Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2013 Article ID 240295 9 pageshttpdxdoiorg1011552013240295
Research ArticleOnline Stochastic Convergence Analysis of the Kalman Filter
Matthew B Rhudy1 and Yu Gu2
1 Department of Mechanical Engineering at Lafayette College Easton PA 18042 USA2Department of Mechanical and Aerospace Engineering Morgantown WV 26506 USA
Correspondence should be addressed to Yu Gu yugumailwvuedu
Received 15 May 2013 Revised 26 September 2013 Accepted 26 September 2013
Academic Editor Ravi Agarwal
Copyright copy 2013 M B Rhudy and Y Gu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper presentsmodifications to the stochastic stability lemmawhich is thenused to estimate the convergence rate andpersistenterror of the linear Kalman filter online without using knowledge of the true state Unlike previous uses of the stochastic stabilitylemma for stability proof this new convergence analysis technique considers time-varying parameters which can be calculatedonline in real-time tomonitor the performance of the filterThrough simulation of an example problem the newmethodwas shownto be effective in determining a bound on the estimation error that closely follows the actual estimation error Different cases ofassumed process and measurement noise covariance matrices were considered in order to study their effects on the convergenceand persistent error of the Kalman filter
1 Introduction
Since its introduction in 1960 the linear Kalman filter (LKF)[1] has been used widely in industry When the LKF isimplemented in real-time applications it is often difficultto quantify the performance of the filter without accessto some reference ldquotruthrdquo Offline simulations can providesome indication of the filter performance however accuratemathematical models are not always available For the LKFthere are two primary sources of error in the estimationinitialization error and stochastic errors due to the processand measurement noise In the early stages of the filter theinitialization error is dominant and it takes some amountof time for the estimated state to converge to the truestate from this incorrect initial state After the initial errorconvergence the errors due to the noise terms remainresulting in ldquopersistentrdquo errors Because of these types of errorthere is a need to analyze the performance of the LKF onlineby quantifying the convergence rate and persistent errorbounds of the real system Such a tool could benefit manysafety or performance critical systems such as the aircrafthealth management system Existing techniques for onlineperformance analysis of the LKF include outlier detection[2] performance reliability prediction [3] and confidence
bounds from the covariance matrix for example see [4]Confidence bounds can also be established through use ofthe Chebyshev inequality [5] although these bounds tend tobe too large for practical use [6] Some other investigationsfor confidence bounds on the Kalman filter consider thenon-Gaussian case using enhancements to the Chebyshevinequality [6] or the Kantorovich inequality [7] The workpresented herein offers a novel onlinemethod formonitoringthe performance of the LKF by providing an upper bound onthe estimation error
This work was inspired by previous investigations ofthe stability and convergence properties of Kalman filtersEarly continuous-time LKF stability work derived conditionsfor stability of the homogeneous (no noise) equations [8]and different causes of divergence [9] For discrete-timesystems first upper and lower bounds were derived for theerror covariance matrix [10] Then it was determined thatstochastic controllability and observability of the systemweresufficient conditions to prove asymptotic stability of thehomogenous equations [11 12] Later after some commentsin [13 14] corrections were provided for the calculations ofthe error covariance bounds [15] This work was expandedto handle singular state transition matrices [16] and considerconvergence properties of the algebraic Riccati equation
2 International Journal of Stochastic Analysis
[17] and parameter identification [18] Lyapunov stabilitymethods were later applied to the LKF equations as an alter-native means to demonstrate stability of the homogeneousequations [19] More recently the conditions for stability ofthe discrete-timeKalman filter for linear time-invariant (LTI)systems were evaluated with respect to perturbations in theinitial error covariance [20] This existing work provided anecessary basis for investigating the convergence and persi-stent error properties of the LKF for stochastic systems
An important and useful tool for analyzing the stochasticstability of a system is the stochastic stability lemma [2122] This lemma has been used to approach the stabilityof the extended Kalman Filter (EKF) [23] and later for ageneral class of nonlinear filters including EKF and unscentedKalman filter (UKF) [24 25] A common problem with exist-ing convergence analysis techniques for nonlinear state esti-mators is extremely loose bounds on the system and noisematrices leading to very conservative andunrealistic require-ments on the initial error and noise of the system [23] Amethod for the relaxation of these conditions for EKF wasconsidered in a related work [26] Using the stochastic sta-bility lemma these works [23 26] perform an Offline pre-diction of the stability of the state estimation This processinvolves the calculation of a convergence rate and persistenterror which establish an upper bound on the estimation error
In addition to its previous uses for nonlinear systemsthe stochastic stability lemma can also be used to establishimportant results for the LKF Since the LKF is an adaptiveprocess even for linear time-invariant (LTI) systems it beco-mes useful to analyze the convergence rate and persistenterror as a function of time Motivated by this idea thestochastic stability lemma is reconsidered here andmodifica-tions are presented within to handle the time-varying natureof the LKF Using this modified stochastic stability lemmathe convergence properties of the LKF are evaluated thusproviding a more realistic bound on the estimation errorDetermining a bound on the estimation error is useful forapplications where a reference ldquotruthrdquo value is not availablefor validationThis technique provides an upper boundon thefilter performance which can be used to represent the worstcase scenario for the LKF estimation results The purposeof this paper is to present the modified stochastic stabilitylemma develop means of calculating an online bound of theestimation error of the LKF quantify the convergence rate ofthe LKF and offer some insight into the effects of differentassumed values of the noise covariance matrices This workalso provides a foundation for future nonlinear stochasticstate estimation convergence analysis
The rest of this paper is organized as follows In Section 2the LKF equations are defined In Section 3 the derivationof the modified stochastic stability lemma is presentedSection 4 utilizes the modified stochastic stability lemmafrom Section 3 to analyze the convergence of the LKFSection 5 presents the convergence analysis of an exampleLKF problem Finally the conclusions are given in Section 4
Throughout this paper sdot denotes the Euclidean normof vectors 119864[119909] is the expected value of 119909 119864[119909 | 119910] is theexpected value of 119909 conditioned on 119910 I denotes an identitymatrix of appropriate dimensions 120582min and 120582max denote the
minimum and maximum eigenvalues of a matrix the matrixinequality A gt B implies that A minus B is positive definite andsimilarly A ge B implies that A minus B is positive semidefinite
2 Linear Kalman Filter Equations
Consider a discrete-time linear stochastic state space systemof the following form
where x is the state vector y is the output vector F and Hare system matrices and wkminus1 and vk are the process andmeasurement noise vectors that are zero-mean white uncor-related and have assumed covariance matrices Q119896minus1 and R119896respectively For this system the LKF can be implementedusing the following standard set of equations [27]
where P is the error covariance matrix and K is the Kalmangainmatrix In order to analyze the convergence of the LKF itis important to understand its error dynamics Defining theerror in the a posteriori state estimate as x119896 = x119896 minus x119896 andsubstituting in for the estimated state from (2) gives
x119896 = x119896 minus F119896minus1x119896minus1 minus K119896 (y119896 minusH119896F119896minus1x119896minus1) (3)
Inserting the output vector definition from (1) gives
x119896 = x119896 minus F119896minus1x119896minus1 minus K119896 (H119896x119896 + k119896 minusH119896F119896minus1x119896minus1) (4)
Collecting terms reduces the error dynamics to
x119896 = (I minus K119896H119896) (x119896 minus F119896minus1x119896minus1) minus K119896k119896 (5)
Substituting the definition of the state vector from (1) leads to
x119896 = (I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1 minus F119896minus1x119896minus1) minus K119896k119896 (6)
Recognizing the state error at 119896 minus 1 reduces the estimationerror dynamics to the following form
x119896 = (I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896k119896 (7)
Remark 1 This form of the estimation error is one possibilityof representing the error dynamics An autoregressive formof quantifying the estimation error is discussed in [28 29]Another interesting possibility for estimation error quantifi-cation is presented in [6]
International Journal of Stochastic Analysis 3
3 Modified Stochastic Stability Lemma
The basis of this convergence analysis is the stochastic stabi-lity lemma [21 22] which is given as follows
Lemma 2 (stochastic stability lemma) If there exists a sto-chastic process V(120577119896) with the following properties
V1 V2 120583 gt 0 0 lt 120572 le 1
V1100381710038171003817100381712057711989610038171003817100381710038172le 119881 (120577119896) le V2
119864 [119881 (120577119896) | 120577119896minus1] minus 119881 (120577119896minus1) le 120583 minus 120572V (120577kminus1)
(8)
then the random variable 120577119896 is exponentially bounded in meansquare with probability one as in
119864 [100381710038171003817100381712057711989610038171003817100381710038172] le
V2V1119864 [1003817100381710038171003817120577010038171003817100381710038172] (1 minus 120572)
119896+120583
V1
119896minus1
sum119894=0
(1 minus 120572)119894 (9)
where 120572 is the convergence rate and V1 V2 and 120583 are constantsThe proof for this lemma is provided in [22] This lemma hasbeen used to determine stability properties of the EKF in [23] Amodified version of this lemma is presented here which includestime-varying parameters
Lemma 3 (modified stochastic stability lemma) Assume thatthere is a stochastic process V(120577119896) and parameters bk v0 120583k gt 0and 0 lt 120572k le 1 such that the following inequalities are satisfiedfor all k
119881 (1205770) le V01003817100381710038171003817120577010038171003817100381710038172 (10)
119864 [119881 (120577119896) | 120577119896minus1] minus 119881 (120577119896minus1) le 120583119896minus1 minus 120572119896minus1119881 (120577119896minus1) (12)
then the random variable 120577119896 is bounded in mean square withprobability one by the following inequality
119864 [100381710038171003817100381712057711989610038171003817100381710038172] le
le 119864 [120583119896minus1 + (1 minus 120572119896minus1) 119881 (120577119896minus1) | 120577119896minus2] (16)
which can be simplified using (15)
119864 [119881 (120577119896) | 120577119896minus2] le 120583119896minus1 + (1 minus 120572119896minus1) 119864 [119881 (120577119896minus1) | 120577119896minus2]
(17)
This method can be applied recursively for 119896 minus 3 119896 minus 4 0thus giving
119864 [119881 (120577119896) | 1205770] le 119881 (1205770)
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(18)
Taking the expectation of this inequality and applying (14)gives
119864 [119881 (120577119896)] le 119864 [119881 (1205770)]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(19)
Taking the expectation of (10) gives
119864 [119881 (1205770)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172] (20)
which can be inserted into (19) thus giving
119864 [119881 (120577119896)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(21)
Similarly for the lower bound the expected value is taken for(11) and the result is used to obtain
119887119896119864 [100381710038171003817100381712057711989610038171003817100381710038172] le 119864 [119881 (120577119896)]
119864 [119881 (120577119896)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(22)
which is rearranged to obtain the final result in (13)
Remark 4 It is important to note the differences betweenLemmas 2 and 3 In Lemma 3 the terms 120583 and 120572 are time-varying quantities whereas for Lemma 2 these terms wereboth considered as constants with respect to the discrete-time 119896 Additionally the bounds for the stochastic processare treated differently The upper bound of the process isconsidered only for the initial time step while the lower
4 International Journal of Stochastic Analysis
bound is considered as a time-varying quantity The useful-ness of Lemma 3 is not for stability analysis but for the onlinemonitoring of convergence and estimation error boundsTheconsideration of time-varying parameters is the key to thefollowing online convergence and error analysis
4 Online Convergence and Error Analysis
This section considers a new approach to analyzing theconvergence and estimation error of the LKF in real-timeUsing Lemma 3 the main result of this paper can be stated
Theorem 5 (Kalman filter convergence theorem) Consider alinear stochastic system using the LKF equations as describedin Section 2 Let the following assumptions hold
(1) The systemmatrixF119896 is nonsingular (invertible) for allk
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(29)
119887119896 = 120582min (Pminus1
119896) (30)
Proof The proof of this theorem is detailed in the followingsections
Remarks (1)The bound in (23) only matters for the assumedinitial covariance matrix Since this has a known value theconstant V0 should be selected as the minimum eigenvalue ofthe inverse of the assumed initial covariance matrix and thisbound will be automatically satisfied
(2) It is worth noting in (24) that if the error covari-ance approaches infinity (divergence) then the term 119887119896 willapproach zero which would lead to an infinite bound on theestimation error thus indicating divergence of the filter asexpected For a stable system however the error covariancematrix has an upper bound which can be determined fromthe stochastic controllability and observability properties ofthe system [11 15]
(3)Theparameters 120572 and 120583 are both functions of the samematrix where 120572 is the minimum eigenvalue and 120583 is the traceof the matrix Since the eigenvalues of this matrix lie between0 and 1 (the a priori covariance is always greater than or equalto the process noise covariance matrix) and recalling that thetrace of a matrix is equal to the sum of its eigenvalues [31]the parameter 120583 will satisfy 0 lt 120572119896minus1 lt 120583119896minus1 lt 119899 for all119896 where 119899 is the number of states in the filter From here itis interesting to note that increasing the parameter 120572 whichcorresponds to the convergence of the stochastic process willin turn also increase the parameter 120583 which corresponds tothe persistent error bound due to noise This introduces atradeoff in convergence and persistent error which can betuned through the selection of the process and measurementnoise covariance matrices
(4) Using Lemma 3 for analysis of the LKF convergenceleads to three important time-varying parameters 120572119896 120583119896and 119887119896 The parameter 120572119896 represents the convergence of thestochastic process as defined in the following section by(31) while the parameter 119887119896 represents the convergence ofthe error covariance The parameter 120583119896 corresponds to thepersistent error bound on the filter due to the process andmeasurement noise That is in (27) it is shown that theinitial error term will vanish as 119896 increases thus leaving theterm containing 120583119896 which contains the persistent responseThis makes sense because as a LKF progresses in timeeventually the performance will converge within a regiondetermined from the process and measurement noise sincethese phenomena do not disappear with time Together thesethree parameters determine a bound on the convergenceand persistent error of the filter using (27) Due to thetime-varying nature of these parameters the bound must bedetermined online and therefore cannot provide an Offlineprediction of the filter convergence as in [23 26]
The proof of Theorem 5 is provided next
41 Defining and Decomposing the Estimation Error AnalysisAs recommended in other works for example [19 23]a candidate Lyapunov function is selected to define thestochastic process using a quadratic form of the estimationerror and inverse error covariance matrix as in
Note that this function is used in the context of Lemma 3 notusing traditional Lyapunov stability theorems therefore it isonly being used as a tool for analyzing the convergence notto prove the stability of the filter Inserting the error dynamicsfrom (7) into this function gives
119881 (x119896) = [(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]119879
times Pminus1119896[(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]
(32)
Taking the conditional expectation with respect to x119896minus1 andusing the assumption that the process and measurementnoise are uncorrelated give
Now the problem of analyzing the LKF estimation error hasbeen divided into three parts the homogeneous problem in(34) the process noise problem in (35) and themeasurementnoise problem in (36) The homogeneous problem considersthe deterministic part of the filter that is no noise The pro-cess and measurement noise problems consider the effects ofthe stochastic uncertainty in the prediction andmeasurementequations respectively Each of these three parts is consideredseparately in the following sections
42 The Homogeneous Problem The homogeneous part ofthe problem is defined by (34) This part of the problem isrelated to the convergence rate of the filter For this part ofthe analysis a bound is desired in the form
Γ119909
119896le (1 minus 120572119896minus1) 119881 (x119896minus1) (37)
This inequality is desired as it is the assumption given by (12)ignoring for now the noise terms and assuming that 120583119896 = 0for all 119896 Substituting in for (31) and (34) gives
x119879119896minus1
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1x119896minus1
le (1 minus 120572119896minus1) x119879
119896minus1Pminus1119896minus1
x119896minus1(38)
This scalar inequality is equivalent to the matrix inequality
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1
(39)
The following relationship can be derived from the LKFequations in (2)
I minus K119896H119896 = P119896Pminus1
119896|119896minus1 (40)
Substituting (40) into (39) gives
F119879119896minus1
Pminus1119896|119896minus1
P119896Pminus1
119896|119896minus1F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1 (41)
Taking the inverse of this inequality gives
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1ge (1 minus 120572119896minus1)
minus1P119896minus1 (42)
Note that this operation requires that the system matrix Fbe nonsingular for all 119896 (assumption (1)) The covariancematrices are invertible because they are positive definite bydefinition Starting from the covariance prediction equationin (2) and rearranging give
P119896minus1 = Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1 (43)
Substituting this equation into the matrix inequality yields
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1
ge (1 minus 120572119896minus1)minus1Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1
(44)
Now the system matrix can be removed from the inequality
P119896|119896minus1Pminus1
119896P119896|119896minus1 ge (1 minus 120572119896minus1)
120572119896minus1I le (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(47)
Therefore the time-varying parameter 120572 can be determinedas the minimum eigenvalue of the matrix as in (28) Fromthe covariance prediction equation in (2) it is clear that the apriori covariance is greater than the process noise covariancematrix therefore 120572 is always between 0 and 1 Note thatincreasing Q will increase 120572 Alternatively increasing R willdecrease120572 If the parameter120572 is selected as in (28) the desiredinequality (37) is satisfied thus satisfying the homogeneouspart of the problem Next the process noise is considered
43 The Process Noise Problem For the process noise prob-lem the quantity of interest is given by (35) Since this is a
6 International Journal of Stochastic Analysis
scalar equation the trace can be taken without changing thevalue
Γ119908
119896= Tr Γ119908
119896 = Tr 119864 [w119879
119896minus1(I minus K119896H119896)
119879Pminus1119896
times (I minus K119896H119896)w119896minus1] (48)
Using the trace property ofmultiplication reordering [31] andremoving the deterministic terms from the expectation yield
Γ119908
119896= Tr (I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) 119864 [w119896minus1w
119879
119896minus1] (49)
Using (40) simplifies the equation to
Γ119908
119896= Tr Pminus1
119896|119896minus1P119896Pminus1
119896|119896minus1119864 [w119896minus1w
119879
119896minus1] (50)
Inserting the covariance update equation from (2) gives
Since the process noise covariance matrix can be chosenfreely for the LKF it is assumed that the assumed processnoise covariance matrix is greater than the actual covarianceof the process noise as in (25)This bound ismotivated by theidea that it is better to assume greater rather than less noisethan there actually is in the system This leads to the boundon the process noise term
While increasing Q was shown to increase the convergencerate in the previous section it is clear here that this increasein convergence comes at the expense of a larger bound on theprocess noise term This selection of Q becomes a tradeoffbetween the convergence and the accuracy of the estimatethat is assuming an unnecessarily large Q will lead to fasterconvergence but larger persistent errors of the filter dueto process noise Next the measurement noise problem isconsidered
44 The Measurement Noise Problem For the measurementnoise problem the quantity of interest is given by (36) Sincethis is a scalar equation the trace can be taken withoutchanging the value
Similarly as for the process noise the assumed measurementnoise covariance matrix is selected as an upper bound onthe actual measurement noise covariance as in (26) whichdetermines the bound for the measurement noise term
From here it is shown that increasing the assumed mea-surement noise covariance matrix R will in fact lead to asmaller bound on the estimation error due to measurementnoise Now that each part of the problem has been consideredseparately the results are combined and Lemma 3 is applied
45 Final Result from theModified Stochastic Stability LemmaCombining the results from the previous sections gives thefollowing inequality
119864 [119881 (x119896) | x119896minus1]
le (1 minus 120572119896minus1) 119881 (x119896minus1)
+I minus (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
P119896|119896minus1 (65)
Then combining the terms gives (29)Thus the inequality in(12) has been satisfied
In order to apply Lemma 3 the inequalities (10) and (11)also need to be satisfiedThese inequalities are guaranteed bythe assumptions (23) and (24) in Theorem 5 Thus the nec-essary conditions for Lemma 3 have been satisfied thereforethe estimation error of the LKF is bounded in mean squarewith probability one and the bound is given by (27) Thiscompletes the proof of Theorem 5 In the following sectiona LKF example is provided to illustrate the usefulness ofTheorem 5 for LKF convergence analysis
5 An Illustrative Example
To demonstrate the convergence analysis method fromSection 4 a simple LKF example is presented This exampleproblem was adapted from Example 51 in [27] to includeprocess noise The system equations are defined in the formof (1) with system matrices defined by
F119896 = F =[[[[
[
1 1198791198792
2
0 1 119879
0 0 1
]]]]
]
H119896 = H = [1 0 0]
(66)
and the true process and measurement noise covariancematrices are given by
119864 [w119896w119879
119896] = 10
minus8I
119864 [v119896v119879
119896] = 10
minus8
(67)
where 119879 is the sampling time which for this example isconsidered to be 002 The initial conditions are assumed tobe
while the true initial state for the system is actually
x0 = [1 05 02]119879 (69)
Note that this considers a case of reasonably large initializa-tion error
In order to apply Theorem 5 certain assumptions needto be satisfied From the definition of F it is clear that thismatrix is invertible Four different cases of assumed processand measurement covariance matrices were considered assummarized in Table 1
It is shown in Table 1 that (25) and (26) are satisfiedNote that these cases vary the assumed noise propertiesnot the actual noise The true noise covariance matrices aregiven by (67) for all cases The value for the initial Lyapunovfunction upper bound V0 is calculated from the assumedinitial covariance matrix with (23) Additionally the valuesfor the time-varying convergence rate 120572119896 noise parameter120583119896 and Lyapunov function bound 119887119896 are defined using (28)(29) and (24) respectivelyThese values are calculated onlineat each time step of the filter Using these equations theconvergence properties can be calculated online with (27)
For the given example the presented convergence analy-sis technique is applied and the results are given as followsSince the initial covariance is the identity matrix V0 = 1 Thetime-varying convergence and error parameters are shown inFigure 1 for each of the considered cases of assumed processand measurement noise covariance
The parameter 120572119896 represents the convergence rate of thestochastic process 120583119896 represents the persistent error of thestochastic process and 119887119896 represents the convergence of theerror covariance From these time-varying parameters thebound on the expected value of the norm of the estimationerror squared can be determined from (27) This bound isverified with respect to the actual estimation error which wasdetermined from simulation as shown in Figure 2
It is shown in Figure 2 that the estimation error doesnot exceed the theoretical bounds The online bounds arerelatively close to the estimation error thus providing areasonable guide to the convergence and steady-state errorof the filter performance This is useful because a referencetruth is not available to evaluate the performance of a filter inmost practical applicationsThis method provides a means ofcalculating an upper bound on the performance of the filterusing only known values from the filtering process
There are some interesting observations to make fromFigures 1 and 2 regarding the different noise covarianceassumptions Case 1 which represents perfect knowledge ofthe simulated noise properties offers a very good approxima-tion to the convergence and persistent error of the example
8 International Journal of Stochastic Analysis
0
001
002
003
0
1
2
02468
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 103
Discrete-time k100 101 102 103
Discrete-time k100 101 102 103
times105
120572k
120583k
b k
Figure 1 LKF example time-varying convergence and error param-eters
Simulation
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 10310minus10
10minus8
10minus6
10minus4
10minus2
100
102
xer
r2
Figure 2 LKF example estimation error with bounds
filter Increasing the assumption on the process noise (Case2) leads to an increase in 120572119896 but also an increase in 120583119896 aspredicted However this increase in assumed process noisesignificantly increased the parameter 119887119896 thus leading to aslowly converging loose bound on the estimation error Asimilar performance bound was seen for Case 4 due to thedominant effect of the parameter 119887119896 however the parameters120572119896 and 120583119896 were similar to Case 1 This makes sense becausethe ratio between the assumed Q and R remained the samefor Cases 1 and 4 For Case 3 increasing the assumedmeasurement noise decreased the parameters 120572119896 and 120583119896as expected but the parameter 119887119896 also decreased furtherdecreasing the convergence of the estimation error This lead
OnlineOffline
Simulation
10minus10
100
1010
xer
r2
Discrete-time k100 101 102 103
Figure 3 LKF example online versus offline estimation errorbounds
to a slower converging bound but a tighter bound on thepersistent error This demonstrates a tradeoff in the selectionof the measurement noise covariance which could be usedfor filter tuning depending on the application and desiredconvergence properties
The predicted estimation error bound from Offline anal-ysis [23 26] using Lemma 2 is also provided as a referenceto demonstrate the effectiveness of using this new onlinemethod To relate the time-varying parameters to previousOffline work using Lemma 2 [23 26] the following relationsare used
120572 = min (120572119896)
120583 = max (120583119896)
V1 = min (119887119896)
(70)
The bound from Case 1 is used for this comparison as shownin Figure 3
While the Offline estimation error bound is valid it isextremely loose and does not provide a realistic portrayalof the convergence of the estimation error This shows thatthe presented online method is useful for more closelydetermining the convergence and persistent error of the LKFbut is limited in that it cannot predict these bounds prior tothe filtering process and it cannot be used for Offline stabilityanalysis
6 Conclusions
This paper presented a modified stochastic stability lemmaand a Kalman filter convergence theorem which are newtools that can be used to quantify the performance ofKalman filters online Through an example it was shownthat this new convergence analysis method is effective indetermining an upper bound on the performance of theLKF Also useful information about the convergence of theparticular LKF algorithm can be calculated This analysisis applied during the filtering process thus providing thecapability for real-time convergence and performance moni-toring Different cases of noise covariance assumptions wereconsidered showing that increasing the assumed process
International Journal of Stochastic Analysis 9
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
[17] and parameter identification [18] Lyapunov stabilitymethods were later applied to the LKF equations as an alter-native means to demonstrate stability of the homogeneousequations [19] More recently the conditions for stability ofthe discrete-timeKalman filter for linear time-invariant (LTI)systems were evaluated with respect to perturbations in theinitial error covariance [20] This existing work provided anecessary basis for investigating the convergence and persi-stent error properties of the LKF for stochastic systems
An important and useful tool for analyzing the stochasticstability of a system is the stochastic stability lemma [2122] This lemma has been used to approach the stabilityof the extended Kalman Filter (EKF) [23] and later for ageneral class of nonlinear filters including EKF and unscentedKalman filter (UKF) [24 25] A common problem with exist-ing convergence analysis techniques for nonlinear state esti-mators is extremely loose bounds on the system and noisematrices leading to very conservative andunrealistic require-ments on the initial error and noise of the system [23] Amethod for the relaxation of these conditions for EKF wasconsidered in a related work [26] Using the stochastic sta-bility lemma these works [23 26] perform an Offline pre-diction of the stability of the state estimation This processinvolves the calculation of a convergence rate and persistenterror which establish an upper bound on the estimation error
In addition to its previous uses for nonlinear systemsthe stochastic stability lemma can also be used to establishimportant results for the LKF Since the LKF is an adaptiveprocess even for linear time-invariant (LTI) systems it beco-mes useful to analyze the convergence rate and persistenterror as a function of time Motivated by this idea thestochastic stability lemma is reconsidered here andmodifica-tions are presented within to handle the time-varying natureof the LKF Using this modified stochastic stability lemmathe convergence properties of the LKF are evaluated thusproviding a more realistic bound on the estimation errorDetermining a bound on the estimation error is useful forapplications where a reference ldquotruthrdquo value is not availablefor validationThis technique provides an upper boundon thefilter performance which can be used to represent the worstcase scenario for the LKF estimation results The purposeof this paper is to present the modified stochastic stabilitylemma develop means of calculating an online bound of theestimation error of the LKF quantify the convergence rate ofthe LKF and offer some insight into the effects of differentassumed values of the noise covariance matrices This workalso provides a foundation for future nonlinear stochasticstate estimation convergence analysis
The rest of this paper is organized as follows In Section 2the LKF equations are defined In Section 3 the derivationof the modified stochastic stability lemma is presentedSection 4 utilizes the modified stochastic stability lemmafrom Section 3 to analyze the convergence of the LKFSection 5 presents the convergence analysis of an exampleLKF problem Finally the conclusions are given in Section 4
Throughout this paper sdot denotes the Euclidean normof vectors 119864[119909] is the expected value of 119909 119864[119909 | 119910] is theexpected value of 119909 conditioned on 119910 I denotes an identitymatrix of appropriate dimensions 120582min and 120582max denote the
minimum and maximum eigenvalues of a matrix the matrixinequality A gt B implies that A minus B is positive definite andsimilarly A ge B implies that A minus B is positive semidefinite
2 Linear Kalman Filter Equations
Consider a discrete-time linear stochastic state space systemof the following form
where x is the state vector y is the output vector F and Hare system matrices and wkminus1 and vk are the process andmeasurement noise vectors that are zero-mean white uncor-related and have assumed covariance matrices Q119896minus1 and R119896respectively For this system the LKF can be implementedusing the following standard set of equations [27]
where P is the error covariance matrix and K is the Kalmangainmatrix In order to analyze the convergence of the LKF itis important to understand its error dynamics Defining theerror in the a posteriori state estimate as x119896 = x119896 minus x119896 andsubstituting in for the estimated state from (2) gives
x119896 = x119896 minus F119896minus1x119896minus1 minus K119896 (y119896 minusH119896F119896minus1x119896minus1) (3)
Inserting the output vector definition from (1) gives
x119896 = x119896 minus F119896minus1x119896minus1 minus K119896 (H119896x119896 + k119896 minusH119896F119896minus1x119896minus1) (4)
Collecting terms reduces the error dynamics to
x119896 = (I minus K119896H119896) (x119896 minus F119896minus1x119896minus1) minus K119896k119896 (5)
Substituting the definition of the state vector from (1) leads to
x119896 = (I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1 minus F119896minus1x119896minus1) minus K119896k119896 (6)
Recognizing the state error at 119896 minus 1 reduces the estimationerror dynamics to the following form
x119896 = (I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896k119896 (7)
Remark 1 This form of the estimation error is one possibilityof representing the error dynamics An autoregressive formof quantifying the estimation error is discussed in [28 29]Another interesting possibility for estimation error quantifi-cation is presented in [6]
International Journal of Stochastic Analysis 3
3 Modified Stochastic Stability Lemma
The basis of this convergence analysis is the stochastic stabi-lity lemma [21 22] which is given as follows
Lemma 2 (stochastic stability lemma) If there exists a sto-chastic process V(120577119896) with the following properties
V1 V2 120583 gt 0 0 lt 120572 le 1
V1100381710038171003817100381712057711989610038171003817100381710038172le 119881 (120577119896) le V2
119864 [119881 (120577119896) | 120577119896minus1] minus 119881 (120577119896minus1) le 120583 minus 120572V (120577kminus1)
(8)
then the random variable 120577119896 is exponentially bounded in meansquare with probability one as in
119864 [100381710038171003817100381712057711989610038171003817100381710038172] le
V2V1119864 [1003817100381710038171003817120577010038171003817100381710038172] (1 minus 120572)
119896+120583
V1
119896minus1
sum119894=0
(1 minus 120572)119894 (9)
where 120572 is the convergence rate and V1 V2 and 120583 are constantsThe proof for this lemma is provided in [22] This lemma hasbeen used to determine stability properties of the EKF in [23] Amodified version of this lemma is presented here which includestime-varying parameters
Lemma 3 (modified stochastic stability lemma) Assume thatthere is a stochastic process V(120577119896) and parameters bk v0 120583k gt 0and 0 lt 120572k le 1 such that the following inequalities are satisfiedfor all k
119881 (1205770) le V01003817100381710038171003817120577010038171003817100381710038172 (10)
119864 [119881 (120577119896) | 120577119896minus1] minus 119881 (120577119896minus1) le 120583119896minus1 minus 120572119896minus1119881 (120577119896minus1) (12)
then the random variable 120577119896 is bounded in mean square withprobability one by the following inequality
119864 [100381710038171003817100381712057711989610038171003817100381710038172] le
le 119864 [120583119896minus1 + (1 minus 120572119896minus1) 119881 (120577119896minus1) | 120577119896minus2] (16)
which can be simplified using (15)
119864 [119881 (120577119896) | 120577119896minus2] le 120583119896minus1 + (1 minus 120572119896minus1) 119864 [119881 (120577119896minus1) | 120577119896minus2]
(17)
This method can be applied recursively for 119896 minus 3 119896 minus 4 0thus giving
119864 [119881 (120577119896) | 1205770] le 119881 (1205770)
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(18)
Taking the expectation of this inequality and applying (14)gives
119864 [119881 (120577119896)] le 119864 [119881 (1205770)]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(19)
Taking the expectation of (10) gives
119864 [119881 (1205770)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172] (20)
which can be inserted into (19) thus giving
119864 [119881 (120577119896)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(21)
Similarly for the lower bound the expected value is taken for(11) and the result is used to obtain
119887119896119864 [100381710038171003817100381712057711989610038171003817100381710038172] le 119864 [119881 (120577119896)]
119864 [119881 (120577119896)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(22)
which is rearranged to obtain the final result in (13)
Remark 4 It is important to note the differences betweenLemmas 2 and 3 In Lemma 3 the terms 120583 and 120572 are time-varying quantities whereas for Lemma 2 these terms wereboth considered as constants with respect to the discrete-time 119896 Additionally the bounds for the stochastic processare treated differently The upper bound of the process isconsidered only for the initial time step while the lower
4 International Journal of Stochastic Analysis
bound is considered as a time-varying quantity The useful-ness of Lemma 3 is not for stability analysis but for the onlinemonitoring of convergence and estimation error boundsTheconsideration of time-varying parameters is the key to thefollowing online convergence and error analysis
4 Online Convergence and Error Analysis
This section considers a new approach to analyzing theconvergence and estimation error of the LKF in real-timeUsing Lemma 3 the main result of this paper can be stated
Theorem 5 (Kalman filter convergence theorem) Consider alinear stochastic system using the LKF equations as describedin Section 2 Let the following assumptions hold
(1) The systemmatrixF119896 is nonsingular (invertible) for allk
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(29)
119887119896 = 120582min (Pminus1
119896) (30)
Proof The proof of this theorem is detailed in the followingsections
Remarks (1)The bound in (23) only matters for the assumedinitial covariance matrix Since this has a known value theconstant V0 should be selected as the minimum eigenvalue ofthe inverse of the assumed initial covariance matrix and thisbound will be automatically satisfied
(2) It is worth noting in (24) that if the error covari-ance approaches infinity (divergence) then the term 119887119896 willapproach zero which would lead to an infinite bound on theestimation error thus indicating divergence of the filter asexpected For a stable system however the error covariancematrix has an upper bound which can be determined fromthe stochastic controllability and observability properties ofthe system [11 15]
(3)Theparameters 120572 and 120583 are both functions of the samematrix where 120572 is the minimum eigenvalue and 120583 is the traceof the matrix Since the eigenvalues of this matrix lie between0 and 1 (the a priori covariance is always greater than or equalto the process noise covariance matrix) and recalling that thetrace of a matrix is equal to the sum of its eigenvalues [31]the parameter 120583 will satisfy 0 lt 120572119896minus1 lt 120583119896minus1 lt 119899 for all119896 where 119899 is the number of states in the filter From here itis interesting to note that increasing the parameter 120572 whichcorresponds to the convergence of the stochastic process willin turn also increase the parameter 120583 which corresponds tothe persistent error bound due to noise This introduces atradeoff in convergence and persistent error which can betuned through the selection of the process and measurementnoise covariance matrices
(4) Using Lemma 3 for analysis of the LKF convergenceleads to three important time-varying parameters 120572119896 120583119896and 119887119896 The parameter 120572119896 represents the convergence of thestochastic process as defined in the following section by(31) while the parameter 119887119896 represents the convergence ofthe error covariance The parameter 120583119896 corresponds to thepersistent error bound on the filter due to the process andmeasurement noise That is in (27) it is shown that theinitial error term will vanish as 119896 increases thus leaving theterm containing 120583119896 which contains the persistent responseThis makes sense because as a LKF progresses in timeeventually the performance will converge within a regiondetermined from the process and measurement noise sincethese phenomena do not disappear with time Together thesethree parameters determine a bound on the convergenceand persistent error of the filter using (27) Due to thetime-varying nature of these parameters the bound must bedetermined online and therefore cannot provide an Offlineprediction of the filter convergence as in [23 26]
The proof of Theorem 5 is provided next
41 Defining and Decomposing the Estimation Error AnalysisAs recommended in other works for example [19 23]a candidate Lyapunov function is selected to define thestochastic process using a quadratic form of the estimationerror and inverse error covariance matrix as in
Note that this function is used in the context of Lemma 3 notusing traditional Lyapunov stability theorems therefore it isonly being used as a tool for analyzing the convergence notto prove the stability of the filter Inserting the error dynamicsfrom (7) into this function gives
119881 (x119896) = [(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]119879
times Pminus1119896[(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]
(32)
Taking the conditional expectation with respect to x119896minus1 andusing the assumption that the process and measurementnoise are uncorrelated give
Now the problem of analyzing the LKF estimation error hasbeen divided into three parts the homogeneous problem in(34) the process noise problem in (35) and themeasurementnoise problem in (36) The homogeneous problem considersthe deterministic part of the filter that is no noise The pro-cess and measurement noise problems consider the effects ofthe stochastic uncertainty in the prediction andmeasurementequations respectively Each of these three parts is consideredseparately in the following sections
42 The Homogeneous Problem The homogeneous part ofthe problem is defined by (34) This part of the problem isrelated to the convergence rate of the filter For this part ofthe analysis a bound is desired in the form
Γ119909
119896le (1 minus 120572119896minus1) 119881 (x119896minus1) (37)
This inequality is desired as it is the assumption given by (12)ignoring for now the noise terms and assuming that 120583119896 = 0for all 119896 Substituting in for (31) and (34) gives
x119879119896minus1
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1x119896minus1
le (1 minus 120572119896minus1) x119879
119896minus1Pminus1119896minus1
x119896minus1(38)
This scalar inequality is equivalent to the matrix inequality
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1
(39)
The following relationship can be derived from the LKFequations in (2)
I minus K119896H119896 = P119896Pminus1
119896|119896minus1 (40)
Substituting (40) into (39) gives
F119879119896minus1
Pminus1119896|119896minus1
P119896Pminus1
119896|119896minus1F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1 (41)
Taking the inverse of this inequality gives
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1ge (1 minus 120572119896minus1)
minus1P119896minus1 (42)
Note that this operation requires that the system matrix Fbe nonsingular for all 119896 (assumption (1)) The covariancematrices are invertible because they are positive definite bydefinition Starting from the covariance prediction equationin (2) and rearranging give
P119896minus1 = Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1 (43)
Substituting this equation into the matrix inequality yields
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1
ge (1 minus 120572119896minus1)minus1Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1
(44)
Now the system matrix can be removed from the inequality
P119896|119896minus1Pminus1
119896P119896|119896minus1 ge (1 minus 120572119896minus1)
120572119896minus1I le (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(47)
Therefore the time-varying parameter 120572 can be determinedas the minimum eigenvalue of the matrix as in (28) Fromthe covariance prediction equation in (2) it is clear that the apriori covariance is greater than the process noise covariancematrix therefore 120572 is always between 0 and 1 Note thatincreasing Q will increase 120572 Alternatively increasing R willdecrease120572 If the parameter120572 is selected as in (28) the desiredinequality (37) is satisfied thus satisfying the homogeneouspart of the problem Next the process noise is considered
43 The Process Noise Problem For the process noise prob-lem the quantity of interest is given by (35) Since this is a
6 International Journal of Stochastic Analysis
scalar equation the trace can be taken without changing thevalue
Γ119908
119896= Tr Γ119908
119896 = Tr 119864 [w119879
119896minus1(I minus K119896H119896)
119879Pminus1119896
times (I minus K119896H119896)w119896minus1] (48)
Using the trace property ofmultiplication reordering [31] andremoving the deterministic terms from the expectation yield
Γ119908
119896= Tr (I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) 119864 [w119896minus1w
119879
119896minus1] (49)
Using (40) simplifies the equation to
Γ119908
119896= Tr Pminus1
119896|119896minus1P119896Pminus1
119896|119896minus1119864 [w119896minus1w
119879
119896minus1] (50)
Inserting the covariance update equation from (2) gives
Since the process noise covariance matrix can be chosenfreely for the LKF it is assumed that the assumed processnoise covariance matrix is greater than the actual covarianceof the process noise as in (25)This bound ismotivated by theidea that it is better to assume greater rather than less noisethan there actually is in the system This leads to the boundon the process noise term
While increasing Q was shown to increase the convergencerate in the previous section it is clear here that this increasein convergence comes at the expense of a larger bound on theprocess noise term This selection of Q becomes a tradeoffbetween the convergence and the accuracy of the estimatethat is assuming an unnecessarily large Q will lead to fasterconvergence but larger persistent errors of the filter dueto process noise Next the measurement noise problem isconsidered
44 The Measurement Noise Problem For the measurementnoise problem the quantity of interest is given by (36) Sincethis is a scalar equation the trace can be taken withoutchanging the value
Similarly as for the process noise the assumed measurementnoise covariance matrix is selected as an upper bound onthe actual measurement noise covariance as in (26) whichdetermines the bound for the measurement noise term
From here it is shown that increasing the assumed mea-surement noise covariance matrix R will in fact lead to asmaller bound on the estimation error due to measurementnoise Now that each part of the problem has been consideredseparately the results are combined and Lemma 3 is applied
45 Final Result from theModified Stochastic Stability LemmaCombining the results from the previous sections gives thefollowing inequality
119864 [119881 (x119896) | x119896minus1]
le (1 minus 120572119896minus1) 119881 (x119896minus1)
+I minus (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
P119896|119896minus1 (65)
Then combining the terms gives (29)Thus the inequality in(12) has been satisfied
In order to apply Lemma 3 the inequalities (10) and (11)also need to be satisfiedThese inequalities are guaranteed bythe assumptions (23) and (24) in Theorem 5 Thus the nec-essary conditions for Lemma 3 have been satisfied thereforethe estimation error of the LKF is bounded in mean squarewith probability one and the bound is given by (27) Thiscompletes the proof of Theorem 5 In the following sectiona LKF example is provided to illustrate the usefulness ofTheorem 5 for LKF convergence analysis
5 An Illustrative Example
To demonstrate the convergence analysis method fromSection 4 a simple LKF example is presented This exampleproblem was adapted from Example 51 in [27] to includeprocess noise The system equations are defined in the formof (1) with system matrices defined by
F119896 = F =[[[[
[
1 1198791198792
2
0 1 119879
0 0 1
]]]]
]
H119896 = H = [1 0 0]
(66)
and the true process and measurement noise covariancematrices are given by
119864 [w119896w119879
119896] = 10
minus8I
119864 [v119896v119879
119896] = 10
minus8
(67)
where 119879 is the sampling time which for this example isconsidered to be 002 The initial conditions are assumed tobe
while the true initial state for the system is actually
x0 = [1 05 02]119879 (69)
Note that this considers a case of reasonably large initializa-tion error
In order to apply Theorem 5 certain assumptions needto be satisfied From the definition of F it is clear that thismatrix is invertible Four different cases of assumed processand measurement covariance matrices were considered assummarized in Table 1
It is shown in Table 1 that (25) and (26) are satisfiedNote that these cases vary the assumed noise propertiesnot the actual noise The true noise covariance matrices aregiven by (67) for all cases The value for the initial Lyapunovfunction upper bound V0 is calculated from the assumedinitial covariance matrix with (23) Additionally the valuesfor the time-varying convergence rate 120572119896 noise parameter120583119896 and Lyapunov function bound 119887119896 are defined using (28)(29) and (24) respectivelyThese values are calculated onlineat each time step of the filter Using these equations theconvergence properties can be calculated online with (27)
For the given example the presented convergence analy-sis technique is applied and the results are given as followsSince the initial covariance is the identity matrix V0 = 1 Thetime-varying convergence and error parameters are shown inFigure 1 for each of the considered cases of assumed processand measurement noise covariance
The parameter 120572119896 represents the convergence rate of thestochastic process 120583119896 represents the persistent error of thestochastic process and 119887119896 represents the convergence of theerror covariance From these time-varying parameters thebound on the expected value of the norm of the estimationerror squared can be determined from (27) This bound isverified with respect to the actual estimation error which wasdetermined from simulation as shown in Figure 2
It is shown in Figure 2 that the estimation error doesnot exceed the theoretical bounds The online bounds arerelatively close to the estimation error thus providing areasonable guide to the convergence and steady-state errorof the filter performance This is useful because a referencetruth is not available to evaluate the performance of a filter inmost practical applicationsThis method provides a means ofcalculating an upper bound on the performance of the filterusing only known values from the filtering process
There are some interesting observations to make fromFigures 1 and 2 regarding the different noise covarianceassumptions Case 1 which represents perfect knowledge ofthe simulated noise properties offers a very good approxima-tion to the convergence and persistent error of the example
8 International Journal of Stochastic Analysis
0
001
002
003
0
1
2
02468
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 103
Discrete-time k100 101 102 103
Discrete-time k100 101 102 103
times105
120572k
120583k
b k
Figure 1 LKF example time-varying convergence and error param-eters
Simulation
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 10310minus10
10minus8
10minus6
10minus4
10minus2
100
102
xer
r2
Figure 2 LKF example estimation error with bounds
filter Increasing the assumption on the process noise (Case2) leads to an increase in 120572119896 but also an increase in 120583119896 aspredicted However this increase in assumed process noisesignificantly increased the parameter 119887119896 thus leading to aslowly converging loose bound on the estimation error Asimilar performance bound was seen for Case 4 due to thedominant effect of the parameter 119887119896 however the parameters120572119896 and 120583119896 were similar to Case 1 This makes sense becausethe ratio between the assumed Q and R remained the samefor Cases 1 and 4 For Case 3 increasing the assumedmeasurement noise decreased the parameters 120572119896 and 120583119896as expected but the parameter 119887119896 also decreased furtherdecreasing the convergence of the estimation error This lead
OnlineOffline
Simulation
10minus10
100
1010
xer
r2
Discrete-time k100 101 102 103
Figure 3 LKF example online versus offline estimation errorbounds
to a slower converging bound but a tighter bound on thepersistent error This demonstrates a tradeoff in the selectionof the measurement noise covariance which could be usedfor filter tuning depending on the application and desiredconvergence properties
The predicted estimation error bound from Offline anal-ysis [23 26] using Lemma 2 is also provided as a referenceto demonstrate the effectiveness of using this new onlinemethod To relate the time-varying parameters to previousOffline work using Lemma 2 [23 26] the following relationsare used
120572 = min (120572119896)
120583 = max (120583119896)
V1 = min (119887119896)
(70)
The bound from Case 1 is used for this comparison as shownin Figure 3
While the Offline estimation error bound is valid it isextremely loose and does not provide a realistic portrayalof the convergence of the estimation error This shows thatthe presented online method is useful for more closelydetermining the convergence and persistent error of the LKFbut is limited in that it cannot predict these bounds prior tothe filtering process and it cannot be used for Offline stabilityanalysis
6 Conclusions
This paper presented a modified stochastic stability lemmaand a Kalman filter convergence theorem which are newtools that can be used to quantify the performance ofKalman filters online Through an example it was shownthat this new convergence analysis method is effective indetermining an upper bound on the performance of theLKF Also useful information about the convergence of theparticular LKF algorithm can be calculated This analysisis applied during the filtering process thus providing thecapability for real-time convergence and performance moni-toring Different cases of noise covariance assumptions wereconsidered showing that increasing the assumed process
International Journal of Stochastic Analysis 9
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
119864 [119881 (120577119896) | 120577119896minus1] minus 119881 (120577119896minus1) le 120583 minus 120572V (120577kminus1)
(8)
then the random variable 120577119896 is exponentially bounded in meansquare with probability one as in
119864 [100381710038171003817100381712057711989610038171003817100381710038172] le
V2V1119864 [1003817100381710038171003817120577010038171003817100381710038172] (1 minus 120572)
119896+120583
V1
119896minus1
sum119894=0
(1 minus 120572)119894 (9)
where 120572 is the convergence rate and V1 V2 and 120583 are constantsThe proof for this lemma is provided in [22] This lemma hasbeen used to determine stability properties of the EKF in [23] Amodified version of this lemma is presented here which includestime-varying parameters
Lemma 3 (modified stochastic stability lemma) Assume thatthere is a stochastic process V(120577119896) and parameters bk v0 120583k gt 0and 0 lt 120572k le 1 such that the following inequalities are satisfiedfor all k
119881 (1205770) le V01003817100381710038171003817120577010038171003817100381710038172 (10)
119864 [119881 (120577119896) | 120577119896minus1] minus 119881 (120577119896minus1) le 120583119896minus1 minus 120572119896minus1119881 (120577119896minus1) (12)
then the random variable 120577119896 is bounded in mean square withprobability one by the following inequality
119864 [100381710038171003817100381712057711989610038171003817100381710038172] le
le 119864 [120583119896minus1 + (1 minus 120572119896minus1) 119881 (120577119896minus1) | 120577119896minus2] (16)
which can be simplified using (15)
119864 [119881 (120577119896) | 120577119896minus2] le 120583119896minus1 + (1 minus 120572119896minus1) 119864 [119881 (120577119896minus1) | 120577119896minus2]
(17)
This method can be applied recursively for 119896 minus 3 119896 minus 4 0thus giving
119864 [119881 (120577119896) | 1205770] le 119881 (1205770)
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(18)
Taking the expectation of this inequality and applying (14)gives
119864 [119881 (120577119896)] le 119864 [119881 (1205770)]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(19)
Taking the expectation of (10) gives
119864 [119881 (1205770)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172] (20)
which can be inserted into (19) thus giving
119864 [119881 (120577119896)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(21)
Similarly for the lower bound the expected value is taken for(11) and the result is used to obtain
119887119896119864 [100381710038171003817100381712057711989610038171003817100381710038172] le 119864 [119881 (120577119896)]
119864 [119881 (120577119896)] le V0119864 [1003817100381710038171003817120577010038171003817100381710038172]
119896minus1
prod119894=0
(1 minus 120572119894)
+
119896minus1
sum119894=0
[
[
120583119896minus119894minus1
119894
prod119895=1
(1 minus 120572119896minus119895)]
]
(22)
which is rearranged to obtain the final result in (13)
Remark 4 It is important to note the differences betweenLemmas 2 and 3 In Lemma 3 the terms 120583 and 120572 are time-varying quantities whereas for Lemma 2 these terms wereboth considered as constants with respect to the discrete-time 119896 Additionally the bounds for the stochastic processare treated differently The upper bound of the process isconsidered only for the initial time step while the lower
4 International Journal of Stochastic Analysis
bound is considered as a time-varying quantity The useful-ness of Lemma 3 is not for stability analysis but for the onlinemonitoring of convergence and estimation error boundsTheconsideration of time-varying parameters is the key to thefollowing online convergence and error analysis
4 Online Convergence and Error Analysis
This section considers a new approach to analyzing theconvergence and estimation error of the LKF in real-timeUsing Lemma 3 the main result of this paper can be stated
Theorem 5 (Kalman filter convergence theorem) Consider alinear stochastic system using the LKF equations as describedin Section 2 Let the following assumptions hold
(1) The systemmatrixF119896 is nonsingular (invertible) for allk
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(29)
119887119896 = 120582min (Pminus1
119896) (30)
Proof The proof of this theorem is detailed in the followingsections
Remarks (1)The bound in (23) only matters for the assumedinitial covariance matrix Since this has a known value theconstant V0 should be selected as the minimum eigenvalue ofthe inverse of the assumed initial covariance matrix and thisbound will be automatically satisfied
(2) It is worth noting in (24) that if the error covari-ance approaches infinity (divergence) then the term 119887119896 willapproach zero which would lead to an infinite bound on theestimation error thus indicating divergence of the filter asexpected For a stable system however the error covariancematrix has an upper bound which can be determined fromthe stochastic controllability and observability properties ofthe system [11 15]
(3)Theparameters 120572 and 120583 are both functions of the samematrix where 120572 is the minimum eigenvalue and 120583 is the traceof the matrix Since the eigenvalues of this matrix lie between0 and 1 (the a priori covariance is always greater than or equalto the process noise covariance matrix) and recalling that thetrace of a matrix is equal to the sum of its eigenvalues [31]the parameter 120583 will satisfy 0 lt 120572119896minus1 lt 120583119896minus1 lt 119899 for all119896 where 119899 is the number of states in the filter From here itis interesting to note that increasing the parameter 120572 whichcorresponds to the convergence of the stochastic process willin turn also increase the parameter 120583 which corresponds tothe persistent error bound due to noise This introduces atradeoff in convergence and persistent error which can betuned through the selection of the process and measurementnoise covariance matrices
(4) Using Lemma 3 for analysis of the LKF convergenceleads to three important time-varying parameters 120572119896 120583119896and 119887119896 The parameter 120572119896 represents the convergence of thestochastic process as defined in the following section by(31) while the parameter 119887119896 represents the convergence ofthe error covariance The parameter 120583119896 corresponds to thepersistent error bound on the filter due to the process andmeasurement noise That is in (27) it is shown that theinitial error term will vanish as 119896 increases thus leaving theterm containing 120583119896 which contains the persistent responseThis makes sense because as a LKF progresses in timeeventually the performance will converge within a regiondetermined from the process and measurement noise sincethese phenomena do not disappear with time Together thesethree parameters determine a bound on the convergenceand persistent error of the filter using (27) Due to thetime-varying nature of these parameters the bound must bedetermined online and therefore cannot provide an Offlineprediction of the filter convergence as in [23 26]
The proof of Theorem 5 is provided next
41 Defining and Decomposing the Estimation Error AnalysisAs recommended in other works for example [19 23]a candidate Lyapunov function is selected to define thestochastic process using a quadratic form of the estimationerror and inverse error covariance matrix as in
Note that this function is used in the context of Lemma 3 notusing traditional Lyapunov stability theorems therefore it isonly being used as a tool for analyzing the convergence notto prove the stability of the filter Inserting the error dynamicsfrom (7) into this function gives
119881 (x119896) = [(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]119879
times Pminus1119896[(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]
(32)
Taking the conditional expectation with respect to x119896minus1 andusing the assumption that the process and measurementnoise are uncorrelated give
Now the problem of analyzing the LKF estimation error hasbeen divided into three parts the homogeneous problem in(34) the process noise problem in (35) and themeasurementnoise problem in (36) The homogeneous problem considersthe deterministic part of the filter that is no noise The pro-cess and measurement noise problems consider the effects ofthe stochastic uncertainty in the prediction andmeasurementequations respectively Each of these three parts is consideredseparately in the following sections
42 The Homogeneous Problem The homogeneous part ofthe problem is defined by (34) This part of the problem isrelated to the convergence rate of the filter For this part ofthe analysis a bound is desired in the form
Γ119909
119896le (1 minus 120572119896minus1) 119881 (x119896minus1) (37)
This inequality is desired as it is the assumption given by (12)ignoring for now the noise terms and assuming that 120583119896 = 0for all 119896 Substituting in for (31) and (34) gives
x119879119896minus1
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1x119896minus1
le (1 minus 120572119896minus1) x119879
119896minus1Pminus1119896minus1
x119896minus1(38)
This scalar inequality is equivalent to the matrix inequality
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1
(39)
The following relationship can be derived from the LKFequations in (2)
I minus K119896H119896 = P119896Pminus1
119896|119896minus1 (40)
Substituting (40) into (39) gives
F119879119896minus1
Pminus1119896|119896minus1
P119896Pminus1
119896|119896minus1F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1 (41)
Taking the inverse of this inequality gives
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1ge (1 minus 120572119896minus1)
minus1P119896minus1 (42)
Note that this operation requires that the system matrix Fbe nonsingular for all 119896 (assumption (1)) The covariancematrices are invertible because they are positive definite bydefinition Starting from the covariance prediction equationin (2) and rearranging give
P119896minus1 = Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1 (43)
Substituting this equation into the matrix inequality yields
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1
ge (1 minus 120572119896minus1)minus1Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1
(44)
Now the system matrix can be removed from the inequality
P119896|119896minus1Pminus1
119896P119896|119896minus1 ge (1 minus 120572119896minus1)
120572119896minus1I le (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(47)
Therefore the time-varying parameter 120572 can be determinedas the minimum eigenvalue of the matrix as in (28) Fromthe covariance prediction equation in (2) it is clear that the apriori covariance is greater than the process noise covariancematrix therefore 120572 is always between 0 and 1 Note thatincreasing Q will increase 120572 Alternatively increasing R willdecrease120572 If the parameter120572 is selected as in (28) the desiredinequality (37) is satisfied thus satisfying the homogeneouspart of the problem Next the process noise is considered
43 The Process Noise Problem For the process noise prob-lem the quantity of interest is given by (35) Since this is a
6 International Journal of Stochastic Analysis
scalar equation the trace can be taken without changing thevalue
Γ119908
119896= Tr Γ119908
119896 = Tr 119864 [w119879
119896minus1(I minus K119896H119896)
119879Pminus1119896
times (I minus K119896H119896)w119896minus1] (48)
Using the trace property ofmultiplication reordering [31] andremoving the deterministic terms from the expectation yield
Γ119908
119896= Tr (I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) 119864 [w119896minus1w
119879
119896minus1] (49)
Using (40) simplifies the equation to
Γ119908
119896= Tr Pminus1
119896|119896minus1P119896Pminus1
119896|119896minus1119864 [w119896minus1w
119879
119896minus1] (50)
Inserting the covariance update equation from (2) gives
Since the process noise covariance matrix can be chosenfreely for the LKF it is assumed that the assumed processnoise covariance matrix is greater than the actual covarianceof the process noise as in (25)This bound ismotivated by theidea that it is better to assume greater rather than less noisethan there actually is in the system This leads to the boundon the process noise term
While increasing Q was shown to increase the convergencerate in the previous section it is clear here that this increasein convergence comes at the expense of a larger bound on theprocess noise term This selection of Q becomes a tradeoffbetween the convergence and the accuracy of the estimatethat is assuming an unnecessarily large Q will lead to fasterconvergence but larger persistent errors of the filter dueto process noise Next the measurement noise problem isconsidered
44 The Measurement Noise Problem For the measurementnoise problem the quantity of interest is given by (36) Sincethis is a scalar equation the trace can be taken withoutchanging the value
Similarly as for the process noise the assumed measurementnoise covariance matrix is selected as an upper bound onthe actual measurement noise covariance as in (26) whichdetermines the bound for the measurement noise term
From here it is shown that increasing the assumed mea-surement noise covariance matrix R will in fact lead to asmaller bound on the estimation error due to measurementnoise Now that each part of the problem has been consideredseparately the results are combined and Lemma 3 is applied
45 Final Result from theModified Stochastic Stability LemmaCombining the results from the previous sections gives thefollowing inequality
119864 [119881 (x119896) | x119896minus1]
le (1 minus 120572119896minus1) 119881 (x119896minus1)
+I minus (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
P119896|119896minus1 (65)
Then combining the terms gives (29)Thus the inequality in(12) has been satisfied
In order to apply Lemma 3 the inequalities (10) and (11)also need to be satisfiedThese inequalities are guaranteed bythe assumptions (23) and (24) in Theorem 5 Thus the nec-essary conditions for Lemma 3 have been satisfied thereforethe estimation error of the LKF is bounded in mean squarewith probability one and the bound is given by (27) Thiscompletes the proof of Theorem 5 In the following sectiona LKF example is provided to illustrate the usefulness ofTheorem 5 for LKF convergence analysis
5 An Illustrative Example
To demonstrate the convergence analysis method fromSection 4 a simple LKF example is presented This exampleproblem was adapted from Example 51 in [27] to includeprocess noise The system equations are defined in the formof (1) with system matrices defined by
F119896 = F =[[[[
[
1 1198791198792
2
0 1 119879
0 0 1
]]]]
]
H119896 = H = [1 0 0]
(66)
and the true process and measurement noise covariancematrices are given by
119864 [w119896w119879
119896] = 10
minus8I
119864 [v119896v119879
119896] = 10
minus8
(67)
where 119879 is the sampling time which for this example isconsidered to be 002 The initial conditions are assumed tobe
while the true initial state for the system is actually
x0 = [1 05 02]119879 (69)
Note that this considers a case of reasonably large initializa-tion error
In order to apply Theorem 5 certain assumptions needto be satisfied From the definition of F it is clear that thismatrix is invertible Four different cases of assumed processand measurement covariance matrices were considered assummarized in Table 1
It is shown in Table 1 that (25) and (26) are satisfiedNote that these cases vary the assumed noise propertiesnot the actual noise The true noise covariance matrices aregiven by (67) for all cases The value for the initial Lyapunovfunction upper bound V0 is calculated from the assumedinitial covariance matrix with (23) Additionally the valuesfor the time-varying convergence rate 120572119896 noise parameter120583119896 and Lyapunov function bound 119887119896 are defined using (28)(29) and (24) respectivelyThese values are calculated onlineat each time step of the filter Using these equations theconvergence properties can be calculated online with (27)
For the given example the presented convergence analy-sis technique is applied and the results are given as followsSince the initial covariance is the identity matrix V0 = 1 Thetime-varying convergence and error parameters are shown inFigure 1 for each of the considered cases of assumed processand measurement noise covariance
The parameter 120572119896 represents the convergence rate of thestochastic process 120583119896 represents the persistent error of thestochastic process and 119887119896 represents the convergence of theerror covariance From these time-varying parameters thebound on the expected value of the norm of the estimationerror squared can be determined from (27) This bound isverified with respect to the actual estimation error which wasdetermined from simulation as shown in Figure 2
It is shown in Figure 2 that the estimation error doesnot exceed the theoretical bounds The online bounds arerelatively close to the estimation error thus providing areasonable guide to the convergence and steady-state errorof the filter performance This is useful because a referencetruth is not available to evaluate the performance of a filter inmost practical applicationsThis method provides a means ofcalculating an upper bound on the performance of the filterusing only known values from the filtering process
There are some interesting observations to make fromFigures 1 and 2 regarding the different noise covarianceassumptions Case 1 which represents perfect knowledge ofthe simulated noise properties offers a very good approxima-tion to the convergence and persistent error of the example
8 International Journal of Stochastic Analysis
0
001
002
003
0
1
2
02468
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 103
Discrete-time k100 101 102 103
Discrete-time k100 101 102 103
times105
120572k
120583k
b k
Figure 1 LKF example time-varying convergence and error param-eters
Simulation
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 10310minus10
10minus8
10minus6
10minus4
10minus2
100
102
xer
r2
Figure 2 LKF example estimation error with bounds
filter Increasing the assumption on the process noise (Case2) leads to an increase in 120572119896 but also an increase in 120583119896 aspredicted However this increase in assumed process noisesignificantly increased the parameter 119887119896 thus leading to aslowly converging loose bound on the estimation error Asimilar performance bound was seen for Case 4 due to thedominant effect of the parameter 119887119896 however the parameters120572119896 and 120583119896 were similar to Case 1 This makes sense becausethe ratio between the assumed Q and R remained the samefor Cases 1 and 4 For Case 3 increasing the assumedmeasurement noise decreased the parameters 120572119896 and 120583119896as expected but the parameter 119887119896 also decreased furtherdecreasing the convergence of the estimation error This lead
OnlineOffline
Simulation
10minus10
100
1010
xer
r2
Discrete-time k100 101 102 103
Figure 3 LKF example online versus offline estimation errorbounds
to a slower converging bound but a tighter bound on thepersistent error This demonstrates a tradeoff in the selectionof the measurement noise covariance which could be usedfor filter tuning depending on the application and desiredconvergence properties
The predicted estimation error bound from Offline anal-ysis [23 26] using Lemma 2 is also provided as a referenceto demonstrate the effectiveness of using this new onlinemethod To relate the time-varying parameters to previousOffline work using Lemma 2 [23 26] the following relationsare used
120572 = min (120572119896)
120583 = max (120583119896)
V1 = min (119887119896)
(70)
The bound from Case 1 is used for this comparison as shownin Figure 3
While the Offline estimation error bound is valid it isextremely loose and does not provide a realistic portrayalof the convergence of the estimation error This shows thatthe presented online method is useful for more closelydetermining the convergence and persistent error of the LKFbut is limited in that it cannot predict these bounds prior tothe filtering process and it cannot be used for Offline stabilityanalysis
6 Conclusions
This paper presented a modified stochastic stability lemmaand a Kalman filter convergence theorem which are newtools that can be used to quantify the performance ofKalman filters online Through an example it was shownthat this new convergence analysis method is effective indetermining an upper bound on the performance of theLKF Also useful information about the convergence of theparticular LKF algorithm can be calculated This analysisis applied during the filtering process thus providing thecapability for real-time convergence and performance moni-toring Different cases of noise covariance assumptions wereconsidered showing that increasing the assumed process
International Journal of Stochastic Analysis 9
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
bound is considered as a time-varying quantity The useful-ness of Lemma 3 is not for stability analysis but for the onlinemonitoring of convergence and estimation error boundsTheconsideration of time-varying parameters is the key to thefollowing online convergence and error analysis
4 Online Convergence and Error Analysis
This section considers a new approach to analyzing theconvergence and estimation error of the LKF in real-timeUsing Lemma 3 the main result of this paper can be stated
Theorem 5 (Kalman filter convergence theorem) Consider alinear stochastic system using the LKF equations as describedin Section 2 Let the following assumptions hold
(1) The systemmatrixF119896 is nonsingular (invertible) for allk
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(29)
119887119896 = 120582min (Pminus1
119896) (30)
Proof The proof of this theorem is detailed in the followingsections
Remarks (1)The bound in (23) only matters for the assumedinitial covariance matrix Since this has a known value theconstant V0 should be selected as the minimum eigenvalue ofthe inverse of the assumed initial covariance matrix and thisbound will be automatically satisfied
(2) It is worth noting in (24) that if the error covari-ance approaches infinity (divergence) then the term 119887119896 willapproach zero which would lead to an infinite bound on theestimation error thus indicating divergence of the filter asexpected For a stable system however the error covariancematrix has an upper bound which can be determined fromthe stochastic controllability and observability properties ofthe system [11 15]
(3)Theparameters 120572 and 120583 are both functions of the samematrix where 120572 is the minimum eigenvalue and 120583 is the traceof the matrix Since the eigenvalues of this matrix lie between0 and 1 (the a priori covariance is always greater than or equalto the process noise covariance matrix) and recalling that thetrace of a matrix is equal to the sum of its eigenvalues [31]the parameter 120583 will satisfy 0 lt 120572119896minus1 lt 120583119896minus1 lt 119899 for all119896 where 119899 is the number of states in the filter From here itis interesting to note that increasing the parameter 120572 whichcorresponds to the convergence of the stochastic process willin turn also increase the parameter 120583 which corresponds tothe persistent error bound due to noise This introduces atradeoff in convergence and persistent error which can betuned through the selection of the process and measurementnoise covariance matrices
(4) Using Lemma 3 for analysis of the LKF convergenceleads to three important time-varying parameters 120572119896 120583119896and 119887119896 The parameter 120572119896 represents the convergence of thestochastic process as defined in the following section by(31) while the parameter 119887119896 represents the convergence ofthe error covariance The parameter 120583119896 corresponds to thepersistent error bound on the filter due to the process andmeasurement noise That is in (27) it is shown that theinitial error term will vanish as 119896 increases thus leaving theterm containing 120583119896 which contains the persistent responseThis makes sense because as a LKF progresses in timeeventually the performance will converge within a regiondetermined from the process and measurement noise sincethese phenomena do not disappear with time Together thesethree parameters determine a bound on the convergenceand persistent error of the filter using (27) Due to thetime-varying nature of these parameters the bound must bedetermined online and therefore cannot provide an Offlineprediction of the filter convergence as in [23 26]
The proof of Theorem 5 is provided next
41 Defining and Decomposing the Estimation Error AnalysisAs recommended in other works for example [19 23]a candidate Lyapunov function is selected to define thestochastic process using a quadratic form of the estimationerror and inverse error covariance matrix as in
Note that this function is used in the context of Lemma 3 notusing traditional Lyapunov stability theorems therefore it isonly being used as a tool for analyzing the convergence notto prove the stability of the filter Inserting the error dynamicsfrom (7) into this function gives
119881 (x119896) = [(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]119879
times Pminus1119896[(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]
(32)
Taking the conditional expectation with respect to x119896minus1 andusing the assumption that the process and measurementnoise are uncorrelated give
Now the problem of analyzing the LKF estimation error hasbeen divided into three parts the homogeneous problem in(34) the process noise problem in (35) and themeasurementnoise problem in (36) The homogeneous problem considersthe deterministic part of the filter that is no noise The pro-cess and measurement noise problems consider the effects ofthe stochastic uncertainty in the prediction andmeasurementequations respectively Each of these three parts is consideredseparately in the following sections
42 The Homogeneous Problem The homogeneous part ofthe problem is defined by (34) This part of the problem isrelated to the convergence rate of the filter For this part ofthe analysis a bound is desired in the form
Γ119909
119896le (1 minus 120572119896minus1) 119881 (x119896minus1) (37)
This inequality is desired as it is the assumption given by (12)ignoring for now the noise terms and assuming that 120583119896 = 0for all 119896 Substituting in for (31) and (34) gives
x119879119896minus1
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1x119896minus1
le (1 minus 120572119896minus1) x119879
119896minus1Pminus1119896minus1
x119896minus1(38)
This scalar inequality is equivalent to the matrix inequality
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1
(39)
The following relationship can be derived from the LKFequations in (2)
I minus K119896H119896 = P119896Pminus1
119896|119896minus1 (40)
Substituting (40) into (39) gives
F119879119896minus1
Pminus1119896|119896minus1
P119896Pminus1
119896|119896minus1F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1 (41)
Taking the inverse of this inequality gives
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1ge (1 minus 120572119896minus1)
minus1P119896minus1 (42)
Note that this operation requires that the system matrix Fbe nonsingular for all 119896 (assumption (1)) The covariancematrices are invertible because they are positive definite bydefinition Starting from the covariance prediction equationin (2) and rearranging give
P119896minus1 = Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1 (43)
Substituting this equation into the matrix inequality yields
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1
ge (1 minus 120572119896minus1)minus1Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1
(44)
Now the system matrix can be removed from the inequality
P119896|119896minus1Pminus1
119896P119896|119896minus1 ge (1 minus 120572119896minus1)
120572119896minus1I le (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(47)
Therefore the time-varying parameter 120572 can be determinedas the minimum eigenvalue of the matrix as in (28) Fromthe covariance prediction equation in (2) it is clear that the apriori covariance is greater than the process noise covariancematrix therefore 120572 is always between 0 and 1 Note thatincreasing Q will increase 120572 Alternatively increasing R willdecrease120572 If the parameter120572 is selected as in (28) the desiredinequality (37) is satisfied thus satisfying the homogeneouspart of the problem Next the process noise is considered
43 The Process Noise Problem For the process noise prob-lem the quantity of interest is given by (35) Since this is a
6 International Journal of Stochastic Analysis
scalar equation the trace can be taken without changing thevalue
Γ119908
119896= Tr Γ119908
119896 = Tr 119864 [w119879
119896minus1(I minus K119896H119896)
119879Pminus1119896
times (I minus K119896H119896)w119896minus1] (48)
Using the trace property ofmultiplication reordering [31] andremoving the deterministic terms from the expectation yield
Γ119908
119896= Tr (I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) 119864 [w119896minus1w
119879
119896minus1] (49)
Using (40) simplifies the equation to
Γ119908
119896= Tr Pminus1
119896|119896minus1P119896Pminus1
119896|119896minus1119864 [w119896minus1w
119879
119896minus1] (50)
Inserting the covariance update equation from (2) gives
Since the process noise covariance matrix can be chosenfreely for the LKF it is assumed that the assumed processnoise covariance matrix is greater than the actual covarianceof the process noise as in (25)This bound ismotivated by theidea that it is better to assume greater rather than less noisethan there actually is in the system This leads to the boundon the process noise term
While increasing Q was shown to increase the convergencerate in the previous section it is clear here that this increasein convergence comes at the expense of a larger bound on theprocess noise term This selection of Q becomes a tradeoffbetween the convergence and the accuracy of the estimatethat is assuming an unnecessarily large Q will lead to fasterconvergence but larger persistent errors of the filter dueto process noise Next the measurement noise problem isconsidered
44 The Measurement Noise Problem For the measurementnoise problem the quantity of interest is given by (36) Sincethis is a scalar equation the trace can be taken withoutchanging the value
Similarly as for the process noise the assumed measurementnoise covariance matrix is selected as an upper bound onthe actual measurement noise covariance as in (26) whichdetermines the bound for the measurement noise term
From here it is shown that increasing the assumed mea-surement noise covariance matrix R will in fact lead to asmaller bound on the estimation error due to measurementnoise Now that each part of the problem has been consideredseparately the results are combined and Lemma 3 is applied
45 Final Result from theModified Stochastic Stability LemmaCombining the results from the previous sections gives thefollowing inequality
119864 [119881 (x119896) | x119896minus1]
le (1 minus 120572119896minus1) 119881 (x119896minus1)
+I minus (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
P119896|119896minus1 (65)
Then combining the terms gives (29)Thus the inequality in(12) has been satisfied
In order to apply Lemma 3 the inequalities (10) and (11)also need to be satisfiedThese inequalities are guaranteed bythe assumptions (23) and (24) in Theorem 5 Thus the nec-essary conditions for Lemma 3 have been satisfied thereforethe estimation error of the LKF is bounded in mean squarewith probability one and the bound is given by (27) Thiscompletes the proof of Theorem 5 In the following sectiona LKF example is provided to illustrate the usefulness ofTheorem 5 for LKF convergence analysis
5 An Illustrative Example
To demonstrate the convergence analysis method fromSection 4 a simple LKF example is presented This exampleproblem was adapted from Example 51 in [27] to includeprocess noise The system equations are defined in the formof (1) with system matrices defined by
F119896 = F =[[[[
[
1 1198791198792
2
0 1 119879
0 0 1
]]]]
]
H119896 = H = [1 0 0]
(66)
and the true process and measurement noise covariancematrices are given by
119864 [w119896w119879
119896] = 10
minus8I
119864 [v119896v119879
119896] = 10
minus8
(67)
where 119879 is the sampling time which for this example isconsidered to be 002 The initial conditions are assumed tobe
while the true initial state for the system is actually
x0 = [1 05 02]119879 (69)
Note that this considers a case of reasonably large initializa-tion error
In order to apply Theorem 5 certain assumptions needto be satisfied From the definition of F it is clear that thismatrix is invertible Four different cases of assumed processand measurement covariance matrices were considered assummarized in Table 1
It is shown in Table 1 that (25) and (26) are satisfiedNote that these cases vary the assumed noise propertiesnot the actual noise The true noise covariance matrices aregiven by (67) for all cases The value for the initial Lyapunovfunction upper bound V0 is calculated from the assumedinitial covariance matrix with (23) Additionally the valuesfor the time-varying convergence rate 120572119896 noise parameter120583119896 and Lyapunov function bound 119887119896 are defined using (28)(29) and (24) respectivelyThese values are calculated onlineat each time step of the filter Using these equations theconvergence properties can be calculated online with (27)
For the given example the presented convergence analy-sis technique is applied and the results are given as followsSince the initial covariance is the identity matrix V0 = 1 Thetime-varying convergence and error parameters are shown inFigure 1 for each of the considered cases of assumed processand measurement noise covariance
The parameter 120572119896 represents the convergence rate of thestochastic process 120583119896 represents the persistent error of thestochastic process and 119887119896 represents the convergence of theerror covariance From these time-varying parameters thebound on the expected value of the norm of the estimationerror squared can be determined from (27) This bound isverified with respect to the actual estimation error which wasdetermined from simulation as shown in Figure 2
It is shown in Figure 2 that the estimation error doesnot exceed the theoretical bounds The online bounds arerelatively close to the estimation error thus providing areasonable guide to the convergence and steady-state errorof the filter performance This is useful because a referencetruth is not available to evaluate the performance of a filter inmost practical applicationsThis method provides a means ofcalculating an upper bound on the performance of the filterusing only known values from the filtering process
There are some interesting observations to make fromFigures 1 and 2 regarding the different noise covarianceassumptions Case 1 which represents perfect knowledge ofthe simulated noise properties offers a very good approxima-tion to the convergence and persistent error of the example
8 International Journal of Stochastic Analysis
0
001
002
003
0
1
2
02468
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 103
Discrete-time k100 101 102 103
Discrete-time k100 101 102 103
times105
120572k
120583k
b k
Figure 1 LKF example time-varying convergence and error param-eters
Simulation
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 10310minus10
10minus8
10minus6
10minus4
10minus2
100
102
xer
r2
Figure 2 LKF example estimation error with bounds
filter Increasing the assumption on the process noise (Case2) leads to an increase in 120572119896 but also an increase in 120583119896 aspredicted However this increase in assumed process noisesignificantly increased the parameter 119887119896 thus leading to aslowly converging loose bound on the estimation error Asimilar performance bound was seen for Case 4 due to thedominant effect of the parameter 119887119896 however the parameters120572119896 and 120583119896 were similar to Case 1 This makes sense becausethe ratio between the assumed Q and R remained the samefor Cases 1 and 4 For Case 3 increasing the assumedmeasurement noise decreased the parameters 120572119896 and 120583119896as expected but the parameter 119887119896 also decreased furtherdecreasing the convergence of the estimation error This lead
OnlineOffline
Simulation
10minus10
100
1010
xer
r2
Discrete-time k100 101 102 103
Figure 3 LKF example online versus offline estimation errorbounds
to a slower converging bound but a tighter bound on thepersistent error This demonstrates a tradeoff in the selectionof the measurement noise covariance which could be usedfor filter tuning depending on the application and desiredconvergence properties
The predicted estimation error bound from Offline anal-ysis [23 26] using Lemma 2 is also provided as a referenceto demonstrate the effectiveness of using this new onlinemethod To relate the time-varying parameters to previousOffline work using Lemma 2 [23 26] the following relationsare used
120572 = min (120572119896)
120583 = max (120583119896)
V1 = min (119887119896)
(70)
The bound from Case 1 is used for this comparison as shownin Figure 3
While the Offline estimation error bound is valid it isextremely loose and does not provide a realistic portrayalof the convergence of the estimation error This shows thatthe presented online method is useful for more closelydetermining the convergence and persistent error of the LKFbut is limited in that it cannot predict these bounds prior tothe filtering process and it cannot be used for Offline stabilityanalysis
6 Conclusions
This paper presented a modified stochastic stability lemmaand a Kalman filter convergence theorem which are newtools that can be used to quantify the performance ofKalman filters online Through an example it was shownthat this new convergence analysis method is effective indetermining an upper bound on the performance of theLKF Also useful information about the convergence of theparticular LKF algorithm can be calculated This analysisis applied during the filtering process thus providing thecapability for real-time convergence and performance moni-toring Different cases of noise covariance assumptions wereconsidered showing that increasing the assumed process
International Journal of Stochastic Analysis 9
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
Note that this function is used in the context of Lemma 3 notusing traditional Lyapunov stability theorems therefore it isonly being used as a tool for analyzing the convergence notto prove the stability of the filter Inserting the error dynamicsfrom (7) into this function gives
119881 (x119896) = [(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]119879
times Pminus1119896[(I minus K119896H119896) (F119896minus1x119896minus1 + w119896minus1) minus K119896v119896]
(32)
Taking the conditional expectation with respect to x119896minus1 andusing the assumption that the process and measurementnoise are uncorrelated give
Now the problem of analyzing the LKF estimation error hasbeen divided into three parts the homogeneous problem in(34) the process noise problem in (35) and themeasurementnoise problem in (36) The homogeneous problem considersthe deterministic part of the filter that is no noise The pro-cess and measurement noise problems consider the effects ofthe stochastic uncertainty in the prediction andmeasurementequations respectively Each of these three parts is consideredseparately in the following sections
42 The Homogeneous Problem The homogeneous part ofthe problem is defined by (34) This part of the problem isrelated to the convergence rate of the filter For this part ofthe analysis a bound is desired in the form
Γ119909
119896le (1 minus 120572119896minus1) 119881 (x119896minus1) (37)
This inequality is desired as it is the assumption given by (12)ignoring for now the noise terms and assuming that 120583119896 = 0for all 119896 Substituting in for (31) and (34) gives
x119879119896minus1
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1x119896minus1
le (1 minus 120572119896minus1) x119879
119896minus1Pminus1119896minus1
x119896minus1(38)
This scalar inequality is equivalent to the matrix inequality
F119879119896minus1(I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1
(39)
The following relationship can be derived from the LKFequations in (2)
I minus K119896H119896 = P119896Pminus1
119896|119896minus1 (40)
Substituting (40) into (39) gives
F119879119896minus1
Pminus1119896|119896minus1
P119896Pminus1
119896|119896minus1F119896minus1 le (1 minus 120572119896minus1)P
minus1
119896minus1 (41)
Taking the inverse of this inequality gives
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1ge (1 minus 120572119896minus1)
minus1P119896minus1 (42)
Note that this operation requires that the system matrix Fbe nonsingular for all 119896 (assumption (1)) The covariancematrices are invertible because they are positive definite bydefinition Starting from the covariance prediction equationin (2) and rearranging give
P119896minus1 = Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1 (43)
Substituting this equation into the matrix inequality yields
Fminus1119896minus1
P119896|119896minus1Pminus1
119896P119896|119896minus1F
minus119879
119896minus1
ge (1 minus 120572119896minus1)minus1Fminus1119896minus1(P119896|119896minus1 minusQ119896minus1) F
minus119879
119896minus1
(44)
Now the system matrix can be removed from the inequality
P119896|119896minus1Pminus1
119896P119896|119896minus1 ge (1 minus 120572119896minus1)
120572119896minus1I le (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
times (Q119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
(47)
Therefore the time-varying parameter 120572 can be determinedas the minimum eigenvalue of the matrix as in (28) Fromthe covariance prediction equation in (2) it is clear that the apriori covariance is greater than the process noise covariancematrix therefore 120572 is always between 0 and 1 Note thatincreasing Q will increase 120572 Alternatively increasing R willdecrease120572 If the parameter120572 is selected as in (28) the desiredinequality (37) is satisfied thus satisfying the homogeneouspart of the problem Next the process noise is considered
43 The Process Noise Problem For the process noise prob-lem the quantity of interest is given by (35) Since this is a
6 International Journal of Stochastic Analysis
scalar equation the trace can be taken without changing thevalue
Γ119908
119896= Tr Γ119908
119896 = Tr 119864 [w119879
119896minus1(I minus K119896H119896)
119879Pminus1119896
times (I minus K119896H119896)w119896minus1] (48)
Using the trace property ofmultiplication reordering [31] andremoving the deterministic terms from the expectation yield
Γ119908
119896= Tr (I minus K119896H119896)
119879Pminus1119896(I minus K119896H119896) 119864 [w119896minus1w
119879
119896minus1] (49)
Using (40) simplifies the equation to
Γ119908
119896= Tr Pminus1
119896|119896minus1P119896Pminus1
119896|119896minus1119864 [w119896minus1w
119879
119896minus1] (50)
Inserting the covariance update equation from (2) gives
Since the process noise covariance matrix can be chosenfreely for the LKF it is assumed that the assumed processnoise covariance matrix is greater than the actual covarianceof the process noise as in (25)This bound ismotivated by theidea that it is better to assume greater rather than less noisethan there actually is in the system This leads to the boundon the process noise term
While increasing Q was shown to increase the convergencerate in the previous section it is clear here that this increasein convergence comes at the expense of a larger bound on theprocess noise term This selection of Q becomes a tradeoffbetween the convergence and the accuracy of the estimatethat is assuming an unnecessarily large Q will lead to fasterconvergence but larger persistent errors of the filter dueto process noise Next the measurement noise problem isconsidered
44 The Measurement Noise Problem For the measurementnoise problem the quantity of interest is given by (36) Sincethis is a scalar equation the trace can be taken withoutchanging the value
Similarly as for the process noise the assumed measurementnoise covariance matrix is selected as an upper bound onthe actual measurement noise covariance as in (26) whichdetermines the bound for the measurement noise term
From here it is shown that increasing the assumed mea-surement noise covariance matrix R will in fact lead to asmaller bound on the estimation error due to measurementnoise Now that each part of the problem has been consideredseparately the results are combined and Lemma 3 is applied
45 Final Result from theModified Stochastic Stability LemmaCombining the results from the previous sections gives thefollowing inequality
119864 [119881 (x119896) | x119896minus1]
le (1 minus 120572119896minus1) 119881 (x119896minus1)
+I minus (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
P119896|119896minus1 (65)
Then combining the terms gives (29)Thus the inequality in(12) has been satisfied
In order to apply Lemma 3 the inequalities (10) and (11)also need to be satisfiedThese inequalities are guaranteed bythe assumptions (23) and (24) in Theorem 5 Thus the nec-essary conditions for Lemma 3 have been satisfied thereforethe estimation error of the LKF is bounded in mean squarewith probability one and the bound is given by (27) Thiscompletes the proof of Theorem 5 In the following sectiona LKF example is provided to illustrate the usefulness ofTheorem 5 for LKF convergence analysis
5 An Illustrative Example
To demonstrate the convergence analysis method fromSection 4 a simple LKF example is presented This exampleproblem was adapted from Example 51 in [27] to includeprocess noise The system equations are defined in the formof (1) with system matrices defined by
F119896 = F =[[[[
[
1 1198791198792
2
0 1 119879
0 0 1
]]]]
]
H119896 = H = [1 0 0]
(66)
and the true process and measurement noise covariancematrices are given by
119864 [w119896w119879
119896] = 10
minus8I
119864 [v119896v119879
119896] = 10
minus8
(67)
where 119879 is the sampling time which for this example isconsidered to be 002 The initial conditions are assumed tobe
while the true initial state for the system is actually
x0 = [1 05 02]119879 (69)
Note that this considers a case of reasonably large initializa-tion error
In order to apply Theorem 5 certain assumptions needto be satisfied From the definition of F it is clear that thismatrix is invertible Four different cases of assumed processand measurement covariance matrices were considered assummarized in Table 1
It is shown in Table 1 that (25) and (26) are satisfiedNote that these cases vary the assumed noise propertiesnot the actual noise The true noise covariance matrices aregiven by (67) for all cases The value for the initial Lyapunovfunction upper bound V0 is calculated from the assumedinitial covariance matrix with (23) Additionally the valuesfor the time-varying convergence rate 120572119896 noise parameter120583119896 and Lyapunov function bound 119887119896 are defined using (28)(29) and (24) respectivelyThese values are calculated onlineat each time step of the filter Using these equations theconvergence properties can be calculated online with (27)
For the given example the presented convergence analy-sis technique is applied and the results are given as followsSince the initial covariance is the identity matrix V0 = 1 Thetime-varying convergence and error parameters are shown inFigure 1 for each of the considered cases of assumed processand measurement noise covariance
The parameter 120572119896 represents the convergence rate of thestochastic process 120583119896 represents the persistent error of thestochastic process and 119887119896 represents the convergence of theerror covariance From these time-varying parameters thebound on the expected value of the norm of the estimationerror squared can be determined from (27) This bound isverified with respect to the actual estimation error which wasdetermined from simulation as shown in Figure 2
It is shown in Figure 2 that the estimation error doesnot exceed the theoretical bounds The online bounds arerelatively close to the estimation error thus providing areasonable guide to the convergence and steady-state errorof the filter performance This is useful because a referencetruth is not available to evaluate the performance of a filter inmost practical applicationsThis method provides a means ofcalculating an upper bound on the performance of the filterusing only known values from the filtering process
There are some interesting observations to make fromFigures 1 and 2 regarding the different noise covarianceassumptions Case 1 which represents perfect knowledge ofthe simulated noise properties offers a very good approxima-tion to the convergence and persistent error of the example
8 International Journal of Stochastic Analysis
0
001
002
003
0
1
2
02468
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 103
Discrete-time k100 101 102 103
Discrete-time k100 101 102 103
times105
120572k
120583k
b k
Figure 1 LKF example time-varying convergence and error param-eters
Simulation
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 10310minus10
10minus8
10minus6
10minus4
10minus2
100
102
xer
r2
Figure 2 LKF example estimation error with bounds
filter Increasing the assumption on the process noise (Case2) leads to an increase in 120572119896 but also an increase in 120583119896 aspredicted However this increase in assumed process noisesignificantly increased the parameter 119887119896 thus leading to aslowly converging loose bound on the estimation error Asimilar performance bound was seen for Case 4 due to thedominant effect of the parameter 119887119896 however the parameters120572119896 and 120583119896 were similar to Case 1 This makes sense becausethe ratio between the assumed Q and R remained the samefor Cases 1 and 4 For Case 3 increasing the assumedmeasurement noise decreased the parameters 120572119896 and 120583119896as expected but the parameter 119887119896 also decreased furtherdecreasing the convergence of the estimation error This lead
OnlineOffline
Simulation
10minus10
100
1010
xer
r2
Discrete-time k100 101 102 103
Figure 3 LKF example online versus offline estimation errorbounds
to a slower converging bound but a tighter bound on thepersistent error This demonstrates a tradeoff in the selectionof the measurement noise covariance which could be usedfor filter tuning depending on the application and desiredconvergence properties
The predicted estimation error bound from Offline anal-ysis [23 26] using Lemma 2 is also provided as a referenceto demonstrate the effectiveness of using this new onlinemethod To relate the time-varying parameters to previousOffline work using Lemma 2 [23 26] the following relationsare used
120572 = min (120572119896)
120583 = max (120583119896)
V1 = min (119887119896)
(70)
The bound from Case 1 is used for this comparison as shownin Figure 3
While the Offline estimation error bound is valid it isextremely loose and does not provide a realistic portrayalof the convergence of the estimation error This shows thatthe presented online method is useful for more closelydetermining the convergence and persistent error of the LKFbut is limited in that it cannot predict these bounds prior tothe filtering process and it cannot be used for Offline stabilityanalysis
6 Conclusions
This paper presented a modified stochastic stability lemmaand a Kalman filter convergence theorem which are newtools that can be used to quantify the performance ofKalman filters online Through an example it was shownthat this new convergence analysis method is effective indetermining an upper bound on the performance of theLKF Also useful information about the convergence of theparticular LKF algorithm can be calculated This analysisis applied during the filtering process thus providing thecapability for real-time convergence and performance moni-toring Different cases of noise covariance assumptions wereconsidered showing that increasing the assumed process
International Journal of Stochastic Analysis 9
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
Since the process noise covariance matrix can be chosenfreely for the LKF it is assumed that the assumed processnoise covariance matrix is greater than the actual covarianceof the process noise as in (25)This bound ismotivated by theidea that it is better to assume greater rather than less noisethan there actually is in the system This leads to the boundon the process noise term
While increasing Q was shown to increase the convergencerate in the previous section it is clear here that this increasein convergence comes at the expense of a larger bound on theprocess noise term This selection of Q becomes a tradeoffbetween the convergence and the accuracy of the estimatethat is assuming an unnecessarily large Q will lead to fasterconvergence but larger persistent errors of the filter dueto process noise Next the measurement noise problem isconsidered
44 The Measurement Noise Problem For the measurementnoise problem the quantity of interest is given by (36) Sincethis is a scalar equation the trace can be taken withoutchanging the value
Similarly as for the process noise the assumed measurementnoise covariance matrix is selected as an upper bound onthe actual measurement noise covariance as in (26) whichdetermines the bound for the measurement noise term
From here it is shown that increasing the assumed mea-surement noise covariance matrix R will in fact lead to asmaller bound on the estimation error due to measurementnoise Now that each part of the problem has been consideredseparately the results are combined and Lemma 3 is applied
45 Final Result from theModified Stochastic Stability LemmaCombining the results from the previous sections gives thefollowing inequality
119864 [119881 (x119896) | x119896minus1]
le (1 minus 120572119896minus1) 119881 (x119896minus1)
+I minus (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
P119896|119896minus1 (65)
Then combining the terms gives (29)Thus the inequality in(12) has been satisfied
In order to apply Lemma 3 the inequalities (10) and (11)also need to be satisfiedThese inequalities are guaranteed bythe assumptions (23) and (24) in Theorem 5 Thus the nec-essary conditions for Lemma 3 have been satisfied thereforethe estimation error of the LKF is bounded in mean squarewith probability one and the bound is given by (27) Thiscompletes the proof of Theorem 5 In the following sectiona LKF example is provided to illustrate the usefulness ofTheorem 5 for LKF convergence analysis
5 An Illustrative Example
To demonstrate the convergence analysis method fromSection 4 a simple LKF example is presented This exampleproblem was adapted from Example 51 in [27] to includeprocess noise The system equations are defined in the formof (1) with system matrices defined by
F119896 = F =[[[[
[
1 1198791198792
2
0 1 119879
0 0 1
]]]]
]
H119896 = H = [1 0 0]
(66)
and the true process and measurement noise covariancematrices are given by
119864 [w119896w119879
119896] = 10
minus8I
119864 [v119896v119879
119896] = 10
minus8
(67)
where 119879 is the sampling time which for this example isconsidered to be 002 The initial conditions are assumed tobe
while the true initial state for the system is actually
x0 = [1 05 02]119879 (69)
Note that this considers a case of reasonably large initializa-tion error
In order to apply Theorem 5 certain assumptions needto be satisfied From the definition of F it is clear that thismatrix is invertible Four different cases of assumed processand measurement covariance matrices were considered assummarized in Table 1
It is shown in Table 1 that (25) and (26) are satisfiedNote that these cases vary the assumed noise propertiesnot the actual noise The true noise covariance matrices aregiven by (67) for all cases The value for the initial Lyapunovfunction upper bound V0 is calculated from the assumedinitial covariance matrix with (23) Additionally the valuesfor the time-varying convergence rate 120572119896 noise parameter120583119896 and Lyapunov function bound 119887119896 are defined using (28)(29) and (24) respectivelyThese values are calculated onlineat each time step of the filter Using these equations theconvergence properties can be calculated online with (27)
For the given example the presented convergence analy-sis technique is applied and the results are given as followsSince the initial covariance is the identity matrix V0 = 1 Thetime-varying convergence and error parameters are shown inFigure 1 for each of the considered cases of assumed processand measurement noise covariance
The parameter 120572119896 represents the convergence rate of thestochastic process 120583119896 represents the persistent error of thestochastic process and 119887119896 represents the convergence of theerror covariance From these time-varying parameters thebound on the expected value of the norm of the estimationerror squared can be determined from (27) This bound isverified with respect to the actual estimation error which wasdetermined from simulation as shown in Figure 2
It is shown in Figure 2 that the estimation error doesnot exceed the theoretical bounds The online bounds arerelatively close to the estimation error thus providing areasonable guide to the convergence and steady-state errorof the filter performance This is useful because a referencetruth is not available to evaluate the performance of a filter inmost practical applicationsThis method provides a means ofcalculating an upper bound on the performance of the filterusing only known values from the filtering process
There are some interesting observations to make fromFigures 1 and 2 regarding the different noise covarianceassumptions Case 1 which represents perfect knowledge ofthe simulated noise properties offers a very good approxima-tion to the convergence and persistent error of the example
8 International Journal of Stochastic Analysis
0
001
002
003
0
1
2
02468
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 103
Discrete-time k100 101 102 103
Discrete-time k100 101 102 103
times105
120572k
120583k
b k
Figure 1 LKF example time-varying convergence and error param-eters
Simulation
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 10310minus10
10minus8
10minus6
10minus4
10minus2
100
102
xer
r2
Figure 2 LKF example estimation error with bounds
filter Increasing the assumption on the process noise (Case2) leads to an increase in 120572119896 but also an increase in 120583119896 aspredicted However this increase in assumed process noisesignificantly increased the parameter 119887119896 thus leading to aslowly converging loose bound on the estimation error Asimilar performance bound was seen for Case 4 due to thedominant effect of the parameter 119887119896 however the parameters120572119896 and 120583119896 were similar to Case 1 This makes sense becausethe ratio between the assumed Q and R remained the samefor Cases 1 and 4 For Case 3 increasing the assumedmeasurement noise decreased the parameters 120572119896 and 120583119896as expected but the parameter 119887119896 also decreased furtherdecreasing the convergence of the estimation error This lead
OnlineOffline
Simulation
10minus10
100
1010
xer
r2
Discrete-time k100 101 102 103
Figure 3 LKF example online versus offline estimation errorbounds
to a slower converging bound but a tighter bound on thepersistent error This demonstrates a tradeoff in the selectionof the measurement noise covariance which could be usedfor filter tuning depending on the application and desiredconvergence properties
The predicted estimation error bound from Offline anal-ysis [23 26] using Lemma 2 is also provided as a referenceto demonstrate the effectiveness of using this new onlinemethod To relate the time-varying parameters to previousOffline work using Lemma 2 [23 26] the following relationsare used
120572 = min (120572119896)
120583 = max (120583119896)
V1 = min (119887119896)
(70)
The bound from Case 1 is used for this comparison as shownin Figure 3
While the Offline estimation error bound is valid it isextremely loose and does not provide a realistic portrayalof the convergence of the estimation error This shows thatthe presented online method is useful for more closelydetermining the convergence and persistent error of the LKFbut is limited in that it cannot predict these bounds prior tothe filtering process and it cannot be used for Offline stabilityanalysis
6 Conclusions
This paper presented a modified stochastic stability lemmaand a Kalman filter convergence theorem which are newtools that can be used to quantify the performance ofKalman filters online Through an example it was shownthat this new convergence analysis method is effective indetermining an upper bound on the performance of theLKF Also useful information about the convergence of theparticular LKF algorithm can be calculated This analysisis applied during the filtering process thus providing thecapability for real-time convergence and performance moni-toring Different cases of noise covariance assumptions wereconsidered showing that increasing the assumed process
International Journal of Stochastic Analysis 9
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
+I minus (P119896|119896minus1 + P119896|119896minus1H119879
119896Rminus1119896H119896P119896|119896minus1)
minus1
P119896|119896minus1 (65)
Then combining the terms gives (29)Thus the inequality in(12) has been satisfied
In order to apply Lemma 3 the inequalities (10) and (11)also need to be satisfiedThese inequalities are guaranteed bythe assumptions (23) and (24) in Theorem 5 Thus the nec-essary conditions for Lemma 3 have been satisfied thereforethe estimation error of the LKF is bounded in mean squarewith probability one and the bound is given by (27) Thiscompletes the proof of Theorem 5 In the following sectiona LKF example is provided to illustrate the usefulness ofTheorem 5 for LKF convergence analysis
5 An Illustrative Example
To demonstrate the convergence analysis method fromSection 4 a simple LKF example is presented This exampleproblem was adapted from Example 51 in [27] to includeprocess noise The system equations are defined in the formof (1) with system matrices defined by
F119896 = F =[[[[
[
1 1198791198792
2
0 1 119879
0 0 1
]]]]
]
H119896 = H = [1 0 0]
(66)
and the true process and measurement noise covariancematrices are given by
119864 [w119896w119879
119896] = 10
minus8I
119864 [v119896v119879
119896] = 10
minus8
(67)
where 119879 is the sampling time which for this example isconsidered to be 002 The initial conditions are assumed tobe
while the true initial state for the system is actually
x0 = [1 05 02]119879 (69)
Note that this considers a case of reasonably large initializa-tion error
In order to apply Theorem 5 certain assumptions needto be satisfied From the definition of F it is clear that thismatrix is invertible Four different cases of assumed processand measurement covariance matrices were considered assummarized in Table 1
It is shown in Table 1 that (25) and (26) are satisfiedNote that these cases vary the assumed noise propertiesnot the actual noise The true noise covariance matrices aregiven by (67) for all cases The value for the initial Lyapunovfunction upper bound V0 is calculated from the assumedinitial covariance matrix with (23) Additionally the valuesfor the time-varying convergence rate 120572119896 noise parameter120583119896 and Lyapunov function bound 119887119896 are defined using (28)(29) and (24) respectivelyThese values are calculated onlineat each time step of the filter Using these equations theconvergence properties can be calculated online with (27)
For the given example the presented convergence analy-sis technique is applied and the results are given as followsSince the initial covariance is the identity matrix V0 = 1 Thetime-varying convergence and error parameters are shown inFigure 1 for each of the considered cases of assumed processand measurement noise covariance
The parameter 120572119896 represents the convergence rate of thestochastic process 120583119896 represents the persistent error of thestochastic process and 119887119896 represents the convergence of theerror covariance From these time-varying parameters thebound on the expected value of the norm of the estimationerror squared can be determined from (27) This bound isverified with respect to the actual estimation error which wasdetermined from simulation as shown in Figure 2
It is shown in Figure 2 that the estimation error doesnot exceed the theoretical bounds The online bounds arerelatively close to the estimation error thus providing areasonable guide to the convergence and steady-state errorof the filter performance This is useful because a referencetruth is not available to evaluate the performance of a filter inmost practical applicationsThis method provides a means ofcalculating an upper bound on the performance of the filterusing only known values from the filtering process
There are some interesting observations to make fromFigures 1 and 2 regarding the different noise covarianceassumptions Case 1 which represents perfect knowledge ofthe simulated noise properties offers a very good approxima-tion to the convergence and persistent error of the example
8 International Journal of Stochastic Analysis
0
001
002
003
0
1
2
02468
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 103
Discrete-time k100 101 102 103
Discrete-time k100 101 102 103
times105
120572k
120583k
b k
Figure 1 LKF example time-varying convergence and error param-eters
Simulation
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 10310minus10
10minus8
10minus6
10minus4
10minus2
100
102
xer
r2
Figure 2 LKF example estimation error with bounds
filter Increasing the assumption on the process noise (Case2) leads to an increase in 120572119896 but also an increase in 120583119896 aspredicted However this increase in assumed process noisesignificantly increased the parameter 119887119896 thus leading to aslowly converging loose bound on the estimation error Asimilar performance bound was seen for Case 4 due to thedominant effect of the parameter 119887119896 however the parameters120572119896 and 120583119896 were similar to Case 1 This makes sense becausethe ratio between the assumed Q and R remained the samefor Cases 1 and 4 For Case 3 increasing the assumedmeasurement noise decreased the parameters 120572119896 and 120583119896as expected but the parameter 119887119896 also decreased furtherdecreasing the convergence of the estimation error This lead
OnlineOffline
Simulation
10minus10
100
1010
xer
r2
Discrete-time k100 101 102 103
Figure 3 LKF example online versus offline estimation errorbounds
to a slower converging bound but a tighter bound on thepersistent error This demonstrates a tradeoff in the selectionof the measurement noise covariance which could be usedfor filter tuning depending on the application and desiredconvergence properties
The predicted estimation error bound from Offline anal-ysis [23 26] using Lemma 2 is also provided as a referenceto demonstrate the effectiveness of using this new onlinemethod To relate the time-varying parameters to previousOffline work using Lemma 2 [23 26] the following relationsare used
120572 = min (120572119896)
120583 = max (120583119896)
V1 = min (119887119896)
(70)
The bound from Case 1 is used for this comparison as shownin Figure 3
While the Offline estimation error bound is valid it isextremely loose and does not provide a realistic portrayalof the convergence of the estimation error This shows thatthe presented online method is useful for more closelydetermining the convergence and persistent error of the LKFbut is limited in that it cannot predict these bounds prior tothe filtering process and it cannot be used for Offline stabilityanalysis
6 Conclusions
This paper presented a modified stochastic stability lemmaand a Kalman filter convergence theorem which are newtools that can be used to quantify the performance ofKalman filters online Through an example it was shownthat this new convergence analysis method is effective indetermining an upper bound on the performance of theLKF Also useful information about the convergence of theparticular LKF algorithm can be calculated This analysisis applied during the filtering process thus providing thecapability for real-time convergence and performance moni-toring Different cases of noise covariance assumptions wereconsidered showing that increasing the assumed process
International Journal of Stochastic Analysis 9
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
Figure 1 LKF example time-varying convergence and error param-eters
Simulation
Discrete-time k
Case 1Case 2
Case 3Case 4
100 101 102 10310minus10
10minus8
10minus6
10minus4
10minus2
100
102
xer
r2
Figure 2 LKF example estimation error with bounds
filter Increasing the assumption on the process noise (Case2) leads to an increase in 120572119896 but also an increase in 120583119896 aspredicted However this increase in assumed process noisesignificantly increased the parameter 119887119896 thus leading to aslowly converging loose bound on the estimation error Asimilar performance bound was seen for Case 4 due to thedominant effect of the parameter 119887119896 however the parameters120572119896 and 120583119896 were similar to Case 1 This makes sense becausethe ratio between the assumed Q and R remained the samefor Cases 1 and 4 For Case 3 increasing the assumedmeasurement noise decreased the parameters 120572119896 and 120583119896as expected but the parameter 119887119896 also decreased furtherdecreasing the convergence of the estimation error This lead
OnlineOffline
Simulation
10minus10
100
1010
xer
r2
Discrete-time k100 101 102 103
Figure 3 LKF example online versus offline estimation errorbounds
to a slower converging bound but a tighter bound on thepersistent error This demonstrates a tradeoff in the selectionof the measurement noise covariance which could be usedfor filter tuning depending on the application and desiredconvergence properties
The predicted estimation error bound from Offline anal-ysis [23 26] using Lemma 2 is also provided as a referenceto demonstrate the effectiveness of using this new onlinemethod To relate the time-varying parameters to previousOffline work using Lemma 2 [23 26] the following relationsare used
120572 = min (120572119896)
120583 = max (120583119896)
V1 = min (119887119896)
(70)
The bound from Case 1 is used for this comparison as shownin Figure 3
While the Offline estimation error bound is valid it isextremely loose and does not provide a realistic portrayalof the convergence of the estimation error This shows thatthe presented online method is useful for more closelydetermining the convergence and persistent error of the LKFbut is limited in that it cannot predict these bounds prior tothe filtering process and it cannot be used for Offline stabilityanalysis
6 Conclusions
This paper presented a modified stochastic stability lemmaand a Kalman filter convergence theorem which are newtools that can be used to quantify the performance ofKalman filters online Through an example it was shownthat this new convergence analysis method is effective indetermining an upper bound on the performance of theLKF Also useful information about the convergence of theparticular LKF algorithm can be calculated This analysisis applied during the filtering process thus providing thecapability for real-time convergence and performance moni-toring Different cases of noise covariance assumptions wereconsidered showing that increasing the assumed process
International Journal of Stochastic Analysis 9
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
noise tends to significantly slow the convergence of the filterand increase the persistent error bound while increasing theassumed measurement noise tends to slow the convergencebut decreases the persistent error bound Future work willinvolve extending this technique to nonlinear systems
Acknowledgments
This work was supported in part by NASA Grants noNNX10AI14G and no NNX12AM56A and NASA West Vir-ginia Space Grant Consortium Graduate Fellowship
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960
[2] H Liu S Shah and W Jiang ldquoOn-line outlier detection anddata cleaningrdquoComputers andChemical Engineering vol 28 no9 pp 1635ndash1647 2004
[3] S Lu H Lu and W J Kolarik ldquoMultivariate performancereliability prediction in real-timerdquo Reliability Engineering andSystem Safety vol 72 no 1 pp 39ndash45 2001
[4] D-J Jwo and T-S Cho ldquoA practical note on evaluatingKalmanfilter performance optimality anddegradationrdquoAppliedMathematics and Computation vol 193 no 2 pp 482ndash5052007
[5] J G Saw M C K Yang and T C Mo ldquoChebyshev inequalitywith estimated mean and variancerdquo The American Statisticianvol 38 no 2 pp 130ndash132 1984
[6] J L Maryak J C Spall and B D Heydon ldquoUse of the Kalmanfilter for inference in state-space models with unknown noisedistributionsrdquo IEEE Transactions on Automatic Control vol 49no 1 pp 87ndash90 2004
[7] J C Spall ldquoThe kantorovich inequality for error analysis of theKalman filter with unknown noise distributionsrdquo Automaticavol 31 no 10 pp 1513ndash1517 1995
[8] R E Kalman ldquoNew methods in Wiener filtering theoryrdquo inProceedings of the 1st Symposium on Engineering Applications ofRandom FunctionTheory and Probability J L Bogdanoff and FKozin Eds Wiley New York NY USA 1963
[9] R J Fitzgerald ldquoDivergence of the Kalman filterrdquo IEEE Trans-actions on Automatic Control vol 16 no 6 pp 736ndash747 1971
[10] H W Sorenson ldquoOn the error behavior in linear minimumvariance estimation problemsrdquo IEEE Transactions on AutomaticControl vol 12 no 5 pp 557ndash562 1967
[11] J J Deyst Jr and C F Price ldquoConditions for asymptotic sta-bility of the discrete minimum-variance linear estimatorrdquo IEEETransactions on Automatic Control vol 13 no 6 pp 702ndash7051968
[12] A H Jazwinski Stochastic Processes and Filtering Theory Aca-demic Press New York NY USA 1970
[13] K L Hitz T E Fortmann and B D O Anderson ldquoA note onbounds on solutions of the Riccati equationrdquo IEEE Transactionson Automatic Control vol 17 no 1 p 178 1972
[14] E Tse ldquoObserver-estimators for discrete-time systemsrdquo IEEETransactions on Automatic Control vol 18 no 1 pp 10ndash16 1973
[15] J J Deyst Jr ldquoCorrection to ldquoConditions for asymptotic stabilityof the discrete minimum-variance linear estimatorrdquordquo IEEETransactions on Automatic Control vol 18 no 5 pp 562ndash5631973
[16] J B Moore and B D O Anderson ldquoCoping with singulartransition matrices in estimation and control stability theoryrdquoInternational Journal of Control vol 31 no 3 pp 571ndash586 1980
[17] S W Chan G C Goodwin and K S Sin ldquoConvergence pro-perties of the Riccati difference equation in optimal filteringof nonstabilizable systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 2 pp 110ndash118 1984
[18] L Guo ldquoEstimating time-varying parameters by the Kalmanfilter based algorithm stability and convergencerdquo IEEE Trans-actions on Automatic Control vol 35 no 2 pp 141ndash147 1990
[19] J L Crassidis and J L Junkins Optimal Estimation of DynamicSystems chapter 5 CRC Press Boca Raton Fla USA 2004
[20] E F Costa and A Astolfi ldquoOn the stability of the recursive Kal-man filter for linear time-invariant systemsrdquo in Proceedingsof the American Control Conference (ACC rsquo08) pp 1286ndash1291Seattle Wash USA June 2008
[21] R G Agniel and E I Jury ldquoAlmost sure boundedness of ran-domly sampled systemsrdquo SIAM Journal on Control vol 9 no 3pp 372ndash384 1971
[22] T-J Tarn and Y Rasis ldquoObservers for nonlinear stochasticsystemsrdquo IEEE Transactions on Automatic Control vol 21 no4 pp 441ndash448 1976
[23] K Reif S Gunther E Yaz and R Unbehauen ldquoStochasticstability of the discrete-time extended Kalman filterrdquo IEEE Tra-nsactions on Automatic Control vol 44 no 4 pp 714ndash728 1999
[24] K Xiong H Y Zhang and C W Chan ldquoPerformance evalua-tion of UKF-based nonlinear filteringrdquo Automatica vol 42 no2 pp 261ndash270 2006
[25] Y Wu D Hu and X Hu ldquoComments on ldquoPerformance eva-luation of UKF-based nonlinear filteringrdquordquo Automatica vol 43no 3 pp 567ndash568 2007
[26] M Rhudy Y Gu and M R Napolitano ldquoRelaxation of initialerror and noise bounds for stability of GPSINS attitude estima-tionrdquo in AIAA Guidance Navigation and Control ConferenceMinneapolis Minn USA August 2012
[27] D SimonOptimal State EstimationWiley NewYork NYUSA2006
[28] J C Spall ldquoThe distribution of nonstationary autoregressiveprocesses under general noise conditionsrdquo Journal of Time SeriesAnalysis vol 14 no 3 pp 317ndash320 1993 (correction vol 14 p550 1993)
[29] J L Maryak J C Spall and G L Silberman ldquoUncertaintiesfor recursive estimators in nonlinear state-space models withapplications to epidemiologyrdquo Automatica vol 31 no 12 pp1889ndash1892 1995
[30] R VHogg JWMcKean andA T Craig Introduction toMath-ematical Statistics Pearson Education 2005
[31] E Kreyszig Advanced Engineering Mathematics Wiley NewYork NY USA 9th edition 2006
[32] F L Lewis Optimal Estimation Wiley New York NY USA1986
