CN107121933A - A kind of stage forecast and smoothing algorithm based on many rate models - Google Patents
A kind of stage forecast and smoothing algorithm based on many rate models Download PDFInfo
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Abstract
The invention provides a kind of stage forecast and smoothing algorithm based on many rate models, it can make full use of and can survey output and can not survey output to correct identification, and identification precision is high, fast convergence rate；It sets up many rate model y (t) using system input data u (t), the system noise v (t) collected, then can survey output y (o through exporting sampler collectionl)；Then according to many rate Model Distinguishes prediction output y (mk), and the variance of computing system variable, it is finally based on the variance of system variable and output y (o can be surveyedl) estimation desired value to prediction output y (mk) be modified, so as to realize to the amendment of output can not be surveyed.
Description
Technical Field
The invention relates to the technical field of parameter identification, in particular to a stage prediction and smoothing algorithm based on a multi-rate model.
Background
The auxiliary model algorithm is firstly proposed in convergence analysis of an auxiliary model identification method of a multivariable system, control theory and application 1997,14(2): 192-.
As in the ARX model, for example,
where y (t) is the measurable output of the system, u (t) is the input of the system and is all measurable, a, b are the parameters to be identified, v (t) is the noise-obeying mean of the system is zero, the variance is a Gaussian distribution, n is the order of the input and output of the model, θyIs a parameter corresponding to the output vector, θuIs a parameter corresponding to the input vector;
by using the proposed aided model identification method, an estimate of the unmeasured output y (tq + i) can be obtained,
whereinIs thatIs estimated by the estimation of (a) a,is thatIs estimated by the estimation of (a) a,is an estimate of the loss output y (tq + i), q is the ratio of the output sample period to the input sample period, i.e. y (tq) is measurable, y (tq + i), i is 1, …, q-1 is not measurable;
after the calculated system output is sampled by an output sampler, part of output data can be lost due to the fact that the sampling period is slow, the lost data, namely, the unpredictable output, is identified by adopting the existing auxiliary model identification method, the unknown output at the tq + i moment can only be predicted by the output before the tq + i, the output y (tq + q) which is measurable by the system cannot be fully utilized to correct the output which is undetectable, the identification precision is low, and the convergence rate of the algorithm can be reduced by estimating system parameters by the unpredictable output with low identification precision.
Disclosure of Invention
In view of the above problems, the present invention provides a stage prediction and smoothing algorithm based on a multi-rate model, which can fully utilize measurable output to correct and identify non-measurable output, and has high identification precision and fast convergence speed.
The technical scheme is characterized by comprising the following algorithm steps:
(S1) establishing a multi-rate model by using the collected system input data u (t) and system noise v (t)
(S2) collecting measurable output y (o) by output samplerl),l＝1,…,β；
(S3), and identifying the predicted output y (m) according to the multi-rate modelk) K is 1, …, α, and calculates the variance of the system variables, and finally, based on the variance of the system variables and the measurable output y (o)l) To the predicted output y (m)k) Correcting to realize correction of the output which cannot be detected;
A(d)＝1-a1d-1-…-and-n
wherein B (d) ═ b1d-1+b2d-2+…+bnd-n
θy＝[a1,…,an]T
θu＝[b1,…,bn]T
θ＝[θy T,θu T]T，
y (t) is the measurable output of the system, v (t) is the noise of the system, and follows a Gaussian distribution with mean zero and variance, A (d) is the polynomial of the system output, B (d) is the polynomial of the system input,is the information vector output at time t,is the information vector input at time t, θyIs a parameter corresponding to the output vector, θuIs the parameter corresponding to the input vector, d is the backshifting operator (d)-1y (t-1)), t represents the number of collected data, and t is 1 or 2 … … N.
It is further characterized in that the system variable variance calculating step comprises:
definition of
WhereinP (t) is the variance of the variable e (t), θhThe parameter θ for the h-th system is estimated from equations (2) and (3)
Then, e (t) ═ e can be obtained from formula (1) and formula (4)t-1,et-2,…,et-n]θy+v(t), (5)
From the formulae (2), (3) and (5)
Where P (t | t-i) ═ Cov (e (t), e (t-i)) is the covariance of e (t) and e (t-i), and is calculated as
In the step (S3), when the predicted output is corrected using the nearest measurable output, the correction of the predicted output includes the steps of:
if m isk+1＝olTo obtain the following distribution
WhereinIs a variable, then obtain
Thereby using a measurable output y (o)l) And predicted outputError pair prediction output ofCorrecting to obtain an output estimation expected value:
in the step (S3), when the predicted output is corrected using the corrected output, the correction of the predicted output includes the steps of:
if m isk+1＝mk+1To obtain the following distribution
Then using the total expectation and total deviation formula to obtain
Wherein
Thereby obtaining
And obtaining an output estimation expected value:
the modification of the prediction output comprises the steps of:
(1.1) collecting input data u (t) and measurable output data y (o)l)，t＝1,…,N,l＝1,…,β；
(1.2) setting ol+1＝mk；
(1.3) construction of information on k-timeVector quantity
(1.4) calculation of the predicted output variable from the equation (3)
(1.5) the variance P (m) of the prediction output is calculated from the expressions (6) and (7), respectivelyk) Sum covariance P (m)k|mk-j)，j＝1,…,n；
(1.6) reacting mk＝mk+1, if mk≠ol+1If so, correcting the predicted output by using the corrected output, and returning to the step (1.3); otherwise, the predicted output is corrected by using the known closest measurable output, namely, the next step is continued;
(1.7) calculated from the expressions (4), (6) and (7), respectivelyP(mk) And P (m)k|mk-j)，j＝1,…,n；
(1.8) let mk＝mk-1, calculating an estimated expected value of the prediction output according to equations (8) and (9)And estimated expected value of system variable variance
(1.9) let m againk＝mk1, if mk≠olThen the predicted output continues to be corrected using the corrected output, i.e. calculated according to equations (10) and (11)Andand returning to the step (1.8); otherwise, continuing the next step;
(1.10) let l ═ l +1, if ol≥oβIf all the unmeasured outputs at all the moments before the last measurable output are corrected, the cycle is terminated; otherwise, returning to the step (1.2).
The invention has the advantages that the method identifies and predicts the output y (m) according to the multi-rate modelk) And then based on the variance of the system variables and the measurable output y (o)l) To the predicted output y (m)k) And correcting to realize correction of the unmeasured output, namely accurately identifying the lost output data of the multi-rate model through the estimated expected value of the predicted output, wherein the accuracy of the identified lost output data is superior to that of the lost output identified by using the conventional auxiliary model method through the estimated expected value of the system variable variance, and the identified predicted output can be used for identifying the system parameters, so that the accuracy of parameter identification is greatly improved.
Drawings
FIG. 1 is a system block diagram of the present invention;
FIG. 2 is a flow chart of the present invention for modifying the identification output;
FIG. 3 is a schematic diagram of a simulation of a prior art aiding model algorithm;
FIG. 4 is a schematic diagram of a simulation of the phase prediction and smoothing algorithm of the present invention.
Detailed Description
As shown in FIG. 1, the phase prediction and smoothing algorithm of the present invention comprises the following steps:
(S1) establishing a multi-rate model by using the collected system input data u (t) and system noise v (t)
(S2) collecting measurable output y (o) by output samplerl),l＝1,…,β；
(S3), and identifying the predicted output y (m) according to the multi-rate modelk) K is 1, …, α, and calculates the variance of the system variables, and finally, based on the variance of the system variables and the measurable output y (o)l) To the predicted output y (m)k) Correcting to realize correction of the output which cannot be detected;
wherein the prediction output y (m)k) I.e., an undetectable output;
A(d)＝1-a1d-1-…-and-n
B(d)＝b1d-1+b2d-2+…+bnd-n
θy＝[a1,…,an]T
θu＝[b1,…,bn]T
θ＝[θy T,θu T]T，
y (t) is the measurable output of the system, v (t) is the noise of the system, and follows a Gaussian white noise with mean zero and variance of 0.1, A (d) is the polynomial of the system output, B (d) is the polynomial of the system input,is the information vector output at time t,is the information vector input at time t, θyIs a parameter corresponding to the output vector, θuIs the parameter corresponding to the input vector, d is the backshifting operator (d)-1y (t-1)), t represents the number of collected data, and t is 1 or 2 … … N.
The system variable variance calculating step comprises:
definition of
WhereinP (t) is the variance of the variable e (t), θhThe parameter θ for the h-th system is estimated from equations (2) and (3)
Then, e (t) ═ e can be obtained from formula (1) and formula (4)t-1,et-2,…,et-n]θy+v(t), (5)
From the formulae (2), (3) and (5)
Where P (t | t-i) ═ Cov (e (t), e (t-i)) is the covariance of e (t) and e (t-i), and is calculated as
In step (S3), when the predicted output is corrected using the nearest measurable output (for example, using measurable output y (tq))The correction of the prediction output comprises the following steps:
if m isk+1＝olTo obtain the following distribution
WhereinIs a variable, then obtain
Thereby using a measurable output y (o)l) And predicted outputError pair prediction output ofAnd (3) correcting to obtain an accurate output estimation expected value:
in step (S3), when the predicted output is corrected by the corrected output (for example, by a measurable output)To correctThe correction of the prediction output comprises the following steps:
if m isk+1＝mk+1To obtain the following distribution
However, unlike the above case, y (m)k+1) Is unknown, we know only that its distribution isThen using the total expectation and total difference formula to obtain
WhereinThereby obtaining
And obtaining an output estimation expected value:
finally, the stage prediction and smoothing algorithm can be obtained, the algorithm can correct the predicted output by using measurable output, so that the non-measurable output can be more accurately estimated, namely, the lost output data of the multi-rate model can be accurately identified by using the formulas (8) and (10); using equations (9) and (11), the accuracy of the identified missing output data is better than the missing output identified using the prior aiding model method; the algorithm provided by the invention can improve the identification precision of the undetectable output, and further utilize the identified undetectable output to identify the system parameter, thereby greatly improving the precision of parameter identification.
The correction of the prediction output specifically comprises the following steps:
as shown in FIG. 2, assuming that the total number of collected data is N, the output y (m) which is not measurable by the multi-rate model is obtained by the stage prediction and smoothing algorithm based on the multi-rate modelk) The identification method comprises the following steps:
(1.1) initializing, assuming that u (t) is 0, y (t) is 0, p (t) is 0,θ＝θh,＝h；
(1.2) collecting input data u (t) and measurable output data y (o)l) T 1, …, N, l 1, …, β, and l is set to 1 again;
(1.3) initialize t 1, let ol+1＝mk；
(1.4) constructing an information vector at time k
(1.5) calculation of predicted output variables from the equation (4)
(1.6) the variance P (m) of the prediction output is calculated from the expressions (6) and (7), respectivelyk) Sum covariance P (m)k|mk-j)，j＝1,…,n；
(1.7) reacting mk＝mk+1, if mk≠ol+1If yes, the corrected output is used for correcting the predicted output, and the step (1.4) is returned; otherwise, the predicted output is corrected by using the known closest measurable output, namely, the next step is continued;
(1.8) is calculated from the expressions (4), (6) and (7), respectivelyP(mk) And P (m)k|mk-j)，j＝1,…,n；That is to say
(1.9) let mk＝mk-1, calculating an estimated expected value of the prediction output according to equations (8) and (9)And estimated expected value of system variable variance
(1.10) let m againk＝mk1, if mk≠olThen the predicted output continues to be corrected using the corrected output, i.e. calculated according to equations (10) and (11)Andand returning to the step (1.9); otherwise, continuing the next step;
(1.11) let l equal l +1 if ol≥oβIf all the undetectable outputs at all the moments before the last measurable output are corrected, the cycle is terminated; otherwise, returning to the step (1.3).
As shown in fig. 3 and 4, where + is an estimation output; the solid line is the true output; it can be seen that the error between the estimated value and the true output value output by the phase prediction and smoothing algorithm of the present invention is smaller than the error between the estimated value and the true value of the existing auxiliary model, i.e., the phase prediction and smoothing algorithm is better than the auxiliary model algorithm.
Claims (5)
1. A stage prediction and smoothing algorithm based on a multi-rate model is characterized by comprising the following algorithm steps:
(S1) establishing a multi-rate model by using the collected system input data u (t) and system noise v (t)
(S2) collecting measurable output y (o) by output samplerl),l＝1,…,β；
(S3), and identifying the predicted output y (m) according to the multi-rate modelk) K is 1, …, α, and calculates the variance of the system variables, and finally, based on the variance of the system variables and the measurable output y (o)l) To the predicted output y (m)k) Correcting to realize correction of the output which cannot be detected;
wherein,
y (t) is the measurable output of the system, v (t) is the noise of the system, and follows a Gaussian distribution with mean zero and variance, A (d) is the polynomial of the system output, B (d) is the polynomial of the system input,is the information vector output at time t,is the information vector input at time t, θyIs a parameter corresponding to the output vector, θuIs the parameter corresponding to the input vector, d is the backshifting operator (d)-1y (t-1)), t represents the number of collected data, and t is 1 or 2 … … N.
2. The multi-rate model based phase prediction and smoothing algorithm of claim 1, wherein the system variable variance calculation step comprises:
definition of
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WhereinP (t) is the variance of the variable e (t), θhThe parameter θ for the h-th system is estimated from equations (2) and (3)
Then, e (t) ═ e can be obtained from formula (1) and formula (4)t-1,et-2,…,et-n]θy+v(t), (5)
From the formulae (2), (3) and (5)
Where P (t | t-i) ═ Cov (e (t), e (t-i)) is the covariance of e (t) and e (t-i), and is calculated as
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3. The multi-rate model based phase prediction and smoothing algorithm of claim 1, wherein the step (S3) of modifying the predicted output when modifying the predicted output with the nearest measurable output comprises the steps of:
if m isk+1＝olTo obtain the following distribution
<mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>)</mo> <mo>=</mo> <mi>N</mi> <mo>(</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> <mo>,</mo> </mrow>
WhereinIs a variable, then obtain
<mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>|</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>=</mo> <mi>y</mi> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>)</mo> <mo>=</mo> <mi>N</mi> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>+</mo> <mi>P</mi> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> </mrow> <mo>)</mo> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>(</mo> <mi>y</mi> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>)</mo> <mo>,</mo> <mi>P</mi> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>-</mo> <msup> <mi>P</mi> <mn>2</mn> </msup> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> </mrow> <mo>)</mo> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>)</mo> <mo>,</mo> </mrow>
Thereby using a measurable output y (o)l) And predicted outputError pair prediction output ofCorrecting to obtain an output estimation expected value:
<mrow> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>y</mi> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mover> <mi>P</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
4. the multi-rate model based phase prediction and smoothing algorithm of claim 1, wherein the step (S3) of modifying the predicted output when the predicted output is modified with the modified output comprises the steps of:
if m isk+1＝mk+1To obtain the following distribution
<mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>)</mo> <mo>=</mo> <mi>N</mi> <mo>(</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> <mo>,</mo> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>|</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>=</mo> <mi>y</mi> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>)</mo> <mo>=</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>+</mo> <mi>P</mi> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <msub> <mi>o</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>(</mo> <mi>y</mi> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>,</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
Then using the total expectation and total deviation formula to obtain
Wherein
Thereby obtaining
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>E</mi> <mrow> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>(</mo> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>E</mi> <mrow> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </msub> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </msub> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>V</mi> <mi>a</mi> <mi>r</mi> <mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>E</mi> <mrow> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>V</mi> <mi>a</mi> <mi>r</mi> <mo>(</mo> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Var</mi> <mrow> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>(</mo> <mrow> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msub> <mi>E</mi> <mrow> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </msub> <mo>(</mo> <mi>P</mi> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>-</mo> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>)</mo> <mo>+</mo> <msub> <mi>Var</mi> <mrow> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>+</mo> <mi>P</mi> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>(</mo> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo>&lsqb;</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mover> <mi>P</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>.</mo> </mrow>
And obtaining an output estimation expected value:
<mrow> <mover> <mi>P</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo>&lsqb;</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mover> <mi>P</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
5. a multi-rate model based phase prediction and smoothing algorithm as claimed in claim 2 or 3 or 4, wherein the modification of the prediction output comprises the steps of:
(1.1) collecting input data u (t) and measurable output data y (o)l)，t＝1,…,N,l＝1,…,β；
(1.2) setting ol+1＝mk；
(1.3) constructing an information vector at the k time
(1.4) calculation of the predicted output variable from the equation (3)
(1.5) the variance P (m) of the prediction output is calculated from the expressions (6) and (7), respectivelyk) Sum covariance P (m)k|mk-j)，j＝1,…,n；
(1.6) reacting mk＝mk+1, if mk≠ol+1If so, correcting the predicted output by using the corrected output, and returning to the step (1.3); otherwise, the predicted output is corrected by using the known closest measurable output, namely, the next step is continued;
(1.7) calculated from the expressions (4), (6) and (7), respectivelyP(mk) And P (m)k|mk-j)，j＝1,…,n；
(1.8) let mk＝mk-1, calculating an estimated expected value of the prediction output according to equations (8) and (9)And estimated expected value of system variable variance
(1.9) let m againk＝mk1, if mk≠olThen the predicted output continues to be corrected using the corrected output, i.e. calculated according to equations (10) and (11)Andand returning to the step (1.8); otherwise, continuing the next step;
(1.10) let l ═ l +1, if ol≥oβIf all the unmeasured outputs at all the moments before the last measurable output are corrected, the cycle is terminated; otherwise, returning to the step (1.2).
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| US5450523A (en) * | 1990-11-15 | 1995-09-12 | Matsushita Electric Industrial Co., Ltd. | Training module for estimating mixture Gaussian densities for speech unit models in speech recognition systems |
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