(202f) Modelling and Predictive Control of a Resonating System

In general, a process demonstrates oscillatory response when it has at least one pair of complex poles, a second order underdamped process is the simplest example of such process. However, the real life processes are mostly nonlinear in nature and control of such oscillating nonlinear system becomes quite a challenging task. Mechanical processes such as crane and trolley and some chemical processes exhibit such nonlinear oscillatory dynamics. On the other hand, a Model Predictive Controller (MPC) is one of the most useful inventions in the field of control engineering that finds its application in the field of robotics, aerospace and also in chemical process industries. MPC is particularly efficient when process constraints are to be handled, such is the case for safety regulations, while implementing control action. Due to its immense prospect, researchers of both academic and industrial field show great interest in MPC resulting various developments in its techniques. Nevertheless, application of MPC for control of nonlinear oscillatory process is still a relatively unexplored area.

The main bottleneck towards implementation of MPC for nonlinear oscillating process is to obtain a reliable model of such process. Various types of modeling techniques (mostly black-box models) have been developed over the years such as NARMAX model, artificial neural networks, fuzzy-logic based models, neuro-fuzzy models, support vector machine and kernel methods of modeling, and wavelet decomposition based models (Billings and Wei, 2005; Aadaleesan, 2007) to name a few. Some common issues considered while developing any type of model listed above include model parsimony, ease of development of the model and the accuracy of the model. A good model is expected to be parsimonious in size, easy to develop and have high level of accuracy. None of the modeling techniques, listed above, possess all these merits simultaneously. Moreover the modeling exercise seldom fails once a pair of complex pole is encountered in the process dynamics. On the other hand, use of orthonormal basis functions (OBF) in black box modeling technique has been quite popular among control practitioners in the last decade. OBF incorporates an approximate knowledge about the dominant dynamics (time constant) of the system into the identification process even when the process has significant time delay. The number of free design parameters of the model can be set with the help of this knowledge and the variance of their estimates can be grossly reduced. This results in an increased robustness and accuracy of the model. There are many orthonormal basis filters used in system identification like Finite Impulse Response (FIR) model, Laguerre filter, and Generalized Orthonormal Basis Filter (GOBF). The simplest structure based upon OBF, which is also most popular is the FIR Model. However, FIR model is non-parsimonious in nature. Laguerre filter models have the ability to approximate linear systems (even with time delay) with a model order lower than the FIR modeling. Use of Laguerre models for mildly nonlinear system is possible with piece-wise linear models. However severe nonlinearities can also be incorporated into the model by introducing static polynomial, ANN or wavelet nonlinearity (Saha, 2004; Aadaleesan et. al., 2007) with appreciable accuracy and fewer model terms. Nevertheless, the application of Laguerre modeling is limited to open loop stable process with real poles and it fails to replicate the resonating systems. This limitation may be overcome by introducing Kautz networks which allow approximation of resonating system behavior. Kautz filter (Wahlberg, 1991) is a more generalized structure of OBF where complex poles are incorporated in the model structure that in turn captures the oscillatory behavior of the process. The recursive nature of Kautz construction makes it easy to compute.

In this work, it is intended to demonstrate that Kautz filters are efficient in modeling complex resonant systems. Very few researchers have reported so far on Kautz model because of its complexity (Wang, 2009; Khan et al., 2011). The system identification problem exercised till date involves frequency model which is of no use regarding its implementation with MPC. Hence, the main challenge of this work is to develop a discrete nonlinear model of Kautz filter that is very easy to compute. In the present work, the discrete time state space realization of Kautz model (Wahlberg, 1991) has been taken as a basis for further research. Multiple structures of linear models were developed in the state space form who differ in the way the state updation is carried out. These models are used for developing MPC routine and their subsequent performance analysis. Two numerical case studies have been conducted. In the first case, a second order linear under damped system has been taken.  The resulting identification problem shows incredible results. The model is not only very easy to formulate but also needs less order or no. of Kautz functions to predict the system response in comparison to the previously reported Kautz models. In the second case study, an electromagnetic suspension system has been considered which is inherently nonlinear in nature that has one real and two complex poles. Although linear Kautz model is capable of capturing process dynamics within a narrow range of perturbation, a nonlinear extension of the model is desired for a larger range of confidence. One simple and very efficient MPC algorithm constituting the aforesaid model(s) has been employed in the two case studies. Simulation has been carried out in Matlab® software in Windows 7 (64 bit processor) domain. Simulation study shows that Kautz-MPC algorithm produces satisfactory control performance. Robustness of model and stability limit of the controller has been analyzed. It is thus inferred that, as the chemical processes often show complex behavior and some resonating characteristics, integration of Kautz model with a MPC algorithm would yield an efficient control law which would enhance closed loop control performance of a process. Further research on extension of Kautz model in nonlinear domain is underway. The results will be reported in the extended abstract and/or full paper.

Literature Cited

Aadaleesan, P., Miglani, N., Sharma, R., Saha, P., 2007. Nonlinear system identification using Wiener type Laguerre—Wavelet network model. Chemical Engineering Science 63, 3932 – 3941.

Billings, S.A., Wei, H.L., 2005. The wavelet-NARMAX representation: a hybrid model structure combining polynomial models with multi-resolution wavelet decompositions. International Journal of System Science 36 (3), 137--152.

Khan B., Rossiter J.A. and Valencia-Palomo G., 2011. Exploiting Kautz functions to improve feasibility in MPC, 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011.

Saha, P., Krishnan, S.H., Rao, V.S.R., Patwardhan, S.C., 2004. Modeling and predictive control of MIMO nonlinear systems using Wiener--Laguerre models. Chemical Engineering Communication 191, 1083--1119.

Wang, L., 2009 . Model Predictive Control System Design and Implementation Using MATLAB. Springer, 3 Apple Hill Drive, Natick, MA 01760-2098, USA.

Wahlberg, B. 1991. Identification of resonant systems using Kautz filters. Proceedings of the 30^th Conference on Decision and Control, Brighton, England, December 1991.