CN111812984B - Model-based robust filtering method for inverter control system - Google Patents

Model-based robust filtering method for inverter control system Download PDF

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CN111812984B
CN111812984B CN202010699871.2A CN202010699871A CN111812984B CN 111812984 B CN111812984 B CN 111812984B CN 202010699871 A CN202010699871 A CN 202010699871A CN 111812984 B CN111812984 B CN 111812984B
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张正江
祝旺旺
戴瑜兴
赵升
闫正兵
黄世沛
王环
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Wenzhou University
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Abstract

The invention provides a robust filtering method based on a model for an inverter control system, which comprises the steps of establishing an inverter control system model containing an external disturbance signal and a measurement noise signal; performing optimal estimation on the measurement feedback signal and the prediction signal to obtain a filtering signal and outputting feedback; and comparing the filtering signal with a set standard sinusoidal signal to obtain a deviation signal which is used as the input of a PID controller, thereby reducing the deviation of the output waveform and increasing the accuracy of the output of the inverter. The method is implemented in such a way that model prediction information and measurement information of the system are comprehensively considered, and optimal estimation is adopted so as to enable the filtered data to be closer to the real state of the process, thereby overcoming the influence of external disturbance and measurement noise on the system performance and the defects of the filtering technology in the inverter control system in the prior art.

Description

Model-based robust filtering method for inverter control system
Technical Field
The invention relates to the technical field of filtering of power electronic systems, in particular to a robust filtering method based on a model for an inverter control system.
Background
The constant current and constant frequency inverter is a conversion device widely used in the field of power electronics, and is more frequently used in power supply systems such as an uninterruptible power supply and the like. Generally speaking, most of the process of supplying power to the ac load cannot leave the inverter circuit, and the efficiency of supplying power to the load is greatly affected by the quality of the inverter control system. With the complexity of working environment and the diversification of production requirements, the optimization of inverter control systems and the improvement of filtering technology gradually draw attention from scholars at home and abroad.
In the process of controlling the inverter system, the change of the system working environment directly affects the stability of the output current, and the output of the inverter often does not reach the expected level due to various types of environmental noise contained in the inverter control system. Ambient noise in the conventional sense is primarily external disturbances, of the type present in most control systems, which can reduce the stability of the inverter output signal and thus cause the output signal to fall short of a desired level; in practice, because each link of the inverter control system needs to be measured and monitored in real time, another form of noise, namely measurement noise, is introduced into the sensor, and the measurement noise and the external disturbance all cause distortion of output signals. In order to reduce the influence of external disturbance and measurement noise on the inverter control system, a corresponding filtering method needs to be introduced into a feedback link of the control system, and the filtered feedback information is closer to a true value, so that the inverter can output relatively accurate voltage or current signals. The filtering techniques which are widely applied in recent years mainly include an exponential robust filter, an extended kalman robust filter and the like, and although the filtering techniques can reduce the influence of external disturbance and measurement noise on a control system to a certain extent, the defects of the filtering techniques are not negligible.
Since the birth of the thyristor in the fifth and sixty years of the twentieth century, the international research and exploration on the inverter have drawn the preface. In the late stage of the seventies of the twentieth century, the rapid development of fully-controlled devices such as gate turn-off thyristors (GTOs), Power bipolar transistors (BJTs), Power-field effect transistors (Power-MOSFETs) and the like has greatly promoted the innovation and application of inverter technology. The research institutions that have made a prominent contribution to the development of this field are mainly focused on the united states and the central european region, including the general electric company in the united states, bell laboratories, SMA companies in germany, and the like.
From the aspect of control technology of inverter control systems, the application of vector control technology, multilevel conversion technology, repetitive control, fuzzy control and other technologies proposed abroad to inverters greatly promotes the development of inverter control systems. The vector control technology was originally proposed by siemens engineers blasthke, and its basic principle is to control the excitation and torque currents of an asynchronous motor by controlling the motor stator current vector, thereby realizing the control of the motor torque. The multi-level concept was proposed by Nabae et al in 1980, and thereafter Bhagwat and Stefanovic further popularized the three-level inverter to multi-levels, thereby providing a new direction for developing high-voltage large-capacity inverters. Repetitive control is proposed by the japanese research group and is a control concept based on the internal model principle, which indicates that in a system, if the controlled signal is a feedback signal for controlling it and there is a controlled model in the feedback loop, the system is considered to be stable in a theoretical sense; compared with a general control system, the repetitive control system adds a repetitive compensator, so that the controller can obtain higher precision in a periodic control process. The proposal of the fuzzy control is attributed to the concept of 'fuzzy set' proposed by Zadeh, the technology is based on the fuzzy mathematical theory, the adaptability and the rationality of the control algorithm are greatly improved by simulating the reasoning and decision process of a human, and the technology has been developed into an important branch of the intelligent control.
The research of domestic inverters is mainly to expand and optimize a control method and a filtering technology of an inverter control system, and expand on the basis of various control technologies such as PID control, dead-beat control, double-loop feedback control, repetitive control, sliding mode variable structure control, neural network control and the like, so that the control effect of the system is effectively improved. In order to solve the application defect of a single control technology, the problems of poor steady-state precision of a PID control algorithm in an inverter power supply system, low dynamic response speed of repeated control and the like are well solved in the research of applying PID control and repeated control to a sine wave inverter power supply by high army and the like. On the other hand, as in most control systems, due to the limitation of the working environment, various disturbances inevitably contained in the inverter system, especially external disturbances commonly existing in the feedback system and measurement noise introduced by the sensor, may largely destroy the working accuracy of the inverter system. Zhang Jianhua and the like consider the external disturbance signal as non-Gaussian distribution when evaluating the performance of the feedback control loop, which means that the external disturbance in the inverter control system is not necessarily simple Gaussian signal, and the non-Gaussian disturbance can not be ignored. Currently, domestic scholars often ignore the existence of measurement noise when designing a series of filtering techniques of a control system, and even if some scholars pay attention to the measurement noise, the corresponding filtering method aiming at the measurement noise is difficult to achieve a good effect. Even so, due to the continuous efforts and innovations of domestic scholars, the researches on inverters in China have realized qualitative leaps from weak to strong, from tracking to approach and finally to occupy the leading position of the world.
As can be seen from the above description, the main problems faced by inverter control systems today are the interference of complex signals and the design problems of the corresponding advanced filtering techniques, as the control theory and method are relatively mature. In most of the above system analysis and design processes, the influence of the measurement noise on the system is not explicitly considered, and in fact, the measurement noise, like the common external disturbance in the system, may reduce the performance of the system. For filtering technologies for measuring noise and external disturbance, such as an exponential robust filter and a Kalman filtering technology, the exponential robust filter has a large delay due to the fact that system model prediction information is not used, and Kalman filtering cannot process interference of non-Gaussian signals. Particularly for the inverter control system, the current research center mainly focuses on the improvement and optimization of the control method and the control strategy, and few people pay attention to the interference situation of the inverter system and the corresponding design of the filtering scheme, which leads to the unexpected output of the inverter, thereby causing great resource and economic loss to the production life.
Therefore, a filtering method for an inverter control system is needed to overcome the possible influence of external disturbance and measurement noise on the system performance and the shortcomings of the filtering technique in the prior art.
Disclosure of Invention
The technical problem to be solved by the embodiments of the present invention is to provide a model-based robust filtering method for an inverter control system, which can overcome the possible influence of external disturbance and measurement noise on system performance and the shortcomings of the filtering technology in the inverter control system in the prior art.
In order to solve the above technical problem, an embodiment of the present invention provides a model-based robust filtering method for an inverter control system, including the following steps:
establishing an inverter control system model containing an external disturbance signal and a measurement noise signal on the basis of an inverter control system which is in closed-loop connection formed by a PID controller, an inverter and a robust filter; wherein the external disturbance signal acts on the inverter output; the measurement noise signal and an actual output signal of the inverter control system model jointly form a measurement feedback signal, and the measurement feedback signal and a prediction signal of the inverter control system model act on the input of the robust filter;
acquiring a measurement feedback signal and a prediction signal, and performing optimal estimation on the acquired measurement feedback signal and the prediction signal to obtain a filtering signal;
and comparing the filtering signal with a preset actual input standard sinusoidal signal of the inverter control system model to obtain a deviation signal as the input of the PID controller so as to reduce the deviation of the output waveform of the inverter control system model.
The step of obtaining the measurement feedback signal and the prediction signal, and performing optimal estimation on the obtained measurement feedback signal and the prediction signal to obtain a filtered signal specifically includes:
according to the measurement feedback signal ym(t) and the prediction signal
Figure BDA0002592629140000041
Obtaining the model of the inverter control system by using a Bayesian formulaThe actual output signal y (t) of type being based on said prediction signal
Figure BDA0002592629140000042
And the measurement feedback signal ym(t) posterior probability distribution:
Figure BDA0002592629140000043
in the formula (1), L (y (t) | ym(t))、
Figure BDA0002592629140000044
Respectively representing the prediction signal based on
Figure BDA0002592629140000045
And said measurement feedback signal ym(t) a probability density function;
carrying out maximum likelihood estimation on the formula (1) to obtain the optimal value of the formula as a filtering signal yf(t), expressed as follows:
Figure BDA0002592629140000046
in the formula (2), the reaction mixture is,
Figure BDA0002592629140000047
representing maximum likelihood estimates derived from said prediction signal
Figure BDA0002592629140000048
And said measurement feedback signal ym(t) a function of interest.
Wherein the method further comprises:
when the measurement noise signal epsilon (t) and the external disturbance signal d (t) are both in Gaussian distribution, the measurement noise signal epsilon (t) is set to be epsilon (kT) -N (0, rho)2) And the external disturbance signal d (t) is d (kT) to N (0, sigma)2)；
In determining the inverter control system modelThe prediction error delta (kT) is the same as the distribution of the external disturbance signal d (kT), and when the variance is different, the prediction error delta (kT) is set to be delta (kT) to N (0, delta)2)；
Substituting the gaussian distribution probability density functions set for the measurement noise signal epsilon (t), the disturbance signal d (t), and the prediction error delta (kT) into the formula (1), and substituting the filtered signal y represented by the formula (2)f(t) is described as
Figure BDA0002592629140000051
Wherein the method further comprises:
when the measurement noise signal epsilon (t) and the external disturbance signal d (t) are both in a pollution normal distribution, let epsilon (kT) be ω ε (kT)1(kT)+(1-ω)ε2(kT) and the disturbance signal d (t) is d (kT) ═ η d1(kT)+(1-η)d2(kT); wherein the content of the first and second substances,
Figure BDA0002592629140000052
1- ω and 1- η represent the posterior probability of gross errors occurring in the measurement noise signal ε (t) and the external disturbance signal d (t), respectively;
substituting the pollution normal distribution probability density functions respectively set by the measurement noise signal epsilon (t) and the external disturbance signal d (t) into the formula (1), and substituting the filtering signal y represented by the formula (2)f(t) is described as
Figure BDA0002592629140000053
Wherein the content of the first and second substances,
Figure BDA0002592629140000054
Figure BDA0002592629140000055
the embodiment of the invention has the following beneficial effects:
the method is based on the traditional inverter control system, the measurement noise and the external disturbance are combined with the traditional inverter control system to establish an inverter control system model, the robust filter is designed according to the measurement feedback signal and the prediction signal of the inverter control system model, so that the output of the robust filter which is closer to the real feedback information is obtained, and finally the deviation between the filtering signal and the set standard sinusoidal signal is used as the input to realize the improvement of the control precision, so that the influence of the external disturbance and the measurement noise on the system performance and the defects of the filtering technology in the inverter control system in the prior art can be overcome.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is within the scope of the present invention for those skilled in the art to obtain other drawings based on the drawings without inventive exercise.
Fig. 1 is a flowchart of a model-based robust filtering method for an inverter control system according to an embodiment of the present invention;
fig. 2 is a model block diagram of an inverter control system including an external disturbance signal and a measurement noise signal in a model-based robust filtering method for the inverter control system according to an embodiment of the present invention;
fig. 3 is a comparison graph of outputs of an inverter control system model using a measurement feedback signal feedback and a filtered filtering signal feedback in a model-based robust filtering method for the inverter control system according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings.
As shown in fig. 1, in an embodiment of the present invention, a proposed model-based robust filtering method for an inverter control system includes the following steps:
step S1, establishing an inverter control system model containing an external disturbance signal and a measurement noise signal on the basis of an inverter control system which is formed by a PID controller, an inverter and a robust filter and is in closed-loop connection; wherein the external disturbance signal acts on the inverter output; the measurement noise signal and an actual output signal of the inverter control system model jointly form a measurement feedback signal, and the measurement feedback signal and a prediction signal of the inverter control system model act on the input of the robust filter;
the specific process is to combine the measured noise and the external disturbance with the traditional inverter control system according to the actual control process of the inverter and the working environment of the inverter, and establish an inverter control system model, as shown in fig. 2. The conventional control system mainly includes a feedback link, a pid (proportion Integration differentiation) controller, a controlled object (inverter), external disturbance, system input and output, and the like.
In FIG. 2, let the model actual input signal be the standard sinusoidal signal rsin(t) filtering the signal y with a robust filterf(t) the difference forms a deviation signal e (t) as an input to a PID controller; the PID controller outputs a control signal u (t) to act on an inverter of a controlled object; d (t) represents an external disturbance signal acting on the inverter output; y issin(t) is an actual output signal of the inverter control system model; epsilon (t) represents a measurement noise signal introduced by the sensor, and the measurement noise signal and the model actual output signal y (t) form a measurement feedback signal ym(t) measuring the compliance p (y) of the noise signal ε (t)m(t) | y (t)), i.e., ∈ (t) p (y)m(t) | y (t)); measuring the feedback signal ym(t) and prediction signal
Figure BDA0002592629140000071
A filtered signal y obtained as an input of the robust filter and filtered by the robust filterf(t) with a standard sineSignal rsin(t) comparing to obtain a deviation signal e (t) as the input of a PID controller to form closed-loop control of the system.
It should be noted that the prediction signal
Figure BDA0002592629140000072
Measurement feedback signal y may be utilized in conjunction with a system modelm(t) obtaining the previous moment information, and setting the previous moment information and the actual output signal y of the inverter control system modelsin(t) obeys the prediction error δ (t)
Figure BDA0002592629140000073
Is distributed, i.e.
Figure BDA0002592629140000074
Step S2, obtaining a measurement feedback signal and a prediction signal, and carrying out optimal estimation on the obtained measurement feedback signal and the prediction signal to obtain a filtering signal;
the specific process is that firstly, according to the measurement feedback signal ym(t) and the prediction signal
Figure BDA0002592629140000075
Obtaining an actual output signal y (t) of the inverter control system model based on the prediction signal using a Bayesian formula
Figure BDA0002592629140000076
And measuring the feedback signal ym(t) posterior probability distribution:
Figure BDA0002592629140000077
in the formula (1), L (y (t) | ym(t))、
Figure BDA0002592629140000078
Respectively representing the prediction signal based on
Figure BDA0002592629140000079
And the measurement feedback signal ym(t) a probability density function;
secondly, maximum likelihood estimation is carried out on the formula (1), and the optimal value of the formula is obtained as a filtering signal yf(t), expressed as follows:
Figure BDA00025926291400000710
in the formula (2), the reaction mixture is,
Figure BDA00025926291400000711
representing the estimated and predicted signal of maximum likelihood
Figure BDA00025926291400000712
And measuring the feedback signal ym(t) a function of interest.
It should be noted that the feedback signal y is measuredm(t) may be represented by ymAnd (t) represents a function of the actual output signal y (t) and the measurement noise signal epsilon (t) of the inverter control system model obtained by the maximum likelihood estimation. Aiming at external disturbance d (t) and measurement noise epsilon (t) under different distributions, corresponding mathematical derivation is combined to prove the output y of the robust filterf(t) has a variance less than the measurement feedback signal ymThe variance of (t), namely the robust filter can play a good filtering effect on an inverter control system affected by external disturbance and measurement noise.
In the embodiment of the present invention, in consideration of the discrete condition of the model shown in fig. 2 (the signal only takes value over the sampling period, i.e., t ═ kT), it can be known from the theory related to inverters and power electronics that an ideal discrete model of a DC-AC converter can be described as follows:
ysin(kT)＝Asin(wkT)
wherein, a represents the amplitude of the sinusoidal signal, ω is the frequency of the sinusoidal signal, T represents the sampling period, and k represents the current sampling step.
The inverse measurement can be found by combining the model of the inverter control system shown in FIG. 2Feeding signal ym(t) the previous time information can be used to predict the signal
Figure BDA0002592629140000081
To obtain a prediction signal
Figure BDA0002592629140000082
The discretization of (a) represents the formula:
Figure BDA0002592629140000083
wherein f (y)m((k-1) T)) represents a group of ymA function of (k-1) T, and δ (kT) represents the prediction bias of the model.
Therefore, the design process of the robust filter can be discussed separately according to two typical distributions, which are as follows:
(I) when the measurement noise signal epsilon (t) and the external disturbance signal d (t) are both in Gaussian distribution, the measurement noise signal epsilon (t) is set to be epsilon (kT) -N (0, rho)2) And the external disturbance signal d (t) is d (kT) to N (0, sigma)2)；
When the prediction error delta (kT) of the inverter control system model is determined to be the same as the distribution of the external disturbance signals d (kT) and the variance is different, the prediction error delta (kT) is set to be delta (kT) -N (0, delta and delta) respectively2)；
Substituting a Gaussian distribution probability density function set for each of the measurement noise signal ε (t), the disturbance signal d (t), and the prediction error δ (kT) into the formula (1), and substituting the filtered signal y represented by the formula (2)f(t) is described as
Figure BDA0002592629140000091
(II) when the measurement noise signal epsilon (t) and the external disturbance signal d (t) are both in pollution normal distribution, setting the measurement noise signal epsilon (t) as epsilon (kT) ═ omega epsilon1(kT)+(1-ω)ε2(kT) and the disturbance signal d (t) is d (kT) ═ η d1(kT)+(1-η)d2(kT); wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002592629140000092
1-omega and 1-eta respectively represent the posterior probability of occurrence of gross errors in the measurement noise signal epsilon (t) and the external disturbance signal d (t);
substituting the pollution normal distribution probability density functions respectively set for the measurement noise signal epsilon (t) and the external disturbance signal d (t) into the formula (1), and substituting the filtering signal y represented by the formula (2)f(t) is described as
Figure BDA0002592629140000093
Wherein the content of the first and second substances,
Figure BDA0002592629140000094
Figure BDA0002592629140000095
and step S3, comparing the filtering signal with a preset actual input standard sinusoidal signal of the inverter control system model to obtain a deviation signal as the input of the PID controller so as to reduce the deviation of the output waveform of the inverter control system model.
The specific process is to use the filtered signal y obtained in step S2f(t) and the set standard sinusoidal signal rsin(t) comparing to obtain a deviation signal e (t) as the input of a PID controller, wherein the output of the system model acts on a PWM generator to control the on and off of a switching tube in an inverter bridge, and the influence of external disturbance and measurement noise on the output of an inverter is effectively reduced and the performance of an inverter control system is improved through the real-time information feedback and deviation correction.
As shown in fig. 3, a comparison graph of outputs of the measurement feedback signal feedback and the filtered signal feedback after filtering is respectively adopted for the inverter control system model. As can be seen from fig. 3, the measurement feedback signal contains measurement noise, which seriously affects the performance of the system, and therefore, the output current of the inverter fluctuates significantly. The robust filter adopts model prediction feedback of the system when correcting the measurement feedback signal, so that the control cost of the system is reduced, and the control precision of the system is effectively improved, thereby enabling the output current of the system to be closer to an ideal standard. The experimental result shows that the accuracy of the output current of the inverter control system can be influenced by the measurement noise feedback and the external disturbance feedback, and the influence of the measurement noise feedback and the external disturbance feedback on the system output can be reduced by adopting the robust filter, so that the system output is more real.
The embodiment of the invention has the following beneficial effects:
the method is based on the traditional inverter control system, the measurement noise and the external disturbance are combined with the traditional inverter control system to establish an inverter control system model, the robust filter is designed according to the measurement feedback signal and the prediction signal of the inverter control system model, so that the output of the robust filter which is closer to the real feedback information is obtained, and finally the deviation between the filtering signal and the set standard sinusoidal signal is used as the input to realize the improvement of the control precision, so that the influence of the external disturbance and the measurement noise on the system performance and the defects of the filtering technology in the inverter control system in the prior art can be overcome.
Those skilled in the art will understand that all or part of the steps in the method according to the above embodiments may be implemented by a program instructing related hardware, where the program may be stored in a storage medium readable by a microprocessor (e.g., a DSP chip, an ARM chip, a single chip, etc.), and the storage medium may be, for example, a ROM/RAM of the microprocessor.
While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.

Claims (3)

1. A model-based robust filtering method for an inverter control system, comprising the steps of:
establishing an inverter control system model containing an external disturbance signal and a measurement noise signal on the basis of an inverter control system which is in closed-loop connection formed by a PID controller, an inverter and a robust filter; wherein the external disturbance signal acts on the inverter output; the measurement noise signal and an actual output signal of the inverter control system model jointly form a measurement feedback signal, and the measurement feedback signal and a prediction signal of the inverter control system model act on the input of the robust filter;
acquiring a measurement feedback signal and a prediction signal, and performing optimal estimation on the acquired measurement feedback signal and the prediction signal to obtain a filtering signal;
comparing the filtering signal with a preset actual input standard sinusoidal signal of the inverter control system model to obtain a deviation signal which is used as the input of the PID controller so as to reduce the deviation of the output waveform of the inverter control system model;
the step of obtaining the measurement feedback signal and the prediction signal, and performing optimal estimation on the obtained measurement feedback signal and the prediction signal to obtain a filtered signal specifically includes:
according to the measurement feedback signal ym(t) and the prediction signal
Figure FDA0003576139100000011
Obtaining an actual output signal y (t) of the inverter control system model based on the prediction signal using a Bayesian formula
Figure FDA0003576139100000016
And the measurement feedback signal ym(t) posterior probability distribution:
Figure FDA0003576139100000012
in the formula (1), L (y (t) | ym(t))、
Figure FDA0003576139100000013
Respectively representing the prediction signal based on
Figure FDA0003576139100000014
And the measurement feedback signal ym(t) a probability density function;
carrying out maximum likelihood estimation on the formula (1) to obtain the optimal value of the formula as a filtering signal yf(t), expressed as follows:
Figure FDA0003576139100000015
in the formula (2), the reaction mixture is,
Figure FDA0003576139100000021
representing maximum likelihood estimates derived from said prediction signal
Figure FDA0003576139100000022
And the measurement feedback signal ym(t) a function of interest.
2. The model-based robust filtering method for an inverter control system of claim 1, wherein the method further comprises:
when the measurement noise signal epsilon (t) and the external disturbance signal d (t) are both in Gaussian distribution, the measurement noise signal epsilon (t) is set to be epsilon (kT) -N (0, rho)2) And the external disturbance signal d (t) is d (kT) to N (0, sigma)2)；
When the prediction error delta (kT) of the inverter control system model is determined to be the same as the distribution of the external disturbance signal d (kT) and the variance is different, the prediction error delta (kT) is set to be delta (kT) -N (0, delta and delta) respectively2)；
Substituting the Gaussian distribution probability density functions set for the measurement noise signal epsilon (t), the external disturbance signal d (t) and the prediction error delta (kT) into the formula (1), and filtering represented by the formula (2)Wave signal yf(t) is described as
Figure FDA0003576139100000023
K＝(1+ρ2δ-2)-1。
3. The model-based robust filtering method for an inverter control system of claim 1, wherein the method further comprises:
when the measurement noise signal epsilon (t) and the external disturbance signal d (t) are both in pollution normal distribution, the measurement noise signal epsilon (t) is set to be epsilon (kT) ═ omega epsilon1(kT)+(1-ω)ε2(kT) and the disturbance signal d (t) is d (kT) ═ η d1(kT)+(1-η)d2(kT); wherein the content of the first and second substances,
Figure FDA0003576139100000024
d1/2(kT)～N((0，δ1/2 2) (ii) a 1- ω and 1- η represent the posterior probability of gross errors occurring in the measurement noise signal ε (t) and the external disturbance signal d (t), respectively;
substituting the pollution normal distribution probability density functions respectively set by the measurement noise signal epsilon (t) and the external disturbance signal d (t) into the formula (1), and substituting the filtering signal y represented by the formula (2)f(t) is described as
Figure FDA0003576139100000025
Figure FDA0003576139100000026
Wherein the content of the first and second substances,
Figure FDA0003576139100000027
Figure FDA0003576139100000031
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