional resistance. represents the basic running resistance related to the train speed, , and is the unknown Davis equation coefficient; Represents the additional resistance encountered by trains running on fixed lines.

3. The method according to claim 1, characterized in that The step S2 is specifically as follows: S201. The formula for approximating the unknown additional resistance using a polynomial function is: (2) in, is the least squares polynomial, which is about of A polynomial of the form: ; is any non-negative real number, , is the polynomial dimension, is the independent variable in the polynomial function; is the reconstruction error, which is given by (3) S202. Combining formulas (1) and (2), the train longitudinal motion dynamics equation under polynomial function approximation is obtained: (4) in, is 1×( n +1)-dimensional lateral quantity, represented by the unknown coefficient An estimated value of for( n +1)×1-dimensional column vector that satisfies , T represents the transpose of the vector.

4. The method according to claim 1, characterized in that: The step S4 is specifically as follows: S401. Constructing Lyapunov function under train adaptive polynomial approximation control V ( t )as follows: ； S402, combining step S401 to derive the Lyapunov function, and get ,Through Lyapunov stability theory, the stability of the closed-loop control system under the ,adaptive polynomial approximation of the train can be verified, that ,under the adaptive polynomial approximation control algorithm, the system has good ,displacement and velocity tracking characteristics, and the actual displacement and velocity ,asymptotically track the expected displacement and velocity.

5. A storage medium, characterized in that: The storage medium includes a stored program, wherein the program executes the method according to any one of claims 1 to 4 when executed.

6. An electronic device, characterized in that: It includes an input unit, a memory, a processor, a feedback device and an output unit; wherein, Memory, used to store computer programs; A processor, configured to execute the method according to any one of claims 1 to 4 by running a program stored in a memory.

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