PRIOR KNOWLEDGE BASED IDENTIFICATION OF TAKAGI-SUGENO-KANG FUZZY MODELS FOR STATIC NONLINEAR SYSTEMS

Open Access
Author:
Tewari, Ashutosh
Graduate Program:
Engineering Science and Mechanics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
October 20, 2008
Committee Members:
  • Dr Mirna Urquidi Macdonald, Dissertation Advisor
  • Mirna Urquidi Macdonald, Committee Chair
  • Joseph Paul Cusumano, Committee Member
  • Akhlesh Lakhtakia, Committee Member
  • Soundar Rajan Tirupatikumara, Committee Member
Keywords:
  • Knowledge-based systems
  • Data driven models
  • Neuro-fuzzy models
  • TSK fuzzy models
  • Artificial Intelligence
Abstract:
Data-driven model of a physical system is a mathematical model that has been built solely using historically available input-output pairs. In such type of models we usually try to identify the mapping function (linear or nonlinear) that transforms inputs from a multivariate space to a univariate output space (Multiple-Input-Single-Output/MISO systems). The data-driven models are especially needed when constructing a physical model (based on first principles) of a process is difficult due to the lack of understanding of the underlying physical phenomenon/phenomena. Although quite popular, majority of the data-driven models have a noticeable disadvantage in terms of their poor interpretability. In other words, although such models may correctly estimate the system outputs for a given set of inputs, the manner in which the outputs are generated does not have any physical interpretation. For this reason, data-driven models are sometimes referred as black-box models. Neuro-fuzzy models are a class of data-driven models that partly alleviates the above mentioned problem of poor interpretability. These hybrid models consist of two components. The Artificial-Neural-Network (ANN) component that enables learning from historical data and Fuzzy Logic component which tries to emulate the process of human reasoning. These models are specially desired in situations when a process output can be explained by a domain expert/experts in terms of linguistic statements. In this thesis, a type of neuro-fuzzy models called Takagi-Sugeno-Kang (TSK) models are studied as a tool for static nonlinear system modeling. In general, identification of any data-driven model involves two important steps 1) the structure identification and 2) the parameter identification. In the 1st step, a parameterized model is constructed either using the prior knowledge or a historical dataset or a combination of both. Thereafter, in the 2nd step of parameter identification the optimal values of the model parameters are estimated. The process of parameter estimation is called training and the dataset for this purpose is referred as training dataset. The general approach of parameter estimation can be summarized as follows. An error measure is defined using the training dataset and subsequently minimized with respect to the unknown model parameters. Therefore, the parameter estimation is merely an nonlinear optimization problem. In this thesis, the focus is only on the parameters estimation of TSK models under the assumption that their structure has been fixed a priori. In order to have consistent and near optimal parameter estimates, a sufficiently large noise-free training dataset is required. However, if we have more parameters than what can be correctly estimated from a training dataset then the performance of the trained model becomes questionable. The input-output datasets generated by majority of real world systems usually have low signal to noise ratio, which further reduces their estimation capability. Hence, it becomes imperative to devise robust parameter estimation techniques that are not significantly affected by the noise in training datasets. There is a group of estimation techniques known as regularization techniques that are commonly used for the identification of over-parameterized models from low quality datasets. The basic idea behind regularization is to impose certain restriction on the parameter values, so that the variance in estimated parameter values can be minimized. Contemporary regularization techniques used for the identification of neuro-fuzzy systems are similar to the ones used for the black-box models such as ANNs. To the knowledge of the author, none of these techniques explicitly uses the qualitative knowledge that can be readily obtained from a domain expert. It is certainly counterintuitive not to use existing knowledge to regularize the parameters values of neuro-fuzzy models. In an attempt to address the aforementioned problem, a regularization technique is proposed which works by incorporating prior knowledge during the parameter estimation of TSK models. The resulting TSK models are named as A-Priori Knowledge-based Fuzzy Models (APKFMs). The foundation of APKFMs is a hypothetical function called knowledge function, which is responsible for keeping a model’s parameters consistent with the domain knowledge. The proposed approach has a two fold regularization effect on the TSK models. Firstly, the incorporation of knowledge function ensures that the parameters do not take physically infeasible values during their estimation. Secondly, the use of knowledge function considerably reduces the number of unknown parameters. Therefore, less number of parameters are needed to be estimated from the available training data leading to consistent parameter estimates. The effectiveness of the APKFMs is shown using two examples of static systems. In the first example, a 2-dimensional nonlinear toy function is constructed and subsequently approximated by an APKFM. In the second example, a real world problem pertaining to the maintenance cost estimation of electricity distribution networks is taken up. In both the examples, the performance of APKFMs is benchmarked against neuro-fuzzy models identified using the three commonly used techniques namely, Global least square estimate, Local least square estimate and Ridge regression. The former is an unregularized parameter estimation technique while the other two represent the state of the art regularization techniques for TSK fuzzy models. The performance of all fuzzy models are evaluated and compared based on their robustness towards the noise present in the training datasets. Different training datasets of varying quality are constructed by controlling the two variables viz. the training dataset size and the signal to noise ratio. Rigorous statistical tests confirm that on an average the robustness of APKFM is superior compared to contemporary neuro-fuzzy models.