Modelling, Simulation and Control of Non-linear Dynamical Systems

An Intelligent Approach Using Soft Computing and Fractal Theory

Patricia Melin,Oscar Castillo

• Publisher: CRC Press
• Year: 2001
• ISBN:
  9781000611960
• Edition: 1
• Topic: Mathematik
• Subtopic: Mathematik Allgemein

These authors use soft computing techniques and fractal theory in this new approach to mathematical modeling, simulation and control of complexion-linear dynamical systems. First, a new fuzzy-fractal approach to automated mathematical modeling of non-linear dynamical systems is presented. It is illustrated with examples on the PROLOG programming la

Chapter 1

Introduction to Modelling, Simulation and Control of Non-Linear Dynamical Systems

We describe in this book new methods for automated modelling and simulation of non-linear dynamical systems using Soft Computing techniques and Fractal Theory. We also describe a new method for adaptive model-based control of nonlinear dynamical systems using a hybrid neuro-fuzzy-fractal approach. Soft Computing (SC) consists of several computing paradigms, including fuzzy logic, neural networks and genetic algorithms, which can be used to produce powerful hybrid intelligent systems. Fractal Theory (FT) provides us with the mathematical tools (like the fractal dimension) to understand the geometrical complexity of natural objects and can be used for identification and modelling purposes. Combining SC techniques with FT tools we can take advantage of the “intelligence” provided by the computer methods (like neural networks) and also take advantage of the descriptive power of fractal mathematical tools. Non-linear dynamical systems can exhibit extremely complex dynamic behavior and for this reason it is of great importance to develop intelligent computational tools that will enable the identification of the best model for a particular dynamical system, then obtaining the best simulations for the system and also achieving the goal of controlling the dynamical system in a desired manner. We also describe in this book the basic methodology to develop prototype intelligent systems that are able to find the best model for a particular dynamical system, then perform the numerical simulations necessary to identify all of the possible dynamical behaviors of the system, and finally achieve the goal of adaptive control using the mathematical models of the system and SC techniques.

As a prelude, we shall provide a brief overview of the existing methodologies for modelling, simulation and control of non-linear dynamical systems and also of our own approach in dealing with these problems.

1.1  Modelling and Simulation of Non-Linear Dynamical Systems

Traditionally, mathematical modelling of dynamical systems has been performed by human experts in the following manner (Jamshidi, 1997): 1) The expert according to his knowledge selects a set of models consider to be appropriate for a specific given problem, 2) Parameter estimation of the models is performed with methods similar to least-squares (using the relevant data available), and 3) The “best” model is selected using the measures of goodness for each of the models. Also, we can say that linear statistical models have been traditionally used as an approximation of real dynamic systems, which is not the best thing to do since many of the mechanical, electrical, biological and chemical systems are intrinsically non-linear in nature. In this work, we achieved automated mathematical modelling by using different Soft Computing techniques (Jang, Sun & Mizutani, 1997). The whole process of modelling starts with a time series (data set), which is used to perform a “Time Series Analysis” to extract the components of the time series (Weigend & Gershenfeld, 1994). Time series analysis can be achieved by traditional statistical methods or by efficient classification methods based on SC techniques, like neural networks or fuzzy logic (Kosko, 1997). In our case, we used fuzzy logic for classification of the time series components. After this time series analysis is performed, the qualitative values of the time series components are used to obtain a set of admissible models for a specific problem, this part of the problem was solved by using a set of fuzzy rules (knowledge base) that simulates the human experts in the domain of application. Finally, the “best” model is selected by comparing the measures of goodness for each of the admissible models considered in the previous step.

The simulation of mathematical models traditionally has been performed by exploring the possible dynamic behaviors, for a specific system, for different parameter values of the model (Rasband, 1990). More recently, it has been proposed to use Artificial Intelligence (Russell & Norvig, 1995) techniques for the simulation of mathematical models (for example, by using expert systems (Badiru, 1992)). In this work, we used SC techniques to automate the simulation of dynamical systems. In particular, we make use of genetic algorithms to generate the “best” set of parameter values for a specific model with respect to the goal of obtaining the most efficient simulation possible. Genetic Algorithms (GA) essentially consist of methods for the optimization of a general function based on the concept of “evolution” (Goldberg, 1989). In our particular case, the problem consisted in specifying the appropriate function to be optimized, with the goal of achieving the most efficient simulation possible, i.e., a simulation that enables the identification of all the possible dynamic behaviors for a specific dynamical system. For the identification of dynamic behaviors we make use of a fuzzy rule base that will identify a particular behavior according to the results of the numerical simulations.

In general, the study of non-linear dynamical systems is very important because most of the physical, electrical, mechanical and biochemical systems can be mathematically represented by models (differential or difference equations) in the time domain. Also, it is well known in Dynamical Systems Theory (Devaney, 1989) that the dynamic behavior of a particular system can range from very simple periodic orbits to the very complicated “chaotic” orbits. Non-linear models may exhibit the chaotic behavior for systems of at least three coupled differential equations or at least one difference equation (Ruelle, 1990). In particular, for the case of real-world dynamical systems the mathematical models needed are of very high dimensionality and in general there is a high probability of chaotic behavior, along with all sorts of different periodic and quasi-periodic behaviors (Castillo & Melin, 1998b). For this reason, it becomes very important to be able to obtain the appropriate mathematical models for the dynamical systems and then to be able to perform numerical simulations of these models (Castillo & Melin, 1997b), since this enables forecasting system’s performance in future time. In this way, automated mathematical modelling and simulation of dynamical systems can contribute to real-time control of these systems, and this is critical in real-world applications (Melin & Castillo, 1998b). Also, an intelligent system for modelling and simulation can be useful in the design of real dynamical systems with certain constraints, since the information obtained by the numerical simulations can be used as a feedback in the process of design. The main contribution of the research work presented in this book is to combine several Soft Computing techniques to achieve automated mathematical modelling and simulation of non-linear dynamical systems using the advantages that each specific technique offers. For example, fuzzy logic (Von Altrock, 1995) was used to simulate the reasoning process of human experts in the process of mathematical modelling and genetic algorithms was used to select the best set of parameter values for the simulation of the best model.

The importance of the results presented in this book can be measured from the scientific point of view and also from the practical (or applications) point of view. First, from the scientific point of view, we consider that this research work is very important because the computer methods for automated mathematical modelling and simulation of dynamic systems that were developed contribute, in general, to the advancement of Computer Science, and, in particular, to the advancement of Soft Computing and Artificial Intelligence because the new algorithms that were developed can be considered “intelligent” in the sense that they simulate human experts in modelling and simulation. From the practical point of view, we consider the results of this research work very important for the areas of Control and Design of dynamical systems. Controlling dynamical systems can be made more easy if we are able to analyze and predict the dynamic evolution of these systems and this goal can be achieved with an intelligent system for automated mathematical modelling and simulation. The design of dynamical systems can be made more easy if we can use mathematical models and their simulations for planning the performance of these systems under different set of design constraints. This last two points are of great importance for the industrial applications, since the control of dynamical systems in real-world plants has to be very precise and also the design of this type of systems for specific tasks can be very useful for industry.

1.2  Control of Non-Linear Dynamical Systems

Traditional control of non-linear dynamical systems has been done by using Classical Linear Control Theory and assuming simple linear mathematical models for the systems. However, it is now well known that non-linear dynamical systems can exhibit complex behavior (and as a consequence are difficult to control) and the most appropriate mathematical models for them are the non-linear ones. Since the complexity of mathematical models for real dynamical systems is very high it becomes necessary to use more advanced control techniques. This is precisely the fact that motivated researchers in the area of Artificial Intelligence (AI) to apply techniques that mimic human experts in the domain of dynamical systems control. More recently, techniques like neural networks and fuzzy logic have been applied with some success to the control of non-linear dynamical systems for several domains of application. However, there also has been some limitations and problems with these approaches when applied to real systems. For this reason, we proposed in this book the application of a hybrid approach for the problem of control, combining neural and fuzzy technologies with the knowledge of the mathematical models for the adaptive control of dynamical systems. The basic idea of this hybrid approach is to combine the advantages of the computer methods with the advantages of using mathematical models for the dynamical systems. In this work, new methods were developed for adaptive control of nonlinear systems using a combination of neural networks, fuzzy logic and mathematical models. Neural networks were used for the identification and control of the dynamical system and fuzzy logic was used to enable the change of mathematical models according to the dynamic state of the system. Also, the information and knowledge contained in the mathematical models was used for the control of the system by using their numerical results as input of the neural networks.

Traditionally, the control of dynamical systems has been performed using the classical methods of Linear Control Theory and also using linear models for the systems (Albertos, Strietzel & Mart, 1997). However, real-world problems can be viewed, in general, as non-linear dynamical systems with complex behavior and because of this, the most appropriate mathematical models for these systems are the non-linear ones. Unfortunately, to the moment, it hasn’t been possible to generalize the results of Linear Control Theory to the case of Non-linear Control due to the complexity of the mathematics that will be required (Omidvar & Elliot, 1997). Of course, this mathematical generalization could still take several years of theoretical and empirical research to be developed. On the other hand, it is possible to use non-linear universal approximators that have resulted from the research in the area of SC to the problem of system identification and control. In particular, SC methodologies like neural networks and fuzzy logic have been applied with some success to problems of control and identification of dynamical systems (Korn, 1995). However, there are also problems where one or both methodologies have failed to achieved the level of accuracy desired in the applications (Omidvar & Elliot, 1997). For this reason, we have proposed in this work the use of a hybrid approach for the problem of non-linear adaptive control,

i. e., we proposed to combine the use of neural networks and fuzzy logic with the use of non-linear mathematical models to achieve the goal of adaptive control. In the following lines we give the general idea of this new approach as well as the reasons why such an approach is a good alternative for non-linear control of dynamical systems.

Neural networks are computational systems with learning (or adaptive) characteristics that model the human brain (Kosko, 1992). Generally speaking, biological neural networks consist of neurons and connections between them and this is modeled by a graph with nodes and arcs to form the computational neural network. This graph along with a computational algorithm to specify the learning capabilities of the system is what makes the neural network a powerful methodology to simulate intelligent or expert behavior (Miller, Sutton & Werbos, 1995). It has been shown, that neural networks are universal approximators, in the sense that they can model any general function to a specified accuracy (Kosko, 1992) and for this reason neural networks have been applied to problems of system identification (Pham & Xing, 1995). Also, because of their adaptive capabilities neural networks have been used to control real-world dynamical systems (Ng, 1997).

Fuzzy Logic is an area of SC that enables a computer system to reason with uncertainty. Fuzzy inference systems consist of a set of “if-then” rules defined over fuzzy sets. Fuzzy sets are relations that can be used to model the linguistic variables that human experts use in their domain of expertise (Kosko, 1992). The main difference between fuzzy sets and traditional (crisp) sets is that the membership function for elements of a fuzzy set can take any value between 0 and 1, and not only 0 or 1. This corresponds, in the real world, to many situations where it is difficult to decide in an unambiguous manner if something belongs or not to a specific class. Fuzzy expert systems, for example, have been applied with some success to problems of control, diagnosis and classification just because they can manage the difficult expert reasoning involved in these areas of application (Korn, 1995). The main disadvantage with fuzzy systems is that they can’t adapt to changing situations. For this reason, it is a good idea to combine both methodologies to have the advantages of neural networks (learning and adaptive capabilities) along with the advantages of fuzzy logic (contain expert knowledge) in solving complex real world problems where this flexibility is needed (Yen, Langar & Zadeh, 1995).

In this work, we have proposed a new architecture for developing intelligent control systems based on the use of neural networks, fuzzy logic and mathematical models, to achieve the goal of adaptive control of non-linear dynamical systems. The mathematical model of a non-linear dynamical system consist of a set of simultaneous non-linear differential (or difference) equations describing the dynamics of the system. The knowledge contained in the model is very important in the process of controlling the system, because it relates the different physical variables and their dependencies (Sueda & Iwamasa, 1995). For this reason, our approach is to combine mathematical models with neural networks and fuzzy logic, to achieve adaptive control of non-linear dynamical systems.

The study of non-linear dynamical systems is very interesting because of the complexity of the dynamics involved in the underlying processes (for example, biological, chemical or electrical) and also because of the implications, in the real world, of controlling industrial processes to maximize production. Real non-linear dynamical systems can have a wide range of possible dynamic behaviors, going from simple periodic orbits (stable) to the very complicated chaotic behavior (Kapitaniak, 1996). Controlling a non-linear dynamical system, avoiding chaotic behavior, is only possible using the mathematical models of the system (Sueda & Iwamasa, 1995). For this reason, model-based control is having great success in the control of complex real-world dynamical systems. In our approach, the neural networks were used for identification and control of the system, fuzzy logic was used to choose between different mathematical models of the system, and the knowledge given by the models was used to avoid specific and dangerous dynamic behaviors.

We consider the work on non-linear control presented in this book very important, from the point of view of Computer Science, because it contributed with new methods to develop intelligent control systems using a new hybrid model-based neuro-fuzzy approach for controlling non-linear dynamical systems. Also, from the point of view of the applications, this work is very important because it contributed with new methods for adaptive non-linear control that could eventually be used in the control of real industrial plants or general dynamical systems, which in turn will result in increased productivity and efficiency for these systems.

Chapter 2

Fuzzy Logic for Modelling

This chapter introduces the basic concepts, notation, and basic operations for fuzzy sets that will be needed in the following chapters. Since research on Fuzzy Set Theory has been underway for over 30 years now, it is practically impossible to cover all aspects of current developments in this area. Therefore, the main goal of this chapter is to provide an introduction to and a summary of the basic concepts and operations that are relevant to the study of fuzzy sets. We also introduce in this chapter the definition of linguistic variables and linguistic values and explain how to use them in fuzzy rules, which ar...