CN108628159A - A kind of PID control method, device, equipment and system - Google Patents

Abstract

This application discloses a PID control method, including: making a difference between the input signal and the output signal of the controlled object to obtain an error signal; judging whether the error signal is greater than a preset error threshold; if so, using the first The PID control algorithm calculates the first PID control quantity as the total control quantity; if not, the second PID control algorithm that cancels integral adjustment is used to calculate the second PID control quantity, and the preset compensation algorithm is used to calculate the compensation control quantity, and the second PID The sum of the control quantity and the compensation control quantity is taken as the total control quantity; the total control quantity is output to the controlled object so as to adjust the output signal of the controlled object. The PID control method provided by the application has fast adjustment speed, high control precision and good control performance. The present application also discloses a PID control device, equipment, system and computer-readable storage medium, which also have the above beneficial effects.

Description

A PID control method, device, equipment and system

technical field

The present application relates to the technical field of automatic control, in particular to a PID control method, device, equipment, system and computer-readable storage medium.

Background technique

PID control has the advantages of simple structure, good stability, and reliable operation. It is still the most commonly used control method in industrial process control.

The classic PID control method includes proportional regulation, integral regulation and differential regulation at the same time. Since the integral regulation often causes problems of overshoot or oscillation, an integral-separated PID control method appears in the prior art. The main principle is that the integral adjustment is not used when the error is large, so as to avoid the control amount being too large due to the accumulation of errors; and the integral adjustment is used when the error is small, so as to use the advantage of the integral adjustment to achieve no static error control. However, this method cancels the function of integral adjustment in the early stage of adjustment, that is, when the error is large, thus reducing the adjustment speed of the system.

It can be seen that what kind of PID control method to use to effectively increase the adjustment speed while reducing overshoot and oscillation and realizing non-difference control is a technical problem to be solved urgently by those skilled in the art.

Contents of the invention

The purpose of the present application is to provide a PID control method, device, equipment, system and computer-readable storage medium, so as to effectively increase the adjustment speed while reducing overshoot and oscillation and realizing non-difference control.

In order to solve the above technical problems, the application provides a PID control method, including:

Make a difference between the input signal and output signal of the controlled object to obtain the error signal;

judging whether the error signal is greater than a preset error threshold;

If so, then adopt the first PID control algorithm with integral adjustment to calculate the first PID control quantity so as to be the total control quantity;

If not, then use the second PID control algorithm that cancels the integral adjustment to calculate the second PID control amount, use the preset compensation algorithm to calculate the compensation control amount, and use the sum of the second PID control amount and the compensation control amount as the total control volume;

outputting the total control amount to the controlled object, so as to adjust the output signal of the controlled object.

Optionally, the preset error threshold is e _f =k ₁ r(t);

Wherein, _ef is the preset error threshold; k ₁ is the preset error threshold coefficient; r(t) is the input signal.

Optionally, the preset error threshold coefficient k ₁ ∈[0,0.2].

Optionally, the calculation of the compensation control amount using a preset compensation algorithm includes:

Calculate the compensation control amount according to u _b = k ₂ u ₀ ;

Wherein, u _b is the compensation control amount; k ₂ is the preset compensation coefficient; u ₀ is the preset basic compensation amount corresponding to the controlled object.

Optionally, the transfer function of the controlled object is

Among them, b ₀ is the constant term of the numerator; b is the constant term of the denominator; a is the coefficient of the primary term of the denominator;

The preset basic compensation amount is u ₀ =b/b ₀ .

Optionally, both the first PID control algorithm and the second PID control algorithm are parameter self-tuning PID control algorithms.

The application also provides a PID control device, comprising:

Obtaining module: used to make a difference between the input signal and output signal of the controlled object to obtain the error signal;

Judging module: used to judge whether the error signal is greater than a preset error threshold;

Calculation module: when the error signal is greater than the preset error threshold, the first PID control algorithm with integral control is used to calculate the first PID control quantity as the total control quantity; when the error signal is not greater than the preset error threshold When the preset error threshold is mentioned above, the second PID control amount is calculated by using the second PID control algorithm that cancels the integral control, and the compensation control amount is calculated by using the preset compensation algorithm, and the second PID control amount and the compensation control amount are calculated. and as said total control amount;

Output module: used to output the total control quantity to the controlled object, so as to adjust the output signal of the controlled object.

The application also provides a PID control device, including:

memory: used to store computer programs;

Processor: used to execute the computer program to implement the steps of any one of the above PID control methods.

The present application also provides a computer-readable storage medium, where a computer program is stored in the computer-readable storage medium, and when the computer program is executed by a processor, it is used to implement the steps of any one of the above-mentioned PID control methods .

The present application also provides a PID control system, including a controlled object and the above-mentioned PID control device.

The PID control method provided by the present application includes: making a difference between the input signal and the output signal of the controlled object to obtain an error signal; judging whether the error signal is greater than a preset error threshold; The PID control algorithm calculates the first PID control quantity so as to be the total control quantity; if not, the second PID control algorithm that cancels the integral adjustment is used to calculate the second PID control quantity, and the preset compensation algorithm is used to calculate the compensation control quantity, and the first PID control quantity is calculated. The sum of the two PID control quantities and the compensation control quantity is used as the total control quantity; the total control quantity is output to the controlled object, so as to adjust the output signal of the controlled object.

It can be seen that, compared with the prior art, the PID control method provided by the present application puts the integral adjustment when the error signal is large, which effectively speeds up the response speed of the system; and when the error signal is small, the integral adjustment is canceled The PID control is compensated, and the static error is effectively eliminated while reducing overshoot and oscillation. It can be seen that the PID control method provided by the present application has fast adjustment speed, high control precision and good control performance. The PID control device, equipment, system and computer-readable storage medium provided by the present application can realize the above-mentioned PID control method, and also have the above-mentioned beneficial effects.

Description of drawings

In order to illustrate the prior art and the technical solutions in the embodiments of the present application more clearly, the following briefly introduces the drawings that need to be used in the description of the prior art and the embodiments of the present application. Of course, the following drawings related to the embodiments of the application describe only a part of the embodiments of the application, and those of ordinary skill in the art can obtain other The accompanying drawings, and other obtained drawings also belong to the protection scope of the present application.

Fig. 1 is the flowchart of a kind of PID control method provided by the present application;

Fig. 2 is the control block diagram of a kind of PID control method provided by the present application;

Fig. 3 is the control effect waveform diagram of the PID control method provided by the present application under the first group of PID control parameters;

Fig. 4 is the control effect waveform diagram of the PID control method provided by the present application under the second group of PID control parameters;

Fig. 5 is the waveform diagram of the control effect of the PID control method provided by the present application under the third group of PID control parameters;

Fig. 6 is the waveform diagram of the control effect of the PID control method provided by the present application under the fourth group of PID control parameters;

Fig. 7 is the waveform diagram of the control effect of the PID control method provided by the present application under the condition that the transfer function of the controlled object is inaccurate;

FIG. 8 is a waveform diagram of the control effect of the PID control method provided by the present application after adjusting the preset compensation coefficient for FIG. 8;

FIG. 9 is a structural block diagram of a PID control device provided by the present application.

Detailed ways

The core of the present application is to provide a PID control method, device, equipment, system and computer-readable storage medium, so as to effectively increase the adjustment speed while reducing overshoot and oscillation and realizing non-difference control.

In order to describe the technical solutions in the embodiments of the present application more clearly and completely, the technical solutions in the embodiments of the present application will be introduced below in conjunction with the drawings in the embodiments of the present application. Apparently, the described embodiments are only some of the embodiments of this application, not all of them. Based on the embodiments in this application, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the scope of protection of this application.

The PID control method provided in this application is specifically a PID control method using integral separation and compensation control. Different from the integral separation in the prior art, the present application does not cancel but puts into integral adjustment when the error is large, so as to make the system have a faster response speed and speed up the adjustment process. On the contrary, the present application specifically cancels the integral adjustment when the error is small, so as to reduce the occurrence of overshoot or oscillation, and at the same time combines the compensation control to compensate the PID control quantity after integral separation, so as to eliminate the steady-state error.

Please refer to Fig. 1, Fig. 1 is the flow chart of a kind of PID control method provided by the present application, mainly comprises the following steps:

Step 1: Make a difference between the input signal and the output signal of the controlled object to obtain the error signal.

Specifically, assuming that the input signal of the control system, that is, the expected output, is r(t), and the output signal is y(t), the error signal can be obtained as e(t)=r(t)-y(t).

Step 2: Judging whether the error signal is greater than a preset error threshold; if yes, proceed to step 3; if not, proceed to step 4.

Specifically, in the PID control method provided in the present application, a preset error threshold e _f is specifically set to determine the relative magnitude of the error signal e(t). When the error signal e(t) is greater than the preset error threshold e _f , it means that the current error signal e(t) is relatively large, and the integral adjustment function needs to be put into use, so step 3 can be entered. When the error signal e(t) is not greater than the preset error threshold e _f , it means that the current error signal e(t) is relatively small, and it is necessary to cancel the integral adjustment function and add compensation control, so that step 4 can be entered.

Of course, the specific value of the preset error threshold e _f mentioned here can be selected and set by those skilled in the art according to actual application conditions, which is not limited in this embodiment of the present application. For example, it is easy to understand that the preset error threshold e _f can specifically be an amount related to the input signal r(t), that is, when the set input signal r(t) increases, it acts as input or cancellation of the integral adjustment function The preset error threshold e _f of the limit value of can also be appropriately increased.

Step 3: Using the first PID control algorithm with integral adjustment to calculate the first PID control quantity as the total control quantity, go to step 5.

Specifically, when it is judged that the error signal e(t) is relatively large and the integral adjustment is put into operation, the specific expression of the first PID control algorithm is

Wherein, u _{pid_1 is} the so-called first PID control quantity; k _p , _ki and k _d are proportional coefficients, integral coefficients and differential coefficients respectively.

Since the input of the integral adjustment can eliminate the static error, at this time, the first PID control variable u _{pid_1} can be directly used as the total control variable u, that is, u=u _{pid_1} .

Step 4: Use the second PID control algorithm that cancels the integral adjustment to calculate the second PID control quantity, use the preset compensation algorithm to calculate the compensation control quantity, use the sum of the second PID control quantity and the compensation control quantity as the total control quantity, and go to step 5 .

Specifically, when it is determined that the error signal e(t) is small and the integral adjustment is canceled, the specific expression of the second PID control algorithm is

Wherein, u _{pid_2 is} the so-called second PID control quantity.

When the error signal e(t) is small, it means that the output signal y(t) of the control system is relatively close to the input signal r(t) at this time, and the difference is small, and it is easier to produce overshoot or oscillation. Canceling the integral adjustment function at this time can effectively reduce the generation of overshoot or oscillation.

In addition, in order to achieve no-difference control in the case of canceling the integral adjustment, this application also introduces a compensation control _{variable ub, and the sum of the compensation control variable ub and the second PID control variable u pid_2 is taken} as the total control variable u:

u = u _{pid_2} + u _b .

Of course, as for the specific calculation method and value of the compensation control amount _ub , those skilled in the art can choose and set it by themselves, which is not limited in this embodiment of the present application.

Step 5: Output the total control quantity to the controlled object, so as to adjust the output signal of the controlled object.

After the total control quantity u is determined, it can be output to the controlled object so that the output signal y(t) is gradually stabilized near the input signal r(t) under its control.

In fact, in practical applications, those skilled in the art can freely adopt various flexible ways to implement the above method. For example, please refer to FIG. 2 , which is a control block diagram of a PID control method provided in the present application. As shown in Figure 2, the integral adjustment switching controller is used to judge whether the error signal e(t) is greater than the preset error threshold e _f , and specifically generate an integral adjustment switching signal x(t):

When the integral adjustment switching signal x(t) is 1, it means that the integral adjustment needs to be put into use; when the integral adjustment switching signal x(t) is 0, it means that the integral adjustment needs to be cancelled. Correspondingly, said first PID control algorithm and second PID control algorithm can be realized by the PID controller in Fig. 2, then its specific expression is

At the same time, the switching signal x(t) of the integral adjustment is also the switching signal of the compensation control. When the integral adjustment switching signal x(t) is 0 to cancel the integral adjustment, the compensation control needs to be activated. The control quantity compensator in Fig. 2 is used to decide whether to output the compensation control quantity _ub according to the integral adjustment switching signal x(t). Therefore, the total control quantity u input to the controlled object can be expressed as:

It can be seen that the PID control method provided by the embodiment of the present application puts into the integral adjustment when the error signal is large, which effectively speeds up the response speed of the system; and when the error signal is small, the PID control after the integral adjustment is cancelled. Compensation effectively eliminates static errors while reducing overshoot and oscillation. It can be seen that the PID control method provided by the present application has fast adjustment speed, high control precision and good control performance.

The PID control method provided by this application, on the basis of the foregoing embodiments:

As a preferred embodiment, the preset error threshold is e _f =k ₁ r(t);

Among them, e _f is the preset error threshold; k ₁ is the preset error threshold coefficient; r(t) is the input signal.

Specifically, as mentioned above, the preset error threshold e _f may be an amount associated with the input signal r(t), and specifically may be a proportional relationship. Those skilled in the art can find a more suitable preset error threshold e _f by adjusting the size of the preset error threshold coefficient k ₁ .

As a preferred embodiment, the error threshold coefficient k ₁ ∈[0,0.2] is preset.

Specifically, on the basis of multiple simulations and experiments, the embodiment of the present application provides a reference range of values for the preset error threshold coefficient k ₁ : k ₁ ∈[0,0.2]. Certainly, those skilled in the art may also select values within other ranges, which are not limited in this application.

As a preferred embodiment, using a preset compensation algorithm to calculate the compensation control amount includes:

Calculate the compensation control amount according to u _b = k ₂ u ₀ ;

Among them, u _b is the compensation control amount; k ₂ is the preset compensation coefficient; u ₀ is the preset basic compensation amount corresponding to the controlled object.

Specifically, when calculating the compensation control amount _ub , it is recommended but not limited to set a preset basic compensation amount u ₀ corresponding to the controlled object, and obtain the compensation control through linear calculation with the preset compensation coefficient k ₂ Measure u _b . Those skilled in the art should know that the controlled objects with different types of transfer functions have different final steady-state errors even when the input signal r(t) is the same and the control parameters are the same. Therefore, the corresponding preset basic compensation amount u ₀ can be set for different controlled objects, and on this basis, a more appropriate compensation control amount u _b can be determined by adjusting the preset compensation coefficient k ₂ to realize no-difference control.

As a preferred embodiment, the transfer function of the controlled object is

Among them, b ₀ is the constant term of the numerator; b is the constant term of the denominator; a is the coefficient of the primary term of the denominator;

The preset basic compensation amount is u ₀ =b/b ₀ .

Specifically, when the controlled object is a second-order system, the form of the corresponding transfer function is At this time, preferably, the preset basic compensation amount u ₀ may be taken as u ₀ =b/b ₀ .

As a preferred embodiment, both the first PID control algorithm and the second PID control algorithm are parameter self-tuning PID control algorithms.

Specifically, for PID control, the values of PID control parameters (proportional coefficient k _p , integral coefficient _ki and differential coefficient k _d ) have important decisive significance for control performance. Therefore, in order to obtain more suitable PID control parameters, in the PID control method provided by the present application, the first PID control algorithm and the second PID control algorithm can be further parameter self-tuning PID control algorithms, so that by Parameter tuning effectively improves control accuracy and control performance.

Please refer to Fig. 3 to Fig. 8, Fig. 3 to Fig. 8 are respectively the control effect wave diagrams of the PID control method provided in this application under different circumstances, and the controlled objects corresponding to them are all second-order systems, and the transfer function is specifically The input signal r(t) is a unit step signal, and some related parameters are set as follows: the preset error threshold coefficient is k ₁ =0.2, and the preset basic compensation amount is u ₀ =4. In addition, the preset compensation coefficient corresponding to FIG. 3 to FIG. 7 is k ₂ =1, and the preset compensation coefficient corresponding to FIG. 8 is k ₂ =1.25.

In addition to the PID control method provided by the present application, Fig. 3 to Fig. 8 also show other control methods including classical PID and existing integral-separation PID for the convenience of comparative analysis.

Specifically, FIG. 3 is a control effect waveform diagram when the first group of PID control parameters is adopted, and the PID control parameters are specifically: k _p =20, _ki =20, k _d =3. As can be seen from Fig. 3, the PID control method with integral separation and compensation control provided by the present application (hereinafter referred to as integral separation+compensation PID) can realize fast response and no overshoot; while there is overshoot in classical PID, At the same time, the existing integral separation type PID not only has a large overshoot, but also has a slow response.

Fig. 4 is a waveform diagram of the control effect when the second group of PID control parameters is adopted, and the PID control parameters are specifically: k _p =20, _ki =10, k _d =0.01. It can be seen from Figure 4 that the classic PID and the existing integral-separated PID cannot make the system output reach the desired position, that is, there is a static error; while the integral-separated + compensation PID provided by this application has an overshoot phenomenon, but the overshoot Adjustment is small, and there is no static error.

Fig. 5 is a waveform diagram of the control effect when the third group of PID control parameters is adopted, and the PID control parameters are specifically: k _p =10, _ki =20, k _d =0.01. As can be seen from Figure 5, the integral separation + compensation PID provided by this application has smaller overshoot compared with the classic PID; while the existing integral separation PID is due to unreasonable parameter setting (k _p is small) And making the integral adjustment completely ineffective is equivalent to PD control.

Fig. 6 is a waveform diagram of the control effect when the fourth group of PID control parameters is adopted, and the PID control parameters are specifically: k _p =5, _ki =50, k _d =1. It can be seen from Figure 6 that at this time, due to the large integral coefficient setting, the integral adjustment in the classic PID has a serious impact on the system stability, and the existing integral-separated PID completely fails the integral adjustment due to unreasonable parameter settings ; while the integral separation + compensation PID provided by this application has achieved a better control effect.

From Figure 3 to Figure 6, it can be seen that under the control of the same set of PID control parameters, the integral separation + compensation PID provided by this application can achieve excellent control performance that is significantly better than the classic PID and the existing integral separation PID .

FIG. 7 is a waveform diagram of the control effect of the PID control method provided by the present application under the condition that the transfer function of the controlled object is out of alignment. Suppose the actual transfer function is of accused persons were mistaken for And use the first set of control parameters corresponding to Figure 3 to adjust, and the obtained control effect is shown in Figure 7. It can be seen from Fig. 7 that the integral separation + compensation PID method using k ₂ =1 provided by the present application has a large static error at this time, so the static error can be eliminated by adjusting the control variable compensation coefficient k ₂ .

FIG. 8 is a control effect waveform diagram of the PID control method provided in the present application after adjusting the preset compensation coefficient with respect to FIG. 7 . Specifically, the preset compensation coefficient is adjusted to k ₂ =1.25. It can be seen from Fig. 8 that by adjusting the preset compensation coefficient k ₂ to improve the effect of compensation control, the static error can be successfully eliminated and no-difference control can be realized.

The PID control device provided by the embodiment of the present application is introduced below.

Please refer to FIG. 9. FIG. 9 is a structural block diagram of a PID control device provided by the present application; including an acquisition module 1, a judgment module 2, a calculation module 3 and an output module 4;

The acquisition module 1 is used to make a difference between the input signal and the output signal of the controlled object to obtain an error signal;

The judging module 2 is used to judge whether the error signal is greater than a preset error threshold;

The calculation module 3 is used to calculate the first PID control quantity by using the first PID control algorithm with integral control as the total control quantity when the error signal is greater than the preset error threshold; when the error signal is not greater than the preset error threshold, use cancel the second PID control algorithm of the integral control to calculate the second PID control quantity, use the preset compensation algorithm to calculate the compensation control quantity, and use the sum of the second PID control quantity and the compensation control quantity as the total control quantity;

The output module 4 is used to output the total control quantity to the controlled object, so as to adjust the output signal of the controlled object.

It can be seen that the PID control device provided by this application puts into integral adjustment when the error signal is large, which effectively speeds up the response speed of the system; and when the error signal is small, the PID control after the integral adjustment is cancelled, Static errors are effectively eliminated while reducing overshoot and oscillation. It can be seen that the PID control device provided by the present application has fast adjustment speed, high control precision and good control performance.

The application also provides a PID control device, including:

memory: used to store computer programs;

Processor: used to execute the computer program to implement the steps of any one of the above PID control methods.

The present application also provides a computer-readable storage medium, where a computer program is stored in the computer-readable storage medium, and when the computer program is executed by a processor, it is used to implement the steps of any one of the above-mentioned PID control methods .

The present application also provides a PID control system, including a controlled object and the above-mentioned PID control device.

The specific implementation manners of the PID control device, equipment, system, and computer-readable storage medium provided in the present application and the PID control method described above can be referred to each other correspondingly, so details are not repeated here.

Each embodiment in the present application is described in a progressive manner, each embodiment focuses on the differences from other embodiments, and the same and similar parts of the various embodiments can be referred to each other. As for the device disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and for the related information, please refer to the description of the method part.

It should also be noted that in this application, relative terms such as "first" and "second" are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between such entities or operations. Furthermore, the term "comprises", "comprises" or any other variation thereof is intended to cover a non-exclusive inclusion such that a process, method, article or apparatus comprising a set of elements includes not only those elements, but also includes elements not expressly listed. other elements of or also include elements inherent in such a process, method, article, or device. Without further limitations, an element defined by the phrase "comprising a ..." does not exclude the presence of additional identical elements in the process, method, article or apparatus comprising said element.

The technical solution provided by the present application has been introduced in detail above. In this paper, specific examples are used to illustrate the principles and implementation methods of the present application, and the descriptions of the above embodiments are only used to help understand the methods and core ideas of the present application. It should be pointed out that those skilled in the art can make some improvements and modifications to the application without departing from the principles of the application, and these improvements and modifications also fall within the protection scope of the claims of the application.

Claims (10)

1. A PID control method, characterized in that, comprising: Make a difference between the input signal and output signal of the controlled object to obtain the error signal; judging whether the error signal is greater than a preset error threshold; If so, then adopt the first PID control algorithm with integral adjustment to calculate the first PID control quantity so as to be the total control quantity; If not, then use the second PID control algorithm that cancels the integral adjustment to calculate the second PID control amount, use the preset compensation algorithm to calculate the compensation control amount, and use the sum of the second PID control amount and the compensation control amount as the total control volume; outputting the total control amount to the controlled object, so as to adjust the output signal of the controlled object. 2. PID control method according to claim 1, is characterized in that, described preset error threshold is e _f =k ₁ r(t); Wherein, _ef is the preset error threshold; k ₁ is the preset error threshold coefficient; r(t) is the input signal. 3. The PID control method according to claim 2, characterized in that the preset error threshold coefficient k ₁ ∈[0,0.2]. 4. The PID control method according to claim 1, wherein said calculating the compensation control quantity using a preset compensation algorithm comprises: Calculate the compensation control amount according to u _b = k ₂ u ₀ ; Wherein, u _b is the compensation control amount; k ₂ is the preset compensation coefficient; u ₀ is the preset basic compensation amount corresponding to the controlled object. 5. PID control method according to claim 4, is characterized in that, the transfer function of described controlled object is Among them, b ₀ is the constant term of the numerator; b is the constant term of the denominator; a is the coefficient of the primary term of the denominator; The preset basic compensation amount is u ₀ =b/b ₀ . 6. The PID control method according to any one of claims 1 to 5, characterized in that, both the first PID control algorithm and the second PID control algorithm are parameter self-tuning PID control algorithms. 7. A PID control device, characterized in that, comprising: Obtaining module: used to make a difference between the input signal and output signal of the controlled object to obtain the error signal; Judging module: used to judge whether the error signal is greater than a preset error threshold; Calculation module: when the error signal is greater than the preset error threshold, the first PID control algorithm with integral control is used to calculate the first PID control quantity as the total control quantity; when the error signal is not greater than the preset error threshold When the preset error threshold is mentioned above, the second PID control amount is calculated by using the second PID control algorithm that cancels the integral control, and the compensation control amount is calculated by using the preset compensation algorithm, and the second PID control amount and the compensation control amount are calculated. and as said total control amount; Output module: used to output the total control quantity to the controlled object, so as to adjust the output signal of the controlled object. 8. A PID control device, characterized in that, comprising: memory: used to store computer programs; Processor: used to execute the computer program to realize the steps of the PID control method according to any one of claims 1 to 6. 9. A computer-readable storage medium, characterized in that, a computer program is stored in the computer-readable storage medium, and when the computer program is executed by a processor, it is used to implement any one of claims 1 to 6. The steps of the PID control method. 10. A PID control system, comprising a controlled object and the PID control device according to claim 8.