CN107145474A - A kind of Kalman filtering algorithm based on autoregression model - Google Patents
A kind of Kalman filtering algorithm based on autoregression model Download PDFInfo
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Abstract
The invention provides a kind of Kalman filtering algorithm based on autoregression model, it can pick out the output data that can not survey of autoregression model, and the precision of parameter identification is high；It can not survey output state x (t) using what system input data u (t), the system mode noise v (t) and t that collect optimal estimation x (t | t) picked out autoregression model, the output state x (t) that can not survey of autoregression model exports y (t) through signal amplifier and output noise w (t) system for setting up autoregression model, then system output y (t) realizes t to that can not survey output state x (t) predicted value x (t | t 1) correction.
Description
Technical field
The present invention relates to the parameter identification technique field of process control, specially a kind of Kalman based on autoregression model
Filtering algorithm.
Background technology
Kalman filtering (Kalman filtering) algorithm is a kind of using linear system state equation, then by being
Data are observed in input and output of uniting, and the algorithm of optimal estimation is carried out to system mode, and due to the noise in system and the shadow of interference
Ring, its optimal estimation is also considered as filtering, and Kalman filtering applies in general to state-space model, and its method is as follows：
State-space model：
X (t)=ax (t-1)+bu (t)+w (t),
Y (t)=cx (t)+v (t),
Wherein x (t) is the output state that can not survey of system, a, and b, c is systematic parameter, and u (t) is the input of system, and y (t) is to be
The output of system, v (t), w (t) is the state-noise and output noise of system respectively, and respectively obey average be zero, variance be δ and
ε Gaussian Profile, using Kalman filtering, obtains the following algorithm for estimating to not measured state x (t)：
X (t | t)=x (t | t-1)+r (t) (y (t)-cx (t | t-1)
X (t | t-1)=ax (t-1 | t-1)+bu (t)
P (t | t-1)=a2P (t-1 | t-1)+δ, P (0 | 0)=10
P (t | t)=[1-r (t) c] P (t | t-1)
Wherein x (t | t) is optimal estimation of the t to state x (t), and P (t | t) it is optimal estimation of the t to state x (t)
Variance；
X (t | t-1) is predicted value of the t to state x (t), is also the estimate to x (t) predicted, and P (t | t-1) it is t
Moment
To the estimate variance of state x (t) predicted values；
X (t-1 | t-1) is the t-1 moment to that can not survey output state x (t-1) optimal estimation, and P (t-1 | t-1) it is the t-1 moment pair
The variance of state x (t-1) optimal estimation；
R (t) is the regulation parameter of t；
In above-mentioned, the output state x (t) that can not survey in state-space model is the unknown state of estimating system, system output y
(t) it is that the state estimated is corrected, it can be seen that traditional kalman filter method can only pick out state space
The state x (t-1) and the u at current time of x (t) only with the previous moment in the unknown state of model, i.e. state-space model
(t) it is relevant, and the input before state and t before the t-1 moment is unrelated.
However as the popularization of computer network, in process control field, custom passes through computer to scientists now
The input of network delivery model and output data, but due to the complexity and unpredictability of network, data transfer mistake can be caused
Lost in journey, and process model is usually to represent that the first model of i.e. system is autoregression model using autoregression model, and
Be not in traditional state model, autoregression model x (t) not only with the state x (t-1) at previous moment and current time
U (t) it is relevant, and relevant thus traditional with the state before the t-1 moment and the input before t be directed to shape
The kalman filter method of state space model is not directly applicable the autoregression model of process control field, and it can not be picked out
Autoregression model can not survey output state data, also just the state that estimated can not be corrected, so as to reduce parameter
The precision of identification.
The content of the invention
In view of the above-mentioned problems, the invention provides a kind of Kalman filtering algorithm based on autoregression model, it can be recognized
Go out the output data that can not survey of autoregression model, and the precision of parameter identification is high.
Its technical scheme is so, it is characterised in that it includes following algorithm steps：
(S1), using collection system input data u (t), system mode noise v (t) and t optimal estimation x (t | t)
Pick out autoregression model can not survey output state
(S2), the output state x (t) that can not survey of autoregression model sets up autoregression mould through signal amplifier and output noise w (t)
System output y (t)=cx (t)+w (t) (2) of type
Then exporting y (t) by system realizes t to that can not survey output state x (t) predicted value x (t | t-1) correction；
Wherein
A (d)=1-a1d-1+…-and-n,
B (d)=b1d-1+b2d-2+…+bnd-m,
D is backward shift operator (d-1Y (t)=y (t-1))
θx=[a1,…,an]T
θu=[b1,…,bm]T
V (t), w (t) are the state-noise and output noise of system respectively, and it is zero to obey average respectively, and variance is δ and ε height
This distribution；A (d), B (d) are autoregression model input u (t) and output x (t) multinomial respectively；It is that autoregression model is defeated
Go out corresponding vector,It is that autoregression model inputs corresponding vector, θxIt isCorresponding parameter vector, θuIt isIt is right
The parameter vector answered, t represents the data amount check collected, t=1,2 ... N.
It is further characterized by, and its algorithm steps also includes：
(1.1), the output state x (t) that can not survey for obtaining autoregression model by formula (1), (2) is exported known to y (t) in system
Under the conditions of distribution function
Wherein Y (t-1)=y (t-1) ..., y (1) }；
When known to system output y (t), p (y (t) | Y (t-1)) constant for known to,
Then formula (3) is equivalent to p (x (t) | y (t), Y (t-1)) ∝ p (y (t) | x (t)) p (x (t) | Y (t-1)) (4)；
(1.2), the distribution function for obtaining system output y (t) by formula (1), (2) calculating is
(1.3), define
E (x (t) | Y (t-1))=x (t | t-1), Var (x (t) | Y (t-1))=Var (x (t)-x (t | t-1))=Var (e (t | t-
1))=P (t | t-1), then obtain
(1.4) logarithmic function F (x (t))=ln [p (y (t) | x (t)) p (x (t) | Y (t-1))], is defined,
Logarithmic function is obtained into x (t | t)=x (t | t-1)+r (t) (y (t)-cx (t | t-1) to x (t) derivations) (7)
(1.5), define
Calculate and obtain according to formula (9), (10)
Wherein P (t | t-i)=Cov (e (t | t), e (t-i | t-i)) represent error e (t | t) and e (t-i | t-i) covariance, i
=1 ..., n；
(1.6) e (t | t)=x (t)-x (t | t)=(1-r (t) c) e (t | t-1)-r (t) v (t), are defined
Obtain P (t | t)=Var (e (t | t))=(1-r (t) c)2P(t|t-1)+r2(t) ε=(1-r (t) c) P (t | t-1) (13)
Wherein P (t | t) is t to state x (t) optimal estimation x (t | t) variance；
So that covariance P (t | t-i) it is derived as
P (t | t-i)=Cov (e (t | t), e (t-i | t-i))
=Cov [(1-r (t) c) (a1e(t-1|t-1)+…+ane(t-n|t-n)+w(t))-r(t)v(t),e(t-i|t-i)]
=(1-r (t) c) [P (t-1 | t-i), P (t-2 | t-i) ..., P (t-i | t-i) ..., P (t-i | t-n)]T(14)；
The identification step that can not survey output state x (t) to autoregression model is：
(2.1), collection system input data u (t)；
(2.2), calculate x (t | t-1) by formula (8), (12), (9) respectively, P (t | t-1), r (t), then respectively by formula (7),
(13) x (t | t) is calculated, P (t | t) calculates P (t | t-i) finally according to formula (14)；
(2.3) t=t+1, is made, if t>N, then terminator, obtain N number of autoregression model can not survey output state x (t),
Otherwise, return to step (2.1).
The beneficial effects of the invention are as follows can pick out autoregression model completely by the optimal estimation x of t (t | t)
Output state x (t) can not be surveyed, t can be achieved to that can not survey the pre- of output state x (t) in the system output y (t) then set up
Measured value x (t | t-1) correction, so as to improve the precision of parameter identification.
Brief description of the drawings
Fig. 1 is the system construction drawing of the present invention；
Fig. 2 is the flow chart of identified parameters algorithm of the present invention；
Fig. 3 is the emulation schematic diagram of the present invention.
Embodiment
As shown in figure 1, the present invention includes following algorithm steps：
(S1), using collection system input data u (t), system mode noise v (t) and t optimal estimation x (t | t)
Pick out autoregression model can not survey output state
(S2), the output state x (t) that can not survey of autoregression model sets up autoregression mould through signal amplifier and output noise w (t)
System output y (t)=cx (t)+w (t) (2) of type
Then exporting y (t) by system realizes t to that can not survey output state x (t) predicted value x (t | t-1) correction；
Wherein
A (d)=1-a1d-1+…-and-n,
B (d)=b1d-1+b2d-2+…+bnd-m,
D is backward shift operator (d-1Y (t)=y (t-1))
θx=[a1,…,an]T
θu=[b1,…,bm]T
X (t) is both the input that can not be surveyed output state, be also output model of autoregression model, is immesurable；
X (t) reaches receiving terminal due to being influenceed by external environment condition and network environment, signal by network transmission to terminal
Inevitably there is interference, therefore the passage output measured in terminal have received the pollution of noise, it is assumed that the output of system is made an uproar
Sound w (t) is that average is 0, and variance is 0.1 white Gaussian noise, while it is 0 that the x (t) of autoregression model, which also receives average, side
Difference is 0.1 white Gaussian noise v (t) influence；
A (d), B (d) are autoregression model input u (t) and output x (t) multinomial respectively；
It is that autoregression model exports corresponding vector,It is that autoregression model inputs corresponding vector, θxIt isIt is right
The parameter vector answered, θuIt isCorresponding parameter vector；Parameter c is the signal parameter of signal amplifier；T represents the number collected
According to number, t=1,2 ... N.
Its algorithm steps also includes：
(1.1), the output state x (t) that can not survey for obtaining autoregression model by formula (1), (2) is exported known to y (t) in system
Under the conditions of distribution function
Wherein Y (t-1)=y (t-1) ..., y (1) }；
When known to system output y (t), p (y (t) | Y (t-1)) constant for known to,
Then formula (3) is equivalent to p (x (t) | y (t), Y (t-1)) ∝ p (y (t) | x (t)) p (x (t) | Y (t-1)) (4)；
(1.2), the distribution function for obtaining system output y (t) by formula (1), (2) calculating is
(1.3), define
E (x (t) | Y (t-1))=x (t | t-1), Var (x (t) | Y (t-1))=Var (x (t)-x (t | t-1))=Var (e (t | t-
1))=P (t | t-1), then obtain
(1.4) logarithmic function F (x (t))=ln [p (y (t) | x (t)) p (x (t) | Y (t-1))], is defined,
Logarithmic function is obtained into x (t | t)=x (t | t-1)+r (t) (y (t)-cx (t | t-1) to x (t) derivations) (7)
(1.5), define
Calculate and obtain according to formula (9), (10)
Wherein P (t | t-i)=Cov (e (t | t), e (t-i | t-i)) represent error e (t | t) and e (t-i | t-i) covariance, i
=1 ..., n；
(1.6) e (t | t)=x (t)-x (t | t)=(1-r (t) c) e (t | t-1)-r (t) v (t), are defined
Obtain P (t | t)=Var (e (t | t))=(1-r (t) c)2P(t|t-1)+r2(t) ε=(1-r (t) c) P (t | t-1) (13)
Wherein P (t | t) is t to state x (t) optimal estimation x (t | t) variance；
So that covariance P (t | t-i) it is derived as
P (t | t-i)=Cov (e (t | t), e (t-i | t-i))
=Cov [(1-r (t) c) (a1e(t-1|t-1)+…+ane(t-n|t-n)+w(t))-r(t)v(t),e(t-i|t-i)]
=(1-r (t) c) [P (t-1 | t-i), P (t-2 | t-i) ..., P (t-i | t-i) ..., P (t-i | t-n)]T (14)。
The Kalman filtering algorithm based on autoregression model of the present invention is finally given, from the above it can be seen that returning certainly
Return the x (t) in model different with the state in state-space model, the x (t) in autoregression model not only with previous moment
X (t-1) is relevant, and relevant with the output at preceding several moment, and such as with above-mentioned x (t-1), x (t-2) ..., x (t-n) has
Close；It can not survey output shape according to formula (7), by what the optimal estimation x of t (t | t) can pick out autoregression model completely
T can be achieved to that can not survey output state x (t) predicted value x (t | t-1) in state x (t), the system output y (t) then set up
Correction, so as to improve the precision of parameter identification；And by formula (13), calculate obtained P (t | t)<P (t | t-1), i.e., originally
The optimal estimation x's (t | t) to state x (t) picked out in the Kalman filtering algorithm based on autoregression model of invention
Variance P (t | t) is less than the variance P (t | t-1) picked out using existing Kalman filtering algorithm, thus the present invention based on from returning
Autoregression model can be efficiently applied to by returning the Kalman filtering algorithm of model, also superior to existing Kalman filtering algorithm, be applicable
Scope is wide.
As shown in Fig. 2 the identification step that can not survey output state x (t) to autoregression model is：
(2.1), initialize, it is assumed that u (t)=0, x (t | t)=0, P (t | t)=10, P (t | t-i)=1, i=1 ..., n, t≤0,δ=δh, ε=εh, make t=1；
(2.2), collection system input data u (t)；
(2.3), build autoregression model and export corresponding vectorAnd autoregression model inputs corresponding vector
(2.4), calculate x (t | t-1) by formula (8), (12), (9) respectively, P (t | t-1), r (t), then respectively by formula (7),
(13) x (t | t) is calculated, P (t | t) calculates P (t | t-i) finally according to formula (14)；
(2.5) t=t+1, is made, if t>N, then terminator, obtain N number of autoregression model can not survey output state x (t),
Otherwise, return to step (2.2).
Output state x (t) can not be really surveyed as shown in figure 3, curve is system, mark "+" (being figure midrange)
It is the optimal estimation x (t | t) to state x (t) calculated using the method for the present invention, it can be seen that the autoregression mould estimated
The output of type can trace into model well and really export.
Claims (3)
1. a kind of Kalman filtering algorithm based on autoregression model, it is characterised in that it includes following algorithm steps：
(S1), using collection system input data u (t), system mode noise v (t) and t optimal estimation x (t | t)
Pick out autoregression model can not survey output state
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(S2), the output state x (t) that can not survey of autoregression model sets up autoregression mould through signal amplifier and output noise w (t)
System output y (t)=cx (t)+w (t) (2) of type
Then exporting y (t) by system realizes t to that can not survey output state x (t) predicted value x (t | t-1) correction；
Wherein
A (d)=1-a1d-1+…-and-n,
B (d)=b1d-1+b2d-2+…+bnd-m,
D is backward shift operator (d-1Y (t)=y (t-1))
θx=[a1,…,an]T
θu=[b1,…,bm]T
V (t), w (t) are the state-noise and output noise of system respectively, and it is zero to obey average respectively, and variance is δ and ε height
This distribution；A (d), B (d) are autoregression model input u (t) and output x (t) multinomial respectively；It is that autoregression model is defeated
Go out corresponding vector,It is that autoregression model inputs corresponding vector, θxIt isCorresponding parameter vector, θuIt isIt is right
The parameter vector answered, t represents the data amount check collected, t=1,2 ... N.
2. a kind of Kalman filtering algorithm based on autoregression model according to claim 1, it is characterised in that its algorithm
Step also includes：
(1.1), the output state x (t) that can not survey for obtaining autoregression model by formula (1), (2) is exported known to y (t) in system
Under the conditions of distribution function
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Wherein Y (t-1)=y (t-1) ..., y (1) }；
When known to system output y (t), p (y (t) | Y (t-1)) constant for known to,
Then formula (3) is equivalent to p (x (t) | y (t), Y (t-1)) ∝ p (y (t) | x (t)) p (x (t) | Y (t-1)) (4)；
(1.2), the distribution function for obtaining system output y (t) by formula (1), (2) calculating is
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<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
(1.3), define
E (x (t) | Y (t-1))=x (t | t-1), Var (x (t) | Y (t-1))=Var (x (t)-x (t | t-1))=Var (e (t | t-
1))=P (t | t-1), then obtain
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
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<mi>Y</mi>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msqrt>
<mrow>
<mn>2</mn>
<mi>&pi;</mi>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>|</mo>
<mi>t</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</msqrt>
</mfrac>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mfrac>
<msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
<mo>-</mo>
<mi>x</mi>
<mo>(</mo>
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<mi>t</mi>
<mo>|</mo>
<mi>t</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mrow>
<mn>2</mn>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>|</mo>
<mi>t</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
(1.4) logarithmic function F (x (t))=ln [p (y (t) | x (t)) p (x (t) | Y (t-1))], is defined,
Logarithmic function is obtained into x (t | t)=x (t | t-1)+r (t) (y (t)-cx (t | t-1) to x (t) derivations) (7)
<mrow>
<mi>r</mi>
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<mi>t</mi>
<mo>)</mo>
</mrow>
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<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
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<mi>t</mi>
<mo>-</mo>
<mn>1</mn>
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</mrow>
<mi>c</mi>
</mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
<mi>P</mi>
<mrow>
<mo>(</mo>
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<mi>t</mi>
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</mrow>
<mo>+</mo>
<mi>&epsiv;</mi>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
(1.5), define
Calculate and obtain according to formula (9), (10)
Wherein P (t | t-i)=Cov (e (t | t), e (t-i | t-i)) represent error e (t | t) and e (t-i | t-i) covariance, i
=1 ..., n；
(1.6) e (t | t)=x (t)-x (t | t)=(1-r (t) c) e (t | t-1)-r (t) v (t), are defined
Obtain P (t | t)=Var (e (t | t))=(1-r (t) c)2P(t|t-1)+r2(t) ε=(1-r (t) c) P (t | t-1) (13)
Wherein P (t | t) is t to state x (t) optimal estimation x (t | t) variance；
So that covariance P (t | t-i) it is derived as
P (t | t-i)=Cov (e (t | t), e (t-i | t-i))
=Cov [(1-r (t) c) (a1e(t-1|t-1)+…+ane(t-n|t-n)+w(t))-r(t)v(t),e(t-i|t-i)]
=(1-r (t) c) [P (t-1 | t-i), P (t-2 | t-i) ..., P (t-i | t-i) ..., P (t-i | t-n)]T (14)。
3. a kind of Kalman filtering algorithm based on autoregression model according to claim 1, it is characterised in that to returning certainly
Returning the identification step that can not survey output state x (t) of model is：
(2.1), collection system input data u (t)；
(2.2), calculate x (t | t-1) by formula (8), (12), (9) respectively, P (t | t-1), r (t), then respectively by formula (7),
(13) x (t | t) is calculated, P (t | t) calculates P (t | t-i) finally according to formula (14)；
(2.3) t=t+1, is made, if t>N, then terminator, obtain N number of autoregression model can not survey output state x (t),
Otherwise, return to step (2.1).
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