CN111258222A - Self-adaptive state estimation method of autoregressive moving average system and closed-loop control system - Google Patents
Self-adaptive state estimation method of autoregressive moving average system and closed-loop control system Download PDFInfo
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Abstract
The application discloses a method and a device for carrying out self-adaptive state estimation on an autoregressive moving average system with additive output noise and control variables and a closed-loop control system. The method comprises the steps of implementing a state space of an autoregressive moving average system with a preset application background of additive output noise and control variables; modeling additive output noise of the autoregressive moving average system by using an L2 norm regular term; estimating the state value and the output noise of the autoregressive moving average system simultaneously by utilizing a regularization least square method; and taking the regularization parameter which is used for adjusting the detection strength of the output noise and corresponds to the minimum error between the sample variance of the estimated residual and the variance of the actual system noise as an optimal regularization parameter. The method and the device can eliminate the negative influence caused by the autocorrelation of the output noise under the condition that the output of the system has the autocorrelation, and provide a method for carrying out self-adaptive unbiased estimation on the state value of the system.
Description
Technical Field
The present application relates to the field of discrete linear system filtering technologies, and in particular, to a method and an apparatus for performing adaptive state estimation on an autoregressive moving average system with additive output noise and control variables, and a closed-loop control system.
Background
In control engineering, an autoregressive moving average system with control variables is widely used to describe actual physical systems. The autoregressive moving average system with control variables is a special discrete time series system and is widely used for modeling and control design of an actual physical system in control engineering. A general autoregressive moving average system with controlled variables can strike a good balance between the complexity of the model and the accuracy of the modeling when describing many real physical systems or dynamic processes. But because it does not take into account the effects of noise acting on the system output, undesirable consequences can occur in many applications such as filtering and false detection. On the basis, an autoregressive moving average system with output noise and control variables is generated.
The kalman filter, which is most applied in discrete linear systems, can be used for the estimation of general autoregressive moving average system states with controlled variables. For a general discrete linear system, the kalman filter is an optimal linear estimator, which provides an estimation value with the least error variance characteristic. However, for an autoregressive moving average system with output noise and control variables, the kalman filter cannot achieve optimal unbiased estimation because the negative effects propagated by the autocorrelation of the system by the non-zero mean output noise cannot be eliminated. The kalman filter can also be interpreted as a maximum likelihood estimation based on gaussian probability distribution, according to which other probability density functions, such as t probability density functions, can also be utilized for designing a state estimator for an autoregressive moving average system with controlled variables in general. It should be noted that these estimators only consider the case where the output of the system can be measured with full accuracy, but do not consider the case where the system output is affected by noise. It is difficult to obtain accurate state estimation results for estimators that do not take output noise into account, given that measurement errors are difficult to completely eliminate. The effects of these estimators in practical applications are therefore still questionable.
From the structure of the autoregressive moving average system, the system output consists of the sum of three regression terms: a regression term on past inputs to the system, a regression term on past outputs from the system, and a white noise sequence. This structure makes it difficult to estimate the state of the autoregressive moving average system. Since an inaccurate measurement value may affect the estimation of a plurality of subsequent time state values. To solve this problem, an autoregressive moving average system with output noise and control variables is proposed. However, how to estimate the state of such systems has not been well studied and solved.
In summary, the problems of the prior art are as follows: existing estimators are designed only for general autoregressive moving average systems, based on the assumption that the output of the system can be measured completely accurately. When the system output is interfered by noise, the state estimation value at the subsequent moment is also seriously interfered due to the autocorrelation of the system output, so that the existing estimator can not provide unbiased and accurate estimation for the state value of the autoregressive moving average system with the output noise. When the output of the system has autocorrelation characteristics, a common estimator, such as a kalman filter, a maximum likelihood estimator based on t distribution, and the like, is difficult to solve the problem that output noise propagates through the autocorrelation characteristics of the system, thereby causing a problem that it is difficult to accurately estimate the state of the autoregressive moving average system with the output noise. Besides, when the statistical variance of the output noise is unknown or has a time-varying characteristic, how to adaptively adjust the estimator parameters is also a problem to be solved.
In view of this, how to provide an accurate state estimation result with adaptive and unbiased characteristics for an autoregressive moving average system with output noise and control variables is a technical problem to be solved in the technical field.
Disclosure of Invention
The application provides a method and a device for carrying out self-adaptive state estimation on an autoregressive moving average system with additive output noise and control variables and a closed-loop control system, and realizes the purpose of providing an accurate state estimation result with self-adaptive unbiased characteristics for the autoregressive moving average system with the output noise and the control variables.
In order to solve the above technical problems, embodiments of the present invention provide the following technical solutions:
one aspect of the embodiments of the present invention provides a method for performing adaptive state estimation on an autoregressive moving average system with additive output noise and a control variable, which is applied to a filter, and includes:
implementing a state space of an autoregressive moving average system with a preset application background of additive output noise and control variables;
modeling additive output noise of the autoregressive moving average system using an L2 norm regularization term;
estimating the state value and the output noise of the autoregressive moving average system simultaneously by utilizing a regularization least square method;
taking the regularization parameter corresponding to the minimum error between the sample variance of the estimated residual error and the variance of the actual system noise as the optimal regularization parameter of the autoregressive moving average system; the regularization parameter is used to adjust the detection strength of the output noise.
Optionally, the implementing the state space of the autoregressive moving average system with the preset application background of additive output noise and control variables includes:
setting an initial autoregressive moving average system as an autoregressive moving average system with additive output noise and control variables, wherein the autoregressive moving average system is as follows:
A(q-1)zk＝B(q-1)uk+C(q-1)εk；
yk＝zk+vk；
and carrying out state space realization on the autoregressive moving average system by using a state space relation, wherein the state space relation is as follows:
xk+1＝Φxk+Γuk+Ω(yk-vk)；
yk＝Hxk+εk+vk；
where k is the sample time index, zk∈RmIs the output of the autoregressive moving average system before being disturbed by noise, yk∈RmFor the autoregressive sliding after noise interferenceAveraging the output of the system; u. ofkInputting the autoregressive moving average system; epsilonkNoise in the autoregressive moving average system is consistent with zero mean and variance of Rk(ii) a gaussian distribution of; v. ofkThe mean value of the output noise of the autoregressive moving average system is vkAnd the variance is DkWhite noise characteristics of (a); u. ofk，εk，vkThe three are not related to each other; q. q.s-1Shifting operator for time index, q-1zk＝zk-1，A(q-1) And C (q)-1) All within the unit circle, A (q)-1)＝1+a1q-1+a2q-2+…+anq-n，B(q-1)＝b1q-1+b2q-2+…+bnq-n；C(q-1)＝1+c1q-1+c2q-1+…+cnq-1(ii) a Phi, gamma, omega and H are known constant matrixes, and x is a real state value of the autoregressive moving average system.
Optionally, the modeling additive output noise of the autoregressive moving average system by using an L2 norm regularization term includes:
modeling additive output noise of the autoregressive moving average system by using a modeling relation, wherein the output noise vkIn accordance with a mean value of
And the variance is DkThe modeling relation is as follows:
wherein f (v)k) For the output noise vkAnd carrying out a normalized L2 norm model.
Optionally, the simultaneously estimating the state value and the output noise of the autoregressive moving average system by using a regularized least square method includes:
calculating a cost function of the autoregressive moving average system based on the L2 norm regularized least squares estimator, the cost function being:
computing
Obtaining a state estimation value and an output noise estimation value of the autoregressive moving average system at the moment k;
in the formula, xkIs the true state value, v, of the autoregressive moving average system at time kkAs output noise of the system, DkFor the output noise vkThe variance of (a) is determined,
for the output noise vkMean value of (a), ykIs the output of the autoregressive moving average system after being interfered by noise;
is the state estimate of the autoregressive moving average system at time k,
and outputting a noise estimation value of the autoregressive moving average system at the k moment.
Optionally, the state estimation value of the autoregressive moving average system at the time k is:
Kk＝Pk|k-1HT(HPk|k-1HT+Rk+Dk)-1；
in the formula (I), the compound is shown in the specification,
is a state prediction value P obtained by the autoregressive moving average system at the moment k according to an estimation value at the moment k-1k|k-1Is composed of
H is a matrix of known constants, RkAn approximation of the variance of the residual is estimated.
Optionally, the output noise estimation value of the autoregressive moving average system at the time k is:
Mk＝Dk(HPk|k-1HT+Rk+Dk)-1；
in the formula (I), the compound is shown in the specification,
is a state prediction value P obtained by the autoregressive moving average system at the k moment according to an estimation value at the k-1 momentk|k-1Is composed of
H is a matrix of known constants, RkAn approximation of the variance of the residual is estimated.
Optionally, the method for calculating the optimal regularization parameter includes:
the optimal regularization parameter if the output noise variance is unknown or varies over time
According to
Calculating to obtain;
if the variance of the output noise is known, taking the variance of the output noise conforming to the white noise characteristic as the regularization parameter;
in the formula, RkFor system internal noise epsilonkThe ideal value of the variance is calculated,
for system internal noise estimation
The variance of (a) is determined,
is a regularization parameter DkTime epsilonkEstimate of (e ∈)kAs noise within the system, DkIs the regularization parameter.
In another aspect, an embodiment of the present invention provides an apparatus for performing adaptive state estimation on an autoregressive moving average system with additive output noise and a control variable, where the apparatus is applied to a filter, and the apparatus includes:
the state space realization module is used for realizing the state space of an autoregressive moving average system with an additive output noise and a control variable in a preset application background;
a noise modeling module for modeling additive output noise of the autoregressive moving average system using an L2 norm regularization term;
the estimation module is used for simultaneously estimating the state value and the output noise of the autoregressive moving average system by utilizing a regularized least square method;
the parameter determining module is used for taking a regularization parameter corresponding to the minimum error between the sample variance of the estimated residual error and the variance of the actual system noise as an optimal regularization parameter of the autoregressive moving average system; the regularization parameter is used for adaptively adjusting the detection strength of the output noise.
An embodiment of the present invention further provides an apparatus for adaptive state estimation of an autoregressive moving average system with additive output noise and a control variable, which is applied to a filter, and includes a processor, which is configured to implement the steps of the method for adaptive state estimation of an autoregressive moving average system with additive output noise and a control variable as described in any one of the preceding items when executing a computer program stored in a memory.
The embodiment of the invention finally provides a closed-loop control system, which comprises a PID controller, an estimator and an autoregressive moving average system;
wherein the PID controller is for generating a control signal to cause the output of the autoregressive moving average system being controlled to track the upper reference signal; the estimator is used for generating a system state estimation value according to the output signal of the autoregressive moving average system and the control signal and sending the system state estimation value to the PID controller as a feedback signal; the autoregressive moving average system is adapted to implement the steps of the method for adaptive state estimation of an autoregressive moving average system with additive output noise and control variables as described in any one of the above when executing a computer program stored in a memory.
The technical solution provided by the present application has the advantage of explicitly modeling and estimating the output noise of an autoregressive moving average system using L2 norm regularization. Because the output noise is explicitly modeled, the adverse effect of autocorrelation of the output noise on the system state estimation value at the subsequent moment can be estimated and reduced, and a more accurate system state estimation result can be provided; modeling additive noise acting on system output by using an L2 norm regularization term, estimating a system state value and output noise simultaneously by using a regularization least square method, and eliminating adverse effects of the output noise on a state estimation value at a subsequent moment through system autocorrelation; in addition, the regularization parameters can be adaptively determined according to the change of the statistical characteristics of the output noise, the problem that the state of an autoregressive moving average system with the output noise and control variables cannot be accurately estimated by a traditional estimation method is solved, the negative influence caused by the autocorrelation of the output noise can be eliminated under the condition that the system output has the autocorrelation, and the adaptive unbiased estimation of the system state value is provided. The method and the device can solve the problem that the condition that the system output is influenced by noise interference in the actual application is not considered in the system design, and can also solve the problem of how to adaptively adjust the parameters of the estimator when the statistical characteristic of the output noise changes along with the time so as to achieve the purpose of optimizing the estimation result.
In addition, the embodiment of the invention also provides a corresponding implementation device and a closed-loop control system for the method for carrying out the self-adaptive state estimation on the autoregressive moving average system with additive output noise and control variables, so that the method has higher practicability, and the device and the closed-loop control system have corresponding advantages.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the disclosure.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions of the related art, the drawings required to be used in the description of the embodiments or the related art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a schematic flow chart of a method for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables according to an embodiment of the present invention;
FIG. 2 shows a system state x according to an embodiment of the present inventionk,1The true value, the kalman filter, the M-estimator and the estimated value of the filter proposed by the present invention are illustrated schematically;
FIG. 3 is a block diagram of an embodiment of an apparatus for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables according to an embodiment of the present invention;
FIG. 4 is a block diagram of another embodiment of an apparatus for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables according to an embodiment of the present invention;
FIG. 5 is a block diagram of an embodiment of a closed loop control system according to the present invention;
FIG. 6 is a schematic diagram of a root mean square error of an estimated value of a Kalman filter under impact noise influence of different occurrence probabilities;
fig. 7 is a schematic diagram of the root mean square error of the estimated value of the filter proposed in the present application under the influence of impact noise with different occurrence probabilities.
Detailed Description
In order that those skilled in the art will better understand the disclosure, the invention will be described in further detail with reference to the accompanying drawings and specific embodiments. It is to be understood that the described embodiments are merely exemplary of the invention, and not restrictive of the full scope of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The terms "first," "second," "third," "fourth," and the like in the description and claims of this application and in the above-described drawings are used for distinguishing between different objects and not for describing a particular order. Furthermore, the terms "comprising" and "having," as well as any variations thereof, are intended to cover non-exclusive inclusions. For example, a process, method, system, article, or apparatus that comprises a list of steps or elements is not limited to only those steps or elements but may include other steps or elements not expressly listed.
Having described the technical solutions of the embodiments of the present invention, various non-limiting embodiments of the present application are described in detail below.
Referring first to fig. 1, fig. 1 is a schematic flowchart of a method for performing adaptive state estimation on an autoregressive moving average system with additive output noise and a control variable according to an embodiment of the present invention, which is applied to a filter and can be used as an estimator of the filter, where the embodiment of the present invention can include the following:
s101: and implementing the state space of the autoregressive moving average system with the preset application background of additive output noise and control variables.
Since the filter is suitable for the filter of the autoregressive moving average system with additive output noise and control variables, the parameters of the autoregressive moving average system need to be preset, namely the application background is preset to the autoregressive moving average system with the output noise and the control variables. The state space implementation of the autoregressive moving average system is also to describe the autoregressive moving average system in the state space for subsequent state estimation.
S102: additive output noise of the autoregressive moving average system is modeled with an L2 norm regularization term.
S103: and simultaneously estimating the state value and the output noise of the autoregressive moving average system by utilizing a regularization least square method.
S104: and taking the regularization parameter corresponding to the minimum error between the sample variance of the estimated residual and the variance of the actual system noise as the optimal regularization parameter of the autoregressive moving average system.
Consider the following single-input single-output autoregressive moving average system:
A(q-1)zk＝B(q-1)uk+C(q-1)εk(1)
yk＝zk+vk(2)
where k is the sample time index, zk∈RmIs the output of the autoregressive moving average system before being disturbed by noise, yk∈RmThe output of the autoregressive moving average system after being interfered by noise is obtained; u. ofkIs an autoregressive moving average system input; epsilonkIs noise in the autoregressive moving average system, and has zero mean and variance of Rk(ii) a gaussian distribution of; v. ofkIs the output noise of the autoregressive moving average system, and is in accordance with the mean value of
And the variance is DkWhite noise characteristics of (a); u. ofk，εk，vkThree are not related to each other, x is fromAnd (4) regressing the real state value of the moving average system. Polynomial A (q)-1)、B(q-1) And C (q)-1) Are respectively given by the following formulas:
A(q-1)＝1+a1q-1+a2q-2+…+anq-n；
B(q-1)＝b1q-1+b2q-2+…+bnq-n；
C(q-1)＝1+c1q-1+c2q-1+…+cnq-1。
q-1shift an operator for a time index, and q-1zk＝zk-1，A(q-1) And C (q)-1) Is within the unit circle to ensure causality and reversibility of the system, phi, gamma, omega and H are known constant matrixes, and the specific value of the matrix can be A (q)-1)、B(q-1) And C (q)-1) Calculated by known methods.
In order to estimate the state of the autoregressive moving average system given by the two formulas (1) and (2), the transfer function of the autoregressive moving average system needs to be converted into a state space equation; wherein (1) A (q) given in the formula-1)zk＝B(q-1)uk+C(q-1)εkThis portion of the transfer function can be represented by the following known state space equation:
xk+1＝Φxk+Γuk+Ωzk(3)
zk＝Hxk+εk(4)
wherein the content of the first and second substances,
Γ＝[b1b2…bn]；
Ω＝[c1-a1c2-a2…cn-an]；
H＝[1 0…0]。
further, in order toWriting an autoregressive moving average system with output noise into a state space equation, and writing y in the equation (2)k＝zk+vkSubstitution into equations (3) and (4) can give:
xk+1＝Φxk+Γuk+Ω(yk-vk) (5)
yk＝Hxk+εk+vk(6)
due to output noise vkIs introduced into the dynamic equation (3) of the system, so that the output noise can propagate through the autocorrelation of the system, thereby having negative influence on the estimated value of the system state at the subsequent time point. Although the conventional kalman filter can provide the minimum variance unbiased estimation for the general linear system, the conventional kalman filter cannot provide the minimum variance unbiased estimation value for the non-general linear system given by the above formula.
Because of the output noise vkIn accordance with a mean value of
And the variance is DkThe white noise characteristic of (a) can be modeled by using a modeling relation formula, where the modeling relation formula can be expressed as:
wherein f (v)k) To output noise vkAnd carrying out a normalized L2 norm model.
Therefore, the L2 norm regular term can be adopted to model the cost function, and the additional increase is added when the cost function is established
This term is used to p vkPenalizing the resulting influence while explicitly estimating vkAnd the influence of the autocorrelation of the system on the subsequent state estimation value is reduced.
Assuming that at time k-1, the filter is producibleGenerating an estimate
And its estimation error covariance matrix
Then, at the time k, the state prediction value obtained by the autoregressive moving average system according to the estimated value at the time k-1 can be obtained according to the estimated value and the system state space equation given in the formula (3)
And
the specific method can be as follows:
furthermore, the predicted value in equation (5) can be used
And the measured value ykFor calculating the latest estimated value at time k
If it will be
Viewed as a random measurement, the true state x of the systemkAnd its predicted value
The relationship between can be expressed as
Wherein deltakThe covariance matrix is represented by P in equation (8) for the prediction error at time kk|k-1Giving out; v is to bekAnd
the error between is recorded as dkThen, then
dkHas a mean of 0 and a variance of Dk(ii) a In combination with the observation model of the system given in equation (6), the following linear regression model can be obtained:
the penalty term can thus be designed separately for the prediction error of each row of the linear regression model in equation (9), i.e. for δk，dk，εkAnd respectively designing a quadratic penalty term to obtain the following cost function of the regularized least square estimator based on the L2 norm:
it is noted that the cost function, which is different from the general state estimation problem, is only in xkFor the independent variable, the cost function given in (10) is given in xkAnd vkIs an independent variable. The same general state estimation and filtering method can only provide the estimation value of the system state
Compared with the prior art, the method provided by the application can simultaneously estimate the state of the system
And output noise
Thus, in the case where the output of the system has an autocorrelation characteristic, it can be eliminatedThe output noise influences the state estimation value at the subsequent moment through the autocorrelation of the system, and further higher estimation precision can be provided. In the above-mentioned cost function, the cost function,
can be regarded as an additionally added regularization term, and DkWhich may be considered a regularization parameter, for adjusting the strength of the detection of the output noise.
(2) Solving the estimation value of the self-adaptive unbiased estimator:
at time k, to obtain
And
it is necessary to find the cost function J in the equation (10)kX corresponding to the minimum valuekAnd vkThe following optimization problem needs to be solved:
by calculating JkRelative to xkAnd vkThe partial derivatives of (c) can be obtained as the following equations (12) and (13):
wherein in the formula (12)
Represents a pair xkPartial derivatives are determined, and in (13)
Represents a pair vkAnd (5) calculating a partial derivative. By reacting a compound of the formula (12)The two partial derivatives of equations (14) and (15) can be further derived by setting the two partial derivatives of equation (13) to 0:
by substituting equation (15) into equation (14) and using the matrix inversion theorem, equation (16) can be obtained:
similarly, substituting equation (14) into equation (15) and using the matrix inversion theorem, equation (17) can be obtained:
the estimated values of the state quantity and the output noise can be obtained by collating expressions (16) and (17), and are given by expressions (18) and (19):
wherein the parameter K in the formulae (14) and (15)kAnd MkThe following expressions are given by the expressions (20) and (21), respectively:
Kk＝Pk|k-1HT(HPk|k-1HT+Rk+Dk)-1(20)
Mk＝Dk(HPk|k-1HT+Rk+Dk)-1(21)
by following a conventional least squares analysis of variance, an estimation error covariance matrix of the state quantities and the output noise estimate can be obtained:
wherein the content of the first and second substances,
is that
The error covariance matrix of (a) is calculated,
is that
Of the error covariance matrix of
And
is that
And
cross covariance matrix of (2).
To demonstrate the stability and unbiasedness of the estimator proposed in this application, a dynamic equation for the estimation error can be established. Before that, the state space equations given in equations (5) and (6) can be first written into compact forms as expressed in equations (26) and (27):
wherein the content of the first and second substances,
the prediction steps in equations (7) and (8) can be represented as equations (28) and (29), respectively, using the compact forms given in equations (26) and (27); expressing the estimated values in expressions (18) and (19) as expression (31); the gains in the expressions (20) and (21) may be expressed as expression (30); the covariance matrix in the equations (22) to (25) is expressed as equation (32):
thus, the error e is estimatedkI.e. true value
And the estimated value
The difference between them can be expressed as (33):
then the error dynamics equation given in equation (34) can be obtained by substituting equations (26) and (31) into equation (33):
it is to be noted that the error dynamic equation given in the formula (34) only sums with the system noise εkAnd output noise vkRelated to the system input ukIs irrelevant. The stability of the error dynamics equation is only related to the homogeneous part of equation (34), i.e. only to the matrix
It is related. And error ekCorresponding error covariance matrix
Then it can be calculated by equation (35):
because of the fact that
Is an error covariance matrix and thus it must satisfy a symmetric and semi-positive condition. In addition, the calculation of equation (35) is independent of the system observed value ykTherefore, the calculation of the error covariance matrix can be completed in advance when the estimator is designed, and the error covariance matrix can be stored for standby, so that the storage space is exchanged for the calculation time.
In verifying stability, polynomial C (q) can be used according to the preconditions-1) Is within the unit circle, and the matrix Φ satisfies the equation in (36):
|I-Φq-1|＝C(q-1) (36)
firstly, based on the relation in the formula (36), the spectrum radius of the matrix phi is less than 1, namely, | phi | < 1 is satisfied. And because of the matrix
All the eigenvalues of the matrix phi are included and the size of the redundant eigenvalues is 0, so the matrix phi
Is also less than 1, i.e. satisfies
Secondly, the matrix is known from the matrix inversion theorem
Satisfies the equation in the formula (37):
due to the fact that
Is a covariance matrix, so it satisfies the condition of symmetry and semi-positive; because of the matrix
Satisfy the conditions of symmetry and positive definite, and further know the matrix
And matrix
With the same non-zero real eigenvalues. So that the matrix
Must be greater than or equal to 1, so that the matrix
Must be less than or equal to 1, i.e. the condition in equation (38) is satisfied:
finally, using the fundamental nature of the product between the matrix norms, the inequality in (39) can be found:
it is thus known that the error system represented by the error dynamic equation (34) is stable.
In the process of verifying the unbiased property of the estimated value, two ends of equation (34) can be expected at the same time, and the following can be obtained:
according to the preconditions, E (. epsilon.)k) Is equal to 0 and
therefore, the desire in the formula (40) can be simplified to the formula (41):
when the initial condition of the estimator satisfies unbiased, i.e. E (E)0) 0 for any k>Time 0, E (E)k) This can be iteratively determined by equation (41), namely:
E(ek)＝0 (42)
when the initial condition of the estimator does not satisfy unbiased, i.e. E (E)0) Not equal to 0, the inequality in (39) can be applied to derive the unbiased nature of the estimate. Norm of matrix
Is noted as α, i.e.:
then, the formula (41) can be used to obtain:
definition of sequences
The following were used:
wherein r is0Is constant and satisfies | | | E (E)0)||≤r0The conditions of (1). By applying equation (44) iteratively, one can deduce:
note the sequence
Satisfies the following conditions:
and when α < 1, the signal is,
satisfy the requirement of
Therefore E (E)k) Will converge to 0 at an exponential rate, i.e.:
the estimated value thus satisfies the unbiased condition in this case as well.
In practical applications, the variance of the output noise is often difficult to measure accurately, and thus how to select the parameter DkIt is very important. Parameter DkControlling the intensity of the output noise suppression if given to DkSet to a smaller value, with respect to v in the cost function (10)kWill be assigned a greater weight, and thus vkWill be small, which means that even if output noise of large amplitude is present, the estimator will detect it as noise of small amplitude; on the contrary, if D is givenkSetting a larger value, then in the cost function with respect to vkWill be assigned a smaller weight, thus vkWill become larger, which at the same time means that even the output y of the system will be largerkBeing measured perfectly accurately, the estimator may also misinterpret the presence of output noise. To find a suitable parameter DkTo correctly detect and suppress the output noise, the estimated value can be estimated
And
expressed as parameter DkA function of, i.e.
And
the residual of the estimated system noise can be obtained by equation (51):
when D is presentkWhen chosen properly, the statistical properties of this residual estimate should be close to those of the true system noise, and thus its variance should be close to Rk. Will be provided with
The variance of (A) is recorded as
When R iskAnd
when the difference value of (D) is minimum, the corresponding DkI.e., the optimal regularization parameter. The best parameter is recorded as
Then
Can be determined by equation (52):
in the formula, RkFor system internal noise epsilonkThe ideal value of the variance is calculated,
for system internal noise estimation (i.e. estimating residual)
The variance of (a) is determined,
representing the regularization parameter set to DkTime epsilonkIs estimated (i.e., the estimated residual), epsilonkAs noise within the system, DkIs a regularization parameter.
It is to be noted here that the above-mentioned,the calculation method of the optimal regularization parameter is different according to different output noises, and if the variance of the output noise is known, the variance when the output noise accords with the white noise characteristic is used as the optimal regularization parameter; optimal regularization parameters if output noise variance is unknown or varies over time
Can be determined by the equation (52).
In the technical solution provided by the embodiment of the present invention, the output noise of the autoregressive moving average system is explicitly modeled and estimated using norm regularization of L2. Because the output noise is explicitly modeled, the adverse effect of autocorrelation of the output noise on the system state estimation value at the subsequent moment can be estimated and reduced, and a more accurate system state estimation result can be provided; modeling additive noise acting on system output by using an L2 norm regularization term, estimating a system state value and output noise simultaneously by using a regularization least square method, and eliminating adverse effects of the output noise on a state estimation value at a subsequent moment through system autocorrelation; in addition, the regularization parameters can be adaptively determined according to the change of the statistical characteristics of the output noise, the problem that the state of an autoregressive moving average system with the output noise and control variables cannot be accurately estimated by a traditional estimation method is solved, the negative influence caused by the autocorrelation of the output noise can be eliminated under the condition that the system output has the autocorrelation, and the adaptive unbiased estimation of the system state value is provided. The method and the device can solve the problem that the condition that the system output is influenced by noise interference in the actual application is not considered in the system design, and can also solve the problem of how to adaptively adjust the parameters of the estimator when the statistical characteristic of the output noise changes along with the time so as to achieve the purpose of optimizing the estimation result.
It should be noted that, in the present application, there is no strict sequential execution order among the steps, and as long as the logical order is met, the steps may be executed simultaneously or according to a certain preset order, and fig. 1 is only an exemplary manner, and does not represent that only the execution order is the order.
The following describes the technical effects obtained by the present application in detail with reference to simulations, and the present application is based on an autoregressive moving average system as shown in the following formula: the following may be included:
zk-0.9zk-1＝0.1uk-1+εk-0.8εk-1；
wherein the system noise epsilonkA gaussian distribution with a mean of 0 and a variance of 0.1 is satisfied.
The initial condition of the system is set to x0The initial condition of the estimator is set to 0
And P0|-10.1. The input to the system being set to a unit step signal, i.e. u k1 is equal to or more than 0 for any k. To make the results statistically significant, all simulation experiments were repeated 1000 times to obtain the root mean square error of the estimates. Wherein, the calculation method of the root mean square error of the estimated value is given by the following formula:
wherein the content of the first and second substances,
indicating the estimation error at time k in the r-th simulation.
To demonstrate the effectiveness of the method proposed in the present application, the output noise v iskFirst set to mean 0 and variance to
Gaussian noise. Table 1 lists the variance σ at different valuesvUnder the interference of Gaussian output noise, when the estimator adopts different parameters D, the root mean square error result of the estimated value in 1000 times of simulation is obtained. It can be seen from the results that a smaller value of the parameter D will result in a cost function for vkWill be assigned a greater weight and is therefore more suitable for outputting the noise variance σvA relatively small case; in contrast, a larger value of the parameter D is more suitable for the output noise variance σvRelatively large cases. When in
The estimator provided by the invention can obtain the minimum estimation error variance. In addition, Table 2 also reports when the system is subjected to upper and lower limits, σ respectivelyvAnd-sigmavThe result when the uniformly distributed output noise influences. σ in this case in order to have the same variance for the uniformly distributed noise and the previously tested gaussian noisev 2Which is 3 times the size of the previous gaussian noise. Table 2 shows results similar to table 1, demonstrating the good performance of the proposed estimator under different types of noise.
TABLE 1 different parameters D and sigmavRoot mean square error (Gaussian output noise) of the resulting estimate in the case
TABLE 2 different parameters D and sigmavRoot mean square error of the resulting estimate (evenly distributed output noise)
In order to further show the capability of the method provided by the application to resist outliers in the output noise, the output noise v is measuredkSet to satisfy a gaussian distribution with a mean of 0 and a variance of 0.1, and when k is 50, at vkAn outlier of-10 magnitude is introduced, namely:
FIG. 2 shows a Kalman filter and the estimator proposed in this application in parameter DkRoot mean square error of the estimates at 0.1, 0.5 and 1. Besides, the parameter DkAlso determined by the adaptive optimal parameter selection method proposed by the present invention, the result of which is also shown in fig. 2. It can be seen in fig. 2 that the proposed estimator is less affected by outlier data and its maximum root mean square error is smaller with respect to the kalman filter. And when the parameter DkWhen given a larger value, e.g. DkEqual to 1, the maximum root mean square error of the estimator is smaller, i.e. less disturbed by outliers, than would otherwise be the case. However, the RMS error at steady state is larger than it otherwise would be because D is the time when D is presentkWhen the value is large, the estimator judges the output noise as large-amplitude noise, and therefore the robustness is exchanged by the accuracy. When the parameter DkWhen given a smaller value, e.g. DkEqual to 0.1, the maximum root mean square error of the estimate is larger than would otherwise be the case, i.e., more disturbed by outliers. The rms error in steady state is now small, in which case the estimator trades robustness for accuracy. When D is presentkWhen the method is determined by the self-adaptive optimal parameter selection method, the estimator gives the maximum root mean square error with the minimum amplitude, and the influence of outlier data can be well resisted.
Table 3 summarizes the maximum RMS error, the mean RMS error, and the calculation time for 1000 simulations when D was varied from 0.01 to 100 (using a computer processor with Intel i7-7700k, and a memory size of 24 GB). It can be seen that the estimator proposed by the present application can provide a smaller maximum root mean square error than the kalman filter, and its computation time is about twice that of the kalman filter. Because the estimated value of the output noise needs to be calculated, the calculation time is relatively longer. In addition, it can be seen that when the value of the parameter D is changed, the calculation time is not changed much. This is because the estimator proposed by the present application has an analytical form of solution, and thus the change of parameters does not affect the computational complexity of the estimator. It can also be seen from table 3 that D, selected by the adaptive parameter selection method, has the smallest maximum root mean square error and the third smallest mean root mean square error in all test cases. This demonstrates that the estimator proposed by the present invention can achieve a good balance between robustness and optimality by selecting a suitable parameter.
TABLE 3 RMS error maximum, mean and calculation time for the obtained estimates for different parameters D
The embodiment of the invention also provides a corresponding device for the method for carrying out the self-adaptive state estimation on the autoregressive moving average system with additive output noise and control variables, thereby further ensuring that the method has higher practicability. Wherein the means can be described separately from the functional module point of view and the hardware point of view. In the following, the apparatus for performing adaptive state estimation on an autoregressive moving average system with additive output noise and a control variable according to an embodiment of the present invention is described, and the apparatus for performing adaptive state estimation on an autoregressive moving average system with additive output noise and a control variable described below and the method for performing adaptive state estimation on an autoregressive moving average system with additive output noise and a control variable described above may be referred to correspondingly.
Referring to fig. 3, based on the angle of functional blocks, fig. 3 is a block diagram of an apparatus for performing adaptive state estimation on an autoregressive moving average system with additive output noise and control variables according to an embodiment of the present invention, in a specific implementation, the apparatus may include:
a state space implementation module 301, configured to implement a state space for an autoregressive moving average system with an additive output noise and a control variable in a preset application context.
A noise modeling module 302 for modeling additive output noise of the autoregressive moving average system using an L2 norm regularization term.
And the estimation module 303 is configured to estimate the state value and the output noise of the autoregressive moving average system simultaneously by using a regularized least square method.
A parameter determining module 304, configured to use a regularization parameter corresponding to minimization of an error between a sample variance of the estimated residual and a variance of actual system noise as an optimal regularization parameter of the autoregressive moving average system; the regularization parameter is used for adaptively adjusting the detection strength of the output noise.
The functions of the functional modules of the apparatus for performing adaptive state estimation on an autoregressive moving average system with additive output noise and control variables according to the embodiments of the present invention may be specifically implemented according to the method in the above method embodiments, and the specific implementation process may refer to the description related to the above method embodiments, and will not be described herein again.
From the above, the embodiment of the invention realizes that an accurate state estimation result with adaptive unbiased characteristics is provided for the autoregressive moving average system with output noise and control variables.
The above mentioned apparatus for adaptive state estimation of an autoregressive moving average system with additive output noise and control variables is described from the functional block perspective, and further, the present application provides an apparatus for adaptive state estimation of an autoregressive moving average system with additive output noise and control variables, which is described from the hardware perspective. Fig. 4 is a block diagram of another apparatus for adaptive state estimation of an autoregressive moving average system with additive output noise and control variables according to an embodiment of the present application. As shown in fig. 4, the apparatus comprises a memory 40 for storing a computer program;
a processor 41 for implementing the steps of the method for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables as mentioned in any of the embodiments above when executing a computer program.
Processor 41 may include one or more processing cores, such as a 4-core processor, an 8-core processor, and so forth. The processor 41 may be implemented in at least one hardware form of a DSP (Digital Signal Processing), an FPGA (Field-Programmable Gate Array), and a PLA (Programmable Logic Array). The processor 41 may also include a main processor and a coprocessor, the main processor is a processor for processing data in an awake state, and is also called a Central Processing Unit (CPU); a coprocessor is a low power processor for processing data in a standby state. In some embodiments, the processor 41 may be integrated with a GPU (Graphics Processing Unit), which is responsible for rendering and drawing the content required to be displayed on the display screen. In some embodiments, processor 41 may further include an AI (Artificial Intelligence) processor for processing computational operations related to machine learning.
In some embodiments, the apparatus for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables may further include a display 42, an input-output interface 43, a communication interface 44, a power supply 45, and a communication bus 46.
Those skilled in the art will appreciate that the configuration shown in FIG. 4 does not constitute a limitation of the apparatus for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables, and may include more or fewer components than those shown, such as sensor 47.
The functions of the functional modules of the apparatus for performing adaptive state estimation on an autoregressive moving average system with additive output noise and control variables according to the embodiments of the present invention may be specifically implemented according to the method in the above method embodiments, and the specific implementation process may refer to the description related to the above method embodiments, and will not be described herein again.
From the above, the embodiment of the invention realizes that an accurate state estimation result with adaptive unbiased characteristics is provided for the autoregressive moving average system with output noise and control variables.
It is understood that, if the method for adaptive state estimation of the autoregressive moving average system with additive output noise and control variables in the above embodiments is implemented in the form of a software functional unit and sold or used as a stand-alone product, it can be stored in a computer readable storage medium. Based on such understanding, the technical solutions of the present application may be substantially or partially implemented in the form of a software product, which is stored in a storage medium and executes all or part of the steps of the methods of the embodiments of the present application, or all or part of the technical solutions. And the aforementioned storage medium includes: a U disk, a removable hard disk, a Read-only Memory (ROM), a Random Access Memory (RAM), an electrically erasable programmable ROM, a register, a hard disk, a removable magnetic disk, a CD-ROM, a magnetic disk, or an optical disk.
Based on this, the embodiment of the present invention further provides a computer readable storage medium, which stores a program for performing adaptive state estimation on an autoregressive moving average system with additive output noise and a control variable, wherein the program for performing adaptive state estimation on the autoregressive moving average system with additive output noise and a control variable is executed by a processor, and the steps of the method for performing adaptive state estimation on the autoregressive moving average system with additive output noise and a control variable are performed as described in any one of the above embodiments.
The functions of the functional modules of the computer-readable storage medium according to the embodiment of the present invention may be specifically implemented according to the method in the foregoing method embodiment, and the specific implementation process may refer to the related description of the foregoing method embodiment, which is not described herein again.
The embodiment of the present invention further provides a closed-loop control system controlled by the PID controller, referring to fig. 5, which may include a PID controller, an estimator, and an autoregressive moving average system;
wherein the PID controller is used for generating a control signal to enable the output of the controlled autoregressive moving average system to track the upper reference signal; the estimator is used for generating a system state estimation value according to an output signal and a control signal of the autoregressive moving average system and sending the system state estimation value to the PID controller as a feedback signal; the autoregressive moving average system is used for implementing the description of the method embodiment for the adaptive state estimation of the autoregressive moving average system with additive output noise and control variables as any one of the above when executing a computer program stored in a memory.
The initial conditions of both the system and the filter remain unchanged, i.e. vkCan be given by:
as shown in FIG. 5, a PID controller can be used to generate the control signal ukSo that the output of the controlled autoregressive moving average system can track the upper reference signal rk. Since the controlled object is a discrete system, a PID controller at discrete time can be used, and the control signal can be expressed as:
wherein the content of the first and second substances,
represents the reference signal rkAnd estimated system output
The difference between them, i.e.
Reference signal rkIs a unit step signal. PID controller parameter set to Kp＝1，Ki0.5 and K d0. Fig. 6 shows the estimated value of the system output and the actual reference signal obtained from 1000 repeated experiments, wherein the average value obtained from 1000 experiments is given by the white curve. As can be seen from fig. 6, the estimated value of the system output obtained by the method proposed by the present application has a small variance, which indicates that the estimator proposed by the present application can provide a more accurate estimated value under the influence of the output noise. Accurately estimating the state and output of the system is critical to a closed loop control system because of the control signal u shown in FIG. 7kThe estimated value of the system output is directly depended on, so that inaccurate estimated value directly causes the situations of reduced performance of the controller, increased energy consumption, increased overshoot and the like. In addition, it can be observed that the estimator proposed by the present application performs significantly better than the kalman filter when disturbed by outlier data, is less affected by outliers and provides a smaller estimation error variance. Especially when D iskWhen the self-adaptive optimal parameter selection method is used for determining, the estimator can hardly be influenced by outliers. The present example demonstrates the good application of the method proposed by the present invention in a practical closed-loop control system.
The functions of the functional modules of the security monitoring system according to the embodiment of the present invention may be specifically implemented according to the method in the above method embodiment, and the specific implementation process may refer to the related description of the above method embodiment, which is not described herein again.
Since the contents of information interaction, execution process, and the like between the units in the autoregressive moving average system are based on the same concept as the method embodiment of the present invention, specific contents may be referred to the description in the embodiment of the present invention, and thus, the details are not repeated herein.
Therefore, the embodiment of the invention realizes the purpose of providing the accurate state estimation result with the self-adaptive unbiased characteristic for the closed-loop control system.
The embodiments are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same or similar parts among the embodiments are referred to each other. The device disclosed by the embodiment corresponds to the method disclosed by the embodiment, so that the description is simple, and the relevant points can be referred to the method part for description.
Those of skill would further appreciate that the various illustrative elements and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both, and that the various illustrative components and steps have been described above generally in terms of their functionality in order to clearly illustrate this interchangeability of hardware and software. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the implementation. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention.
The method, apparatus and closed-loop control system for adaptive state estimation of an autoregressive moving average system with additive output noise and control variables provided by the present application are described in detail above. The principles and embodiments of the present invention are explained herein using specific examples, which are presented only to assist in understanding the method and its core concepts. It should be noted that, for those skilled in the art, it is possible to make various improvements and modifications to the present invention without departing from the principle of the present invention, and those improvements and modifications also fall within the scope of the claims of the present application.
Claims (10)
1. A method for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables, applied to a filter, comprising:
implementing a state space of an autoregressive moving average system with a preset application background of additive output noise and control variables;
modeling additive output noise of the autoregressive moving average system using an L2 norm regularization term;
estimating the state value and the output noise of the autoregressive moving average system simultaneously by utilizing a regularization least square method;
taking the regularization parameter corresponding to the minimum error between the sample variance of the estimated residual error and the variance of the actual system noise as the optimal regularization parameter of the autoregressive moving average system; the regularization parameter is used for adaptively adjusting the detection strength of the output noise.
2. The method for adaptive state estimation of an autoregressive moving average system with additive output noise and control variables according to claim 1, wherein the performing state space implementation of the autoregressive moving average system with the predetermined application context of additive output noise and control variables comprises:
setting an initial autoregressive moving average system as an autoregressive moving average system with additive output noise and control variables, wherein the autoregressive moving average system is as follows:
A(q-1)zk＝B(q-1)uk+C(q-1)εk；
yk＝zk+vk；
and carrying out state space realization on the autoregressive moving average system by using a state space relation, wherein the state space relation is as follows:
xk+1＝Φxk+Γuk+Ω(yk-vk)；
yk＝Hxk+εk+vk；
where k is the sample time index, zk∈RmIs the output of the autoregressive moving average system before being disturbed by noise, yk∈RmIs the output of the autoregressive moving average system after being interfered by noise; u. ofkInputting the autoregressive moving average system; epsilonkNoise in the autoregressive moving average system is consistent with zero mean and variance of Rk(ii) a gaussian distribution of; v. ofkThe mean value of the output noise of the autoregressive moving average system is
And the variance is DkWhite noise characteristics of (a); u. ofk，εk，vkThe three are not related to each other; q. q.s-1Shifting operator for time index, q-1zk＝zk-1，A(q-1) And C (q)-1) All within the unit circle, A (q)-1)＝1+a1q-1+a2q-2+…+anq-n，B(q-1)＝b1q-1+b2q-2+…+bnq-n；C(q-1)＝1+c1q-1+c2q-1+…+cnq-1(ii) a Phi, gamma, omega and H are known constant matrixes, and x is a real state value of the autoregressive moving average system.
3. The method for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables according to claim 2, wherein said modeling the additive output noise of the autoregressive moving average system with an L2 norm regularization term comprises:
modeling additive output noise of the autoregressive moving average system by using a modeling relation, wherein the output noise vkIn accordance with a mean value of
And the variance is DkThe modeling relation is as follows:
wherein f (v)k) For the output noise vkAnd carrying out a normalized L2 norm model.
4. The method of any one of claims 1 to 3, wherein the simultaneous estimation of the state value and the output noise of the autoregressive moving average system with additive output noise and control variables by using a regularized least squares method comprises:
calculating a cost function of the autoregressive moving average system based on the L2 norm regularized least squares estimator, the cost function being:
computing
Obtaining a state estimation value and an output noise estimation value of the autoregressive moving average system at the moment k;
in the formula, xkIs the true state value, v, of the autoregressive moving average system at time kkAs output noise of the system, DkFor the output noise vkThe variance of (a) is determined,
for the output noise vkMean value, ykIs the output of the autoregressive moving average system after being interfered by noise;
is the state estimate of the autoregressive moving average system at time k,
and outputting a noise estimation value of the autoregressive moving average system at the k moment.
5. The method for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables according to claim 4, wherein the state estimation value of the autoregressive moving average system at time k is:
Kk＝Pk|k-1HT(HPk|k-1HT+Rk+Dk)-1；
in the formula (I), the compound is shown in the specification,
is a state prediction value P obtained by the autoregressive moving average system at the moment k according to an estimation value at the moment k-1k|k-1Is composed of
H is a matrix of known constants, RkAn approximation of the variance of the residual is estimated.
6. The method for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables according to claim 4, wherein the output noise estimate of the autoregressive moving average system at time k is:
Mk＝Dk(HPk|k-1HT+Rk+Dk)-1；
in the formula (I), the compound is shown in the specification,
is a state prediction value P obtained by the autoregressive moving average system at the k moment according to an estimation value at the k-1 momentk|k-1Is composed of
H is a matrix of known constants, RkAn approximation of the variance of the residual is estimated.
7. The method of adaptive state estimation for an autoregressive moving average system with additive output noise and control variables according to any of claims 1 to 3, wherein the method of calculating the optimal regularization parameter comprises:
the optimal regularization parameter if the output noise variance is unknown or varies over time
According to
Calculating to obtain;
if the variance of the output noise is known, taking the variance of the output noise conforming to the white noise characteristic as the optimal regularization parameter;
in the formula, RkFor system internal noise epsilonkThe ideal value of the variance is calculated,
for system internal noise estimation
The variance of (a) is determined,
is a regularization parameter DkTime epsilonkEstimate of (e ∈)kAs noise within the system, DkIs the regularization parameter.
8. An apparatus for adaptive state estimation for an autoregressive moving average system with additive output noise and control variables, applied to a filter, comprising:
the state space realization module is used for realizing the state space of an autoregressive moving average system with an additive output noise and a control variable in a preset application background;
a noise modeling module for modeling additive output noise of the autoregressive moving average system using an L2 norm regularization term;
the estimation module is used for simultaneously estimating the state value and the output noise of the autoregressive moving average system by utilizing a regularized least square method;
the parameter determining module is used for taking a regularization parameter corresponding to the minimum error between the sample variance of the estimated residual error and the variance of the actual system noise as an optimal regularization parameter of the autoregressive moving average system; the regularization parameter is used for adaptively adjusting the detection strength of the output noise.
9. An apparatus for adaptive state estimation of an autoregressive moving average system with additive output noise and a control variable, applied to a filter, comprising a processor for implementing the steps of the method for adaptive state estimation of an autoregressive moving average system with additive output noise and a control variable according to any one of claims 1 to 7 when executing a computer program stored in a memory.
10. A closed-loop control system is characterized by comprising a PID controller, an estimator and an autoregressive moving average system;
wherein the PID controller is for generating a control signal to cause the output of the autoregressive moving average system being controlled to track the upper reference signal; the estimator is used for generating a system state estimation value according to the output signal of the autoregressive moving average system and the control signal and sending the system state estimation value to the PID controller as a feedback signal; the autoregressive moving average system is adapted to carry out the steps of a method of adaptive state estimation of an autoregressive moving average system with additive output noise and control variables according to any one of claims 1 to 7 when executing a computer program stored in a memory.
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113625552A (en) * | 2021-08-16 | 2021-11-09 | 西南大学 | Method and device for carrying out robust state estimation on state-limited nonlinear system |
| WO2024031887A1 (en) * | 2022-08-09 | 2024-02-15 | 浙江中控技术股份有限公司 | Model identification method and apparatus for industrial control system, device and storage medium |
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