CN111381498B - Expectation maximization identification method of multi-sensor based on multi-rate variable time-lag state space model - Google Patents

Expectation maximization identification method of multi-sensor based on multi-rate variable time-lag state space model Download PDF

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CN111381498B
CN111381498B CN202010155240.4A CN202010155240A CN111381498B CN 111381498 B CN111381498 B CN 111381498B CN 202010155240 A CN202010155240 A CN 202010155240A CN 111381498 B CN111381498 B CN 111381498B
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顾亚
高津津
朱培逸
刘继承
鲁明丽
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Changshu Institute of Technology
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Abstract

The invention discloses an expectation maximization identification method of a multi-sensor based on a multi-rate variable-time-lag state space model, wherein the multi-sensor expresses the structure of the variable-time-lag state space model by utilizing a multi-sampling rate form; for states with time-varying skewSpatial systems with unknown time lags of hidden Markov models, i.e. at time TiThe time lag variable λ is only in its past information and has only N discrete values; determining a multi-rate state space system with time lag, and determining parameters of a state space model by determining the value of a hidden time lag variable of each sampling time and determining a transition probability matrix and a prior probability of a hidden Markov model. The method is suitable for parameter estimation of other linear and nonlinear systems, is also suitable for parameter estimation problems of multivariable systems, multi-modal systems and pseudo linear systems, and can provide a reliable system modeling method for data prediction and controller design based on the models.

Description

Expectation maximization identification method of multi-sensor based on multi-rate variable time-lag state space model
Technical Field
The invention relates to an expectation maximization identification method based on a variable time-lag state space system, and belongs to the field of complex industrial process modeling and identification.
Background
The need for plant safety, environmental regulations, plant economics and effective process monitoring in complex industrial processes is one of the main reasons that has prompted the development of more advanced state estimation methods. State estimators are tools that infer the state of a process from available metrics and are important tools in enabling efficient process monitoring, diagnostics, and control. Most process monitoring systems rely on the assumption that the state of the system is definitely available, but in reality most process states cannot be measured directly or are accompanied by considerable noise values in the measurement process. Kalman filtering, extended kalman filtering, unscented kalman filtering are currently accepted techniques for continuous value or continuous state dynamic estimation, but cannot be applied directly to discrete value or discontinuous state estimation.
In fact, many of the problems to be solved in complex industrial systems are discrete or discontinuous, and some actual controlled objects contain time lag, and the estimation of a state space system with uncertainty is a mixed state estimation problem. In order to estimate both continuous and discontinuous states simultaneously, an Expectation Maximization (EM) algorithm is proposed. In general, in the presence of bounded system and measurement noise, it is desirable that the maximization estimation method can solve the hybrid state estimation problem, which is an effective optimization strategy to estimate unknown variables or parameters.
Although expectation maximization estimation algorithms are well studied in simultaneous estimation of continuous and discontinuous states, simultaneous estimation of state and uncertainty time lags has been rarely reported for multi-rate systems. Multiple sensor systems improve estimation performance over a single sensor system, however, the sampling rate of process measurements may vary among the different sensors that make up a multi-rate system. Skew inevitably exists in many practical processes and has a significant impact on control performance. In conventional state estimation, it is generally assumed that the time lag of the system is known or constant. However, in reality, it is often unknown and time-varying. To this end, we propose an expectation-maximization estimation algorithm that estimates both the parameters of the multi-rate system and the discrete time-lag sequence.
Disclosure of Invention
1. Objects of the invention
The invention aims to provide a multi-rate time-lag system identification method based on an expectation maximization estimation algorithm so as to achieve high accuracy of system parameter and time-lag identification.
2. The technical scheme adopted by the invention
The invention discloses an expectation maximization identification method of a multi-sensor based on a multi-rate variable time-lag state space model,
the response of the actual sensor always has time lag, and the multi-sensor utilizes a multi-sampling rate form to express the structure of a variable time lag state space model;
for state space systems with time-varying time lags, the unknown time lags are hidden Markov models, i.e., at time TiThe time lag variable λ is only in its past information and has only N discrete values;
determining a multi-rate state space system with time lag, and determining parameters of a state space model by determining the value of a hidden time lag variable of each sampling time and determining a transition probability matrix and a prior probability of a hidden Markov model.
Furthermore, the structure of the variable time lag state space model is expressed by using the form of multiple sampling rates, and the concrete structure of the model
Expression: x is the number oft+1＝Axt+but+ωt, (1)
Figure BDA0002403797330000021
The state variable model represents the dynamic characteristic of a system and comprises a state equation and an output equation; wherein x istIs an unknown state vector, the state is a physical or non-physical motion state of the system, including position, velocity, acceleration, voltage, current; u. oftIs an input;
at unknown time lag
Figure BDA0002403797330000022
Under the condition of (1), the response of an actual sensor and a network control system have uncertain transmission time lag; y istThe system is irregular sampling output, a series of observations can be generated on the system in the control process, the observations comprise the observations of the sensor or the system behaviors needing to be controlled, and the observations needing to be carried out are taken as the system output; the system is only at time T ═ TiΔ T is known, T is a time greater than the span of T, Δ T is the time interval, the time lag varies at different sampling instants; a is an element of Rn×n，b∈Rn×1And c ∈ Rn×1Is the unknown parameter to be estimated; omegatIs process noise, vtIs the measurement noise; these two types of noise are assumed to follow independent and identical gaussian distributions: omegat～L(U,Ω)，
Figure BDA0002403797330000031
Omega is a covariance matrix and is used as a basis,σ2is variance, and assumes the parameters U-0 and μ -0;
for state space systems with time-varying time lags, the time lags are unknown
Figure BDA0002403797330000032
For hidden Markov models, i.e. at time TiThe time lag variable λ is only in its past information and has only N discrete values; thus, the prior probability α of the time-lag sequencemnAnd transition probability betamnIs defined as:
Figure BDA0002403797330000033
Figure BDA0002403797330000034
wherein i is 2,3, N is a real number, and m, N ∈ {1,2, …, d } is a real number;
the parameter theta is composed of local state space model parameters A, b and c and time-lag transition probability betamnAnd an initial skew distribution alphamnThe composition can be expressed as Θ ═ { a, b, c, α, β }.
Further, solving by maximizing the likelihood function of the unknown parameter:
defining the Q function as a log-likelihood function given to all potential variables or data
Figure BDA0002403797330000035
The expected, a posteriori probability can be obtained,
Figure BDA0002403797330000036
is a joint density function of the complete data set, using Bayesian characteristic decomposition, the final expression of the Q function is
Figure BDA0002403797330000037
Wherein the content of the first and second substances,C1is a constant number of times, and is,
Figure BDA0002403797330000038
is to xtIn the expectation that the position of the target is not changed,i(m) is
Figure BDA0002403797330000039
The distribution of the gaussian component of (a) is,
Figure BDA00024037973300000310
is time lag
Figure BDA00024037973300000311
The probability of (a) of (b) being,
Figure BDA00024037973300000312
is composed of
Figure BDA00024037973300000313
Log function of probability, log [ p (x)t|xt-1,Θ)]Being a log function of the probability of state x,
Figure BDA00024037973300000314
is time lag
Figure BDA00024037973300000315
Probability of (log a)mnIs alphamnLog function of (log beta)mnIs betamnIn order to calculate parameter estimation, a gradient algorithm is adopted on the Q function aiming at unknown parameters; take Q (theta | theta)k) The gradient with respect to parameter a is equal to zero:
Figure BDA0002403797330000041
an expression for parameter a can thus be obtained:
Figure BDA0002403797330000042
wherein Q is a Q function, AkFor parameter estimation of parameter A at instant k, bkFor the parameter estimation of the parameter b at the instant k,μt-1is the noise vtThe expectation is that.
Take Q (Θ | Θ) relative to bk) And set it to zero:
Figure BDA0002403797330000043
by calculating its partial derivative, an iterative value of the parameter b can be obtained in the following way:
Figure BDA0002403797330000044
wherein Q is a Q function, utAnd mut-1Is the noise vtAt times t and t-1.
The iteration value of the parameter c is
Figure BDA0002403797330000045
Wherein the content of the first and second substances,i(m) is
Figure BDA0002403797330000046
The distribution of the gaussian component of (a) is,
Figure BDA0002403797330000047
is time lag
Figure BDA0002403797330000048
The probability of (a) of (b) being,
Figure BDA0002403797330000049
at a time TiIs then outputted from the output of (a),
Figure BDA00024037973300000410
for the Q function at time Ti-the value of m is selected from the group,
Figure BDA00024037973300000411
as noise vtAt time Ti-a desire for m.
In making a prior probability alphamnAnd transfer ofProbability betamnIn the calculation of (2), we need to introduce a Lagrange multiplier LαAnd LβTaking Q as amnAnd betamnThe partial derivative of (a) of (b),
Figure BDA00024037973300000412
Figure BDA0002403797330000051
wherein the content of the first and second substances,i(m) is
Figure BDA0002403797330000052
In that
Figure BDA0002403797330000053
The probability distribution under the conditions of the probability distribution,i-1(n) is
Figure BDA0002403797330000054
In that
Figure BDA0002403797330000055
The probability distribution under the conditions of the probability distribution,
Figure BDA0002403797330000056
is time lag
Figure BDA0002403797330000057
The probability of (c).
Further, 1) determining each sampling time λtThe value of the hidden skew variable of (1);
2) determining a transition probability matrix beta for a hidden Markov modelmnAnd a prior probability amn(m,n∈{1,2,...,d})：
Figure BDA0002403797330000058
Figure BDA0002403797330000059
3) Determining parameters { A, b, c } of the state space model;
the system identification in equations (1) and (2) is solved by an EM algorithm; observable variable CobsAnd unknown variable CmisIs composed of
Figure BDA00024037973300000510
Figure BDA00024037973300000511
Wherein the variable C can be observedobsFor the set of inputs u and outputs y, the unknown variable CmisIs a set of time lags λ and states x;
the parameter theta is composed of local state space model parameters A, b and c and time-lag transition probability betamnAnd an initial skew distribution alphamnThe composition can be expressed as Θ ═ { a, b, c, α, β }.
3. Advantageous effects adopted by the present invention
(1) The invention derives a robust EM algorithm to identify a multi-rate state space system with time lag under the framework of expectation maximization. Aiming at the problem that the measured data has uncertain time lag due to random interference and some uncertain factors, an EM estimation algorithm is adopted to research the identification problem of an uncertain time lag and time-varying time lag system under a complex mechanism and a complex working condition, so that the stability and the modeling precision of a complex industrial process system are ensured, and the convergence speed of the algorithm is improved.
(2) The invention considers the random time lag in the signal transmission process by providing the probability distribution of the time lag of each sampling moment, and simultaneously carries out Markov delay sequence estimation and parameter estimation. By correctly expressing the estimation problem in the framework of expectation maximization, the unknown system states and parameters are estimated interactively. The invention summarizes the key identification problem which is urgently needed to be solved in the field, and provides an improved and new identification method (which can process the identification modeling theory and method of a time-delay system with a complex structure) aiming at the uncertain time delay and time-varying time delay existing in the complex industrial process and based on the process mechanism characteristics and the data information of measurable variables, thereby improving the convergence speed of the algorithm and further applying the new method to other process control fields.
(3) The method is suitable for parameter estimation of other linear and nonlinear systems, is also suitable for the parameter estimation problem of multivariable systems, multi-modal systems and pseudo linear systems, can provide a reliable system modeling method for data prediction and controller design based on the model, and has important theoretical research value and practical application prospect.
Drawings
FIG. 1 is a flow chart of the EM algorithm of the present invention;
FIG. 2 is an iteration graph of the Q equation of the present invention;
FIG. 3 is a parameter estimation of the present invention;
FIG. 4 is a graph of a time-lag transition probability matrix estimation of the present invention;
FIG. 5 is a time lag prior probability estimation graph of the present invention;
FIG. 6 is a schematic diagram of multi-rate sampling input and output according to the present invention;
FIG. 7 is a cross-validation result of the estimation model of the present invention;
FIG. 8 shows the self-verification result of the estimation model of the present invention.
Detailed Description
The technical solutions in the examples of the present invention are clearly and completely described below with reference to the drawings in the examples of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without inventive step, are within the scope of the present invention.
The present invention will be described in further detail with reference to the accompanying drawings.
Examples
The structure of a variable time lag state space model is expressed by using a multi-sampling rate form, and a specific expression of the model is as follows:
xt+1＝Axt+but+ωt, (1)
Figure BDA0002403797330000071
the state variable model represents the dynamic characteristic of a system and comprises a state equation and an output equation; wherein x istIs an unknown state vector, the state is a physical or non-physical motion state of the system, including position, velocity, acceleration, voltage, current; u. oftIs an input;
at unknown time lag
Figure BDA0002403797330000072
Under the condition of (1), the response of an actual sensor and a network control system have uncertain transmission time lag; y istThe system is irregular sampling output, a series of observations can be generated on the system in the control process, the observations comprise the observations of the sensor or the system behaviors needing to be controlled, and the observations needing to be carried out are taken as the system output; the system is only at time T ═ TiΔ T is known, T is a time greater than the span of T, Δ T is the time interval, the time lag varies at different sampling instants; a is an element of Rn×n，b∈Rn×1And c ∈ Rn×1Is the unknown parameter to be estimated; omegatIs process noise, vtIs the measurement noise; these two types of noise are assumed to follow independent and identical gaussian distributions: omegat～L(U,Ω)，
Figure BDA0002403797330000073
Omega is the covariance matrix, sigma2Is variance, and assumes the parameters U-0 and μ -0;
for the state space system represented by equations (1) and (2) with time-varying time lags, the time lags are unknown
Figure BDA0002403797330000074
As Hidden Markov Models (HMM), i.e. at time TiThe time lag variable λ is only in its past information and has only N discrete values. Thus, the prior probability α of the time-lag sequencemnAnd transition probability betamnIs defined as:
Figure BDA0002403797330000075
Figure BDA0002403797330000076
where i ═ 2,3, …, N is a real number'm, and N ∈ {1, 2.
Our objective is to determine a multi-rate state space system with time lags, which includes the following unknown variables and parameters:
1) determining each sampling time lambdatThe value of the hidden skew variable of (1);
2) determining a transition probability matrix beta for a hidden Markov modelmnAnd a prior probability amn(m,n∈{1,2,…,d})：
Figure BDA0002403797330000081
Figure BDA0002403797330000082
3) The parameters { A, b, c } of the state space model are determined.
The system identification in equations (1) and (2) is solved by the EM algorithm. Observable variable CobsAnd unknown variable CmisIs composed of
Figure BDA0002403797330000083
Figure BDA0002403797330000084
Wherein the variable C can be observedobsFor the set of inputs u and outputs y, the unknown variable CmisIs a set of time lags λ and states x.
The parameter theta is composed of local state space model parameters A, b and c and time-lag transition probability betamnAnd an initial skew distribution alphamnThe composition can be expressed as Θ ═ { a, b, c, α, β }.
From the perspective of the EM algorithm, this problem can be solved by maximizing the likelihood function of the unknown parameters. The derivation process is discussed in detail below.
The EM algorithm will handle incomplete data problems by computing Maximum Likelihood (ML) estimates of the parameters. The ML estimate of the unknown parameters is usually found using the conventional ML method by directly maximizing the likelihood function of the incomplete data. Introducing some proper missing variables or hidden variables into the EM algorithm, calculating and maximizing the conditional expectation of the complete data likelihood function to derive the ML estimation of the parameters, and greatly simplifying the parameter estimation problem.
Defining the Q function as a log-likelihood function given to all potential variables or data
Figure BDA0002403797330000085
The expected, a posteriori probability can be obtained,
Figure BDA0002403797330000086
is a joint density function of the complete data set, using Bayesian characteristic decomposition, the final expression of the Q function is
Figure BDA0002403797330000091
Wherein, C1Is a constant number of times, and is,
Figure BDA0002403797330000092
is to xtIn the expectation that the position of the target is not changed,i(m) is
Figure BDA0002403797330000093
The distribution of the gaussian component of (a) is,
Figure BDA0002403797330000094
is time lag
Figure BDA0002403797330000095
The probability of (a) of (b) being,
Figure BDA0002403797330000096
is composed of
Figure BDA0002403797330000097
Log function of probability, log [ p (x)t|xt-1,Θ)]Being a log function of the probability of state x,
Figure BDA0002403797330000098
is time lag
Figure BDA0002403797330000099
Probability of (log a)mnIs alphamnLog function of (log beta)mnIs betamnFor calculating the parameter estimation, a gradient algorithm should be applied on the Q function for the unknown parameters. Take Q (theta | theta)k) The gradient with respect to parameter a is equal to zero:
Figure BDA00024037973300000910
an expression for parameter a can thus be obtained:
Figure BDA00024037973300000911
wherein Q is a Q function, AkFor parameter estimation of parameter A at instant k, bkFor parameter estimation of the parameter b at the instant k, μt-1Is the noise vtThe expectation is that.
Take Q (Θ | Θ) relative to bk) And set it to zero:
Figure BDA00024037973300000912
by calculating its partial derivative, an iterative value of the parameter b can be obtained in the following way:
Figure BDA00024037973300000913
wherein Q is a Q function, utAnd mut-1Is the noise vtAt times t and t-1.
The iteration value of the parameter c is
Figure BDA00024037973300000914
Wherein the content of the first and second substances,i(m) is
Figure BDA0002403797330000101
The distribution of the gaussian component of (a) is,
Figure BDA0002403797330000102
is time lag
Figure BDA0002403797330000103
The probability of (a) of (b) being,
Figure BDA0002403797330000104
at a time TiIs then outputted from the output of (a),
Figure BDA0002403797330000105
for the Q function at time Ti-the value of m is selected from the group,
Figure BDA0002403797330000106
as noise vtAt time Ti-a desire for m.
In making a prior probability alphamnAnd transition probability betamnIn the calculation of (2), we need to introduce a Lagrange multiplier LαAnd LβTaking Q as amnAnd betamnThe partial derivative of (a) of (b),
Figure BDA0002403797330000107
Figure BDA0002403797330000108
wherein the content of the first and second substances,i(m) is
Figure BDA0002403797330000109
In that
Figure BDA00024037973300001010
The probability distribution under the conditions of the probability distribution,i-1(n) is
Figure BDA00024037973300001011
In that
Figure BDA00024037973300001012
The probability distribution under the conditions of the probability distribution,
Figure BDA00024037973300001013
is time lag
Figure BDA00024037973300001014
The probability of (c).
Since the EM algorithm is susceptible to initialization, selecting an effective initial value for an unknown vector is a key pre-step to ensure effective implementation of the EM algorithm. The basic goal of the EM algorithm is to obtain accurate parameter estimates by finding the highest probability under certain stopping conditions. The key step is to find the most suitable parameter estimate and the full iteration EM algorithm with the appropriate initial values. The most common initialization strategy is a random strategy, i.e. trying different sets of random initial values for the parameters and then selecting the set that gives the maximum likelihood function. When the initial value is within the unit circle, its stability can be ensured.
The model of the present invention is applicable to a tank system consisting of three separate tanks. The water outlet at the bottom of the water tank is provided with a control valve for regulating water outlet. At the bottom of the apparatus, a water tank is provided as a reservoir for the entire system. The three cans have different cross sections. Under the action of gravity, water is pumped from the water storage tank into the top tank and sequentially passes through the upper tank, the middle tank and the lower tank. Each water tank is provided with a liquid level measuring device, and the liquid level height is measured through hydraulic pressure. The invention takes the inlet flow as an input variable and the water level of the third water tank as the system output. In the experimental process, the water inlet flow and the water outlet flow of the system are firstly adjusted, so that the liquid level of the system reaches the balance position. A random multi-level signal is then superimposed on the system input. The entire system is considered to be a discrete model. Fig. 7 and 8 show the effect of the model in the three-tank experiment, where the solid line is the true measurement output and the dotted line is the model output. It can be seen that the model output is well matched with the actual measurement output, which shows that the identification model can reflect the dynamic characteristics of the process, and the validity of the EM algorithm is proved.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (2)

1. A multi-sensor expectation maximization identification method based on a multi-rate variable time-lag state space model is characterized by comprising the following steps:
the response of the actual sensor always has time lag, and the multi-sensor utilizes a multi-sampling rate form to express the structure of a variable time lag state space model;
for state space systems with time-varying time lags, the unknown time lags are hidden Markov models, i.e., at time TiThe time lag variable λ is only in its past information and has only N discrete values;
determining a multi-rate state space system with time-varying time lag, and determining parameters of a state space model by determining the value of a hidden time lag variable of each sampling time and determining a transition probability matrix and a prior probability of a hidden Markov model;
the structure of a variable time lag state space model is expressed by using a multi-sampling rate form, and a specific expression of the model is as follows:
xt+1＝Axt+but+ωt，(1)
Figure FDA0002711469540000011
the state variable model represents the dynamic characteristic of a system and comprises a state equation and an output equation; wherein x istIs an unknown state vector, the state is a physical or non-physical motion state of the system, including position, velocity, acceleration, voltage, current; u. oftIs an input;
at unknown time lag
Figure FDA0002711469540000012
Under the condition of (1), the response of an actual sensor and a network control system have uncertain transmission time lag; y istThe system is irregular sampling output, a series of observations can be generated on the system in the control process, the observations comprise the observations of the sensor or the system behaviors needing to be controlled, and the observations needing to be carried out are taken as the system output; the system is only at time T ═ TiΔ T is known, T is a time greater than the span of T, Δ T is the time interval, the time lag varies at different sampling instants; a is an element of Rn ×n，b∈Rn×1And c ∈ Rn×1Is the unknown parameter to be estimated; omegatIs process noise, vtIs the measurement noise; these two types of noise are assumed to follow independent and identical gaussian distributions: omegat～L(U，Ω)，
Figure FDA0002711469540000014
Omega is the covariance matrix, sigma2Is variance, and assumes the parameters U-0 and μ -0;
for state space systems with time-varying time lags, the time lags are unknown
Figure FDA0002711469540000013
For hidden Markov models, i.e. at time TiThe time lag variable λ is only in its past information and has only N discrete values; thus, the prior probability α of the time-lag sequencemnAnd transition probability betamnIs defined as:
Figure FDA0002711469540000021
Figure FDA0002711469540000022
wherein, i ═ 2, 3., N is a real number, and m, N ∈ {1, 2., d } is a real number;
(ii) a Solving by maximizing the likelihood function of the unknown parameters:
defining the Q function as a log-likelihood function given to all potential variables or data
Figure FDA0002711469540000023
The expected, a posteriori probability can be obtained,
Figure FDA0002711469540000024
is a joint density function of the complete data set, and the parameter theta is composed of local state space model parameters A, b and c and time-lag transition probability betamnAnd an initial skew distribution alphamnComposition, which can be expressed as Θ ═ { a, b, c, α, β };
using Bayesian property decomposition, the final expression of the Q function is
Figure FDA0002711469540000025
Wherein, C1Is a constant number of times, and is,
Figure FDA0002711469540000026
is to xtIn the expectation that the position of the target is not changed,i(m) is
Figure FDA0002711469540000027
The distribution of the gaussian component of (a) is,
Figure FDA0002711469540000028
is time lag
Figure FDA0002711469540000029
The probability of (a) of (b) being,
Figure FDA00027114695400000210
is composed of
Figure FDA00027114695400000211
Log function of probability, log [ p (x)t|xt-1，Θ)]Being a log function of the probability of state x,
Figure FDA00027114695400000212
is time lag
Figure FDA00027114695400000213
Probability of (log a)mnIs alphamnLog function of (log beta)mnIs betamnIn order to calculate parameter estimation, a gradient algorithm is adopted on the Q function aiming at unknown parameters; take Q (theta | theta)k) The gradient with respect to parameter a is equal to zero:
Figure FDA00027114695400000214
an expression for parameter a can thus be obtained:
Figure FDA0002711469540000031
wherein Q is a Q function, AkFor parameter estimation of parameter A at instant k, bkFor parameter estimation of the parameter b at the instant k, μt-1Is the noise vt(iii) a desire;
take Q (Θ | Θ) relative to bk) And set it to zero:
Figure FDA0002711469540000032
by calculating its partial derivative, an iterative value of the parameter b can be obtained in the following way:
Figure FDA0002711469540000033
wherein Q is a Q function, utAnd mut-1Is the noise vtExpectation at time t and t-1;
the iteration value of the parameter c is
Figure FDA0002711469540000034
Wherein the content of the first and second substances,i(m) is
Figure FDA0002711469540000035
The distribution of the gaussian component of (a) is,
Figure FDA0002711469540000036
is time lag
Figure FDA0002711469540000037
The probability of (a) of (b) being,
Figure FDA0002711469540000038
at a time TiIs then outputted from the output of (a),
Figure FDA0002711469540000039
for the Q function at time Ti-the value of m is selected from the group,
Figure FDA00027114695400000310
as noise vtAt time Ti-a desire for m;
in making a prior probability alphamnAnd transition probability betamnIn the calculation of (2), we need to introduce a Lagrange multiplier LαAnd LβTaking Q as amnAnd betamnThe partial derivative of (a) of (b),
Figure FDA00027114695400000311
Figure FDA00027114695400000312
wherein the content of the first and second substances,i(m) is
Figure FDA00027114695400000313
In that
Figure FDA00027114695400000314
The probability distribution under the conditions of the probability distribution,i-1(n) is
Figure FDA00027114695400000315
In that
Figure FDA00027114695400000316
The probability distribution under the conditions of the probability distribution,
Figure FDA0002711469540000041
is time lag
Figure FDA0002711469540000042
The probability of (c).
2. The multi-sensor expectation-maximization identification method based on the multi-rate variable-time-lag state space model according to claim 1, characterized in that:
1) determining each sampling time lambdatThe value of the hidden skew variable of (1);
2) determining a transition probability matrix beta for a hidden Markov modelmnAnd a prior probability amn，m，n∈{1，2，...，d}：
Figure FDA0002711469540000043
Figure FDA0002711469540000044
3) Determining parameters { A, b, c } of the state space model;
the system identification in equations (1) and (2) is solved by an EM algorithm; observable variable CobsAnd unknown variable CmisIs composed of
Figure FDA0002711469540000045
Figure FDA0002711469540000046
Wherein the variable C can be observedobsFor the set of inputs u and outputs y, the unknown variable CmisIs a set of time lags λ and states x;
the parameter theta is composed of local state space model parameters A, b and c and time-lag transition probability betamnAnd an initial skew distribution alphamnThe composition can be expressed as Θ ═ { a, b, c, α, β }.
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