Design and Analysis of Bayesian Model Predictive Controller

Transcription

1 Computer and Informaton Scence; Vol. 7, No. 3; 014 ISSN E-ISSN Publshed by Canadan Center of Scence and Educaton Desgn and Analyss of Bayesan Model Predctve Controller Yjan Lu 1, Wexng Qan 1 & Lmng D 1 1 School of Electrcal and Automaton Engneerng, Nanjng Normal Unversty, Nanjng, Chna Correspondence: Yjan Lu, School of Electrcal and Automaton Engneerng, Nanjng Normal Unversty, Nanjng, Chna. Tel: E-mal: 63055@njnu.edu.cn Receved: June 9, 014 Accepted: June 16, 014 Onlne Publshed: July 5, 014 do: /cs.v7n3p58 URL: The research s supported by the the Jangsu Government Scholarshp for Overseas Studes (013), the Natonal Natural Scence Foundaton of Chna under Grant and ,the Innovaton Program of Shangha Muncpal Educaton Commsson under Grant 14ZZ087, the Pujang Talent Plan of Shangha Cty, Chna under Grant 14PJ and the Natural Scence Foundaton of Jangsu Provnce of Chna under Grant BK Abstract In ths artcle, a novel predctve controller based on a Bayesan nferrng nonlnear model (BMPC) s presented and analyzed. In the constructon of the BMPC, the Bayesan nferrng model s selected as the predctve model wth the characterstcs of on-lne tracng ablty to the actual controlled object. The nonlnear programmng method called the steepest gradent s set as the recedng horzon optmzaton algorthm of the BMPC. The on-lne controller output s obtaned usng ths method. The convergence analyss of the proposed BMPC s gven and the examples (nonmnmum phase and nonlnear objects) are selected to valdate the performance of the BMPC. The smulaton results show that wth the help of the presented BMPC algorthm, the closed loop control system demonstrates the abltes of ant-dsturbance and robustness. Keywords: Bayesan predctve control, nonlnear system, modelng, recedng horzon optmzaton 1. Introducton The model predctve control (MPC) method, derved from the practce ndustral applcaton, has been wdely used n many ndustral felds such as petroleum ndustry, chemcal ndustry and pharmaceutcal manufacturng ndustry (Armando, 013; Anders, 013; Al, 01). The reason for the success of the MPC s that the MPC method owns three characterstcs, namely, predctve model, feedback correcton and rollng optmzaton. Among them, the predctve model s not lmted to any specal model framework. And any model, whch can provde the ablty to predct the system output, all can be used as the predctve model. So the extensty of the predctve model contrbutes to the research prosperty of the MPC (Aswan, 013; Rodrguez, 013; Gselsson, 013). The predctve model plays an essental role to the predctve accuracy of the MPC. In the past decades, the theory of lnear model predctve control based on lnear predctve model has already been fully researched. But n ndustral control, the majorty of controlled objects possess many knds of nonlnear characterstc. So the research of nonlnear model predctve control s very meanngful and practcal. Untl now, there s no unform model descrbng method for all knds of nonlnear systems n nonlnear model predctve research. Commonly, three desgn deas are presented tll now. The frst dea s to make the nonlnear model lnearzaton and then to solve the problem based on the developed lnear predctve model control method (Chrstofdes, 013; Lncoln, 013; Tung, 013). The second dea s to obtan the nonlnear system model usng dentfcaton method and converted the controller desgn problem nto nonlnear programmng ssue. And then the control varable s optmzed wth the ad of some recedng horzon methods. The nonlnear frst-prncples modelng method s also been used (Mesbah, 010). In the second research dea of nonlnear model predctve control, the fuzzy model and neural network (Xangje, 013; Hazl, 014; Karm, 013) are two knds of frequently used nonlnear models. But the desgns of membershp functons and fuzzy rules all have a great nfluence on the predctve accuracy. In the desgn of neural network model, the number of hdden layers and nodes and the dynamc or statc neural structure are all 58

2 Computer and Informaton Scence Vol. 7, No. 3; 014 determnedd by desgner experence. All n all, the fuzzy model and neural network are all preset and use the obtaned data sample to tran the model structure parameters. From another perspectve, the nonlnear characterstcs are exhbted through the nput and output data. So we should focus on the data drectly nn the desgn of nonlnear predctve model. The modelng method based on statstcal theory s a knd of drect modelng soluton, whch utlzes the sample data to obtan the relatonshp between the nput and output of the research system. The Bayesan Gauss neural network based on Bayesan nferrng method drectly utlzes the nput and output data to dentfy the system nonlnear relatonshp. The former research (Lu, 010) shows that the Bayesan Gauss neural network demonstrates the on-lne modelng and tracng abltes. The novelty of ths artcle s to present a knd of Bayesan model predctve controller (BMPC). The Bayesan Gauss neural network s selected as the predctve nonlnear model. And the desgn procedures of the BMPCC and performance analyss are also gven n detals. Ths artcle s organzed as follows. Secton descrbes the structure of the proposed BMPC and detal desgn procedures of the BMPC. Secton 3 descrbed the experments of the BMPC on the control of the nonmnmum phase and nonlnear objects. The results and dscusson are also gven at secton 4 and 5. The concluson s then presented at secton 6 fnally.. Desgn of Bayesan Model Predctve Controller.1 Structure of the BMPC System The abstract structure of the Bayesan model predctve control system s shown n fgure 1. Fgure 1. Structure of Bayesan model predctve control system The components of the BMPC system nclude four blocks of the controlled object, Bayesan Gauss predctve model, model correcton and nonlnear programmng. u denotes the output of the controllerr at the moment of th k ; yr ( k d) and yˆ( k d) are the d step reference nput and system predctve output respectvely. yp ( k d) s the d th step corrected system output. Because there are many dsturbances n actual ndustral process control, the establshed system model has much m uncertanty nevtably. In the desgn of predctve model control system, the uncertanty should be corrected. The correcton method adopts the followng algorthm. yr ( k d) yk ˆ( d) e ( k ) (1) Where e( k) y( k) yˆ ( k) denotes the predctve error at the moment of k. Usng the correcton method, the uncertanty s compensated to some extent and ths mechansm m also enables the model predctve control to become a feedback control system.. Bayesan Gauss Predctve Model The Bayesan Gauss nferrng model s adopted as the nonlnear predctve model n ths paper. The structure of the Bayesan Gauss nferrng model ( Lu, 010)s shown n fgure. The nput and output of nonlnear system are denoted as ( u, y ( )), 1,..., N and N s the maxmum sample number. th At the k sample moment, X k [ u( k 1), u( k ) ),..., uk ( m), y( k 1), y( k ),..., yk ( n)] s the nput vector of the Bayesan nferrng model. m and n are the related coeffcents about the nput and output varables respectvely. Based on the bayesan formula and gaussan assumpton, the above bayesan nferrng model can be realzed through the followng equatons (Lu, 010) ). ŷy ˆ( k) ( k) N 1 59 y () ()

3 Computer and Informaton Scence Vol. 7, No. 3; 014 e N ( k) 1 T ( X k X) D( Xk X) 0 D dag[ d 11,,..., d d ] mm (5) Where, n above equaton, X denotes the hstory Bayesan nput vector smlar to X k. s varance varable accordng to the sample vector X. The D s called threshold matrx. D s dagonal matrx (mm m n 1) wth the same dmensonn as the number of the elements of nput vector X. The tranng algorthms of the Bayesan nferrng model are not the pont of ths artcle. So the detals of o the algorthms of the Bayesan nferrng model can be found n the former work (Lu, 010). k (3) (4) Fgure. Illustraton of Bayesan nferrng model structure.3 Recedng Horzon of Control Varable The most classcal characterstc of model predctve control s that the control varable u (k) should be obtaned through the on-lne recedng horzon method. The objectve functon s selected as the followng formula. P Jk [ y ( k j) y ( k d) ] [ Uk ( 1)] j 1 r Where P and M denote the predctve and control tme horzons; Uk uk uk ( 1) ; yr ( k ) s the th desred system output at the tme moment of k and determned by the followng equaton. j y ( k j) c y( k) (1 c ) rk r where r s the reference nput sgnal; c s called flexble coeffcent desgned accordng to the controlled object. From the constructon of the Bayesan model predctve control, J (k) s the nonlnear functon about the uk ( ), 1,..., M. Therefore, the optmzaton of the control varables s converted to the mnmzaton of o the 60 p M 1 (6) (7)

4 Computer and Informaton Scence Vol. 7, No. 3; 014 objectve functon J (k). Namely, n the feasble control feld of[ umn, u max ], the optmzaton problem s to obtan a feasble soluton U * [ u * ( k),..., u * ( k M)] T to make the J ( k) mnmum. In the mplementaton of control, only the u * ( k) s used to act on the controlled object at the k tme moment. Then at the tme moment k 1, the optmzaton progress s repeated as the moment k. And n ths way, the whole dynamc process of the controlled object s optmzed. The above recedng horzon of u * ( k ) belongs to the nonlnear programmng problem wth constrant condtons. In ths work, we choose the steepest gradent method as the recedng horzon algorthm. Takng the comprehensve consderaton of the tranng of the Bayesan nferrng predctve model and the workng flow of the BMPC, we gve the mplementaton procedures of the BMPC method. (1) Obtan the sample data of the controlled object. Determne the nput and output vector of the Bayesan nferrng model. Utlze the PSO algorthm (Vahd, 013; Cabrerzo, 013) to tran the parameters of the threshold matrx D. () Determne the wdth of the sldng wndow N and valdate the on-lne predctve ablty of the obtaned Bayesan nferrng model. (3) Based on the traned Bayesan nferrng model, the mplementaton of the BMPC s set as follows. a) Obtan the current output yk and then compute the Bayesan nferrng model output yk ˆ. The error ek s computed as yk yk ˆ. b) Utlze the Bayesan nferrng model to computer the yk ˆ, 1,..., P; accordng to the formula 1, compute the corrected predctve output yp ( k ) ; at the same tme, compute the desred system output yr ( k ) accordng to the formula 7. c) Use the steepest gradent method to solve the mnmzaton of the objectve functon J ( k ) and the optmzed control varable u * ( k ) s obtaned. d) Take the u * ( k) to mplement on the controlled object. Then go to the step a) to contnue the next optmzaton procedure at the tme moment k Convergence Analyss of Bayesan Model Predctve Controller The objectve functon J ( k ) s descrbed as the vector form. T T J( k) [ Y Y ] [ Y Y ] U U (8) r p r p where, Y [ ( 1,..., )] T r yr k yr k P, Y [ ( 1,..., )] T P yp k yp k P, U [ U ( k),..., U ( k M 1)]. Let E Yr YP, J ( k ) s the functon aboutu. Accordng to the steepest gradent method (Zhang, 010), the optmal control law s lsted as follows. T T T Ek ( 1) Uk Uk ( 1) Ek ( 1) 1 Uk (9) where, denotes the optmzed step length, 0. Accordng to the above formula T Ek ( 1) T Ek ( 1) Uk Uk 1 Obtan the one frst order dervatve of the J (k) J ( k) T E( k 1) U( k) T U( k) Ek ( 1) Uk k U( k) k k The approxmaton of the frst order dervatve s set as J ( k) T E( k 1) U( k) T U( k) Ek ( 1) Uk k U( k) k k Take the formula 10 nto the above equaton and the obtan the followng formula \begn{equaton} (10) (11) (1) 61

5 Computer and Informaton Scence Vol. 7, No. 3; 014 J( k) ( )[ U( k) T U ( k)] U ( k ) In order to ensure the convergence of the optmzaton algorthm, the objectve functonn J ( k ) should be non-ncreasng functon. From the above analyss, Jk s satsfed. The convergence of the BMPC s proven. 4. Smulaton Case Studes To nvestgate the performance of the Bayesan model predctvee controller, the experments are conducted on the closed loop control of two knds of controlled object, namely the nonmnmum phase and nonlnear objects. In the experments, the control effectveness, ant-dsturbance ablty and robust performance are observed. 4.1 Descrpton of the Controlled Objects Nonmnmum Phase Object In the feld of electrcal systems, there are many nonmnmum controlled objects such as the thermal power generatonn systems, the hydraulc generator system, etc. Because of the large phase lag, the nonmnmum systems are hard to control wth the slow response of system output. The nonmnmum system wth the followng transform functon s chosen as the frst expermental object s Y() s () (11.75 s)(1 8.5 s) U s 4.1. Nonlnear Object There are many knds of nonlnear characterstcss n ndustral process. The followng dfference equaton descrbes the second expermental controlled object as a nonlnear object example. y y( k) (0.8 ( k1) 0.5 e ) y( k 1) ( 1) ( e y k yk ( )) uk ( 1) 0. uk 0.1 u( k1) u( k) ek (15) where, e ( k) denotes the whte gaussan nose wth zero mean and varance value Smulaton Procedures 4..1 Tranng and Testng of Bayesan Predctve Model Frst, t s needed to obtan the sample data for the tranng and testng of the Bayesan predctve models. In the experments, the control varable u( k) adopts the one mean and the varance value 0. and the 1000 groups of output sample data y( k ) are shown n fgure 3. The former 500 groups of data are used for the tranng of o the Bayesan predctve models and the next 500 groups of data are for the valdaton of the predctve effectveness of the traned models. (13) (14) Fgure 3. Sample data for system output 6

6 Computer and Informaton Scence Vol. 7, No. 3; 014 In the off-lnalgorthms are selected as: the swarm number S 0, c1 0.5, c 0.5. The structure of the bayesan predctve model for the nonmnmum phase object s set as the nput vector X [ u( k), u( k 1), y( k 1)] T and the output yk. Accordng to the choce of the structure, theree s 3 parameters needed to be optmzed n the threshold D. tranng of the Bayesan predctve models, the parameters n the partcle swarm optmzaton The structure of the bayesan predctve model for the nonlnear controlled object s selected as the nput vector X [ u( k), uk ( 1), yk ( 1), yk ( )] T and four parameters n the threshold matrx D are to be optmzed. After the tranng of the partcle swarm optmzaton, the threshold matrx D of the Bayesan predctve model m for the nonmnmum phase object s then obtaned D d ag[6.5656, 3.490, ]. And the threshold matrx D for the nonlnear object s set as D dag[ , , , ]. The wdth of o the sldng wndow N s selected as 3. Accordng to the next 500 groups of testng sample, the on-lne predctve output curves are shown n fgure 4. Fgure 4. On-lne predcton of Bayesan nferrng model 4.. Bayesan Model Predctve Control In the second procedure, the above obtaned Bayesan predctve models are used to construct the closed model m predctve control of the nonmnmumm object and the nonlnear object. In the two closed loop controls, the reference sgnals are all constant value. In the control of the nonmnmum system, the relevant parameters are set as the predctve horzon P 6, M 6 and the flexble coeffcent c 0.6. In order to observe the ant-dsturbance ablty and robust performance of o the Bayesan model predctve control system, the random dsturbances wth ampltude 0.1 are added at the moments of the k 60 and the k 10.Then the control curve of the Bayesan model predctve controller s shown n fgure 5. 63

7 Computer and Informaton Scence Vol. 7, No. 3; 014 Fgure 5. Control result of nonmnmum phase system based on BMPC In the control of the nonlnear object, the relevant parameters are set as the predctve horzon P 6, M 6 and the flexble coeffcent c 0.4. The random dsturbances wth ampltude 0.05 are added at the moments of o the k 60 and the k 10. Then the control curve of the Bayesan model predctve controllerr s shown n fgure 6. Fgure 6. Control result of nonlnear system based on BMPC 5. Results Analyss From the above experments, t can be seen that the tranng of the Bayesan predctve s very smple and the parameters needed to be optmzed are merely the ones n the threshold matrx D. When the Bayesan predctve model s used for the on-lne predcton, t can capture the nonlnear characterstcs quckly and have hgh predcton accuracy. The reason for ts characterstc s that the Bayesan predctve model can update ts structure usng on-lne sample data. In the closed loop of Bayesan model predctve control, the steepest gradent method ensures the convergence * ablty of recedng horzon algorthm. Therefore, the control varable U at every step can obtan the optmzed soluton. From the analyss of the experments, the Bayesan model predctvee control obtans desred control effectveness. When the closed control system have strong dsturbance, the Bayesan model predctve control system also acheves stablty becausee the Bayesan predctve model can capture the nonlnear characterstcs quckly. 64

8 Computer and Informaton Scence Vol. 7, No. 3; Concluson In ths note, the Bayesan model predctve control system s desgned based on the Bayesan nferrng model. The partcle swarm optmzaton algorthm s used to the off-lne tranng of the Bayesan nferrng model. In the on-lne mplementaton of the Bayesan nferrng model, the sldng wndow method s utlzed to acheve the structure updatng so as to capture the nonlnear characterstcs of system quckly. The recedng horzon optmzaton n the desgn of the Bayesan model predctve controller adopts the steepest gradent method, whch ensures the convergence ablty of the proposed model predctve control algorthm. And the experments mplemented on the nonmnmum phase and nonlnear systems show that the Bayesan model predctve control system owns good control effect, ant-dsturbance ablty and robust performance. Acknowledgements The authors would lke to thank the support of the Natonal Natural Scence Foundaton of Chna (No , No ) and the Jangsu Government Scholarshp for Overseas Studes (013). References Al, M., Zoltan, K. N., Adre, E. H., Herman, J. K., & Paul M. V. den Hof. (01). Nonlnear model-based control of a sem-ndustral batch crystallzer usng a populaton balance modelng framework. Control Systems Technology, IEEE Transactons on, 0(5), Anders Wllersrud et al. (013). Short-term producton optmzaton of offshore ol and gas producton usng nonlnear model predctve control. Journal of Process Control, 3(), Armando, D., Assandr et al. (013). Nonlnear parametrc predctve temperature control of a dstllaton column. Control Engneerng Practce, 1(1), Aswan et al. (013). Provably safe and robust learnng-based model predctve control. Automatca, 49(5), Cabrerzo et al. (013). A method based on PSO and granular computng of lngustc nformaton to solve group decson makng problems defned n heterogeneous contexts. European Journal of Operatonal Research, 30(3), Chrstofdes et al. (013). Dstrbuted model predctve control: A tutoral revew and future research drectons. Computers & Chemcal Engneerng, 51, Gselsson et al. (013). Accelerated gradent methods and dual decomposton n dstrbuted model predctve control. Automatca, 49(3), Karm Salahshoor et al. (013). Stablzaton of gas-lft ol wells by a nonlnear model predctve control scheme based on adaptve neural network models. Engneerng Applcatons of Artfcal Intellgence, 6(8), Lncoln, F. L., Moro et al. (013). A mxed-nteger model predctve control formulaton for lnear system. Computer and Chemcal Engneerng, 55, Lu Yjan et al. (010). Modelng of hydraulc turbne system based on a bayesan-gaussan neural network drven by sldng wndow data. Journal of Zhejang Unversty. Scence. C, 11(1), Mesbah, A., Landlust, J., AEM H., HJM K., Jansens, P. J., & PMJ Van den Hof. (010). A model-based control framework for ndustral batch crystallzaton processes. Chemcal Engneerng Research and Desgn, 88(9), Hazl, O. et al. (014). Fuzzy model predctve control of dc-dc converters. In AETA 013: Recent Advances n Electrcal Engneerng and Related Scences, Rodrguez et al. (013). State of the art of fnte control set model predctve control n power electroncs. Industral Informatcs, IEEE Transactons on, 9(), Tung Le et al. (013). Lnear-quadratc model predctve control for urban traffc networks. Transportaton Research Part C: Emergng Technologes, 36, Vahd, F., & Gholam Al, M. (013). An mprovement n RBF learnng algorthm based on PSO for real tme applcatons. Neurocomputng, 111,

9 Computer and Informaton Scence Vol. 7, No. 3; 014 Xangje, L., & Xaobng, K. (013). Nonlnear fuzzy model predctve teratve learnng control for drum-type boler-turbne system. Journal of Process Control, 3(8), Zhang, L. (010). Optmzaton Method. Bejng: Scence Press. Copyrghts Copyrght for ths artcle s retaned by the author(s), wth frst publcaton rghts granted to the journal. Ths s an open-access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton lcense ( 66