A method and apparatus for sensorless control of an ac motor

Abstract

The application discloses a position-sensorless control method and device for an alternating current motor, and the method comprises the following steps: firstly, constructing an active flux linkage model; then, constructing a voltage model and a current model based on the active flux linkage model, and constructing a hybrid active flux linkage observer based on the voltage model and the current model; then, outputting a rotor electric angle based on the hybrid active flux linkage observer, and then processing the rotor electric angle to obtain a rotor position and a rotor speed; finally, inputting the rotor position and the rotor speed into a controller, so that the controller controls the alternating current motor. The application establishes a unified position-sensorless control method suitable for a permanent magnet synchronous motor and an asynchronous motor, and the method has stronger universality, and through hybrid active flux linkage observation and extended state observation, the position observation precision and overall stability during low-speed and wide-speed domain operation are improved.

Description

Technical Field

[0001] This invention belongs to the field of AC motor technology, specifically relating to a sensorless control method and device for AC motors. Background Technology

[0002] Currently, high-performance drive systems for AC motors require precise rotor position and speed information to achieve closed-loop control. However, existing mechanical encoders for measuring rotor position suffer from high cost, large size, and limited reliability. Therefore, sensorless control has significant engineering application value. Existing sensorless control strategies can be broadly divided into two categories: methods based on salient pole effect / signal injection and methods based on the basic motor model. The first method is suitable for low-speed regions but reduces DC bus voltage utilization and introduces additional torque ripple and noise. The second method is widely used in medium- and high-speed ranges, but under low-speed conditions, the back EMF amplitude decreases and the signal-to-noise ratio decreases, which can easily lead to unstable position observation.

[0003] Therefore, how to improve the applicable speed range of sensorless control methods and enhance their robustness is a technical problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0004] The purpose of this invention is to solve the technical problems of existing technologies that cannot be directly applied to wide speed ranges and have low robustness.

[0005] To achieve the above technical objectives, in one aspect, the present invention provides a sensorless control method for AC motors, the method comprising:

[0006] Construct an active magnetic flux linkage model;

[0007] A voltage model and a current model are constructed based on the active flux linkage model, and a hybrid active flux linkage observer is constructed based on the voltage model and the current model.

[0008] The rotor electrical angle is output based on the hybrid active flux linkage observer, and then the rotor electrical angle is processed to obtain the rotor position and rotational speed.

[0009] The rotor position and speed are input to the controller so that the controller can control the AC motor.

[0010] Furthermore, the active flux linkage model specifically includes:

[0011] ;

[0012] In the formula, It is an active magnetic flux linkage. It is a permanent magnet synchronous motor. It is an asynchronous motor. For stator flux linkage, It is a quadrature axis inductor. For stator current, It is a direct-axis inductor. For direct-axis stator current, It is a permanent magnet flux linkage. For rotor electrical angle, For rotor inductance, For magnetizing inductance, This is the equivalent leakage inductance on the stator side.

[0013] Furthermore, the voltage model is specifically shown below:

[0014] ;

[0015] In the formula, The rate of change of active flux linkage under the voltage model. For stator resistance, The rate of change of stator current. This is the actual output voltage of the inverter.

[0016] Furthermore, the current model is specifically shown below:

[0017] ;

[0018] In the formula, The rate of change of active flux linkage under the current model. The rate of change of active flux linkage, The rotor's electric angular velocity, For a matrix, , This represents the active flux linkage under the current model.

[0019] Furthermore, the hybrid active flux observer is specifically described below:

[0020] ;

[0021] In the formula, For observation error, This is the estimated value of the active flux linkage. These are observations of active flux linkage under the current model. For the rate of change of the first-order extended observation, It is a second-order extended state variable. The rate of change of the observed active flux linkage under the voltage model. and All are observer gains. For the rate of change of the second-order extended observation, This is the reference voltage output by the controller.

[0022] The method further includes calculating the final compensation voltage and adding the final compensation voltage to the reference voltage output by the controller as the final value. .

[0023] Furthermore, the step of processing the rotor electrical angle output by the hybrid active flux linkage observer to obtain the rotor position and rotational speed specifically includes:

[0024] The active flux linkage is determined based on the hybrid active flux linkage observer;

[0025] The active magnetic flux was determined to be in Components of the axis and in Components of the axis ;

[0026] Will and The rotor electrical angle is obtained by performing an arctangent treatment.

[0027] The rotor position and rotational speed are obtained based on the rotor electrical angle.

[0028] On the other hand, the present invention also provides a sensorless control device for an AC motor, the device comprising:

[0029] A construction module is used to construct an active flux linkage model, and based on the active flux linkage model, construct a voltage model and a current model, and then based on the voltage model and the current model, construct a hybrid active flux linkage observer;

[0030] An active flux linkage observer is used to output the rotor electrical angle, and then the rotor electrical angle is processed to obtain the rotor position and speed, which are then sent to the controller;

[0031] A controller is used to control an AC motor based on its rotor position and speed.

[0032] This invention provides a sensorless control method and apparatus for AC motors. Compared with existing technologies, this method first constructs an active flux linkage model; then, based on the active flux linkage model, it constructs a voltage model and a current model, and based on the voltage model and current model, it constructs a hybrid active flux linkage observer; next, it outputs the rotor electrical angle based on the hybrid active flux linkage observer, and then processes the rotor electrical angle to obtain the rotor position and speed; finally, it inputs the rotor position and speed to the controller so that the controller can control the AC motor. This application establishes a unified sensorless control method applicable to permanent magnet synchronous motors and asynchronous motors, which has stronger versatility, and improves the position observation accuracy and overall stability during low-speed and wide-speed-range operation through hybrid active flux linkage observation and extended state observation. Attached Figure Description

[0033] To more clearly illustrate the technical solutions in the embodiments or prior art of this specification, the drawings used in the description of the embodiments or prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this specification. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0034] Figure 1 The diagram shown is a schematic flowchart of a sensorless control method for an AC motor provided in an embodiment of this specification.

[0035] Figure 2 The diagram shown is a structural schematic of a sensorless control device for an AC motor provided in an embodiment of this specification. Detailed Implementation

[0036] To enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0037] like Figure 1 The diagram shown is a schematic flowchart of a sensorless control method for AC motors provided in the embodiments of this specification. Although this specification provides the method operation steps or device structure shown in the following embodiments or figures, based on convention or without creative effort, the method or device may include more or fewer operation steps or module units after partial combination. In steps or structures where there is no necessary causal relationship in logic, the execution order of these steps or the module structure of the device are not limited to the execution order or module structure shown in the embodiments or figures of this specification. When the method or module structure is applied in actual devices, servers, or terminal products, it can be executed sequentially or in parallel according to the method or module structure shown in the embodiments or figures (e.g., in a parallel processor or multi-threaded processing environment, or even in a distributed processing or server cluster implementation environment).

[0038] The sensorless control method for AC motors provided in the embodiments of this specification can be applied to various AC motors. The core of this application lies in: using active flux linkage as a unified modeling object, constructing a hybrid observer combining voltage and current models, and introducing extended state observation and inverter nonlinear adaptive compensation mechanisms on this basis, thereby achieving stable sensorless operation of AC motors such as asynchronous motors and permanent magnet synchronous motors over a wide speed range. Figure 1 As shown, the method specifically includes the following steps:

[0039] Step S101: Construct an active magnetic flux linkage model.

[0040] Specifically, this application establishes a unified active flux linkage model for AC motors, including permanent magnet synchronous motors and asynchronous motors. The stator flux linkage of the AC motor satisfies:

[0041] (1)

[0042] Where d is the differential symbol and t is time, equation (1) can be expressed as:

[0043] (2)

[0044] Based on the above equation, the active flux linkage model can also be expressed as:

[0045]

[0046] In the formula, It is an active magnetic flux linkage. For a permanent magnet synchronous motor, Equation 3 represents the active flux linkage model corresponding to the permanent magnet synchronous motor. For an asynchronous motor, Equation 4 represents the active flux linkage model corresponding to the asynchronous motor. For stator flux linkage, It is a quadrature axis inductor. For stator current, It is a direct-axis inductor. For direct-axis stator current, It is a permanent magnet flux linkage. For rotor electrical angle, For rotor inductance, For magnetizing inductance, The equivalent leakage inductance on the stator side is used to collect various parameters such as stator current and quadrature axis inductance through sampling units or sensors in existing technologies. This application does not impose specific restrictions on the sampling units or sensors.

[0047] Based on the above, this application unifies permanent magnet synchronous motors and asynchronous motors into the same active flux linkage analysis framework.

[0048] Step S102: Construct a voltage model and a current model based on the active flux linkage model, and construct a hybrid active flux linkage observer based on the voltage model and the current model.

[0049] Specifically, within the same active analysis framework, voltage and current models of the active flux linkage are established separately. The voltage model is shown below:

[0050] (5)

[0051] In the formula, The rate of change of active flux linkage under the voltage model. For stator resistance, The rate of change of stator current. This is the actual output voltage of the inverter.

[0052] The current model is shown below:

[0053] (6)

[0054] In the formula, The rate of change of active flux linkage under the current model. The rate of change of active flux linkage, The rotor's electric angular velocity, For a matrix, , This represents the active flux linkage under the current model.

[0055] Since the voltage model contains an integral element, it suffers from a critical stability problem, while the current model is more stable in the low-speed region. This application fuses the two to construct an active flux linkage observer, as shown below:

[0056] (7)

[0057] In the formula, For observation error, This is the estimated value of the active flux linkage. These are observations of active flux linkage under the current model. For the rate of change of the first-order extended observation, It is a second-order extended state variable. The rate of change of the observed active flux linkage under the voltage model. and All are observer gains. For the rate of change of the second-order extended observation, This is the reference voltage output by the controller.

[0058] Among them, active flux linkage and the like are conventional definitions. By fitting equation (6) into the standard extended state observer algorithm, equation (7) can be obtained, which is the active flux linkage observer in this application. Except for the parameters calculated by the recorded formula described in this application, the other parameters can be calculated in the existing scheme. The active flux linkage observer in this application uses the extended state variables to absorb parameter perturbations, sampling errors and external disturbances, thereby improving the robustness of the system and the observation stability under low-speed conditions.

[0059] It should be noted that line voltage sensors are typically not configured in actual systems; therefore, this invention uses the reference voltage output by the controller. It replaces the measured quantum voltage in the flux linkage calculation. However, the reference voltage... With the actual output voltage of the inverter Errors caused by inverter nonlinearity exist, therefore, corresponding compensation terms must be introduced in the observer and controller. To address this problem, this invention sets up an inverter nonlinearity adaptive compensation module to estimate the equivalent voltage error caused by dead-time effect and device nonlinearity online, and generate an initial compensation voltage. :

[0060] (8)

[0061] In the formula, To compensate for the estimated voltage amplitude, This is a vector composed of the sign functions of the three-phase currents. Updated online. It can achieve dynamic compensation for the nonlinear error of the inverter.

[0062] To avoid the overcompensation problem that easily occurs in traditional dead-zone compensation, this invention further introduces a nonlinear shaping function to smooth and correct the compensation voltage waveform. The expression is:

[0063] (9)

[0064] In the formula, The shape parameter of the integer function. This represents the stator phase current. Based on this shaping function, the final compensation voltage can be obtained. :

[0065] (10)

[0066] The final compensation voltage is superimposed on the reference voltage output by the controller to obtain the final modulation voltage command. This method can balance compensation capability and waveform smoothness, thereby reducing current harmonics, torque ripple, and additional losses caused by inaccurate compensation.

[0067] This invention updates the shaping parameters online based on the characteristic harmonics of the stator current. First, the stator current angle is defined. :

[0068] (11)

[0069] In the formula, Stator current Axial components, Stator current Axial components.

[0070] Then, the d-axis current in the stator current orientation coordinate system is obtained. Then, its 6th, 12th, and 18th harmonic components are extracted to construct the shape parameter update law:

[0071] (12)

[0072] Where c1, c2, and c3 are weighting coefficients. Using this method, the shape of the compensation waveform can be automatically adjusted according to the harmonic level, further optimizing the compensation effect.

[0073] Step S103: Based on the rotor electrical angle output by the hybrid active flux observer, the rotor electrical angle is then processed to obtain the rotor position and rotational speed.

[0074] In this embodiment of the application, the step of processing the rotor electrical angle output by the hybrid active flux linkage observer to obtain the rotor position and rotational speed specifically includes:

[0075] The active flux linkage is determined based on the hybrid active flux linkage observer;

[0076] The active magnetic flux was determined to be in Components of the axis and in Components of the axis ;

[0077] Will and The rotor electrical angle is obtained by performing an arctangent treatment.

[0078] The rotor position and rotational speed are obtained based on the rotor electrical angle.

[0079] Specifically, each component can be calculated using methods commonly used in the field. Regarding position and rotational speed extraction, this application extracts the rotor electrical angle based on the component of the active magnetic flux in the stationary coordinate system, i.e.:

[0080] (13)

[0081] Furthermore, a smooth speed estimate is output by combining the T2S phase-locked loop. This position extraction structure improves the stability and anti-interference capability of phase extraction, and finally feeds the estimated rotor position and speed back to the controller, forming a complete sensorless closed-loop control system. This is a common solution in the field. The rotor electrical angle obtained after arctangent is passed through a second-order phase-locked loop to output a relatively smooth speed and rotor position. The second-order phase-locked loop is equivalent to a low-pass filter. The rotor position and rotor speed can also be obtained by processing the rotor electrical angle in other ways, and there are no specific limitations on this.

[0082] Step S104: Input the rotor position and speed to the controller so that the controller can control the AC motor.

[0083] Specifically, this application does not impose any particular limitation on the controller; any type of controller in the art can realize the function of controlling the AC motor according to the rotor position and speed.

[0084] This application proposes a unified sensorless control technology solution applicable to both asynchronous motors and permanent magnet synchronous motors. This solution involves unified active flux linkage modeling, hybrid active flux linkage observation, extended state disturbance absorption, inverter nonlinear adaptive compensation, and phase-locked loop position extraction. It can effectively improve the stability, observation accuracy, and engineering applicability of the system in a wide speed range, especially in low-speed conditions.

[0085] Based on the above-described sensorless control method for AC motors, one or more embodiments of this specification also provide a platform or terminal for sensorless control of AC motors. This platform or terminal may include devices, software, modules, plug-ins, servers, clients, etc., using the methods described in the embodiments of this specification, combined with necessary hardware implementation devices. Based on the same innovative concept, the systems in one or more embodiments provided in this specification are as described in the following embodiments. Since the implementation schemes and methods for solving the system problem are similar, the specific system implementation in the embodiments of this specification can refer to the implementation of the aforementioned methods. Repeated descriptions will not be repeated. The terms "unit" or "module" used below can refer to a combination of software and / or hardware that achieves a predetermined function. Although the systems described in the following embodiments are preferably implemented in software, hardware implementation, and a combination of software and hardware, are also possible and contemplated.

[0086] Specifically, Figure 2 This is a schematic diagram of the module structure of one embodiment of the sensorless control device for AC motors provided in this specification, as shown below. Figure 2 As shown, the sensorless control device for AC motors provided in this specification includes:

[0087] The construction module 201 is used to construct an active flux linkage model, and construct a voltage model and a current model based on the active flux linkage model, and then construct a hybrid active flux linkage observer based on the voltage model and the current model;

[0088] An active flux linkage observer 202 is used to output the rotor electrical angle, and then process the rotor electrical angle to obtain the rotor position and speed and send them to the controller;

[0089] Controller 203 is used to control the AC motor according to the rotor position and speed.

[0090] It should be noted that the system described above may include other implementation methods based on the description of the corresponding method embodiments. The specific implementation methods can be referred to the description of the corresponding method embodiments above, and will not be elaborated here.

[0091] This application also provides an electronic device, including:

[0092] processor;

[0093] Memory used to store the processor's executable instructions;

[0094] The processor is configured to perform the methods provided in the embodiments described above.

[0095] The electronic device provided in this application stores executable instructions of the processor in a memory. When the processor executes the executable instructions, it first constructs an active flux linkage model; then, based on the active flux linkage model, it constructs a voltage model and a current model, and based on the voltage model and current model, it constructs a hybrid active flux linkage observer; next, it outputs the rotor electrical angle based on the hybrid active flux linkage observer, and then processes the rotor electrical angle to obtain the rotor position and speed; finally, it inputs the rotor position and speed to the controller so that the controller controls the AC motor. This application establishes a unified sensorless control method applicable to permanent magnet synchronous motors and asynchronous motors, which is more versatile, and improves the position observation accuracy and overall stability during low-speed and wide-speed-range operation by using hybrid active flux linkage observation and extended state observation.

[0096] The foregoing has described specific embodiments of this specification. Other embodiments are within the scope of the appended claims. In some cases, the actions or steps recited in the claims may be performed in a different order than that shown in the embodiments and may still achieve the desired result. Furthermore, the processes depicted in the drawings do not necessarily require the specific or sequential order shown to achieve the desired result. In some embodiments, multitasking and parallel processing are possible or may be advantageous.

[0097] The methods or apparatus described in the embodiments provided in this specification can implement business logic through a computer program and record it on a storage medium. The storage medium can be read and executed by a computer to achieve the effects of the solutions described in the embodiments of this specification, such as:

[0098] Construct an active magnetic flux linkage model;

[0099] A voltage model and a current model are constructed based on the active flux linkage model, and a hybrid active flux linkage observer is constructed based on the voltage model and the current model.

[0100] The rotor electrical angle is output based on the hybrid active flux linkage observer, and then the rotor electrical angle is processed to obtain the rotor position and rotational speed.

[0101] The rotor position and speed are input to the controller so that the controller can control the AC motor.

[0102] The storage medium can include physical devices for storing information, typically digitizing the information and then storing it using electrical, magnetic, or optical methods. The storage medium can include: devices that store information using electrical energy, such as various types of memory, like RAM and ROM; devices that store information using magnetic energy, such as hard disks, floppy disks, magnetic tapes, magnetic core memory, bubble memory, and USB flash drives; and devices that store information using optical methods, such as CDs or DVDs. Of course, there are other readable storage media, such as quantum memories and graphene memories.

[0103] The embodiments in this specification are not limited to conforming to industry communication standards, standard computer resource data update and data storage rules, or the situations described in one or more embodiments of this specification. Slightly modified implementations based on certain industry standards or custom methods or embodiments can also achieve the same, equivalent, or similar, or predictable, implementation effects as described above. Embodiments that utilize these modified or modified methods for data acquisition, storage, judgment, and processing still fall within the scope of optional implementations of the embodiments in this specification.

[0104] The controller can be implemented in any suitable manner. For example, it can take the form of a microprocessor or processor and a computer-readable medium storing computer-readable program code (e.g., software or firmware) executable by the (micro)processor, logic gates, switches, application-specific integrated circuits (ASICs), programmable logic controllers, and embedded microcontrollers. Examples of controllers include, but are not limited to, the following microcontrollers: ARC 625D, Atmel AT91SAM, Microchip PIC18F26K20, and Silicon Labs C8051F320. A memory controller can also be implemented as part of the control logic of the memory. Those skilled in the art will also recognize that, in addition to implementing the controller in purely computer-readable program code form, the same functionality can be achieved by logically programming the method steps to make the controller take the form of logic gates, switches, ASICs, programmable logic controllers, and embedded microcontrollers. Therefore, such a controller can be considered a hardware component, and the means included therein for implementing various functions can also be considered as structures within the hardware component. Alternatively, the means for implementing various functions can be considered as both software modules implementing the method and structures within the hardware component.

[0105] The device embodiments described above are merely illustrative. For example, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or plug-ins may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some interfaces, indirect coupling or communication connection between devices or units, and may be electrical, mechanical, or other forms.

[0106] These computer program instructions can also be loaded onto a computer or other programmable resource data updating device, causing a series of operational steps to be performed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable device for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0107] The various embodiments in this specification are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, system embodiments are basically similar to method embodiments, so the description is relatively simple; relevant parts can be referred to the descriptions in the method embodiments. In the description of this specification, the terms "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of this specification. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described can be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification and the features of different embodiments or examples.

[0108] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A sensorless control method for AC motors, characterized in that, The method includes: Construct an active magnetic flux linkage model; A voltage model and a current model are constructed based on the active flux linkage model, and a hybrid active flux linkage observer is constructed based on the voltage model and the current model. The rotor electrical angle is output based on the hybrid active flux linkage observer, and then the rotor electrical angle is processed to obtain the rotor position and rotational speed. The rotor position and speed are input to the controller so that the controller can control the AC motor.

2. The sensorless control method for AC motors as described in claim 1, characterized in that, The active flux linkage model specifically includes: ； In the formula, It is an active magnetic flux linkage. It is a permanent magnet synchronous motor. It is an asynchronous motor. For stator flux linkage, It is a quadrature axis inductor. For stator current, It is a direct-axis inductor. For direct-axis stator current, It is a permanent magnet flux linkage. For rotor electrical angle, For rotor inductance, For magnetizing inductance, This is the equivalent leakage inductance on the stator side.

3. The sensorless control method for AC motors as described in claim 2, characterized in that, The voltage model is as follows: ； In the formula, The rate of change of active flux linkage under the voltage model. For stator resistance, The rate of change of stator current. This is the actual output voltage of the inverter.

4. The sensorless control method for AC motors as described in claim 3, characterized in that, The specific current model is as follows: ； In the formula, The rate of change of active flux linkage under the current model. The rate of change of active flux linkage, The rotor's electric angular velocity, For a matrix, , This represents the active flux linkage under the current model.

5. The sensorless control method for AC motors as described in claim 4, characterized in that, The hybrid active flux linkage observer is specifically shown below: ； In the formula, For observation error, This is an estimate of the active flux linkage. These are observations of active flux linkage under the current model. For the rate of change of the first-order extended observation, It is a second-order extended state variable. The rate of change of the observed active flux linkage under the voltage model. and All are observer gains. For the rate of change of the second-order extended observation, This is the reference voltage output by the controller.

6. The sensorless control method for AC motors as described in claim 5, characterized in that, The method further includes calculating the final compensation voltage and adding the final compensation voltage to the reference voltage output by the controller as the final value. .

7. The sensorless control method for AC motors as described in claim 1, characterized in that, The process of obtaining the rotor position and speed by processing the rotor electrical angle output from the hybrid active flux linkage observer specifically includes: The active flux linkage is determined based on the hybrid active flux linkage observer; The active magnetic flux was determined to be in Components of the axis and in Components of the axis ; Will and The rotor electrical angle is obtained by performing an arctangent treatment. The rotor position and rotational speed are obtained based on the rotor electrical angle.

8. A sensorless control device for an AC motor, characterized in that, The device includes: A construction module is used to construct an active flux linkage model, and based on the active flux linkage model, construct a voltage model and a current model, and then based on the voltage model and the current model, construct a hybrid active flux linkage observer; An active flux linkage observer is used to output the rotor electrical angle, and then the rotor electrical angle is processed to obtain the rotor position and speed, which are then sent to the controller; A controller is used to control an AC motor based on its rotor position and speed.

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