Parameter Estimation for Dynamic System using Unscented Kalman filter

Transcription

1 Parameter Estmaton for Dynamc System usng Unscented Kalman flter Jhoon Seung 1,a, Amr Atya F. 2,b, Alexander G.Parlos 3,c, and Klto Chong 1,4,d* 1 Dvson of Electroncs Engneerng, Chonbuk Natonal Unversty, South Korea 2 Department of Electroncs Engneerng, Calro Unversty, Egypt 3 Department of Mechancal Engneerng, exas A&M Unversty, USA 4 Advanced Research Center for Electroncs and nformaton, JBNU, Jeonju, South Korea a seungjhoon@gmal.com, b amr@alumn.caltech.edu, c aparlos@tamu.edu, d* ktchong@jbnu.ac.k Abstract. In ths paper, we present a method of determnng the parameters of a dynamc system usng state estmate flter. State estmate flters such as Extended Kalman flter and the Unscented Kalman flter are wdely used to estmate the status n robot and GPS navgaton systems. he method of determnng the parameters s dffcult because of non-lnearty and complexty of the dynamc systems. he status and parameters are smultaneously estmated to decde the parameter value based on the measurement data set of the system. he work was appled to a coupled tank. he mentoned system was modeled and then setup for computer smulatons and experments purposes to test the performance of the estmaton process. Keywords: Parameter Estmaton, Extended Kalman Flter, Unscented Kalman Flter, Coupled ank. 1 Introducton Most of the mechancal and electrcal systems currently used n ndustry can be represented as second-order dynamcs model comprsed of mass, damper, sprng, nductances, resstors and capactors for convenence and ease of control. However, the parameters of a real system have measurement nose due to the dffcultes of physcal measurement. A change n parameter values can also happen due to the deteroraton of the system and sudden change of the envronment. It s for these reasons that we determne the optmal parameters of the system usng flters such as the extended kalman flter (EKF) or the unscented kalman flter (UKF). A coupled tank was used for the expermental tests. It was constructed wth a NI DAQ board and smulated n MALAB to apply the estmate flters on a real system. Computer smulaton for parameter estmaton was performed usng the dynamc equaton derved for the system. he results were analyzed wth varous mplementatons of EKF and UKF sutable for non-lnear system. IS 213, ASL Vol. 23, pp , 213 SERSC

2 Proceedngs, he d Internatonal Conference on Informaton Scence and echnology he unscented kalam flters and the coupled tank along wth the non-lnearlty of the system are descrbed n Secton 2. he computer smulaton results and analyss are outlned n Secton 3. Secton 4 of Concluson dscusses the estmaton results and the conclusons drawn. 2 Background he estmaton method s wdely used n engneerng applcatons such as for the state of a dynamc system, the atttude of a robot, navgaton system, etc. he estmate flters used for estmatng the state and parameters of a dynamc system are descrbed n ths secton. 2.1 Unscented Kalman Flter he UKF based on U wthout the lnearzaton process could effectvely be appled to non-lnear systems. And then U has outstandng approxmaton n nonlnear equaton. xk = f( xk 1, k 1) + wk 1 (1) z = hx (, k) + v (2) k k k It gernerally has a better performance than EKF, because t s not necessary to calculate the jacoban matrx of system equaton and then shows the mprovng approach for non-lnear system [1]. he UKF algorthm s as follows; Intalze wth: xˆ = Ex [ ] (3) P = E[( x xˆ )( x xˆ ) ] (4) For k {1,, } Calculate sgma ponts: me update: K 1 xˆ k 1 xˆ k 1 n Pk 1 xˆ k 1 n Pk 1 χ = [, + ( + λ), ( + λ) ] (5) χ = F[ χ k 1] (6) x ˆ ( m) = w kk 1 = (7) = ( c) [ χ ˆ 1 ][ kk χ ˆ 1] kk (8) Ζ H[ χ ] (9) P = w x x 241

3 Parameter Estmaton for Dynamc System usng Unscented Kalman flter z ˆ ( m) = w Ζ kk 1 = (1) Measurement update equatons: P zz ( c) 1 [ 1 ˆ 1 ][ 1 ˆ 1] kk = w kk z kk kk z kk = (11) xz P () c 1 [ 1 ˆ 1 ][ 1 ˆ 1] kk = w χ kk xkk Z kk zkk = (12) K P xz zz 1 kk 1 ( ) (13) x ˆ kk x ˆ K ( z k zˆ ) (14) P zz kk K ( ) K (15) where Pk s state error covarance, ( m) ( c), w w are the weghts. xz Pk s correlaton error covarance, Fg. 1 Coupled ank system at rght sde and block dagram of coupled tank at left sde. 2.2 System he dynamc system and the state and parameter estmaton process are descrbed n ths chapter. he dynamc system s setup n order to obtan the system measurement data. he equaton for the system s derved va mathematcal methods. he derved equaton s transformed nto state space equaton form. he dynamc system tested s the coupled tank. he system has second order ordnary dfferental equaton (ODE). he orfce coeffcent parameter s mportant for the stablty and control of the system. he coupled tank shown n Fg.1 s manufactured by Quanser nnovate. Educate Co. he system was constructed to control the pump but s used to deal wth parameter 242

4 Proceedngs, he d Internatonal Conference on Informaton Scence and echnology estmaton n our experments [2]. he numercal model of Coupled tank s comprsed of an nput and output functon s as follows; he nflow quantty of water n the tank 2 s same as outflow quantty of water n the tank 1. herefore, the total equaton that s ncludng the nput from the pump s derved. dh1 q1- c1 h1 - A1 = (16) dt dh2 c1 h1 - c2 h2 - A2 = (17) dt q = kv (18) where k s [cm/volts] constants, and v s the pump nput, and A 1, A 2 denote the nsde areas of tanks. In order to transform the equaton adapted the flters, dh we are defned as 1 A1 Ah 1 1 dt = & dh, 2 A2 = Ah& 2 2. And then t arrangs the ether dt sdes of equaton from the equatons (16) and (17), s gven by (19). A h& = kv - c h Ah& = c h - c h (19) he state space matrx form s obtaned to apply to the flters. é ê kv - c h ù ú 1 1 éh & ù 1 A ê 1 ú = êh & ë2úû c1 h1 - c2 h2 ê A ú ë 2 û (2) he parameter of the dynamc system s represented n able 1. Most of parameter s drectly measured value or the ntal value when t s constructed. able 1. Parameters of Coupled tank Descrpton Sym. Val. Unt Cross-sectonal areas of tank 1 A cm Cross-sectonal areas of tank 2 A cm Pumpng rate q 5 Volts Flow constant k 6 cm / Volts Orfce coeffcent of tank 1 c 1 5 Orfce coeffcent of tank 2 c

5 Parameter Estmaton for Dynamc System usng Unscented Kalman flter 3 Experments 3.1 State and Parameter Estmaton he state and parameter estmaton result s shown n Fg.2. he estmaton error at the ntal part of the process s emphaszed as shown. he algorthms for EKF and UKF estmate the state value on the measurement data. he measurement error n ank 1 occurred when the pumped water dropped nto the tank. In Fg.2, each sold lne s the measurement data, the results are represented as sold-sold lne for EKF and sold-dot lne for UKF. he unexpected state error happened due to the ntal water pressure dropped from the pump. Accordng to that error, the estmaton value that s the orfce coeffcent ntally ncreased but stablzed afterward through recursve update. Fg. 2 Result of State Estmaton for Coupled ank on left sde and Result of Estmaton for Orfce Coeffcent of ank 1 and ank 2 on rght sde. 3.3 Comparson he estmaton error and results are analyzed, calculated and compared usng mathematcal methods. he estmaton performance s analyzed usng objectve results of the processng tme of the flters. Based on the computer smulaton and expermental results of EKF and UKF from the prevous secton, the performance of each s analyzed through ther RMSE. able 2 denoted the result of the parameter estmaton of coupled tank system. where value s the estmated value, RMS s calculated after the converged tme at reference value. able 2. Result of Parameter Estmaton for Coupled ank EKF (RMS) UKF (RMS) Unknown parameter Value Error Value Error c c

6 Proceedngs, he d Internatonal Conference on Informaton Scence and echnology he experment results for Couple ank s shown n able 3. able 3. Result of Parameter Estmaton for Coupled ank Unknown parameter EKF (RMS) UKF (RMS) Value Error Value Error c c Concluson For ths research, models were derved, dynamc systems were constructed and the computer smulaton and expermental tests were performed. he measurement of data systems were collected through NI DAQ Board and lnked to MALAB Smulnk. he state and parameters are estmated n order to analyze the performance of extended kalman flter and unscented kalman flter. he orfce coeffcents of a coupled tank that affects system stablty were estmated for ths research. As the results of the accuracy of estmaton, UKF shows a outstandng performance for estmatng the state and parameter. On the other hand, the results of EKF estmaton have a lot of estmaton error because of lnearzaton of state equaton. hs research was able to verfy the accuracy and relablty of UKF. Acknowledgements hs work was supported by the Natonal Research Foundaton of Korea (NRF) grant funded by the Korea government (MES) (No ) and (No ) References 1. S. Juler, J. Uhlmann, "A new extenson of the Kalman flter to nonlnear systems", n: Proceedngs of the 1997 SPIE AeroSense Symposum, SPIE, Vol. 368, pp , englsh&pcat_code=exp-spe&prod_code=s13-tanks&tmpl=1 245