4. Aqueous two-phase extraction of phenolic compounds from Sedum dendroideum with
4.2 Materials and methods
The theory of Kalman filtering starts from the assumption that the nature of the signal of interest is stochastic and that it is generated by passing a white stochastic process (white noise) through a linear dynamical system [7–9]. Contrary to the Wiener theory in which a linear dynamic system is described by an input–output relation, i.e. by a corresponding transfer function in the complex domain, here the system is represented in time domain by a model in a state space. The quoted model encompasses a dynamic state equation (which in the case of continual signals represents a vectorial linear differential equation of the first order, while in the case of discrete signals this equation becomes a vectorial linear difference equation of the first order) and an algebraic equation of the system output. In the discrete case these two equations can be written in the following manner
x kð þ 1Þ ¼ Fx kð Þ þ Gx kð Þ; x 0ð Þ; ð1:57Þ y kð Þ ¼ Hx kð Þ þ v kð Þ; ð1:58Þ where
1) xðkÞ—state vector of the linear discrete system in a discrete moment tk¼ kT (T—sampling period)
2) xð0Þ—unknown initial condition with a mathematical expectance m0 and a covariance matrix P0, i.e.
E x 0f ð Þg ¼ m0; E ðx 0ð Þ m0Þ x 0ð ð Þ m0ÞT
¼ P0: ð1:59Þ
3) xðkÞ—white stochastic process with a zero mean value which excites the linear model (1.57), the so-called noise of the state or of the process, i.e.
Efx 0ð Þg ¼ 0; E x kð ÞxTð Þj
¼ Q kð Þdk;j ð1:60Þ for each k; j¼ 1; 2; . . ., where dk;j is Kronecker delta symbol (dk;j¼ 0 for k 6¼ j and (dk;j¼ 1 for k ¼ j), and Q ðkÞ represents the covariance matrix for noise.
4) vðkÞ—white stochastic noise with a zero mean value, which represents additive noise in the output state Eq. (1.58), i.e. measurement noise, i.e.
E v 0f ð Þg ¼ 0; E v k ð ÞvTð Þj
¼ R kð Þdk;j ð1:61Þ for each k; j¼ 1; 2; . . . where dk;jis Kronecker delta symbol, and RðkÞ represents covariance matrix for this noise.
5) F, G and H given matrices of corresponding dimensions, which in a general case may also depend on the time index k. If these matrices are constant, while covariance matrices of noise QðkÞ and R ðkÞ also do not depend on the time index k, i.e.QðkÞ ¼ Q and R ðkÞ ¼ R, the considered model is time-invariant or stationary.
6) Further we assume that the vectorial stochastic variables xð0Þ, x ðkÞ and v ðkÞ are mutually non-correlated, so that
E x kð ÞvTð Þj
¼ 0; E x k ð Þ x 0½ ð Þ m0T
¼ 0; E v k ð Þ x 0½ ð Þ m0T
¼ 0 ð1:62Þ for each k; j¼ 1; 2; . . ..
Let us note that the dynamic Eq. (1.57) represents a model-generator of the state vector, i.e. the physical mechanism generating the components of the state vector as the physical variables of interest, while the algebraic Eq. (1.58) describes the mechanism of measurement (observation) of the output signal, taking into account the sensor inaccuracy itself, as expressed through additive noise. Thus the output signal of interest is generated as a linear combination of the components of the state vector, which is additionally contaminated by measurement noise.
The Kalman filter itself represents a recursive numerical algorithm generating an estimation of the immeasurable state vector in the current discrete moment, with a time index k, based on the available estimation of the state vector in the preceding discrete moment, with the time index k 1, and the newly obtained measurement in the given discrete moment, yðkÞ. If we introduce the following notation
^x k kð j Þ ¼ ^xkð Þþ estimation of the state vector xðkÞ in the moment k, after the last measurement, yðkÞ has been performed;
^x k kð j 1Þ ¼ ^xkð Þ estimation of the state vector xðkÞ in the moment k, before the last measurement, yðkÞ has been performed,
then the estimation ^xkðþÞ can be formed based on the predictor–corrector algo-rithm representing a linear combination of the previously known estimation, ^xkðÞ and the newly obtained measured information, yðkÞ, i.e.
^
xkð Þ ¼ Kþ ð Þ^xk kð Þ þ K k ð Þy kð Þ ð1:63Þ where KðkÞ and K kð Þ are unknown matrices dependent on the time moment k.
These matrices represent free parameters in the algorithm (1.63) and they should be chosen so that the estimation (1.63) gives the best possible approximation of the immeasurable random vector variable xðkÞ. Thus posed problem imposes the requirement to define criteria or indicators for the appraisal of the quality of the estimation. Similar to the well-known criteria for the appraisal of the quality of measuring equipment, like its accuracy and precision, the terms are introduced here of unbiased estimation and the total variance of estimation error. If we introduce the corresponding notation for estimation errors as
~
xkð Þ ¼ x kþ ð Þ ^xkð Þ;þ ð1:64Þ
~
xkð Þ ¼ x k ð Þ ^xkð Þ; ð1:65Þ it is said then that the estimation ^xkðþÞ of the state vector x ðkÞ is unbiased if its mean value (mathematical expectation) is equal to zero, i.e.
E ~fxkð Þþg ¼ 0 ) E ^xf kð Þþg ¼ E x kf ð Þg: ð1:66Þ On the other hand, the corresponding covariance matrices of estimation errors are defined as
Pkð Þ ¼ E ~xþ kð Þ~xþ Tkð Þþ
; ð1:67Þ
Pkð Þ ¼ E ~x kð Þ~x Tkð Þ
; ð1:68Þ
so that the total variance of the estimation error is defined as the sum of diagonal elements of the PkðþÞ matrix, since each diagonal element of this matrix repre-sents the variance of the estimation error for the corresponding component of the estimated state vector. In this way the total variance of the estimation error in the moment k, denoted as r2kðþÞ, is given as the trace of the matrix (1.67), i.e.
r2kðþÞ ¼ Trace Pf kðþÞg ð1:69Þ The condition (1.66) corresponds to the term of accuracy, and the condition (1.69) to the term of precision in the measurement technique. Namely, since the estimation of the state vector is a stochastic variable and since it is based on noisy measurements, the condition (1.66) points out that the mean (expected) value of
this estimation will be equal to the mean (expected) value of the immeasurable and estimated stochastic state vector itself (the so-called accuracy condition), while the actual realizations of the estimation will be in some neighborhood around this mean value, this neighborhood being smaller with a smaller value of the criterion (1.69), i.e. if the precision is higher. Bearing this in mind, the unknown weighted matrix in the estimation algorithm (1.63) can be determined from the condition of unbiased estimation (1.66) and from the condition of minimum criterion (1.69), and thus such estimation is denoted as the estimation of minimum variance of error or a discrete Kalman filter. Let us note too that ^xkðþÞ is also called the filtered estimation, and ^xkðÞ is the single-step prediction [7–9]. Based on (1.58), (1.63) and (1.64) we may further write
~
xkð Þ ¼ x kþ ð Þ Kð Þ^xk kð Þ K k ð Þ Hx k½ ð Þ þ v kð Þ þ K½ ð Þx kk ð Þ Kð Þx kk ð Þ;
where the last bracketed addend is included artificially and its value is zero, from which it follows, taking into account (1.65), that
~
xkð Þ ¼ I Kþ ½ ð Þ K kk ð ÞHx kð Þ þ Kð Þ~xk kð Þ K k ð Þv kð Þ: ð1:70Þ Since it is assumed that E vðkÞf g ¼ 0 and if it is also assumed that the previous estimation was unbiased, i.e. E ~fxkðÞg ¼ 0, the condition (1.66) will be satisfied for each xðkÞ only for
I Kð Þ K kk ð ÞH ¼ 0 ) Kð Þ ¼ I K kk ð ÞH: ð1:71Þ Replacing (1.71) into (1.63) and (1.70) it is obtained that
^xkð Þ ¼ ^xþ kð Þ þ K k ð Þ y k½ ð Þ H^xkð Þ; ð1:72Þ
~
xkð Þ ¼ I K kþ ½ ð ÞH~xkð Þ K k ð Þv kð Þ: ð1:73Þ Incorporating (1.73) and the definition of the covariance matrix of estimation (filtration) error into (1.67), it follows further that
Pkð Þ ¼ Pþ kð Þ K k ð ÞHPkð Þ P kð ÞH TKTð Þk þ K kð Þ HP kð ÞH Tþ R
KTð Þ;k ð1:74Þ
where R¼ E v ðkÞvf TðkÞg. When deriving the solution (1.74) it was taken into account that
E ~ xkð Þv Tð Þk
¼ 0; E v k ð Þ~xTkð Þ
¼ 0; ð1:75Þ
since the error ~xkðÞ depends on the realization of state noise x ðiÞ, i¼ 1; 2; . . .; k 1, which is assumed to be uncorrelated with measurement noise vðkÞ. By replacing (1.74) into (1.69), we obtain the expression for the total var-iance of the estimation (filtration) error.
r2kðþÞ ¼ Trace Pf kðÞg 2Trace KðkÞHPf kðÞg þ Trace KðkÞ HP kðÞHTþ R
KTðkÞ
ð1:76Þ
The last expression was derived utilizing the fact that the second addend in (1.74) represents the transposed third addend and using the property of the trace function that Trace Af g ¼ Trace A T
. The optimum matrix of amplification of the algorithm K kð Þ corresponds to the minimal total variance of the estimation error and is determined from the necessary condition for the minimum of the criterion (1.76)
or2kð Þþ
oK kð Þ ¼ 0 ð1:77Þ
The application of the operator of partial differentiation to (1.76) and the cal-culation of the corresponding partial derivatives requires the knowledge of the following rules for the differentiation of the matrix trace, as a scalar expression, over a matrix argument [7]
o
oATrace BACf g ¼ BTCT; ð1:78Þ o
oATrace AB T
¼ 2AB for B ¼ BT: ð1:79Þ
Applying the partial differentiation operator to (1.76), and using the condition (1.77), it is obtained that
2 o
oKðkÞTrace KðkÞHPf kð Þg
þ o
oKðkÞTrace KðkÞ HP kð ÞH Tþ R KTð Þk
¼ 0:
ð1:80Þ
If we further apply the rule (1.78) to the first term in (1.80), with B¼ I;
A¼ K ðkÞ and C ¼ HPkðÞ, and the rule (1.79) to the second term in (1.80), with A¼ K ðkÞ and B ¼ HPkðÞ HTþ R, the relation (1.80) reduces to
or2kð Þk
oK kð Þ¼ 2Pkð ÞH Tþ 2K kð Þ HP kð ÞH Tþ R
¼ 0 ð1:81Þ
from which we obtain the optimum amplification matrix K kð Þ ¼ Pkð ÞH T HPkð ÞH Tþ R 1
: ð1:82Þ
If the right side of the expression (1.81) is again partially differentiated, applying the rule (1.78) to the second addend in (1.81), also choosing B¼ I, A¼ K ðkÞ and C ¼ HPkðÞ HTþ R, it can be written
o2r2kð Þk
oK kð Þ2 ¼ 2 HP kð ÞH Tþ R
0; ð1:83Þ
from which we conclude that the optimum solution (1.82) corresponds to the minimum of the scalar criterion (1.69). By replacing the optimum amplification (1.82) into the expression (1.74), we obtain the expression for the covariance matrix of estimation (filtration) error
Pkð Þ ¼ Pþ kð Þ P kð ÞH T HPkð ÞH Tþ R 1
HPkð Þ; ð1:84Þ or, if we introduce relation (1.82) into the expression (1.84),
Pkð Þ ¼ I K kþ ½ ð ÞHPkð Þ: ð1:85Þ Relations (1.72), (1.82) and (1.84) or (1.85) define the estimation correction step based on measurements or the estimation (filtration) step in a discrete Kalman filter. Obviously the realization of this step assumes that prior to it the prediction step was realized, i.e. that the values ^xkðÞ and PkðÞ are known. Let us note that the variable
~y kð Þ ¼ y kð Þ H^xkð Þ ð1:86Þ is denoted as the measurement residual or the innovation. Namely, based on the output Eq. (1.58) it is possible to estimate (predict) its expected value before the signal itself is measured. Since measurement noise is assumed to have a zero mean value, and in the moment k immediately before the signal yðkÞ is measured the prediction ^xkðÞ of the immeasurable random state xðkÞ is known, the expected value or prediction of the output in the moment k is
^
y kð Þ ¼ H^xkð Þ: ð1:87Þ
In this way, the complete new measurement in the moment k, yðkÞ, does not introduce any new information about the system, since before the measurement itself we already predicted its value (1.87) according to the measurement model (1.58), so that the only new information in the measurement is its residual or innovation (1.86). Further according to (1.58), (1.65) and (1.86) we obtain
~yðkÞ ¼ H~xkðÞ þ v ðkÞ; ð1:88Þ from which we conclude that the mean (expected) value of the residual is Ef~yðkÞg ¼ 0. This follows from the assumption that measurement noise v ðkÞ has a zero mean value, EfvðkÞg ¼ 0, and that the prediction ^xkðÞ is an unbiased estimation of the state, i.e. Ef~xðkÞg ¼ 0. The covariance matrix of the residual (which represents variance in the case of one-dimensional signal) is given by the expression
S kð Þ ¼ E ~y k ð Þ~yTð Þk
¼ HPkð ÞH Tþ R ð1:89Þ
To derive the expression (1.89) we used the relations (1.75) and (1.88). Simi-larly to measurement noise vðkÞ, the residual ~y ðkÞ also represents white noise with a zero mean value and with a corresponding covariance matrix SðkÞ in (1.89) (noise covariance matrix vðkÞ is R) [7–9].
The prediction of the system may be regarded as its estimation (filtration) when a measurement is unavailable (in that case the covariance matrix of measurement noise R! 1, so that Kalman amplification in (1.82) is equal to zero, i.e. K¼ 0).
Further, if we include KðkÞ ¼ 0 in (1.72) and (1.85), it follows that
^xkð Þ ¼ ^xþ kð Þ K kð Þ¼0
Pkð Þ ¼ Pþ kð Þ K kð Þ¼0
The prediction ^xkðÞ itself represents the extrapolation of the state estimation (filtration) in the moment k 1, ^xk1ðþÞ, to the moment k, but immediately before the signal yðkÞ is measured. This extrapolation may be done based on the model (1.57) itself, bearing in mind that the expected value of noise xðkÞ is equal to zero.
Then according to (1.57), we may write
^
xkðÞ ¼ F^xk1ðþÞ: ð1:90Þ
Relation (1.90) is derived from (1.57) by replacing the state vector xðkÞ by
^xkðÞ, and the state vector in the previous moment xðk 1Þ with ^xk1ðþÞ, simultaneously neglecting state noise xðkÞ. The prediction error, ~xkðÞ in (1.65), can be written as
~
xkð Þ ¼ Fx k ð Þ þ Gx kð Þ F^xk1ð Þ ¼ F~xþ k1ð Þ þ Gx kþ ð Þ
so that the covariance matrix of the prediction error, Pkð Þ in (1.68), is given by Pkð Þ ¼ FP k1ð ÞFþ Tþ GQGT: ð1:91Þ While deriving the expression (1.91) we took into account that the cross-cor-relation is
E ~ xk1ð Þxþ Tð Þk
¼ 0; E x k ð Þ~xTk1ð Þþ
¼ 0 ð1:92Þ
since the filtration error ~xk1ð Þ depends on noise realization x iþ ð Þ; 0\i k 2, and the assumption is that the discrete array x if ð Þg is white noise and that it is uncorrelated in time, i.e. the momentary realization of this noise x kð Þ is not correlated with its prior realizations x if ð Þ; i\kg. Since according to the lemma on matrix inversion the expression (1.84) may be written in the form
P1k ð Þ ¼ Pþ 1k ð Þ þ H TR1H; ð1:93Þ the expression for the amplification matrix of the Kalman filter (1.82) reduces to
K kð Þ ¼ P kð ÞPþ 1k ð Þþ
Pkð ÞH T HPkð ÞH Tþ R 1
¼ Pkð Þ Pþ 1k ð Þ þ H TR1H
Pkð ÞH T HPkð ÞH Tþ R 1
¼ Pkð ÞHþ T Iþ R1HPkð ÞH T
HPkð ÞH Tþ R
1
¼ Pkð ÞHþ T R1Rþ R1HPkð ÞH T
HPkð ÞH Tþ R
1
from where it stems that
K kð Þ ¼ Pkð ÞHþ TR1: ð1:94Þ The expression (1.94) shows that the value of Kalman filter amplification K depends on the size of the covariance matrix of estimation error P, which represents a figure of merit of the system state estimation quality, and on the size of the covariance matrix of measurement noise R, which describes the accuracy of the output signal measurement in the system. A high value of noise, i.e. a high R, and a small filtration error, i.e. low P, show that the residual ~y is mostly the consequence of measurement noise, so that the filter imparts small weight, i.e.
small amplification K to this residual, which does not bear important information about the estimated state, so that the estimated (filtered) state will be close to its prediction. On the other hand, low measurement noise (low R) and a large error in the system state estimation (high P) show that the residual contains a significant information about the estimation error, so that the filter, through its large ampli-fication K, performs weighting of the residual ~y in a significantly larger amount compared to the prediction of the state vector. Also, the size of the covariance matrix of the prediction error in (1.91) depends directly on the size of the covariance matrix of state noise Q (mean power of state noise). A high Q shows inadequacy of the signal model in the state space, while low Q results in a small error covariance matrix, which shows that the model in the state space represents an adequate approximation of the real system of interest. This further shows that the matrix of filter amplification is proportional to Q and inversely proportional to R, i.e.
K QR1 ð1:95Þ
The function of a Kalman filter may be represented by the following Table (Table1.1).
High filter amplification K points out to its wide bandwidth and faster filter response to excitation in the form of measured signal, i.e. the measurement residual. However, a high filter amplification simultaneously means a smaller degree of noise reduction in the system. On the other hand, smaller filter ampli-fication means its smaller bandwidth and slower response to the excitation in the form of measurement signal.
For the implementation of the complete recursive algorithm (Kalman filter) it is necessary to adopt starting (initial) values ^x0ð Þ, Pþ 0ð Þ, corresponding to a timeþ index of k¼ 0. If we adopt
^
x0ð Þ ¼ E x 0þ f ð Þg ¼ m0; P0ð Þ ¼ Pþ 0¼ E x 0 ð ð Þ m0Þ x 0ð ð Þ m0ÞT
; ð1:96Þ such choice ensures that the prediction ^x1ð Þ is unbiased.
Indeed, since according to (1.90)
^
x1ð Þ ¼ F^x 0ð Þþ ð1:97Þ
if (1.96) is replaced into (1.97), it follows that
^x1ð Þ ¼ Fm 0¼ FE x 0f ð Þg ¼ E Fx 0f ð Þg ð1:98Þ Expression (1.98) may be expanded by the zero term GE x 0f ð Þg, from which we obtain, according to (1.57), the condition for the unbiased prediction
^
x1ð Þ ¼ E Fx 0 f ð Þ þ Gx 0ð Þg ¼ E x 1f ð Þg: ð1:99Þ The unbiased prediction ^x1ð Þ further implies, according to (1.70) and (1.71), unbiased filtration ^x1ð Þ, and this further results in unbiased ^xþ 2ð Þ, etc., and using induction we conclude that the estimation of the state will be unbiased in each moment k. The equations of a digital Kalman filter for a time invariant system model are shown in Table1.2.
Table1.2 shows that, in the case of a time-invariant system model, the cal-culation of the covariance matrix of the prediction and estimation (filtration) error proceeds independently on the calculation of the prediction and estimation of the system state, so that this calculation can be done prior the filter implementation in the real time itself, so that the designer may estimate if the values of parameters in the model of the filter were adopted in an adequate manner (the parameters of the filter are matrices F, G, H in the model of the system, the initial values m0and P0, as well as noise statistics Q and R). To verify practically the design of a Kalman filter it is necessary if the estimation of the state at a measurement sample of finite width is in agreement (i.e. consistent) with the theoretical assumptions. The sta-tistical criteria for the analysis of the filter consistency are
• estimation (filtration) errors, whose elements should represent random variables with a zero mean value and a zero expected (mean) amplitude, in accordance with the value of the square root from the corresponding diagonal element of covariance matrix of estimation error, Pkð Þ.þ
Table 1.1 Heuristic description of Kalman filter function
Condition Amplification K Filter parameters
Confidence in model (prediction)
Low • P low (adequate model)
• R high (bad measurement) Confidence in residual
(signal measurement)
High • P low (adequate model);
• R low (good measurement)
• P high (inadequate model)
• residual (innovation) ~y kð Þ, which should satisfy the same assumptions as the error ~xkð Þ, and it is only necessary to replace the error covariance, Pþ kð Þ, withþ the corresponding residual covariance, S kð Þ.
• residual (innovation), which should represent white stochastic process.
The last two criteria can be tested in real time, during the operation of the filter itself, while the first criterion, although the most important one, can be applied only in a simulated experiment, since the real error, i.e. the system state, is not known in reality [11]. The application of the quoted criteria is based on the theory of statistical decision, i.e. the hypothesis testing [11-13], and the reader is referred to literature in order to become acquainted with the topic in more detail.
Let us also note that the important properties of the Kalman filter are the following:
• Kalman filter is a linear function of the current measurement, y kð Þ.
• the estimation of the system state ^xkð Þ explicitly depends only on the currentþ measurement y kð Þ, while its dependence on the previous measurements Yk1¼
y 0ð Þ; y 1ð Þ; . . .; y k 1ð Þ
f g reflects only through their influence to the prediction,
^ xkð Þ.
• covariance matrices of the prediction errors, Pkð Þ, and the estimation errors, Pkð Þ, can be calculated in advance, before the implementation of the filterþ itself, for the case of a time-invariant system model.
• the assumptions that the noise of state, x kð Þ, and of measurement, v kð Þ, are white and mutually uncorrelated can be diminished and replaced by the assumptions about the correlated (colored) state noise, correlated measurement Table 1.2 Flow diagram of the algorithm of digital Kalman filtration
1. Initialization
• read constant matrices: F, G, H, m0, P0, Q, R
• set initial values of the estimated state vector and covariance matrix of estimation error:
^x0ð Þ ¼ mþ 0; P0ð Þ ¼ Pþ 0
2. For each k¼ 1; 2; . . ., perform:
2.1. prediction step, consisting in the calculation of:
• system state prediction: ^xkð Þ ¼ F^x k1ð Þþ
• covariance matrix of prediction error: Pkð Þ ¼ FP k1ð ÞFþ Tþ GQGT
• system output prediction: ^y kð Þ ¼ H^xkð Þ 2.2. Estimation (filtration) step, consisting from:
• output signal measurement, y kð Þ
• calculation of residual: ~y kð Þ ¼ y kð Þ ^y kð Þ
• calculation of covariance matrix of residual: S kð Þ ¼ HPkð ÞH Tþ R
• calculation of amplification matrix: K kð Þ ¼ Pkð ÞH TS1ð Þk
• calculation of state estimation: ^xkð Þ ¼ ^xþ kð Þ þ K k ð Þ~y kð Þ
• calculation of covariance matrix of estimation (filtration) error: Pkð Þ ¼ I K kþ ð ð ÞHÞPkð Þ 3. Increment the iteration (time) counter k by 1 and repeat the procedure starting from the step 2.
noise and mutually correlated noises; such assumption require certain modifi-cation of the filter equation [7, 9–11].
• if the stochastic variables x 0ð Þ, x kð Þ and v kð Þ have Gaussian (normal) distri-bution, then the conditional function of the state probability, x kð Þ, when the measurements are given up to the current moment, k, Yk¼ y 0f ð Þ; y 1ð Þ; . . .;
y kð Þg is Gaussian (normal) and its expected value is E x k ð Þ Y k
¼ ^xkð Þþ and the covariance matrix is
E x kð Þ E x k ð Þ Y k
x kð Þ E x k ð Þ Y k
Yk
¼ Pkð Þþ
In other words, in the quoted case a Kalman filter generates a recursive con-ditional mathematical expectation, which represents an optimal estimation of the state vector in the sense of minimal possible covariance matrix of the estimation error, which reaches the Cramér-Rao lower bound [13–15].
• if the stochastic variables x 0ð Þ, x kð Þ and v kð Þ are not Gaussian, then the Kalman filter is optimal only within the class of linear filters, in the sense of minimal covariance of the estimation error within the said class of filters.
• by replacing (1.84) into (1.91) the vectorial difference equation is obtained
Pkþ1ð Þ ¼ F P kð Þ P kð ÞH T HPkð ÞH Tþ R 1
HPkð Þ
n o
FTþ GQGT ð1:100Þ which is called the Riccati equation; in the case of a time-invariant system, the solution of the Riccati equation converges asymptotically (k! 1) to a finite solution P if the model is observable within the state space [7, 9–11, 16]; the condition of the model observability implies that the information about the immeasurable system states is contained in the measured output, which ensures that the estimation error remains limited; if additionally the model is in the state space and controllable, the solution P in equilibrium or stochastic equilibrium state is also unique; the controllability condition ensures that state noise, as a random excitation signal, acts to all components of the state vector, which prevents the convergence to zero of the covariance matrix of estimation error P, i.e. the sta-tionary solution P will be positively definite matrix with all eigenvalues positive [7, 9–11, 16].
• In a stochastic stationary state the amplification matrix of Kalman filter K is constant, and the Kalman filter reduces to a Wiener optimal filter [7, 9, 10, 16].