Identification of a PWA Model of a Batch Reactor for Model Predictive Control
The complex hybrid and nonlinear nature of many processes that are met in practice causes problems with both structure modelling and parameter identification; therefore, obtaining a model that is suitable for MPC is often a difficult task. The basic idea of this paper is to present an identification method for a piecewise affine (PWA) model based on a fuzzy clustering algorithm. First we introduce the PWA model. Next, we tackle the identification method. We treat the fuzzy clustering algorithm, deal with the projections of the fuzzy clusters into the input space of the PWA model and explain the estimation of the parameters of the PWA model by means of a modified least-squares method. Furthermore, we verify the usability of the proposed identification approach on a hybrid nonlinear batch reactor example. The result suggest that the batch reactor can be efficiently identified and thus formulated as a PWA model, which can eventually be used for model predictive control purposes.
Batch reactor, fuzzy clustering, hybrid systems, identification, nonlinear systems, PWA systems.[1] R. Babuˇska. Fuzzy Modelling for Control. KAP, 1998.
[2] R. Babuˇska and H. B. Verbruggen. An overview of fuzzy modelling for
control. Control Engineering Practice, 4(11):1593-1606, 1996.
[3] A. Bemporad. Efficient conversion of mixed logical dynamical systems
into an equivalent piecewise affine form. IEEE Trans. Automatic Control,
49(5):832-838, 2004.
[4] A. Bemporad and M. Morari. Control of systems integrating logic,
dynamics and constraints. Automatica, 35(3):407-427, 1999.
[5] D. Girimonte and R. Babuˇska. Structure for nonlinear models with
mixed discrete and continuous inputs: a comparative study. In Proc.
of IEEE International Conf. on system, Man and Cybernetics, pages
2392-2397, 2004.
[6] W. P. M. H. Heemels, B. De Schutter, and A. Bemporad. Equivalence
of hybrid dynamical models. Automatica, 37(7):1085-1091, 2001.
[7] G. Karer. Prediktivno vodenje hibridnih sistemov. PhD thesis, Fakulteta
za elektrotehniko, Univerza v Ljubljani, 2009.
[8] G. Karer, G. Muˇsiˇc, I. ˇ Skrjanc, and B. Zupanˇciˇc. Hybrid fuzzy modelbased
predictive control of temperature in a batch reactor. Computers
and Chemical Engineering, 31:1552-1564, 2007.
[9] E. C. Kerrigan and D. Q. Mayne. Optimal control of constrained,
piecewise affine systems with bounded disturbances. In Proc. 41st IEEE
Conference on Decision and Control, Las Vegas, 2002.
[10] D. Q. Mayne and S. Rakovi'c. Model predictive control of constrained
piecewise affine discrete-time systems. International Journal of Robust
and Nonlinear Control, 13(3):261-279, 2003.
[11] R. Palm and D. Driankov. Fuzzy switched hybrid systems - modelling
and identification. In Proc. of the 1998 IEEE/ISCI/CIRA/SAS Joint Conf.,
Gaithersburg MD, pages 130-135, 1998.
[12] B. Potoˇcnik, G. Muˇsiˇc, and B. Zupanˇciˇc. A new technique for translating
discrete hybrid automata into piecewise affine systems. Math. comput.
model. dyn. syst., 10(1):41-57, 2004.
[13] B. Potoˇcnik, G. Muˇsiˇc, and B. Zupanˇciˇc. Model predictive control of
discrete time hybrid systems with discrete inputs. ISA Transactions,
44(2):199-211, 2005.
[14] Y. Qin and L.-M. Jia. Fuzzy hybrid control and its application in
complex combustion processes. In 2002 IEEE International Conference
on Artificial Intelligence Systems (ICAIS-02), page 78. IEEE, 2002.
[15] D. S'aez, C. E. Cort'es, and A. N'u˜nez. Hybrid adaptive predictive control
for the multi-vehicle dynamic pick-up and delivery problem based on
genetic algorithms and fuzzy clustering. Computers and Operations
Research, 35(11):3412 - 3438, 2008.
[16] I. ˇ Skrjanc and D. Matko. Fuzzy predictive functional control in the state
space domain. Journal of Intelligent and Robotic Systems, 31:283-297,
2001.
[17] E. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE
Transactions on Automatic control, 26(2):346-358, 1981.
[18] M. Sugeno and K. Tanaka. Successive identification of a fuzzy model
and its application to prediction of a complex system. Fuzzy Sets and
Systems, 42:315-334, 1991.
[19] T. Takagi and M. Sugeno. Fuzzy identification of systems and its application
to modelling and control. IEEE Trans. System Man Cybernet.,
15:116-132, 1985.
[20] A. Van der Schaft and H. Schumacher. An introduction to hybrid
dynamical systems. Lecture Notes in Control and Information Sciences,
251:v-vii, 1999.
[21] H. S. Witsenhausen. A class of hybrid-state continuous time dynamic
systems. IEEE Trans. on Automatic Control, 11(2):161-167, 1966.
Batch reactor, fuzzy clustering, hybrid systems, identification, nonlinear systems, PWA systems.[1] R. Babuˇska. Fuzzy Modelling for Control. KAP, 1998.
[2] R. Babuˇska and H. B. Verbruggen. An overview of fuzzy modelling for
control. Control Engineering Practice, 4(11):1593-1606, 1996.
[3] A. Bemporad. Efficient conversion of mixed logical dynamical systems
into an equivalent piecewise affine form. IEEE Trans. Automatic Control,
49(5):832-838, 2004.
[4] A. Bemporad and M. Morari. Control of systems integrating logic,
dynamics and constraints. Automatica, 35(3):407-427, 1999.
[5] D. Girimonte and R. Babuˇska. Structure for nonlinear models with
mixed discrete and continuous inputs: a comparative study. In Proc.
of IEEE International Conf. on system, Man and Cybernetics, pages
2392-2397, 2004.
[6] W. P. M. H. Heemels, B. De Schutter, and A. Bemporad. Equivalence
of hybrid dynamical models. Automatica, 37(7):1085-1091, 2001.
[7] G. Karer. Prediktivno vodenje hibridnih sistemov. PhD thesis, Fakulteta
za elektrotehniko, Univerza v Ljubljani, 2009.
[8] G. Karer, G. Muˇsiˇc, I. ˇ Skrjanc, and B. Zupanˇciˇc. Hybrid fuzzy modelbased
predictive control of temperature in a batch reactor. Computers
and Chemical Engineering, 31:1552-1564, 2007.
[9] E. C. Kerrigan and D. Q. Mayne. Optimal control of constrained,
piecewise affine systems with bounded disturbances. In Proc. 41st IEEE
Conference on Decision and Control, Las Vegas, 2002.
[10] D. Q. Mayne and S. Rakovi'c. Model predictive control of constrained
piecewise affine discrete-time systems. International Journal of Robust
and Nonlinear Control, 13(3):261-279, 2003.
[11] R. Palm and D. Driankov. Fuzzy switched hybrid systems - modelling
and identification. In Proc. of the 1998 IEEE/ISCI/CIRA/SAS Joint Conf.,
Gaithersburg MD, pages 130-135, 1998.
[12] B. Potoˇcnik, G. Muˇsiˇc, and B. Zupanˇciˇc. A new technique for translating
discrete hybrid automata into piecewise affine systems. Math. comput.
model. dyn. syst., 10(1):41-57, 2004.
[13] B. Potoˇcnik, G. Muˇsiˇc, and B. Zupanˇciˇc. Model predictive control of
discrete time hybrid systems with discrete inputs. ISA Transactions,
44(2):199-211, 2005.
[14] Y. Qin and L.-M. Jia. Fuzzy hybrid control and its application in
complex combustion processes. In 2002 IEEE International Conference
on Artificial Intelligence Systems (ICAIS-02), page 78. IEEE, 2002.
[15] D. S'aez, C. E. Cort'es, and A. N'u˜nez. Hybrid adaptive predictive control
for the multi-vehicle dynamic pick-up and delivery problem based on
genetic algorithms and fuzzy clustering. Computers and Operations
Research, 35(11):3412 - 3438, 2008.
[16] I. ˇ Skrjanc and D. Matko. Fuzzy predictive functional control in the state
space domain. Journal of Intelligent and Robotic Systems, 31:283-297,
2001.
[17] E. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE
Transactions on Automatic control, 26(2):346-358, 1981.
[18] M. Sugeno and K. Tanaka. Successive identification of a fuzzy model
and its application to prediction of a complex system. Fuzzy Sets and
Systems, 42:315-334, 1991.
[19] T. Takagi and M. Sugeno. Fuzzy identification of systems and its application
to modelling and control. IEEE Trans. System Man Cybernet.,
15:116-132, 1985.
[20] A. Van der Schaft and H. Schumacher. An introduction to hybrid
dynamical systems. Lecture Notes in Control and Information Sciences,
251:v-vii, 1999.
[21] H. S. Witsenhausen. A class of hybrid-state continuous time dynamic
systems. IEEE Trans. on Automatic Control, 11(2):161-167, 1966.
@article{"International Journal of Information, Control and Computer Sciences:49797", author = "Gorazd Karer and Igor Skrjanc and Borut Zupancic", title = "Identification of a PWA Model of a Batch Reactor for Model Predictive Control", abstract = "The complex hybrid and nonlinear nature of many processes that are met in practice causes problems with both structure modelling and parameter identification; therefore, obtaining a model that is suitable for MPC is often a difficult task. The basic idea of this paper is to present an identification method for a piecewise affine (PWA) model based on a fuzzy clustering algorithm. First we introduce the PWA model. Next, we tackle the identification method. We treat the fuzzy clustering algorithm, deal with the projections of the fuzzy clusters into the input space of the PWA model and explain the estimation of the parameters of the PWA model by means of a modified least-squares method. Furthermore, we verify the usability of the proposed identification approach on a hybrid nonlinear batch reactor example. The result suggest that the batch reactor can be efficiently identified and thus formulated as a PWA model, which can eventually be used for model predictive control purposes.
", keywords = "Batch reactor, fuzzy clustering, hybrid systems, identification, nonlinear systems, PWA systems.", volume = "5", number = "5", pages = "415-6", }
