Journal of Mechanical Engineering and Automation (): 6- DOI: 593/jjmea Cramér-Rao Bounds for Estimation of Linear System oise Covariances Peter Matiso * Vladimír Havlena Czech echnical University in Prague Faculty of Electrical Engineering Department of Control Engineering Abstract he performance of Kalman filter depends directly on the noise covariances which are usually not nown and need to be estimated Several estimation algorithms have been published in past decades but the measure of estimation quality is missing he Cramér-Rao bounds represent limitation of quality of parameter estimation that can be obtained from given data In this article he Cramér-Rao bounds for noise covariance estimation of linear time-invariant stochastic system will be derived wo different linear system models will be considered Further the performance of earlier published methods will be discussed according to the Cramér-Rao bounds he analogy between the Cramér-Rao bounds and the Riccati equation will be pointed out Keywords Cramér-Rao bounds Kalman Filter oise covariance estimation Introduction he Cramér-Rao (CR) bounds represent a limitation of the estimation quality of unnown parameters from the given data If the variance of the estimates reaches CR bounds it can be stated that the estimation algorithm wors optimally In this paper we will derive CR bounds for the estimation of noise covariances of a linear stochastic system he noise covariances are tuning parameters of Kalman filter and the filter performance depends on them However they are not nown in general and hence tuning of a Kalman filter remains a challenging tas he very first methods for noise covariance estimation were published in 7s by Mehra[] Recently new methods were proposed by Odelson et al [3] and further modified by Rajamani et al[45] and Aesson et al[6] A maimum lielihood approach was described in[7] In Section 5 these two approaches will be compared to the CR bounds and their performance will be discussed In the last part of the paper relationship between the Riccati equation and the CR bounds will be pointed out Deriving of the noise covariance CR bounds is based on[8] and[9] Additional information about CR bounds and Bayesian estimation can be found in[] Cramér-Rao Bounds * Corresponding author: petermatiso@felcvutcz (Peter Matiso) Published online at http://journalsapuborg/jmea Copyright Scientific & Academic Publishing All Rights Reserved his paper was presented at the 8 th IFAC World Congress Italy [online] http://wwwifac-papersonlinenet/detailed/545html he tas is to estimate a vector of parameters from a set of measured data y y y y he upper inde emphasizes that vector y contains data samples he conditioned probability density function (pdf) p( y ) is assumed to be nown Further the nabla operator is defined as () he Fisher information matri (FIM) J( ) for parameters is defined as ( ) ln J p y y () where operator y is the conditioned epected value defined as f( ) f( ) p( y ) dy (3) y Alternatively we can define a posterior FIM (and posterior CR bounds) using joint probability p( y ) instead of the conditioned one where p( y ) p( y ) p Probability function p represents prior information about the parameter vector Assume having an unbiased estimate of parameter vector ˆ obtained by an arbitrary method When the true value of parameter vector is compared to the unbiased estimated set the following inequality holds [8] where J represents the Cramér-Rao bound Inequality A B means that A B is a positive semidefinite matri Based on (4) the following statement can be (4)
Journal of Mechanical Engineering and Automation (): 6-7 concluded Assuming that CR bound is reachable any estimation algorithm woring optimally (in the sense of the smallest covariance of obtained estimates) must give estimates whose variance is equal to the Cramér-Rao bound If the CR bound is reachable then the optimal result can be achieved by a maimum lielihood approach Consider a scalar linear stochastic system a v (5) y c e where and y represent the state and the measured output respectively Stochastic inputs v and e are described by normal distributions vt () Q et () R (6) Using the notation of[9] the logarithms of the conditioned probability of the state measurements and unnown parameters can be epressed as y py i i ln p ln i (7) ln p ln i p i i he following matrices are used for FIM calculation K ln p (8) K ln p (9) K ln p () K ln p () K ln p () K ln p (3) L y ln p y (4) L ln p y y (5) L ln p y y (6) Further recursive formulas for FIM of the state and unnown parameters are defined as J J L K J K K (7) J K K K where J K J K (8) and K K J J L J J L (9) () Initial conditions of the recursive algorithm are set as J K J K ln p () J A ln p θ θ θ θ where K and A represent prior information If no prior probability function is nown then A K he final form of FIM of the state and parameters is J J J () J J 3 Cramér-Rao Bounds for oise Covariances Estimation Consider a scalar stochastic system given by (5) where a he noise sources are not correlated and are defined by (6) he probabilities used in (7) can be defined as p Q a Q (3) p y R c R ow matrices ()-(6) can be calculated using the following general integral formulas assuming e d ( ) e d ( ) ( ) e d e d 3 4 Further logarithm of the Gaussian function is given by ln ( P) ln ln P (5) P CR bounds of Q R estimates are to be found considering a scalar system with two unnown parameters he nabla operator in this case is of the form (6) Q R where Q R At the first step the arguments of the epectation operators in ()-(6) are calculated as follows (4)
8 Peter Matiso et al: Cramér-Rao Bounds for Estimation of Linear System oise Covariances a a ln p Q (7) a ln p Q (8) a 3 ln p Q Q (9) ln p y y c (3) 3 R R et matrices ()-(6) are obtained using integral formulas (4) as K (3) i K i (3) K Q (33) L (34) L R (35) A ln p (36) It can be seen from (3)-(35) that several of the matrices are zero which significantly simplifies formula (7) In par- ticular matri J in () is zero and this fact allows us to calculate FIM for unnown covariances independently on the states he resulting FIM can be written as J J L K (37) where J represents the prior information about unnown parameters Q R If no prior information is available it can be set to zero From the above equations and the form of matrices in (3)-(35) it can be seen that FIM is calculated for each covariance independently Formula for FIM is of the form J J Q (38) R where it can be assumed that J A particular closed form solution to recursive formula (38) is given by J Q R 4 Cramér-Rao Bounds for a System in Innovation Form (39) In Section a stochastic system was defined using (5) he system has two independent sources of uncertainty the process noise and the measurement noise It is assumed the process and the measurement noises to be normal with zero mean and covariances Q R respectively In non-scalar case variables Q R are matrices and contain redundant information oise covariances represent properties of the noise sources Considering a higher order system allows us to model the same number of noise sources as the sum of numbers of states and outputs Matri Q would have the same dimension as the number of states and matri R would have a dimension equal to the number of outputs However from the observed output data only the number of noise sources equal to the number of outputs can be recognized herefore we can alternatively use the so-called innovation form of stochastic system that represents a minimal form[] or [] he disadvantage of such formulation comes from the fact that we loose the physical bacground of the noise sources and their structure However in many cases we only have output data at hand and no other additional information about the stochastic inputs In such case the innovation form of the stochastic system can be used a K (4) y c where K is a steady state Kalman gain and () is the stationary innovation process hen ( ) R (4) he Cramér-Rao bounds for K R ε estimation are derived using the same formulas as for the system (5) Analogously to formulas (7)-(3) we define a 3 a KR ln p a (4) a KR a KR ln p (43) a KR ln p a a 4 3 K RK RK (44) a a 3 3 RK R RK
Journal of Mechanical Engineering and Automation (): 6-9 ln p y c y (45) 3 R R he epected values are calculated using integral formulas (4) Matrices ()-(6) are of the form K (46) K (47) K KR K (48) KR R L (49) L (5) R he formula for FIM calculation is of the form (37) Substituting (48) and (5) to (37) leads to the resulting formula K KR J (5) KR R variance of the estimates is calculated he dependence of the variance of the estimated Q-parameters on the number of used data samples is shown in Figure J(Q) - 8 6 4 8 6 4 4 6 8 4 Figure Cramér-Rao bound for Q estimation and variance of estimated Q using ALS and Maimum lielihood method In Figure an analogous result for R estimation is shown We can state that ML method converges faster to the CR bound than ALS ML algorithm gives better estimates in case of covariance Q than ALS even with very small sets of data (around 5 samples) 9 8 7 6 CR bound ML method ALS method CR bound ML method ALS method 5 Comparison of the Methods for oise Covariance Estimation J(R) - 5 4 3 In the previous sections we derived the CR bounds for noise covariances estimation In this chapter a comparison of ALS (Autoregressive least squares) method and maimum lielihood method will be presented he ALS method [3-5] uses the assumption that estimated autocorrelation function calculated from the finite set of data converges to the true autocorrelation function In[7] it is shown that the assumption of fast convergence is not correct and this fact can significantly limit the quality of the estimates In the same paper a maimum lielihood approach is demonstrated using a simple maimum seeing method We can compare performance of both approaches according to the CR bounds he ALS algorithm prior setting contains the optimal Kalman filter gain and the maimum lag for autocorrelation calculation d 5 he ML method searches the maimum in the interval to he grid is logarithmic and contains 8 8 points Consider a system of the form (5) where a 5 c Q and R he estimation algorithm uses time samples and is repeated 3 times hen the resulting 4 6 8 4 Figure Cramér-Rao bound for R estimation and variance of estimated R using ALS and Maimum lielihood method 6 Relationship between Cramér-Rao Bounds and Riccati Equation In this section we will point out the relationship between Riccati equation (RE) and CR bounds It is well nown that under some conditions Kalman filter is optimal It was stated in the previous section that CR bounds represent a limit of estimation quality herefore it can be epected that RE should converge to CR bounds for state estimation of a linear system Using formulas (8)-() and (4) the CR bounds for state estimation can be easily obtained ichavsý et al (998) We have
Peter Matiso et al: Cramér-Rao Bounds for Estimation of Linear System oise Covariances K A Q A K A Q K Q L C R C (5) ow the recursive formula for FIM calculation Šimandl et al () is of the form J K K J K L K (53) Using (5) we obtain resulting formula J Q Q A J A Q A C R C A Q (54) Riccati equation is of the form P AP A AP C CP C R CP A Q (55) It can be stated that P and matri Moreover if initial values J converge to the same P J then trajectories of P and J are the same his fact proves the optimality of Kalman filter Equivalency of equations (54) and (55) is proved in Appendi 7 Conclusions In this paper Cramér-Rao bounds for the noise covariance estimation were derived for two different stochastic models From (3)-(35) it can be seen that matrices K K and L are zero that means that the information between state and noise covariances is not correlated It can be concluded that the covariances and the states can be estimated independently of each other Another observation can be done on matrices (33) and (35) It can be seen that there is no correlation between information about the covariances hat means that CR bounds can be calculated for each covariance separately Considering limit cases Q converging to zero and R being arbitrary large CR bound for Q estimation will converge to zero either It means that there eists a method that can estimate Q up to an arbitrary level of accuracy CR bounds can be used as a quality measure for all newly proposed estimating algorithms We have compared a maimum lielihood approach with ALS method of Odelson et al (5) according to the CR bounds Based on these eamples (Figure and Figure ) it can be observed that the CR bounds for the noise covariances converge to zero relatively fast even for small sets of data ( to 4 samples) In Section 6 we pointed out a relationship between Fisher information matri recursive formula and Riccati equation Equivalence of the equations is proved in Appendi We concluded that this is another proof of optimality of Kalman filter APPEDIX Consider two recursive equations (56) and (57) that are Fisher information matri recursive equation and discrete Riccati equation Both equations are equivalent and P J for sufficient large not necessarily J Q Q A J A Q AC R C A Q (56) P AP A AP C CP C R CP A Q (57) he right side of equation (56) is modified using substitution J P Q Q A J A Q AC R C A Q Q A A QA P A Q AC R C A Q Q A A QA P A Q AC R C P A Q AC R C A Q Q A A QA P C R C P A Q AC R C A Q ow both sides of the equation are inverted so the left side is P and the right side is of the form Q A A QA P C R C P AQ ACR C AQ QA P A Q A C R C P CR C A P CR C AQ A A Q QA P A Q A C R C AP CR C A Q Further an inversion lemma (58) will be applied on the epression A BD C A A B D CA B CA (59) he final formula is of the form P CR C P AP A AP C CP C R CP A Q which is eactly Riccati equation ACKOWLEDGEMES his wor was supported by the Czech Science Foundation grant o P3//353 REFERECES [] Mehra RK (97) On the identification of variances and adaptive Kalman filtering IEEE rans on Automat Control AC 5 pp 75 84 [] Mehra R K (97) Approaches to Adaptive Filtering IEEE
Journal of Mechanical Engineering and Automation (): 6- rans on Automat Control AC 7 pp 693 698 [3] Odelson B J Rajamani M R and Rawlings J B (5) A new autocovariance least squares method for estimating noise covariances Automatica Vol 4 pp 33 38 [4] Rajamani M R and Rawlings J B (8) Estimation of the disturbance structure from data using semidefinite programming and optimal weighting Automatica Vol 45 pp 7 [5] Rajamani M R and Rawlings J B [online] http://jbrwwwc hewiscedu/software/als [6] Aesson B M Jorgensen J B Poulsen K and Jorgensen S B (8) A generalized autocovariance least squares method for Kalman filter tuning Journal of Process control Vol 8 pp 769 779 [7] Matiso P and Havlena V () oise covariances estimation for Kalman filter tuning Proceedings of IFAC AL- COSP urey [8] ichavsý P Muravchi C H and ehorai A (998) Posterior Cramér-Rao Bounds for Discrete-ime onlinear Filtering IEEE rans on Signal Processing Vol 46 pp 386-396 [9] Šimandl M Královec J and ichavsý P () Filtering predictive and smoothing Cramér-Rao bounds for discrete-time nonlinear dynamic systems Automatica Vol 37 pp 73 76 [] Candy J V (9) Bayesian Signal Processing John Wiley & Sons Inc USA [] Ljung L (987) System Identification he heory for he User Prentice Hall USA [] Kailath Sayed A H and Hassibi B () Linear Estimation Prentice Hall USA [3] Anderson B D O and Moore J (979) Optimal filtering Dover Publications USA [4] Matiso P (9) Estimation of covariance matrices of the noise of linear stochastic systems Diploma thesis Czech echnical University in Prague [5] Matiso P and Havlena V () Cramér-Rao bounds for estimation of linear system noise covariances Proceedings of the 8 th IFAC World Congress Italy [6] Papoulis A (99) Probability Random Variables and Stochastic Processes hird edition Mc Graw-Hill Inc USA