(188t) Identification of Piecewise Autoregressive Exogenous (PWARX) Model Using Efficient Optimization Algorithm

Authors: 
Yang, Y., California State University Long Beach
An efficient identification approach for piecewise autoregressive exogenous (PWARX) model is presented in this study. Specifically, we focus on a subclass, called hyperplane autoregressive exogenous (HHARX) model, which is able to approximate general nonlinear process as a sum of hinge function accurately and ensure the continuity of model even in a piecewise affine form. In fact, the ability and superiority of HHARX model have been shown in many literatures [1,2,3]. It receives great attentions from academia due to its simple structure, arbitrary accuracy and suitability for model predictive control (MPC). Hence, this empirical model is especially useful for the control and optimization of complex nonlinear process, for which first principle model is difficult to obtain.

However, the computational method for HHARX model identification has not been well-developed yet. For example, the widely used expectation maximization (EM) based algorithm is concerned with maximum likelihood estimation (MLE) but does not take the process continuity into account. Baesd on the same reason, the clustering-based method also cannot be used to build HHARX model. Among all existing methods, the mixed-integer programming (MIP) has been shown as a very promising technique for identifying all hinging function simultaneously to achieve a global optimal solution [1], but limited in the case where not too much data is available. In order to build HHARX model, the formulation proposed in Ref. [1] has to introduce many binary variables, proportional to the number of data and hyperplanes, such that even a state-of-the-art solver cannot handle it.

To build HHARX model for process control with big-data setting, a novel optimization approach is presented to identify hinging function sequentially. This scheme enables mixed-integer programming to solve an identification problem with relatively large-scale data set, and obtain a satisfied solution instead of the global optimum. In each step, only one hinging hyperplane is optimized to minimize the error between predictive and real outputs. Parameters for other hyperplanes are unchanged, such that only a few binary variables have to be considered. Using warm start, the modeling error is reduced continuously and finally a suboptimal solution will be founded promptly. Moreover, we also show that some practical issues in the general piecewise system identification problem, including switching frequency, missing output data and specified steady state, can be addressed efficiently within the same framework. Finally, the efficiency and accuracy of the proposed computational scheme for system identification are demonstrated by studying two nonlinear processes and comparing our results with conventional approaches.

Reference

[1] Roll J, Bemporad A, Ljung L. Identification of piecewise affine system via mixed-integer programming. Automatica. 2004;40:37-50.

[2] Breiman L. Hinging hyperplanes for regression, classification and function approximation. IEEE Transactions on Information Theory. 1993;39:999-1013.

[3] Pucar P, Sjoberg J. On the hinge-finding algorithm for hinging hyperplanes. IEEE Transactions on Information Theory. 1998;44:3310-3319.