Robust Adaptive Estimators for Nonlinear Systems

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1 Abdul Wahab Hamimi Fadziati Binti and Katebi Reza () Robust adaptive estimators for nonlinear systems. In: Conference on Control and Fault-olerant Systems (Sysol) his version is available at Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise eplicitly stated on the manuscript Copyright and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please chec the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaing activities or any commercial gain. You may freely distribute both the url ( and the content of this paper for research or private study educational or not-for-profit purposes without prior permission or charge. Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.u he Strathprints institutional repository ( is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs epose data about those outputs and enable the management and persistent access to Strathclyde's intellectual output.

2 Robust Adaptive Estimators for onlinear Systems H. F. Wahab and R. Katebi Industrial Control Centre Department of Electrical and Electronic Engineering University of Strathclyde G QE Glasgow UK. Department of Electrical Electronic & System Engineering Universiti Kebangsaan Malaysia 46 UKM Bangi Selangor Malaysia {r.atebi hamimi.abdul-wahab}@strath.ac.u Abstract his paper is concerned with the development of new adaptive nonlinear estimators which incorporate adaptive estimation techniques for system noise statistics with the robust H technique. hese include Etended H Filter (EHF) State Dependent H Filter (SDHF) and Unscented H Filter (UHF). he new filters are aimed at compensating the nonlinear dynamics as well as the system modeling errors by adaptively estimating the noise statistics and unnown parameters. For comparison purposes this adaptive technique has also being applied to the Kalman-based filter which include etended Kalman filter (EKF) state dependent Kalman filter (SDKF) and Unscented Kalman filter (UKF). he performance of the proposed estimators is demonstrated using a two-state Van der Pol oscillator as a simulation eample. I. IRODUCIO Filter tuning which is usually performed manually by a trial and error method can be relatively difficult as it is not simple to designate the right noise covariance matrices Q R []. Although robust estimation approaches could handle uncertainties from modeling errors and system noises the mean square error (MSE) of the output increases for H filter []. As a remedy a significant body of research has been committed to deal with this issue through adaptive filtering. Different schemes of traditional adaptive approaches for noise covariance estimation have been derived in a variety of methods; Bayesian [ ] maimum lielihood (ML)[4 5] covariance matching [6 7] and correlation techniques [8- ]. Other schemes that fall under off-line category that are available to estimate the noise statistics include subspace method [ ] and time series approach [4]. he Bayesian adaptive filter recursively obtain the a posteriori probability distribution function of the states and a vector of unnowns [5] while maimum lielihood method attempt to compute the unnown covariances by maimizing a lielihood function such as joint marginal and conditional estimates [5]. Both of these developments however are very computationally demanding. On the other hand covariance matching techniques generate the consistency between the covariances of the state estimate residuals and their theoretical covariances by increasing or decreasing the covariance of the state noises [7]. Meanwhile in the correlation method the output of the system is being correlated either directly or after a nown linear operation on it [8] using autocorrelation function of the output or the autocorrelation function of the innovation. he common disadvantage of all these on-line estimation is that they have been designed for the linear systems only. In this paper the Jazwinsi [7] approach for adaptive estimation is used to improve the performance of nonlinear filters. he system modeling errors and errors due to the neglected nonlinearities from linearization can be compensated by adaptively estimating the noise statistics and unnown parameters using the adaptive filters. he remainder of this paper is structured as follows: Section II presents a brief formulation of robust-based H filter. Section III describes an adaptive scheme that can find Q and R in real time even for nonlinear dynamics and observations building on the ideas of Jazwinsi. he comparison between the proposed filters is performed by simulation studies in Section IV. A general conclusion ends the paper. II. ROBUS-BASED H FILERS FORMULAIO he following discrete-time nonlinear equations are adopted: f ( u ) w () y h( ) v () n where is the state vector m y is the observation vector. w( ) ~ ( Q( )) and v( ) ~ ( R( )) are process noise and measurement noise. A. Etended H filter (EHF) Suppose the standard nonlinear discrete-time system as in equation ()-(). Following is a complete solution to the discrete H filtering [6]: f ( u ) () P F P F Q (4) z L (5) K P H H P H R (6) K y H (7)

3 H P P P H L R P e L where F and nonlinear functions f and h. (8) H denotes the Jacobian matrices of the I with I is an identity matri of appropriate dimension. he matri Re is defined as R H Re P H L I L (9) he purpose of this filter is to find the estimation of z using the measurements of y such [7]: sup w vl L e j j P j j Q j j R w v () where is a given scalar. In order to adjust the filter performance two weighting positive scalars Q and R chosen by the designer and can be select practically as the covariance matrices of the process and measurement noises need to be tuned. he act as a tuning parameter to control the tradeoff between H performance and minimum variance performance where the etended H filter reduces to the etended Kalman filter when. Previously proposed by [8 9] for discretetime linear system a method to adjust to its minimum is adopted [7] where P P H R H I () he above terms yields ma eig P H R H () where the maimum eigenvalue of the matri A ma eig( A). herefore can be opt as represented by ma eig P H R H () where is a scalar larger than one. B. State Dependent H filter (SDHF) Aiming to combine the advantages of both State dependent filter (SDF) and EHF SDHF employs a state-dependent model where the equation can be represented by statedependent coefficient (SDC) form as A( ) B( ) u w (4) y C ( ) v y respectively where A( ) B( ) and C ( ) are the state y (5) dependent matrices. Defining z C (6) z Cz I where I is an identity matri of appropriate dimension. his filter then employs the H design technique to estimate the system state and given by A( ) B( ) u (7) P A( ) P A( ) Q (8) K P C ( ) C ( ) P C ( ) R (9) y y y K y C ( ) () y C ( ) P P P C ( ) C R P y y z e Cz where R Cy ( ) Re P Cy ( ) C z I C z Similar method to EHF should be used to find the value of. C. Unscented H filter (UHF) UHF integrates the advantages of both UKF and EHF. he algorithms of this filter are given by where: ime update: () () ( i) *( i) i... n () *( i) ( np ) i i... n *( ni) np i i... n ( i) ( i) ( i) f ( u t ) n i (4) () i n (5) ( i) ( i) n n ii P Q (6) Measurement update: ( i) ( i ) y h t (7) y n n () i y n (8) i ( i) ( i) Py y y y y n (9) ii

4 n ( i) ( i) P y y () y n i Approimating the measurement covariance P H P H y and cross-correlation Py P H using the statistical linear error propagation [] the filtered estimates and the remaining UHF equations can be rewritten as where y y P R P y y () P y P P Py P R e P R e R P y P y Py I Py () () Using the equation () in EHF and approimate crosscorrelation covariance P P H the value of can be derive [7]: y y y ma eig P P P R P P (4) III. OISE ADAPIVE ESIMAOR FORMULAIO he estimators presented in the following section are nonlinear adaptive algorithms which are revised from the linear adaptive algorithm. his section presents the incorporation of the proposed adaptive filtering algorithms with the robust-based filters presented earlier for more enhanced nonlinear filtering algorithms. he objective of the integrated adaptive nonlinear filters is to tae into consideration the erroneous time-varying noise statistics of dynamical systems as well as to compensate the nonlinearity effects neglected by linearization. Generally in different conditions Q and R are changing. hese noise covariances reflect the uncertainties or discrepancies between the assumed dynamic model and the actual re-entry phenomena. herefore determining the suitable values of R and Q plays an important role to obtain a converged filter []. he residuals of the Kalman-based filter should be a zero-mean white noise process if it is based on a fully and ideally tuned model. If the residuals are not white noise the filter does not operate optimally and this implies the poor design. A good way to verify whether the filter needs tuning is to monitor the residuals. he wor presented herein is motivated by Jazwinsi [7 ] who treated the case of a scalar Gaussian measurement noise with a single predicted residual processed by an adaptive filter. A scheme for updating Q which appears to have such adaptive features is presented here. he predicted residuals are defined to be r( l) y( l) y( l) y( ) l (5) By use of the Kalman filter relations this equation may be written as r( l) H( l) F( l ) ( ) ( ) l H ( l) F( l i) w( i ) v( l) i It follows directly that E r( l) r( m) H ( l) F( l ) P( ) F ( i ) H ( m) l H ( l) F( l i) Q( i l) i F ( m i) H ( m) R( l) (6) (7) o generate consistency between actual covariance (residuals) and their theoretical covariances (statistics): r r (8) Assuming the process noise covariance by a scalar parameter q and epressly Q qi equation (7) and (8) will yield: r H P H qh H R (9) for the case where one residual is used. hen q may be recursively formed according to: q r r q H H if q otherwise (4) he estimator of equation (4) is of little statistical significance since it is based on a single residual. However by employing a sliding window to compute the sample mean for predicted residuals; the problem of using a single residual is overcome. Jazwinsi (Jazwinsi 969) demonstrates that for the subsequent sample mean: r mr / l Rl (4) and attain the following estimator by maimizing ( m r ) : where and q mr mr q if q S otherwise m r q S F P F S S S S S S... S S (4) (4)

5 S H F / l l l Rl S H F S / l l l Rl / l R H (44) he length of the moving window employed to update q has to be choose wisely. he algorithm will place more emphasis on fitting the incoming data than fitting the previous data and it might never converge if the window size is too small. On the other hand if the window size is too large then the algorithm will fail to fit promptly to new situations. In this study deciding the right window length is done manually. he estimate of the plant noise variance to be used in the adaptive filter is then Q q I (45) his algorithm taes care of the problem of filter divergence and stiffness observed when the filter becomes overconfident and the impact of incoming observations is very limited. he Jazwinsi algorithm detects such behaviour and action is taen in order to modify the sensitivity of the filter by inflating the system noise variance in a suitable manner. We can also obtain the following estimator for covariance R where: mr mr q if q r S otherwise (46) he results of Jazwinsi [7] approach from the adaptive estimation field will be embedded to improve the eisting algorithms with nonlinear filters presented in section II and III. Using the integrated filters the system modeling errors and errors due to the neglected nonlinearities from linearization can be compensated by adaptively estimating the noise statistics and unnown parameters heoretically an adaptive filter can estimate both the system and the measurement errors. However it is not easy to distinguish between errors in Q and R; thus maing adaptive filtering algorithms that attempt to update both the system and the measurement noises are not robust. Since the measurement noise statistics are relatively recognized compared to the system model error the adaptive estimation of the process noise covariance Q is considered in this paper. Equations (4) - (45) are integrated with the robust-based filter presented in section II and alman-based filter (because of limited spaces these algorithms are not shown here) where new adaptive nonlinear filters are proposed and named as Adaptive Etended H filter (AEHF) Adaptive State Dependent H filter (ASDHF) and Unscented H filter (AUHF) Adaptive Etended Kalman filter (AEKF) Adaptive State Dependent Kalman Filter (ASDKF) and Adaptive Unscented Kalman filter (AUKF). IV. UMERICAL SIMULAIO o eemplify the performance improvement of the proposed adaptive robust and Kalman-based filter techniques over the usual standard filters the Van der Pol oscillator is considered to investigate the performance of the filters and also to compare the robustness of the estimator due to noise errors using MALAB/Simulin facilities. Consider the following discrete-time model of the Van der Pol Oscillator: ( 9 ) where.5 is the sampling time and. Let the output y be given by y v w (47) (48) he Jacobian for the process and measurement are defined as: F ( 9 (49) H while state dependent coefficient (SDC) are as follow: h A( ) B( ) 9 C( ) D( ) (5) Process noise with a covariance of I and measurement noise with a covariance of.5i is added to the system states and measurements respectively. he initial state is. he initial state estimate is chosen as chosen o 6 and the initial state covariance matri P o and o the value of Q and R are chosen as 5 I. I and.5 respectively In order to validate the effect of new adaptive filters in the case of which process and measurement noises are not really nown incorrect values noise covariance are introduced to these methods to eamine their potential to etract the real values. he adaptive filter with estimator equation (4) - (45) are simulated on a noisy measurement. he length of the sliding window of the adaptive noise algorithms was set to time steps. o confirm these results Monte Carlo simulations with 5 runs are eecuted. he mean square error (MSE) are defined as follows: MC MSE (5) MC j ( j)

6 where MC is the number of Monte Carlo simulations. Figure -Figure shows the performance comparison of the filters in terms of their mean of MSEs. Figure and Figure 4 plot the MSEs for 5 random runs when using the standard nonlinear filters and the new adaptive nonlinear filters for incorrect process noise. It shows clearly that the proposed new adaptive method is the best choice for practical cases in which the real values of noise variances are unnown. Mean of MSEs Mean of MSEs EKF SDKF UKF EHF SDHF UHF Standard Adaptive Filters Figure. Performance comparison of the filters in terms of their mean of MSEs for EKF SDKF UKF EHF SDHF UHF Standard Adaptive Filters Figure. Performance comparison of the filters in terms of their mean of MSEs for V. COCLUSIO he simulations indicate that under incorrect noise information the new adaptive nonlinear filters provide a more accurate estimate than that of the standard nonlinear filters (EKF SDKF UKF EHF SDHF UHF). It is also shown that the adaptive robust-based H filters (AEHF ASDHF AUHF) are more robust than the usual adaptive alman-based filters (AEKF ASDKF AUKF). herefore less noise contaminated residuals can be obtained to design a more accurate fault detection and isolation system. REFERECES [] A. H. Jazwinsi. (97). Stochastic Processes and Filtering heory. 64. [] C. G. Hilborn and D. G. Lainiotis "Optimal Estimation in the Presence of Unnown Parameters" Systems Science and Cybernetics IEEE ransactions on vol. 5 pp [] D. L. Alspach et al. "A Bayesian solution to the problem of state estimation in an unnown noise environment " International Journal of Control vol. 9 pp // 974. [4]. Bohlin "Four Cases of Identification of Changing Systems" Computational Methods for Modeling of onlinear Systems vol. 6 p [5] R. Kashyap "Maimum lielihood identification of stochastic linear systems" Automatic Control IEEE ransactions on vol. 5 pp [6] K. Myers and B. apley "Adaptive sequential estimation with unnown noise statistics" Automatic Control IEEE ransactions on vol. pp [7] A. H. Jazwinsi "Adaptive filtering" Automatica vol. 5 pp [8] R. Mehra "Approaches to adaptive filtering" Automatic Control IEEE ransactions on vol. 7 pp [9] R. Mehra "On the identification of variances and adaptive Kalman filtering" Automatic Control IEEE ransactions on vol. 5 pp [] B. Carew and P. Belanger "Identification of optimum filter steady-state gain for systems with unnown noise covariances" Automatic Control IEEE ransactions on vol. 8 pp [] B. J. Odelson et al. "A new autocovariance least-squares method for estimating noise covariances" Automatica vol. 4 pp [] M. Viberg "Subspace-based methods for the identification of linear time-invariant systems" Automatica vol. pp [] P. Van Overschee and B. De Moor "A unifying theorem for three subspace system identification algorithms" in American Control Conference pp vol.. [4] G. E. P. Bo et al. ime series analysis: forecasting and control vol. 74: Wiley. [5] B. J. Odelson "Estimating disturbance covariances from data for improved control performance" UIVERSIY OF WISCOSI. [6] G. A. Einice and L. B. White "Robust etended Kalman filtering" Signal Processing IEEE ransactions on vol. 47 pp [7] W. Li and Y. Jia "H-infinity filtering for a class of nonlinear discrete-time systems based on unscented transform" Signal Processing vol. 9 pp. -7. [8] S. Xuemin and D. Li "A dynamic system approach to speech enhancement using the H filtering algorithm" Speech and Audio Processing IEEE ransactions on vol. 7 pp [9] D. Labarre et al. "Dual H Algorithms for Signal Processing- Application to Speech Enhancement" Signal Processing IEEE ransactions on vol. 55 pp [] G. Sibley et al. "he iterated sigma point alman filter with applications to long range stereo" in Proceedings of Robotics: Science and Systems 6 pp [] A. Mohamed and K. Schwarz "Adaptive Kalman filtering for IS/GPS" Journal of Geodesy vol. 7 pp [] A. Jazwinsi and A. Bailie "Adaptive filtering Interim report" ASA echnical Reports (March 967)967.

7 EKF AEKF.5.4 SDKF ASDKF UKF AUKF MSE of.5.4 MSE of.. MSE of MSE of EHF AEHF MSE of SDHF ASDHF MSE of UHF AUHF Figure. MSE for state across 5 random runs 7 6 EKF AEKF. SDKF ASDKF 6 5 UKF AUKF MSE of 5 4 MSE of MSE of MSE of.5.5 EHF AEHF MSE of SDHF ASDHF MSE of UHF AUHF Figure 4. MSE for state across 5 random runs