Development of Box–Jenkins type time series models by combining conventional and orthonormal basis filter approaches Lemma D. Tufa a , M. Ramasamy a, * , Sachin C. Patwardhan b , M. Shuhaimi a a Chemical Engineering Department, Universiti Teknologi PETRONAS, 31750 Tronoh, Perak, Malaysia b Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India article info Article history: Received 20 March 2009 Received in revised form 16 July 2009 Accepted 18 July 2009 Keywords: Orthonormal basis filter System identification Simulation models Prediction models Multi-step-ahead prediction abstract A unified scheme for developing Box–Jenkins (BJ) type models from input–output plant data by combin- ing orthonormal basis filter (OBF) model and conventional time series models, and the procedure for the corresponding multi-step-ahead prediction are presented. The models have a deterministic part that has an OBF structure and an explicit stochastic part which has either an AR or an ARMA structure. The pro- posed models combine all the advantages of an OBF model over conventional linear models together with an explicit noise model. The parameters of the OBF–AR model are easily estimated by linear least square method. The OBF–ARMA model structure leads to a pseudo-linear regression where the parameters can be easily estimated using either a two-step linear least square method or an extended least square method. Models for MIMO systems are easily developed using multiple MISO models. The advantages of the proposed models over BJ models are: parameters can be easily and accurately determined without involving nonlinear optimization; a prior knowledge of time delays is not required; and the identification and prediction schemes can be easily extended to MIMO systems. The proposed methods are illustrated with two SISO simulation case studies and one MIMO, real plant pilot-scale distillation column. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction A turning point in the history of process control is the successful introduction of model predictive controllers (MPC) in process industries [1,2]. MPC has been widely accepted in process indus- tries due to many of their advantages in realizing an efficient con- trol performance especially in multivariable systems. There exist a number of MPC implementations currently each differing from other in terms of how the MPC problem has been formulated, the type of model used for prediction and the techniques used in solv- ing the optimization problem. A complete design of MPC includes the necessary mechanism for obtaining the best possible model, which captures the dynamics fully and allows the prediction to be calculated [3,4]. Models with large number of parameters (non-parsimonious) require higher computational resources dur- ing online implementation and pose a challenging task in estimat- ing the model parameters during the model development stage, whereas, models with inconsistent parameters result in inaccurate predictions. Both the above problems negatively affect the perfor- mance of the MPC applications. Among the various linear model structures, step response, finite impulse response (FIR) and auto regressive with exogenous input (ARX) model structures are most commonly used in process indus- tries due to the simplicity in determining the model parameters. However, large input–output data set is required to minimize var- iance errors in model parameters. Step response and FIR models are non-parsimonious and conventional ARX models suffer from the problem of inconsistency in parameters. In addition, to develop the traditional linear model such as ARX, auto regressive moving average with exogenous input (ARMAX) and Box–Jenkins (BJ), a prior knowledge of time delays in the system is required. Estima- tion of model parameters is one of the most important issues in system identification. The parameters of FIR, ARX and step re- sponse models can be easily estimated using linear least square method whereas the estimation of ARMAX and BJ model parame- ters is essentially a nonlinear optimization problem. Orthonormal basis filter (OBF) models can be considered as a generalization of FIR models where the delays in FIR are replaced with more complex rational functions [5]. OBF models allow incor- poration of a priori knowledge of system dynamics into the model, and due to this, they can accurately capture the dynamics with a fewer number of parameters. Unlike ARX models, OBF models do not have the parameter inconsistency problem. OBF models are parsimonious in parameters compared to FIR and step response models [5–7]. The parameters of OBF models can be easily deter- mined using linear least square method and time delays can also be easily estimated and incorporated into the models [5,7]. Engineering applications of orthonormal parameterizations emerged in the 1970s in the areas of digital filter structures 0959-1524/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2009.07.009 * Corresponding author. Tel.: +60 5 368 7585; fax: +60 5 365 6176. E-mail address: marappagounder@petronas.com.my (M. Ramasamy). Journal of Process Control 20 (2010) 108–120 Contents lists available at ScienceDirect Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont