CN109240085B - non-Gaussian system dynamic data correction and system control performance optimization method - Google Patents

non-Gaussian system dynamic data correction and system control performance optimization method Download PDF

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CN109240085B
CN109240085B CN201811199510.0A CN201811199510A CN109240085B CN 109240085 B CN109240085 B CN 109240085B CN 201811199510 A CN201811199510 A CN 201811199510A CN 109240085 B CN109240085 B CN 109240085B
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任密蜂
张雯
陈荣辉
张旭霞
梁艳
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Taiyuan University of Technology
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Abstract

The invention relates to a method for optimizing system control performance, in particular to a method for correcting dynamic data of a non-Gaussian system and optimizing system control performance, which solves the problems that the data measurement is not close to a real value after being corrected and the optimization effect of a control strategy is not obvious under the condition that the wind power generation process is influenced by non-Gaussian process noise and non-Gaussian measurement noise, and comprises the following steps: firstly, describing a system model under non-Gaussian disturbance; secondly, deducing an iterative formula in the problem description by utilizing an EM algorithm; thirdly, solving the corrected output by using an iterative formulay r (ii) a And fourthly, selecting the performance indexes based on the statistical information to obtain an optimal control law. The advantages are that: 1. considering the influence of non-Gaussian random noise on the system; 2. the dynamic characteristics of the process are considered, and the actual process is better expressed; 3. the calculation efficiency is high, and the requirements of the actual industrial process are met; 4. and (3) fully depicting the random characteristic of the non-Gaussian random quantity by adopting entropy statistical information, and establishing a tracking control index.

Description

non-Gaussian system dynamic data correction and system control performance optimization method
Technical Field
The invention relates to a system control performance optimization method, in particular to a non-Gaussian system dynamic data correction and system control performance optimization method.
Background
In practical systems, random disturbances are widespread, especially in industrial processes, such as random factors of wind speed variations in wind power systems, random factors of measurement errors of sensor measurement data.
At present, the correction of the measurement noise of a control system is optimized by controlling the mean value and the variance of corrected data based on the assumption that the measurement noise is gaussian. The measurement error of the actual industrial process is usually non-gaussian, and if the measurement error is corrected by using a method under the traditional gaussian assumption, the optimization control effect is not obvious.
A control strategy can achieve good control results, and it is an important premise that accurate measurement data is obtained in the control process, and for a system containing measurement noise, even if an existing optimal performance index is constructed, the control effect is not reliable. In industrial processes, the measurement data inevitably contains errors (including random errors and gross errors) and the measurement noise is often non-gaussian. Gross errors can be quite large, and both errors can adversely affect process control for systems that require accurate state and measurement estimates. Therefore, in order to ensure that the system maintains a good control effect, more accurate measurement data should be provided, i.e. the raw measurement data should be corrected to be as close to the real data as possible. At present, the correction of the measurement noise of a control system is optimized by controlling the mean value and the variance of corrected data based on the assumption that the measurement noise is gaussian. Such an optimization strategy has the following drawbacks: 1. the control theory for stochastic control systems does not explicitly consider the measurement noise present in the actual sensor device, but even if it is considered, it is discussed on the assumption that it is gaussian distributed; 2. in the conventional data correction method, the dynamic characteristics of the process are not taken into consideration, and therefore, the accessibility of the set value cannot be ensured; 3. in terms of non-gaussian random control, in the existing scientific achievements, the set value is usually taken as a constant value or a preset reference track, but in fact, the set value/reference track needs to be set according to the actual situation, for example, the randomness of the wind speed is a motion track which changes at any time, so that the processing scheme needs to be further considered.
Because the above-mentioned drawbacks and problems in the prior art are difficult to solve, it is necessary to design a method for optimizing the dynamic data correction and system control performance of a non-gaussian system.
Disclosure of Invention
The technical problem solved by the invention is as follows: aiming at the problems that the data measurement is not close to the true value after being corrected and the optimization effect of the control strategy is not obvious under the condition that the wind power generation process is influenced by non-Gaussian process noise (random wind speed) and non-Gaussian measurement noise, the method for correcting the dynamic data of the non-Gaussian system and optimizing the control performance of the system is provided.
The invention is realized by the following operation steps: the method for correcting the dynamic data of the non-Gaussian system and optimizing the control performance of the system comprises the following operation steps:
step one, describing a system model under non-Gaussian disturbance:
in a univariate system, the output at time t can be decomposed into two parts: predictable portion
Figure GDA00030313371000000211
And an unpredictable portion (δ (t)):
Figure GDA0003031337100000021
where ω (t) is the process noise, δ (t) and the measurement noise ε (t) are set to be the Gaussian mixture model, and the probability density function is as follows:
Figure GDA0003031337100000022
Figure GDA0003031337100000023
in the formula, g [. C]Is the density of a Gaussian distribution ckAnd fnIs taken as the mean value of the average value,
Figure GDA0003031337100000024
and
Figure GDA0003031337100000025
is the variance;
measurement output ymThe relation to the real output y is:
ym(t)＝y(t)+ε(t)，
corrected output yrThe posterior distribution of (t) is output by measurement ym(t) and predictable portion
Figure GDA0003031337100000026
Giving out; according to BayesCriterion, yr(t) posterior distribution and ymLikelihood function of (t) and
Figure GDA0003031337100000027
is proportional to the product of the likelihood functions of (a):
Figure GDA0003031337100000028
Figure GDA0003031337100000029
and L [ y ]m(t)|yr(t)]The calculation formula is as follows:
Figure GDA00030313371000000210
Figure GDA0003031337100000031
wherein, the mean value of each Gaussian component satisfies the following relation:
Figure GDA0003031337100000032
p(yr(t)) is yr(t), which is a constant, then:
Figure GDA0003031337100000033
based on the maximum a posteriori distribution principle, yrThe estimated value of (t) can be obtained by the following formula:
Figure GDA0003031337100000034
i.e. to maximize
Figure GDA0003031337100000035
And L [ y ]m(t)|yr(t)]To obtain the optimum correction signal yr(t) of (d). For measurement data y containing non-Gaussian noisem(t) directly using the maximum likelihood estimation method to find the optimum correction signal yr(t), it is obvious that the problem of too large calculation amount will be faced, so an implicit variable which cannot be observed is defined firstly, a probability model depending on the implicit variable is established, and parameter maximum likelihood estimation is searched in the probability model, namely, the EM algorithm is adopted to finish correcting the signal yr(t) solving;
step two, deducing an iterative formula in the problem description by utilizing an EM algorithm:
(1) and (3) defining hidden variables, writing a log-likelihood function of the complete data:
imagine observation data ym(ym1,ym2,…,ymJ) Is generated as follows: first according to the probability akSelecting the kth Gaussian distribution partial model g (y)m|θk) (ii) a Then according to the probability distribution g (y) of the k-th partial modelm|θk) Generating data ym(ii) a At this time, observation data ymjJ ═ 1,2, …, J, known; observation data of reaction ymjData from the K-th partial model is unknown, K1, 2jkWhich is defined as follows:
Figure GDA0003031337100000036
obtaining observation data in the same way
Figure GDA0003031337100000037
Corresponding hidden variable
Figure GDA0003031337100000038
N ═ 1,2,., N is defined as follows:
Figure GDA0003031337100000041
the complete data are:
Figure GDA0003031337100000042
from this, the likelihood function of the full data is written:
Figure GDA0003031337100000043
wherein the content of the first and second substances,
Figure GDA0003031337100000044
then, the log-likelihood function of the full data is:
Figure GDA0003031337100000045
(2) e, determining a Q function in the EM algorithm;
and (3) expecting the log-likelihood function of the complete data to obtain a Q function:
Figure GDA0003031337100000046
calculation of E (γ)jk),
Figure GDA0003031337100000047
Are respectively marked as
Figure GDA0003031337100000048
Figure GDA0003031337100000051
The same principle is that:
Figure GDA0003031337100000052
Figure GDA0003031337100000053
is to observe data y under the current model parametersmjThe probability from the kth partial model becomes the partial model k for the observation data ymjThe responsivity of (a);
Figure GDA0003031337100000054
is to observe data under the current model parameters
Figure GDA0003031337100000055
The probability from the nth partial model is the n pairs of observation data of the partial model
Figure GDA0003031337100000056
The responsivity of (a);
according to the above calculation, the method can obtain
Figure GDA0003031337100000057
(3) M steps of the EM algorithm: maximizing the Q function to obtain an iterative formula;
the M steps of iteration are maximum value of Q function to parameter, namely model parameter of new iteration is solved:
Figure GDA0003031337100000058
by using
Figure GDA0003031337100000059
Denotes theta(i+1)Each parameter of (a); to find
Figure GDA00030313371000000510
Only the Q function needs to be respectively paired with lambdak,ηk,μn,ρnObtaining a deviation derivative and making the deviation derivative be 0; to find
Figure GDA00030313371000000511
Is at Σ ak＝1，∑bnThe partial derivative is calculated and made 0 under the condition of 1, and the result is as follows:
Figure GDA0003031337100000061
Figure GDA0003031337100000062
Figure GDA0003031337100000063
Figure GDA0003031337100000064
Figure GDA0003031337100000065
Figure GDA0003031337100000066
obtaining an iterative formula for solving each parameter; repeating the above calculations until the log-likelihood function values no longer change significantly;
step three, solving and correcting the output y by using the iterative formular：
The gaussian components have the following relationship:
Figure GDA0003031337100000067
solving the equation set can obtain the optimal corrected output yr；
Selecting performance indexes based on statistical information, which specifically comprises the following steps:
because the process is affected by non-Gaussian random interference, more general statistical information except the mean and the variance is adopted for measuring the randomness; in order to characterize the tracking performance and the interference suppression performance, the average value of the squared tracking error and the entropy of the tracking error are minimized; furthermore, the control energy is also minimized; thus, the controller is obtained by minimizing the following performance indicators:
Figure GDA0003031337100000071
here, a general measure of uncertainty, i.e., entropy, is used instead of variance in a gaussian system; considering the calculation efficiency, the selected entropy measure is quadratic arbitrary entropy
Figure GDA0003031337100000072
As is known from the definition of Information Potential (IP),
Figure GDA0003031337100000073
the quadratic arbitrary entropy is a monotonic function of the quadratic information potential, so the maximization of the quadratic arbitrary entropy is equivalent to the minimization of the information potential, and a random gradient method is adopted, namely, the optimal control method is realized.
Compared with the prior art, the invention has the following advantages:
(1) because the actual industrial process is inevitably influenced by random noise and is generally non-Gaussian random noise, the influence caused by the random noise is considered when the distribution of the measurement noise and the process noise is determined, and the method has more generality and practical significance than the method in which the influence of the noise is ignored or the noise is assumed to obey Gaussian distribution in the prior art;
(2) the invention fully considers the dynamic characteristics of the process and better expresses the actual process;
(3) on the premise of reasonable initial parameter selection, the method has higher calculation efficiency and meets the requirements of the actual industrial process;
(4) the invention adopts entropy statistical information to fully depict the random characteristic of non-Gaussian random quantity and establishes a tracking control index.
Drawings
FIG. 1 is a flow chart of the single step algorithm of the present invention.
FIG. 2 is a graph comparing the calibration output of the method of the present invention with that of the prior art. The horizontal axis is time, the vertical axis is output response of the non-Gaussian system, and the three tracks respectively represent a set value, output corrected by the existing Gaussian method and output corrected by the non-Gaussian method in the invention. It is demonstrated by the figure that the system output after the method is adopted is obviously superior to the existing method in accuracy and rapidity.
Detailed Description
The process of the invention is described below with reference to specific examples: for a typical wind energy conversion system, the equivalent load of a permanent magnet synchronous generator is made up of a constant inductance LsAnd a variable resistor RsAre connected in parallel. Under the application example, the method for correcting the dynamic data of the non-Gaussian system and optimizing the control performance of the system comprises the following operation steps:
step one, the permanent magnet synchronous generator model can be described as follows:
Figure GDA0003031337100000081
wherein R is stator resistance, udAnd uqVoltages of d-and q-components of the stator, LdAnd LqInductances of the stator, i, being d-and q-components, respectivelydAnd iqCurrents, phi, of d-and q-components of the stator, respectivelymIs the magnetic flux (which is a constant), p is the number of pole pair pairs, ΩGIs the rotational speed of the generator, JhIs the total inertia of the turbine, called the high speed shaft, RtIs the turbine radius, i is the gear ratio, u-RsIs a control input, y ═ ΩGIs the system output;
the state space model of the permanent magnet synchronous generator based on the wind power generation system is established as follows:
Figure GDA0003031337100000082
wherein:
Figure GDA0003031337100000083
Figure GDA0003031337100000084
in order to realize real-time control, a state space model of the system is discretized:
Figure GDA0003031337100000085
wherein x isk∈R3×1Is a state vector, and all variables in F (-) and G (-) are continuous, bounded, and differentiable to the first order, ωkIs a non-gaussian perturbation of random wind speed;
step two, obtaining the measurement output y after the influence of the measured noise in the control processmThe steps of correcting it are as follows:
for measurement data y containing non-Gaussian noisemDirectly using maximum likelihood estimation to obtain its correction signal yrIt is obvious that the problem of too large calculation amount is faced, so that firstly, defining an invisible variable which cannot be observed, establishing a probability model depending on the invisible variable, and searching parameter maximum likelihood estimation in the probability model, namely, adopting EM algorithm to complete ymAnd solving the correction signal.
(1) Initialization parameter lambdak,ηk,μn,ρn,ak,bn,ymj,
Figure GDA0003031337100000091
(2) E, step E: calculating the responsivity according to the current model parameters;
(3) and M: the model parameters for a new iteration are calculated using the following iteration formula:
Figure GDA0003031337100000092
Figure GDA0003031337100000093
Figure GDA0003031337100000094
Figure GDA0003031337100000095
Figure GDA0003031337100000096
Figure GDA0003031337100000101
(4) repeating the step (2) and the step (3) until convergence;
(5) the final parameter a is obtainedk,bn,λk,μnIs estimated value of
Figure GDA0003031337100000102
By
Figure GDA0003031337100000103
A system of linear equations can be obtained:
Figure GDA0003031337100000104
solving the equation set to obtain corrected output yrIs afterContinuous system dynamic optimization and control provide measurement data closer to the true value;
selecting performance indexes based on statistical information, which specifically comprises the following steps:
because the process is affected by non-Gaussian random interference, more general statistical information except the mean and the variance is adopted for measuring the randomness; in order to characterize the tracking performance and the interference suppression performance, the average value of the squared tracking error and the entropy of the tracking error are minimized; furthermore, the control energy is also minimized; thus, the controller is obtained by minimizing the following performance indicators:
Figure GDA0003031337100000105
here, a general measure of uncertainty, i.e., entropy, is used instead of variance in a gaussian system; considering the calculation efficiency, the selected entropy measure is quadratic arbitrary entropy
Figure GDA0003031337100000106
As is known from the definition of Information Potential (IP),
Figure GDA0003031337100000107
the quadratic arbitrary entropy is a monotonic function of the quadratic information potential, so the maximization of the quadratic arbitrary entropy is equivalent to the minimization of the information potential, and a random gradient method is adopted, namely, the optimal control method is realized.

Claims (1)

1. A non-Gaussian system dynamic data correction and system control performance optimization method is characterized in that: the method comprises the following operation steps:
step one, describing a system model under non-Gaussian disturbance:
in a univariate system, the output at time t can be decomposed into two parts: predictable portion
Figure FDA0003031337090000011
And an unpredictable portion (δ (t)):
Figure FDA0003031337090000012
where ω (t) is the process noise, δ (t) and the measurement noise ε (t) are set to be the Gaussian mixture model, and the probability density function is as follows:
Figure FDA0003031337090000013
Figure FDA0003031337090000014
in the formula, g [. C]Is the density of a Gaussian distribution ckAnd fnIs taken as the mean value of the average value,
Figure FDA0003031337090000015
and
Figure FDA0003031337090000016
is the variance;
measurement output ymThe relation to the real output y is:
ym(t)＝y(t)+ε(t)，
corrected output yrThe posterior distribution of (t) is output by measurement ym(t) and predictable portion
Figure FDA0003031337090000017
Giving out; according to Bayesian criterion, yr(t) posterior distribution and ymLikelihood function of (t) and
Figure FDA0003031337090000018
is proportional to the product of the likelihood functions of (a):
Figure FDA0003031337090000019
Figure FDA00030313370900000110
and L [ y ]m(t)|yr(t)]The calculation formula is as follows:
Figure FDA00030313370900000111
Figure FDA00030313370900000112
wherein, the mean value of each Gaussian component satisfies the following relation:
Figure FDA0003031337090000021
p(yr(t)) is yr(t), which is a constant, then:
Figure FDA0003031337090000022
based on the maximum a posteriori distribution principle, yrThe estimated value of (t) can be obtained by the following formula:
Figure FDA0003031337090000023
i.e. to maximize
Figure FDA0003031337090000024
And L [ y ]m(t)|yr(t)]To obtain the optimum correction signal yr(t) of (d). For measurement data y containing non-Gaussian noisem(t) finding the optimum by directly using the maximum likelihood estimation methodCorrection signal y ofr(t), it is obvious that the problem of too large calculation amount will be faced, so an implicit variable which cannot be observed is defined firstly, a probability model depending on the implicit variable is established, and parameter maximum likelihood estimation is searched in the probability model, namely, the EM algorithm is adopted to finish correcting the signal yr(t) solving;
step two, deducing an iterative formula in the problem description by utilizing an EM algorithm:
(1) and (3) defining hidden variables, writing a log-likelihood function of the complete data:
imagine observation data ym(ym1,ym2,…,ymJ) Is generated as follows: first according to the probability akSelecting the kth Gaussian distribution partial model g (y)m|θk) (ii) a Then according to the probability distribution g (y) of the k-th partial modelm|θk) Generating data ym(ii) a At this time, observation data ymjJ ═ 1,2, …, J, known; observation data of reaction ymjData from the K-th partial model is unknown, K1, 2jkWhich is defined as follows:
Figure FDA0003031337090000025
obtaining observation data in the same way
Figure FDA0003031337090000026
Corresponding hidden variable
Figure FDA0003031337090000027
It is defined as follows:
Figure FDA0003031337090000028
the complete data are:
Figure FDA0003031337090000029
from which a likelihood function of the complete data is written：
Figure FDA0003031337090000031
Wherein the content of the first and second substances,
Figure FDA0003031337090000032
then, the log-likelihood function of the full data is:
Figure FDA0003031337090000033
(2) e, determining a Q function in the EM algorithm;
and (3) expecting the log-likelihood function of the complete data to obtain a Q function:
Figure FDA0003031337090000034
calculation of E (γ)jk),
Figure FDA0003031337090000035
Are respectively marked as
Figure FDA0003031337090000036
Figure FDA0003031337090000037
The same principle is that:
Figure FDA0003031337090000041
Figure FDA0003031337090000042
is to observe data y under the current model parametersmjThe probability from the kth partial model becomes the partial model k for the observation data ymjThe responsivity of (a);
Figure FDA0003031337090000043
is to observe data under the current model parameters
Figure FDA0003031337090000044
The probability from the nth partial model is the n pairs of observation data of the partial model
Figure FDA0003031337090000045
The responsivity of (a);
according to the above calculation, the method can obtain
Figure FDA0003031337090000046
(3) M steps of the EM algorithm: maximizing the Q function to obtain an iterative formula;
the M steps of iteration are maximum value of Q function to parameter, namely model parameter of new iteration is solved:
Figure FDA0003031337090000047
by using
Figure FDA0003031337090000048
Denotes theta(i+1)Each parameter of (a); to find
Figure FDA0003031337090000049
Only the Q function needs to be respectively paired with lambdak,ηk,μn,ρnObtaining a deviation derivative and making the deviation derivative be 0; to find
Figure FDA00030313370900000410
Is at Σ ak＝1，∑bnThe partial derivative is calculated and made 0 under the condition of 1, and the result is as follows:
Figure FDA00030313370900000411
Figure FDA00030313370900000412
Figure FDA0003031337090000051
Figure FDA0003031337090000052
Figure FDA0003031337090000053
Figure FDA0003031337090000054
obtaining an iterative formula for solving each parameter; repeating the above calculations until the log-likelihood function values no longer change significantly;
step three, solving and correcting the output y by using the iterative formular：
The gaussian components have the following relationship:
Figure FDA0003031337090000055
solving the equation set to obtain the optimal corrected equationOutput yr；
Selecting performance indexes based on statistical information, which specifically comprises the following steps:
because the process is affected by non-Gaussian random interference, more general statistical information except the mean and the variance is adopted for measuring the randomness; in order to characterize the tracking performance and the interference suppression performance, the average value of the squared tracking error and the entropy of the tracking error are minimized; furthermore, the control energy is also minimized; thus, the controller is obtained by minimizing the following performance indicators:
Figure FDA0003031337090000056
here, a general measure of uncertainty, i.e., entropy, is used instead of variance in a gaussian system; considering the calculation efficiency, the selected entropy measure is quadratic arbitrary entropy
Figure FDA0003031337090000057
As is known from the definition of Information Potential (IP),
Figure FDA0003031337090000061
the quadratic arbitrary entropy is a monotonic function of the quadratic information potential, so the maximization of the quadratic arbitrary entropy is equivalent to the minimization of the information potential, and a random gradient method is adopted, namely, the optimal control method is realized.
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