Adaptive Estimation and Control based on Linear and Bilinear Neurofuzzy Parametric Models
Yiannis S. Boutalis and Manolis A. Christodoulou
Department of Electrical and Computer Engineering
Democritus University of Thrace
67100 Xanthi, Greece
Email: [email protected] , [email protected]
Abstract
In systems that change with time, the derivation of adaptive estimation and control methodologies is essential. When the system is multi variable, nonlinear and its model is totally or partially unknown the problem becomes even more challenging. The definition of adaptive systems as systems that change to conform to new or changed circumstances, has been used to label approaches and techniques in a variety of areas. In adaptive control, the following specific definition given by Ioannou in [1] outlines its main characteristics: Adaptive control is the combination of a parameter estimator, which generates parameter estimates online, with a control law in order to control classes of plants whose parameters are completely unknown and/or could change with time in an unpredictable manner. The choice of the parameter estimator, the choice of the control law, and the way they are combined leads to different classes of adaptive control schemes. The contribution of Ioannou and coworkers in the field of Adaptive Estimation and Control is essential. His work on Robust Adaptive Control and his tutorial books ([1],[2]) provide a complete source of adaptive estimation and robust control approaches. Moreover, the work in [3], that uses high order neural networks to construct the required parametric model has inspired the authors of this paper in developing their work. Artificial neural networks and adaptive fuzzy systems constitute a reliable choice for modeling unknown systems, since they can be considered as universal approximators. In this sense, they can approximate any smooth nonlinear function to any prescribed accuracy in a convex compact region, provided that sufficient hidden neurons and training data or fuzzy rules are available. Recently, the combination of artificial neural networks and adaptive fuzzy systems has led to the creation of new approaches, fuzzy-neural or neuro-fuzzy approaches that capture the advantages of both fuzzy logic and neural networks and intend to approach systems in a more successful way. This presentation, reports on recent research results of its authors related to a Neuro-Fuzzy approach. It is based on the development of a new adaptive recurrent neuro-fuzzy approximation scheme, which is used for system identification and the construction of a number of controllers with guaranteed stability and robustness. The central idea in the development of the new approximation scheme is an alternative description of a classical dynamical fuzzy system, which allows its approximation by high order neural networks (HONNs), a point that constitutes a innovative element and distinguishes it from other existing Neurofuzzy approaches. More specifically, fuzzy-recurrent high order neural networks (F-RHONN’s) [4] are proposed for the identification of nonlinear dynamical systems assuming an affine in the control form. The system is approximated by a recurrent model using two independent fuzzy sub-systems, each one being associated with different parts of the affine model. The underlying fuzzy models are of Mamdani-type (\cite{Mamdani1976}) assuming a standard defuzzification procedure such as centroid of the area or weighted average. However, the proposed approximation depends on the fact that fuzzy rules could be identified with the help of High Order Neural Networks (HONN) in conjunction with the centers determined from the output fuzzy partitioning. The advantage of using HONN is that the nonlinear identification model is linear with respect to the tunable parameters, leading to a linear parametric model that allows the estimation algorithm to achieve a global minimum in the estimation error function. There are two core ideas in the proposed method, which differentiate it from the already existing ones in the international literature: (i) A number of high order neural networks are specialized to work around some fuzzy output membership function centers, which leads to the separation of the system to neurofuzzy sub-systems and (ii) the introduction of a novel method called parameter hopping. This new method replaces the known from the literature projection method [2]. It is used in order to restrict the weights and avoid drifting of their values to infinity. Based on this approach and the linear parametric modeling, a number of works have followed in ([5],[6]) where a NF approach for the indirect [5] and direct [6] control of square unknown systems has been introduced, assuming only parameter uncertainty. The case of parametric and dynamic uncertainty has been treated in [7]. In these approaches, the a-priori experts information required to build the underlying fuzzy sub-systems is minimized, since all the information related to the if part of the underlying fuzzy rules is approximated by appropriate HONNs, trained by sampled system data. However, the centers of the fuzzy output variables partitions have to be provided by the experts, or be evaluated by off-line techniques based on gathered system data. The necessity for a-priori knowledge of the centers can be overcome if we allow the parametric model to become more complex and assume a bilinear form. In this case, a HONN-based NF controller can be proposed and used for the indirect control of nonlinear dynamical systems under the presence of neural weights and centers uncertainties. The nonlinear system can be considered again that is initially approximated using two independent fuzzy subsystems, where the underlying fuzzy models are of Mamdani type. Using proper indicator functions, every fuzzy subsystem is in the sequel approximated from a group of HONN’s, each one being associated with a family of fuzzy rules. The fuzzy output partitions of the initial fuzzy subsystems remain in the final model but they are also on-line estimated based on sampled data. This way, the parameters to be estimated are the HONN weights and the centers of the initial fuzzy output partitions. The resulting parametric model is bilinear with respect to the unknown parameters leading to the development of appropriate bilinear estimation algorithms. This approach has the advantage that it is almost free from a-priori experts’ information since the only requirement is an estimate of the signs of the initial fuzzy output centers, which can be obtained with the help of off-line procedures on sampled data or with the help of human experience. Therefore, it is not vulnerable to experts’ mistakes. Moreover, the NF model is composed of groups of HONNs, each one being specialized in approximating a part of the entire system, rendering the approximation more effective. Adaptation algorithms are derived based on the matrix to matrix bilinear form using Lyapunov stability analysis on the dynamic error equations. The control law is designed based on the estimated neuro-fuzzy model and aims at adaptively regulating the system states to zero. The weight updating algorithm guarantees that both the identification error and the system states go to zero exponentially fast, while keeping all signals in the closed loop bounded.
References
[1] P. Ioannou and B. Fidan, Adaptive control tutorial. SIAM:Advances in Design and Control Series, 2006. [2] P. A. Ioannou and J. Sun, Robust Adaptive Control. Englewood Cliffs, New Jersey: Prentice-Hall, 1996. [3] E. Kosmatopoulos, M. Polycarpou, M. Christodoulou, and P. Ioannou, “High-order neural network structures for identification of dynamical systems,” IEEE Transactions on Neural Networks, vol. 6, no. 2, pp. 422–431, 1995. [4] D. Theodoridis, Y. Boutalis, and M. Christodoulou, “Dynamical recurrent neuro-fuzzy identification schemes employing switching parameter hopping,” International Journal of Neural Systems, vol. 22, no. 2, p. 16 pages, 2012. [5] E. Mamdani, “Advances in the linguistic synthesis of fuzzy controllers,” Int. Journal of Man-Machine Studies, vol. 8, no. 6, pp. 669–678, 1976. [6] Y. S. Boutalis, D. C. Theodoridis, and M. A. Christodoulou, “A new neuro fds definition for indirect adaptive control of unknown nonlinear systems using a method of parameter hopping,” IEEE Transactions on Neural Networks, vol. 20, no. 4, pp. 609–625, 2009. [7] D. C. Theodoridis, Y. S. Boutalis, and M. A. Christodoulou, “Direct adaptive control of unknown nonlinear systems using a new neuro-fuzzy method together with a novel approach of parameter hopping,” Kybernetica, vol. 45, no. 3, pp. 349–386, 2009. [8] D. Theodoridis, Y. Boutalis, and M. Christodoulou, “Indirect adaptive control of unknown multi variable nonlinear systems with parametric and dynamic uncertainties using a new neuro-fuzzy system description,” International Journal of Neural Systems, vol. 20, no. 2, pp. 129– 148, 2012.Biographies
Yiannis Boutalis received the diploma of Electrical Engineer in 1983 from Democritus University of Thrace (DUTH), Greece and the PhD degree in Electrical and Computer Engineering in 1988 from the Computer Science Division of National Technical University of Athens, Greece. Since 1996, he serves as a faculty member, at the Department of Electrical and Computer Engineering , DUTH, Greece, where he is currently an Associate Professor and director of the Automatic Control Systems lab. Recently, he was also a Visiting Professor for research cooperation at Friedrich-Alexander University of Erlangen-Nuremberg, Germany, chair of Automatic Control. He served as an assistant visiting professor at University of Thessaly, Greece, and as a visiting professor in Air Defence Academy of General Staff of airforces of Greece. He also served as a researcher in the Institute of Language and Speech Processing (ILSP), Greece, and as a managing director of the R&D SME Ideatech S.A, Greece, specializing in pattern recognition and signal processing applications. His current research interests are focused in the development of Computational Intelligence techniques with applications in Control, Pattern Recognition, Signal and Image Processing Problems. Dr. Boutalis teaches undergraduate and postgraduate lessons in the fields of Automatic Control, Robotis and Computational Intelligence Techniques. He has written 3 books (in Greek) for supporting his teaching work and he is co-author in a book (ISBN-10: 363937391X) and 13 book chapters of books published by international houses. He has published more than 140 papers in international scientific Journals and proceedings of international conferences.
Manolis Christodoulou (S’78–M’82–SM’89) was born in Kifissia, Greece, in 1955. He received the diploma degree (EE’78) from the National Technical University of Athens, Greece, the M.S. degree (EE’79) from the University of Maryland, College Park the engineer degree (EE’82) from the University of Southern California, Los Angeles, and the Ph.D. degree (EE’84) from the Democritus University, Thrace, Greece. He joined The Technical University of Crete, Greece in 1988, where he was until recently a Professor of Control. He has been a Visiting Professor at Georgia Tech, Syracuse University, the University of Southern California, Tufts University, Victoria University and the Massachusetts Institute of Technology. He has authored and co-authored more than 200 journal articles, book chapters, books, and conference publications in the areas of control theory and applications, robotics, factory automation, computer integrated manufacturing in engineering, neural networks for dynamic system identification and control, in the use of robots for minimally invasive surgeries and recently in systems biology. Dr. Christodoulou is the organizer of various conferences and sessions of IEEE and IFAC and guest editor in various special issues of International Journals. He is managing and cooperating on various research projects in Greece, in the European Union and in collaboration with the United States. He has held many administrative positions such as the Vice Presidency of the Technical University of Crete, as Chairman of the office of Sponsored research and as a member of the board of governors of the University of Peloponnese. He is a member of the Technical Chamber of Greece. He has been active in the IEEE CS society as the founder and first Chairman of the IEEE Control Systems Society Greek Chapter, which received the 1997 Best Chapter of the Year Award and as the founder of the IEEE Mediterranean Conference on Control and Automation, which became an annual event. Dr Christodoulou received the MCA Founders award in 2005. He is a member of the board of governors of the Mediterranean Control Association since 1993.
