PhD thesis
 
Research on Hybrid Black-Box Modeling for
Nonlinear Systems and Its Application
 
Jinglu Hu

Abstract
 
          Linear system theory is very well developed and there exist many results which can be applied to obtained linear models. On the other hand, most of real processes are nonlinear to some extent. If no physical insight is available and linear approximative models are not good enough, one has to use nonlinear black-box models. The existing nonlinear black-box models (neural networks, adaptive fuzzy systems, etc.), however, do not contain those linearity properties required by linear system theory, so that the results based on linear system theory can not be applied to the obtained nonlinear black-box models. The motivation of this thesis is intended to develop a black-box modeling scheme, with which the techniques based on well developed linear system theory could be extended to nonlinear systems. A hybrid black-box modeling scheme is proposed. Investigations are made to do system identification, system analysis and control design of nonlinear systems under the framework of linear system theory based on the new hybrid modeling scheme.
 

         A black-box model is a standard flexible structure which can be used to approximate a large variety of different systems. In this thesis, a new black-box model structure is proposed by incorporating a group of certain nonlinear structures into a linear model structure. A general nonlinear system is first expressed in a linear structure whose coefficients consist of constant parameters and nonlinear terms. Then a group of certain nonlinear nonparametric models (NNMs) (neural networks, adaptive fuzzy systems, etc.) are incorporated into the linear structure by using them to represent the nonlinear terms. In this way, we obtain a hybrid model structure which provides more freedoms  so that particular effort can be made to find a better compromise between the model flexibility and the model simplicity by using knowledge information efficiently. The obtained hybrid model is equipped with linear structure, flexibility and simplicity.
 

         Parameter estimates are usually based on criterion minimization. When a model includes a noise model part, the criterion function is not always unimodal, even though the model is built to be linear in the parameters. In order to solve such multimodality problem, a hybrid identification method using Genetic Algorithms (GAs) is considered. Particular compromises provided by optimization-based methods and GAs are obtained through introducing a new GA operator named as `development' inspired by the fact that living beings adapt themselves to their environment. The proposed hybrid method combines the reliability properties of the GAs with the accuracy of optimization-based method, while requiring a computation time only slightly higher than the latter. Furthermore, the hybrid identification is typically suitable for solving the multimodal problem resulted from noise models.
 

        One of the most challenging problems is to do control design and system analysis of nonlinear systems using the techniques based on the linear system theory. Since the proposed hybrid black-box model has the required linearity properties, it enables us to solve this challenging problem. First as an example of system analysis, a fault detection scheme based on the use of Kullback discrimination information (KDI) for model discrimination is extended to nonlinear systems. Two ways are considered. One is robust fault detection like approach. A two-step identification algorithm is suggested to identify the proposed hybrid model in such a way that the results give a best linear approximation of the system and the estimate of the modeling error due to nonlinear undermodeling. Then KDI-based robust fault detection scheme is applied. The second is multi-model based approach, where the proposed model is used as an interpolation based Multi-ARMAX-model consisting of several local linear ARMAX models. The fault detection is then performed by applying the KDI to discriminate the identified local ARMAX models. Next as an example of control design, a robust STR adaptive controller is designed for general nonlinear stochastic systems in a similar way to the linear stochastic control theory, based on the use of a hybrid quasi-ARMAX predictor. For such purpose, the hybrid quasi-ARMAX modeling scheme is modified so that the obtained hybrid quasi-ARMAX model is linear not only in the parameters to be adjusted but also in the one-step past input variable, which is favorable to deriving a control law directly.