CN113220048B - Boiler temperature adjusting method and system based on numerical differentiation - Google Patents

Boiler temperature adjusting method and system based on numerical differentiation Download PDF

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CN113220048B
CN113220048B CN202110605520.5A CN202110605520A CN113220048B CN 113220048 B CN113220048 B CN 113220048B CN 202110605520 A CN202110605520 A CN 202110605520A CN 113220048 B CN113220048 B CN 113220048B
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temperature
boiler
differentiator
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CN113220048A (en
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王飚
霍梁
郑乐
林少军
柯吉
武飞
赵微微
杨雨仪
马雨庆
董伊媚
唐必成
赵醒
秦裕德
吴浩
杨航
崔冰晶
吕亚泽
陈东瑞
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Changan University
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    • G05CONTROLLING; REGULATING
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    • G05D23/20Control of temperature characterised by the use of electric means with sensing elements having variation of electric or magnetic properties with change of temperature

Abstract

The invention discloses a boiler temperature adjusting method and system based on numerical differentiation, the calculated amount of a differentiator realized by a numerical differentiation algorithm based on Newton interpolation polynomial is small, the differentiator realized by the method has high accuracy of a differentiation link, good frequency characteristic, good effect of inhibiting high-frequency noise, simple structure and easy realization. Because an ideal differentiator does not exist in actual life and is not easy to realize physically, and temperature sampling points in a boiler system are often distributed equidistantly, the invention provides a numerical differentiation algorithm based on a first derivative of Newton interpolation polynomial, aiming at realizing better differentiation effect by selecting proper interpolation points and sampling step length.

Description

Boiler temperature adjusting method and system based on numerical differentiation
Technical Field
The invention belongs to the field of boiler temperature control, and relates to a boiler temperature adjusting method and system based on numerical differentiation.
Background
In the boiler temperature control system, the temperature is an inertial system having a delay characteristic as a controlled object. A differentiator is needed in the traditional boiler temperature control PID control method, and the problem of numerical differentiation is that a plurality of scholars at home and abroad are researching the subject from the middle of the last century to the present day, and abundant scientific research achievements are obtained. The differential at a node is expressed approximately in discrete data of a function, commonly referred to as a numerical differential. The main reasons why many scholars pay attention to the numerical differentiation problem are two, one is that the numerical differentiation problem is an ill-defined problem typically in the Hadamard sense, i.e. a small error may cause a large error in the numerical result during the measurement process. The second is that in many practical application problems, the derivative of the function needs to be calculated by some discrete data points, for example, the problem of boundary identification in image processing, the determination of peaks in chemical spectra, the solution of the Abel integral equation, and some problems in mathematical and physical equations, etc., so that the algorithm for studying the numerical differentiation problem is meaningful from both theoretical research and practical application aspects. One of the solutions in the prior art is an interpolation method, and when the interpolation method is applied to the differentiator, the differentiator is designed based on the algorithm, so that the structure is complex, the calculated amount is large when the differentiator is actually used in a boiler temperature control system, and high-frequency noise cannot be well filtered. The second existing technical scheme is a taylor expansion method, and the taylor stack development method has a complex design differentiator structure, so that the problem of complex calculation also exists in the practical use of a boiler temperature control system, and the control effect of high-frequency noise on the boiler temperature control system is greatly influenced.
Therefore, the differentiator in the prior art is complex in structure, large in calculation amount in actual use, and obvious in influence of high-frequency noise on the control effect of the differentiator, so that the differentiator has the problems of poor control effect and high maintenance and operation cost.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a boiler temperature adjusting method and system based on numerical differentiation to solve the problems of complex realization process, large calculated amount and poor control effect of a differentiator for boiler temperature control in the prior art.
In order to achieve the purpose, the invention adopts the following technical scheme to realize the purpose:
a boiler temperature adjusting method based on numerical differentiation comprises the following steps:
step 1, acquiring an actual temperature value of a boiler in real time;
step 2, comparing a temperature set value of the boiler with an actual value of the temperature to obtain an error value of real-time water temperature;
step 3, the PID controller adjusts and controls the output quantity according to the error of the real-time water temperature; a transfer function of an N-point discrete differentiator is introduced into a differentiation link in the PID controller; n is a natural number in 2-7;
and 4, adjusting the water quantity according to the output quantity until the error value of the real-time water temperature is 0, and finishing the adjustment.
The invention is further improved in that:
preferably, in step 3, N is 3.
Preferably, the temperature actual value is collected through a temperature sensor; in step 2, the temperature set value is compared with the actual value of the temperature by a comparator.
Preferably, in step 3, the obtaining process of the transfer function is to perform equidistant sampling on the error value of the real-time water temperature to obtain N sampling values, and perform differential calculation on the N sampling values to obtain the transfer function of the N-point discrete disperser.
Preferably, when N is 3, the transfer function is:
Figure BDA0003093964540000031
preferably, when N is 4, the transfer function is:
Figure BDA0003093964540000032
preferably, in step 4, the actuator of the boiler is a valve.
A numerical derivative based boiler temperature adjustment system comprising:
the acquisition module is used for acquiring the actual temperature value of the boiler in real time;
the comparison module is used for comparing a temperature set value of the boiler with an actual value of the temperature to obtain an error value of real-time water temperature;
the PID controller is used for adjusting and controlling the output quantity according to the error of the real-time water temperature;
and the adjusting module is used for adjusting the water quantity according to the output quantity until the error value of the real-time water temperature is 0, and the adjustment is finished.
Compared with the prior art, the invention has the following beneficial effects:
the invention discloses a boiler temperature adjusting method based on numerical differentiation, which adopts a numerical differentiation algorithm based on Newton interpolation polynomial to realize smaller calculated amount of a differentiator, and the differentiator realized by the method has the advantages of high accuracy of a differentiation link, good frequency characteristic, good effect of inhibiting high-frequency noise, simple structure and easy realization. Because an ideal differentiator does not exist in actual life and is not easy to realize physically, and temperature sampling points in a boiler system are often distributed equidistantly, the invention provides a numerical differentiation algorithm based on a first derivative of Newton interpolation polynomial, aiming at realizing better differentiation effect by selecting proper interpolation points and sampling step length.
The invention also discloses a boiler temperature adjusting system based on numerical differentiation, which comprises an acquisition module, a comparison module, a PID controller and an adjusting module, wherein the differentiator is finally introduced into the PID controller through the system, the calculated amount is small, the accuracy of the system is high, the frequency characteristic is good, the effect of inhibiting high-frequency noise is good, the structure is simple, and the realization is easy.
Drawings
Fig. 1 shows the frequency characteristic of an ideal differentiator (curve 1) and the frequency characteristic of a two-to-four-point discrete differentiator (curves 2-4) when the sampling step Δ t is 0.1 s;
fig. 2 shows the frequency characteristics of the two-point discrete differentiator (curves 1 and 2) when the sampling step Δ t is 1s and Δ t is 0.1 s;
FIG. 3(a) is a plot of the root trace of a two to four point discrete differentiator; FIG. 3 (b) is a diagram of a root locus of a five-to seven-point discrete differentiator
FIG. 4 shows the three-point discrete differentiator and the equivalent time constant T when Δ T is 1sнFrequency characteristics of actual differentiation of 0.25 s;
FIG. 5 is a flow chart of a boiler temperature control system;
FIG. 6 is a model diagram of a boiler temperature control system;
FIG. 7 is a graph of the frequency characteristics of a novel PID controller employing a discrete differentiator;
FIG. 8 is a graph comparing frequency characteristics of the system of the present invention using conventional PID with novel PID;
FIG. 9 is a diagram of PID control using a discrete differentiator and system response using a feedforward-feedback control strategy;
FIG. 10 is a graph showing the effect of gradually adjusting the water temperature by the boiler temperature control system.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings:
in the description of the present invention, it should be noted that the terms "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer", etc., indicate orientations or positional relationships based on the orientations or positional relationships shown in the drawings, and are only for convenience of description and simplicity of description, but do not indicate or imply that the device or element being referred to must have a particular orientation, be constructed and operated in a particular orientation, and thus, should not be construed as limiting the present invention; the terms "first," "second," and "third" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance; furthermore, unless expressly stated or limited otherwise, the terms "mounted," "connected," and "connected" are to be construed broadly and encompass, for example, both fixed and removable connections; they may be connected directly or indirectly through intervening media, or they may be interconnected between two elements. The specific meanings of the above terms in the present invention can be understood in specific cases to those skilled in the art.
The boiler temperature control system belongs to a temperature and flow system, so the system has a large inertia characteristic, wherein temperature sampling points are mostly distributed at equal intervals, and the differentiator is constructed by adopting an equidistant Newton interpolation method for increasing sampling step length by considering the factors so as to adjust the water temperature in time and improve the dynamic characteristic of the system. The newton interpolation polynomial in equation (1) enables "on-line" control using only the previous samples. That is, the current boiler water temperature y0Is based on the sampling time t of the current temperature sensor0And past time t0-Δt,t0-2Δt,...t0-k Δ t observed boiler water temperature y0,y-1,y-2...y-kIs estimated.
Figure BDA0003093964540000051
The formula (1) adopts an equidistant Newton interpolation method to construct an interpolation function, and in order to simplify the interpolation function, a variable is introduced
Figure BDA0003093964540000052
(where Δ t is the sampling step size or sampling period for the temperature sensor to detect the boiler temperature).
The current water temperature estimated value of the boiler can be obtained by the formula (1)
Figure BDA0003093964540000053
At t0The first derivative at time (u ═ 0) is:
Figure BDA0003093964540000054
wherein, DeltamThe forward difference operator represents the change amount of the discrete boiler sampling temperature on the discrete node:
Δy-1=y0-y-1 (3)
Figure BDA0003093964540000061
Figure BDA0003093964540000062
Figure BDA0003093964540000063
Figure BDA0003093964540000064
Figure BDA0003093964540000065
the numerical differential algorithm formula when two to seven boiler temperature sampling points are selected can be obtained from the formula (2). Table 1 lists the equidistant forward difference formula based on the first derivative of newton's interpolation.
Table 1 selects equidistant forward difference formula of two to seven temperature nodes
Figure BDA0003093964540000066
The boiler temperature control system adopts a PID controller, and the control effect is analyzed by comparing the frequency characteristics of an ideal differentiator and a discrete differentiator in the boiler temperature control system.
Construction of discrete differentiators
According to the formula (2), laplace transform is performed on each formula in table 1, so that the transfer functions of the corresponding discrete differentiators can be obtained, and the discrete differentiators are constructed, wherein the corresponding transfer functions of the discrete differentiators are shown in table 2. A PID controller in a boiler temperature control system is used as a differential link, discrete point sampling is carried out on boiler temperature errors, and then differential action is carried out. The PID controller with the introduced discrete differentiator can inhibit high-frequency noise in temperature detection, and has a simple structure and easy realization.
TABLE 2 transfer function of three-to seven-point discrete differentiator
Figure BDA0003093964540000071
W in the above table*(j ω) represents the transfer function of the discrete differentiator for each point, ω represents the frequency of the discrete differentiator, j is the imaginary unit, and Δ t represents the sampling interval.
Frequency characteristics and root locus of discrete differentiator
The discrete differentiator is constructed by selecting different numbers of boiler temperature sampling points and different temperature sampling step lengths delta t, and the influence of the number of the sampling points and the sampling step lengths delta t on the effect of the discrete differentiator is analyzed.
In fig. 1, the amplitude-frequency characteristic and the phase-frequency characteristic of two-point, three-point, four-point discrete differentiators and the amplitude-frequency characteristic and the phase-frequency characteristic of an ideal differentiator are shown when the sampling step Δ t is 0.1 s. Fig. 2 shows the frequency characteristics of the two-point discrete differentiator at different sampling steps Δ t equal to 0.1s and Δ t equal to 1 s.
In fig. 1, the amplitude-frequency characteristic of an ideal differentiator crosses a point (1rad/s,0db), the amplitude increases with the frequency, and an ideal differentiation element has an amplification effect on noise in a high-frequency region. Both to four-point discrete differentiators pass through a point (1rad/s,0dB) and in the high frequency region (frequencies greater than 30rad/s) the amplitude does not increase with increasing frequency but stabilizes between 0-50 dB.
Fig. 1 illustrates that the discrete differentiator realizes the differential characteristic of the ideal differential at a low frequency while reducing the amplification of noise by the differential in a high frequency region.
In the amplitude-frequency characteristic diagram of fig. 2, curve 1 has a smaller amplitude than curve 2 in the high frequency region, i.e., the amplitude is smaller when Δ t takes 1s compared to 0.1s in the high frequency region, which shows that the characteristic of the discrete differentiator for the high frequency noise amplification is more attenuated when the sampling step Δ t takes a larger value.
FIG. 3 shows the root locus of a two-point-seven-point discrete differentiator and is labeled
Figure BDA0003093964540000081
Figure BDA0003093964540000082
A point on the time root locus. Wherein
Figure BDA0003093964540000083
Frequency and phase frequency characteristic pole corresponding to 'algorithm bandwidth' of discrete differentiator
Figure BDA0003093964540000084
The mark points on the root trajectory curve are respectively: ". corresponds to
Figure BDA0003093964540000085
"×" corresponds to
Figure BDA0003093964540000086
"" corresponds to
Figure BDA0003093964540000087
From the point of view of figure 3,it can be seen that as the number of sampling points in the algorithm increases, the high frequency transmission coefficient increases. And, the two-point differentiator is in the "algorithm bandwidth"
Figure BDA0003093964540000088
The error is the largest and the transmission coefficient is the smallest in the high frequency region. Seven-point differentiators in "algorithm bandwidth"
Figure BDA0003093964540000089
The error is smallest and the transmission coefficient is largest in the high frequency region.
In addition, comparing fig. 3(a) and (b), when the number of sampling points exceeds 4, it is observed that the discrete differentiator appears on the left side of the s-domain in the root locus. The discrete differentiator constructed at this time does not meet the characteristics of the differential element. The discrete differentiator designed herein takes a maximum of 4 sample points when used for control purposes. Since too many sampling points lead to a complication of the algorithm and the accuracy of the discrete differentiator is not significantly improved over the ideal differentiator within the considered "algorithm bandwidth". It is not suggested to further increase the number of sampling points of the discrete differentiator.
Comparison of discrete differentiators with actual differentiators
In actual industrial application, an ideal differential link does not exist, and an actual differential link has certain inertia, so that the actual differential link adopted by the traditional PID controller in a boiler temperature system can be realized by connecting the inertia link and the ideal differential link in series.
Figure BDA0003093964540000091
Tн-the equivalent time constant of the actual differential element. T isнThis is obtained by the following simple relationship: the actual differentiation element should be equal to the modulo length of the discrete differentiator when the frequency ω → ∞.
Can obtain T corresponding to two-point, three-point and four-point discrete differentiators in turnн
The equivalent time constant of the actual differential element is determined by the following relation:
Figure BDA0003093964540000092
where k is 2, 4, and 6.667, which are coefficients corresponding to two-point, three-point, and four-point discrete differentiators.
In order to more effectively evaluate the properties of the discrete differentiator in the high frequency region, the frequency characteristics of the discrete differentiator are compared with the frequency characteristics of the actual differentiation element having the above-mentioned transfer function.
Fig. 4 shows the frequency characteristic of the three-point discrete differentiator when Δ T is 1s and the equivalent time constant T obtained by equation (4)нContinuous differential element (3) frequency characteristic of 0.25 s:
table 3 compares the standard deviation estimates of the input and output signals of the actual differentiator and the discrete differentiator (with the ideal differentiator as a reference) under different temperature noise. Wherein, the sampling step length Δ t of the discrete differentiator is 1 s. Equivalent time constant of actual differentiator
Figure BDA0003093964540000093
Analyzing table 3, it can be seen that under the interference of different temperature noises, the standard deviation estimated value of the input and output signals of the discrete differentiator is smaller than that of the actual differentiator. The characteristic of the discrete differentiator that can suppress the amplification of the differentiation to the high frequency noise in the high frequency region is explained.
TABLE 3 estimation of standard deviation of input and output signals of discrete differentiators under different noises and corresponding actual differentiators
Figure BDA0003093964540000101
Example (b):
the present invention uses discrete differentiator to replace the traditional PID control differential link to obtain a new PID controller with different structure and applies it to the boiler temperature control system, the structure of the boiler temperature control system is shown in figure 5, the controlled object in the system is a heat exchanger with obvious inertia, the present embodiment uses two inertia links in series to realize the characteristic.
The boiler temperature control system is modeled as in fig. 6. The system belongs to a closed loop system, and the controlled quantity is temperature. The system consists of a comparator, a PID controller, a three-position relay, an actuator, a heat exchanger and a temperature sensor. When a user has a hot water use demand, an actuator (valve) of the boiler temperature control system is opened, and the system enters a water storage stage. The heat exchanger heats the water in the water tank, and a temperature sensor in the boiler temperature control system is responsible for feeding back the actual water temperature to the system in real time. And obtaining an error value between the expected water temperature set by the user and the actual water temperature through the processing of the comparator, and sending the error value into the PID controller. The PID controller adjusts and controls the output quantity, generally voltage, according to the water temperature error. The control output is sent to the actuator (valve) via a three-position relay to regulate the water quantity. When the water temperature reaches the set value of the user, the error of the water temperature of the system is 0, and the hot water requirement of the user is met. It is worth noting that after the user consumes the hot water in the water tank, the actuator can act immediately to perform water replenishing operation, at this time, due to the water consumption of the user, the water temperature can change, so that a water temperature error exists, and at this time, the boiler temperature control system can repeat the above process to adjust the water temperature. The three-position relay with the dead zone characteristic can effectively reduce the action times of the actuator and inhibit continuous small oscillation caused by the quantization of the output quantity of the PID controller.
Since the water temperature system is a large inertia system, the hot water consumption of the user is large. Therefore, the disturbance in the boiler temperature control system is large, and a differential link in the PID controller in the design is realized by adopting a discrete differentiator, and the method comprises the following steps of:
step 1, acquiring a temperature set value and an actual value of the temperature of the boiler in real time.
And 2, sampling the water temperature error value sent by the comparator at equal intervals.
And 3, selecting three boiler temperature sampling points, and calculating by using the three-point numerical differential algorithm formula introduced above to further obtain a transfer function of the three-point discrete differentiator:
Figure BDA0003093964540000111
and 4, introducing the obtained transfer function of the three-point discrete differentiator into a differentiation link of the PID controller, wherein the PID controller is the PID controller adopting the discrete differentiator.
Compared with the conventional PID controller, the PID controller adopting the discrete differentiator designed by the method is faster in water temperature regulation, stronger in disturbance rejection and better in water temperature regulation capacity, and can better meet the hot water requirement of a user.
Fig. 7 shows the frequency characteristics of the novel PID controller to which the discrete differentiator is applied, wherein the differential element structure of the PID controller is: 1) the actual differentiation element 2) is a discrete differentiator with an increased sampling step.
Fig. 8 shows the frequency characteristics of the system using the conventional PID and the novel PID, the differential link of the conventional PID controller uses the actual differential in fig. 7, and the novel PID controller uses the three-point discrete differentiator in fig. 7.
From the results of fig. 7 and 8, it can be known that the PID controller implemented by the actual differential element can reduce the high frequency amplification of the signal, and compared with the PID controller using the actual differential, the high frequency attenuation of the PID controller using the algorithm in this document is significant.
The section also carries out comparative simulation under the PID control strategy and the feedforward-feedback control strategy of the boiler temperature control system adopting the algorithm. Wherein the noise is selected as 20KHz sine signal, and the PID controller is a conventional PID in continuous time form.
FIG. 9 shows the PID control of the system using discrete differentiator and the system response of the system using feedforward-feedback control strategy, and it can be seen that the overshoot and oscillation of the algorithm are smaller compared with the feedforward-feedback composite control, and finally the steady state of 50 ℃ is achieved. The experimental result proves that the PID realized by discrete differentiation can realize higher control quality than feedforward-feedback composite control. Meanwhile, by adopting the algorithm, the structure of the system is simpler, and the filtering effect on high-frequency noise can be realized even if a feedforward filtering link is not provided.
FIG. 10 is a graph showing the effect of the boiler temperature control system gradually adjusting the water temperature without external disturbance. Wherein, the differential link of the curve 1 corresponding to the PID controller in the system is the actual differential, and the differential link of the curve 2 corresponding to the PID controller in the system is the two-point discrete differentiator for increasing the sampling step length, as can be seen from fig. 10, the control effects of the two are basically the same. The water temperature in the system was 56.99 ℃ at 400 seconds and 57 ℃ at 1000 seconds, and the system was always in a steady state. The algorithm can ensure that the transient response of the system is not overshot and the control effect is good.

Claims (6)

1. A boiler temperature adjusting method based on numerical differentiation is characterized by comprising the following steps:
step 1, acquiring an actual temperature value of a boiler in real time;
step 2, comparing the temperature set value of the boiler with the actual value of the temperature to obtain an error value of the real-time water temperature;
step 3, the PID controller adjusts and controls the output quantity according to the error of the real-time water temperature; a transfer function of an N-point discrete differentiator is introduced into a differentiation link in the PID controller; n is a natural number in 2-7;
when N is 4, the transfer function is:
Figure FDA0003677327270000011
delta t is the sampling step length or sampling period of the temperature sensor for detecting the temperature of the boiler;
and 4, adjusting the water quantity according to the output quantity until the error value of the real-time water temperature is 0, and finishing the adjustment.
2. The boiler temperature adjusting method based on numerical differentiation according to claim 1, characterized in that in step 1, the temperature sensor is used to collect the actual temperature value; in step 2, the temperature set value is compared with the actual value of the temperature by a comparator.
3. The numerical differentiation-based boiler temperature adjustment method according to claim 1, wherein in step 3, the transfer function is obtained by sampling the error value of the real-time water temperature at equal intervals to obtain N sampled values, and performing differential calculation on the N sampled values to obtain the transfer function of the N-point discrete disperser.
4. A method for boiler temperature adjustment based on numerical differentiation according to claim 3 characterized in that when N is 3, the transfer function is:
Figure FDA0003677327270000012
Δ t is the sampling step or sampling period for the temperature sensor to detect the boiler temperature.
5. The method as claimed in claim 1, wherein in step 4, the actuator of the boiler is a valve.
6. A boiler temperature adjustment system for implementing the numerical differentiation based boiler temperature adjustment method of claim 1, comprising:
the acquisition module is used for acquiring the actual temperature value of the boiler in real time;
the comparison module is used for comparing a temperature set value of the boiler with an actual value of the temperature to obtain an error value of real-time water temperature;
the PID controller is used for adjusting and controlling the output quantity according to the error of the real-time water temperature; a transfer function of an N-point discrete differentiator is introduced into a differentiation link in the PID controller; n is a natural number in 2-7;
when N is 4, the transfer function is:
Figure FDA0003677327270000021
delta t is the sampling step length or sampling period of the temperature sensor for detecting the temperature of the boiler;
and the adjusting module is used for adjusting the water quantity according to the output quantity until the error value of the real-time water temperature is 0, and the adjustment is finished.
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