Preface

Chapter 7 **An Introduction to Ensemble-Based Data Assimilation Method**

Youmin Tang, Zheqi Shen and Yanqiu Gao

Chapter 8 **Conditions for Optimality of Singular Controls in Dynamic**

Jenq-Lang Wu, Chee-Fai Yung and Tsu-Tian Lee

Chapter 9 **Simultaneous H ∞ Control for a Collection of Nonlinear Systems**

Lidia María Belmonte, Rafael Morales, Antonio Fernández-

Christos K. Volos, Hector E. Nistazakis, Ioannis M. Kyprianidis,

Chapter 12 **Synchronization Phenomena in Coupled Birkhoff-Shaw Chaotic**

**Systems Using Nonlinear Controllers 293**

Ioannis N. Stouboulos and George S. Tombras

Chapter 13 **Design of Dynamic Output Feedback Laws Based on Sums of**

Kenta Hoshino, Daisuke Sonoda and Jun Yoneyama

Chapter 14 **Could the Stock Return be a Leading Indicator of the Economic Growth in the Depression? Analysis Based on Nonlinear**

**Systems with Retarded Control 195** Misir J. Mardanov and Telman K. Melikov

**in the Earth Sciences 153**

**VI** Contents

**Section 2 Control and Applications of Nonlinear Dynamical Systems 193**

**in Strict-Feedback Form 227**

**Mechanical Systems 243** Le Anh Tuan and Soon-Geul Lee

**MIMO System 265**

Chapter 10 **Nonlinear Feedback Control of Underactuated**

Chapter 11 **Nonlinear Cascade-Based Control for a Twin Rotor**

Caballero and José Andrés Somolinos

**Squares of Polynomials 319**

**Dynamic Panel Model 337**

Lee Yuan-Ming and Wang Kuan-Min

This book is a research monograph to detail recent developments of nonlinear systems and control. The book selected a collection of nonlinear systems, which range from quantum me‐ chanics, chaotic systems to nonlinear dynamical systems such as unmanned vehicle plat‐ form and 3D crane system. More interestingly, the book covers not just scientific and engineering problems but also earth data assimilation and economic development model‐ ing, that is, from theories to applications in real-world issues. This book has a couple of ana‐ lytic tools for nonlinear control problems, such as feedback linearization, partial differential equation, Frobenius theorem, Lyapunov theory and its exponents, Nikiforov-Uvarov meth‐ od, eigenvalue/eigenvector, pseudocomposition (predictor-corrector), or stochastic methods including Kalman filtering. Many practical systems or natural phenomena are nonlinear, so from a scientist or engineer's perspective, we interpret the system and represent the system using scientific approaches, including physics-based analysis and chemical or social analy‐ sis, where the tools are basically mathematics. Then, they interpret the system characteristics and try to represent the system using scientific approaches to get comprehensive results of such nonlinear or complex systems. If the scientific structure and analysis are sound, the system or phenomenon may have unique solutions, or it would not be a surprise if no solu‐ tion exists nor any tools or solutions. During the last couple of decades, serious efforts using estimation and many control systems with those analysis tools have helped to solve many complex nonlinear systems. Here we would like to narrow down the contents to some non‐ linear dynamical systems and their controls.

Solving nonlinear system is a daunting challenge, while analyzing nonlinear system is not easy because no universal solution is available, but the authors in this book have demon‐ strated what would be the system responses and behaviors in their fields, what would be the appropriate analysis tools, or how to get reasonable results in terms of convergence (re‐ gion of attraction)—those are the kinds of concerns they want to solve first. As the system goes more complicated or mixed, more analysis tools are needed to understand the system due to its mutual interaction or independent action. Furthermore, variables are getting big‐ ger, sophisticated, or specialized, and sometimes, only one of analytic tools is not sufficient to describe. Thanks to the development of hardware/software of electronics with faster com‐ puting technology, detailed analysis tools have been helping researchers or scientists to get more data or be able to solve higher-order, multidimensional, or more complex nonlinear control problems and real-world issues. Thus, nonlinear analysis tools, simulation, or exper‐ imental methods got to be advanced, extended to a wide and deep as well as quantitatively digitalized measurement system that helps to acquire more useful (visualized) data due to some phenomenal researchers. Hence, many nonlinear systems are resolved alongside the development of tools and control methods. On the other hand, linear system analysis and a set of tools such as linearization or partial differential equations are also used to analyze many nonlinear systems in a piecewise linear manner.

The book consists mainly of two parts as follows: the first section includes design, analysis, methods, and techniques of nonlinear system and the second section includes controls and applications of nonlinear systems. The following are brief outlines of each chapter.

In the first section, "Nonlinear Systems: Design, Analysis, and Estimation Methods," the au‐ thors in Chapter 1 present that quasilinearization technique can be simplified to partial dif‐ ferential equation (PDE) while decomposing time domain into smaller subintervals by applying spectral allocation method and extending up to boundaries with continuity condi‐ tion. In Chapter 2, the authors suggest an algorithm to find the change of coordinates and feedback for partial feedback linearization that are useful tools to convert nonlinear control system into partial linearization with feedback if the system has full rank and is involutive by Frobenius theorem. The authors in Chapter 3 investigate quantum characteristics of sin‐ gular potential of a particle using Nikiforov-Uvarov method and solved analytically the full wave functions with the evaluation of eigenfunctions with eigenvalues. Chapters 4 to 7 deal with estimation methods using Kalman filtering based on stochastic approach while solving nonlinear systems using iterative approach. The authors in Chapter 4 provide discrete-time nonlinear stochastic control problem solving iteratively using model-based optimal control by adding adjustable parameters to a continuous stirred-tank reactor model. In Chapter 5, the authors compare classical methods with multipoint iterative approaches such as compo‐ sition of known methods, weight function procedure, and pseudocomposition to solve non‐ linear systems. A couple of different iterative Kalman filtering algorithms with simulation examples is provided with two comparative studies in terms of state accuracies, estimation errors, and convergence where ISRCDKF provides the most improved state accuracies than the other techniques. A very thoughtful review of ensemble-based estimation methods is present in Chapter 7 where the authors provided many analyses, derivations, and discus‐ sions of Kalman filtering and particle filtering approaches. More importantly, the authors put more weights on those solutions to high dimensional systems in earth sciences where the novelty of this chapter lies in.

In the second section, "Control and Applications of Nonlinear Dynamical Systems," we have selected a couple of control approaches such as optimal, nonlinear, and output feed‐ back. These methods with analytic tools are applied to nonlinear dynamical systems to solve practical nonlinear control problems. In Chapter 8, the authors deal with optimal control problem with retarded control of singular situation in which they first optimize the condi‐ tions and obtain necessary conditions, design optimal solution, and then apply the control‐ ler to Legendre equation to demonstrate the results. The authors in Chapter 9 present simultaneous H-ꝏ control for nonlinear system under strict-feedback form, which is more challenging, but they used backstepping approach based on systematically control storage functions (CSF). Chapter 10 provides a control approach over three-dimensional overhead crane system using two separated subsystems, actuated and unactuated, in which the actu‐ ated subsystem is used to linearize nonlinear feedback states, whereas the unactuated sub‐ system is combined with the linear system. In Chapter 11, the authors develop dynamic model of twin-rotor helicopter, manufactured by Feedback Instruments Inc., which is a non‐ linear cascaded, coupled structure. They also developed its control algorithm, which carries out two electrical and mechanical parts and provided numerical results. In Chapter 12, the authors design a nonlinear controller to target antisynchronization/synchronization states

based on Birkhoff-Shaw nonautonomous chaotic coupled systems, and the stability of the systems is ensured by Lyapunov exponent theorem. With electronic circuit that models the coupling scheme, the authors are to verify the feasibility of their proposed design. Numeri‐ cal methods based on linear matrix inequalities (LMI) provide solutions to stabilization of nonlinear control problems in Chapter 13. With the design of output feedback laws based on the sum-of-squares (SoS) decomposition with state-dependent LMI, nonlinear polynomial systems can be stabilized via generalization as well as provide suitable analysis for stability through Lyapunov analysis. In the final chapter of this book, the authors try to analyze a practical nonlinear modeling problem. The authors examine thoroughly whether the stock return could be a leading indicator of economic growth in the depression period. In order to analyze nonlinear phenomena between economic development and stock return, a nonlin‐ ear dynamic data model is constructed with new current depth of recession indicator, espe‐ cially fluctuations of stock returns, which are highly correlated with economic activities. They propose that the stock return can considerably explain the economic growth in the re‐ cession period according to the country's development level and business cycle stages.

set of tools such as linearization or partial differential equations are also used to analyze

The book consists mainly of two parts as follows: the first section includes design, analysis, methods, and techniques of nonlinear system and the second section includes controls and

In the first section, "Nonlinear Systems: Design, Analysis, and Estimation Methods," the au‐ thors in Chapter 1 present that quasilinearization technique can be simplified to partial dif‐ ferential equation (PDE) while decomposing time domain into smaller subintervals by applying spectral allocation method and extending up to boundaries with continuity condi‐ tion. In Chapter 2, the authors suggest an algorithm to find the change of coordinates and feedback for partial feedback linearization that are useful tools to convert nonlinear control system into partial linearization with feedback if the system has full rank and is involutive by Frobenius theorem. The authors in Chapter 3 investigate quantum characteristics of sin‐ gular potential of a particle using Nikiforov-Uvarov method and solved analytically the full wave functions with the evaluation of eigenfunctions with eigenvalues. Chapters 4 to 7 deal with estimation methods using Kalman filtering based on stochastic approach while solving nonlinear systems using iterative approach. The authors in Chapter 4 provide discrete-time nonlinear stochastic control problem solving iteratively using model-based optimal control by adding adjustable parameters to a continuous stirred-tank reactor model. In Chapter 5, the authors compare classical methods with multipoint iterative approaches such as compo‐ sition of known methods, weight function procedure, and pseudocomposition to solve non‐ linear systems. A couple of different iterative Kalman filtering algorithms with simulation examples is provided with two comparative studies in terms of state accuracies, estimation errors, and convergence where ISRCDKF provides the most improved state accuracies than the other techniques. A very thoughtful review of ensemble-based estimation methods is present in Chapter 7 where the authors provided many analyses, derivations, and discus‐ sions of Kalman filtering and particle filtering approaches. More importantly, the authors put more weights on those solutions to high dimensional systems in earth sciences where

In the second section, "Control and Applications of Nonlinear Dynamical Systems," we have selected a couple of control approaches such as optimal, nonlinear, and output feed‐ back. These methods with analytic tools are applied to nonlinear dynamical systems to solve practical nonlinear control problems. In Chapter 8, the authors deal with optimal control problem with retarded control of singular situation in which they first optimize the condi‐ tions and obtain necessary conditions, design optimal solution, and then apply the control‐ ler to Legendre equation to demonstrate the results. The authors in Chapter 9 present simultaneous H-ꝏ control for nonlinear system under strict-feedback form, which is more challenging, but they used backstepping approach based on systematically control storage functions (CSF). Chapter 10 provides a control approach over three-dimensional overhead crane system using two separated subsystems, actuated and unactuated, in which the actu‐ ated subsystem is used to linearize nonlinear feedback states, whereas the unactuated sub‐ system is combined with the linear system. In Chapter 11, the authors develop dynamic model of twin-rotor helicopter, manufactured by Feedback Instruments Inc., which is a non‐ linear cascaded, coupled structure. They also developed its control algorithm, which carries out two electrical and mechanical parts and provided numerical results. In Chapter 12, the authors design a nonlinear controller to target antisynchronization/synchronization states

applications of nonlinear systems. The following are brief outlines of each chapter.

many nonlinear systems in a piecewise linear manner.

VIII Preface

the novelty of this chapter lies in.

Hence, this book is the culmination of their research and efforts. I hope it will be a good reference for the researchers and that it could provide good insights to obtain the solution for practical problems. Also it is an honor to edit these phenomenal papers. Special thanks go to InTechOpen for the opportunity and Edi Lipović, the Publishing Process Manager.

#### **Dongbin Lee, Ph.D.**

Assistant Professor/Director of Oregon Tech Robotics Lab, Adviser to Oregon Tech Unmanned Systems Club MMET Dept, Oregon Institute of Technology (Oregon Tech), Klamath Falls, Oregon, United States of America

**Nonlinear Systems: Design, Analysis, and Estimation Methods**

## **Solving Nonlinear Parabolic Partial Differential Equations Using Multidomain Bivariate Spectral Collocation Method Solving Nonlinear Parabolic Partial Differential Equations Using Multidomain Bivariate Spectral Collocation Method**

Motsa Sandile Sydney, Samuel Felix Mutua and Shateyi Stanford Shateyi Stanford

Motsa Sandile Sydney, Samuel Felix Mutua and

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64600

#### **Abstract**

In this study we introduce the multidomain bivariate spectral collocation method for solving nonlinear parabolic partial differential equations (PDEs) that are defined over large time intervals. The main idea is to reduce the size of the computational domain at each subinterval to ensure that very accurate results are obtained within shorter computational time when the spectral collocation method is applied. The proposed method is based on applying the quasi-linearization technique to simplify the nonlinear partial differential equation (PDE) first. The time domain is decomposed into smaller nonoverlapping subintervals. Discretization is then performed on both time and space variables using spectral collocation. The approximate solution of the PDE is obtained by solving the resulting linear matrix system at each subinterval independently. When the solution in the first subinterval has been computed, the continuity condition is used to obtain the initial guess in subsequent subintervals. The solutions at different subintervals are matched together along a common boundary. The examples chosen for numerical experimentation include the Burger's-Fisher equation, the Fitzhugh-Nagumo equation and the Burger's-Huxley equation. To demonstrate the accuracy and the effectiveness of the proposed method, the computational time and the error analysis of the chosen illustrative examples are presented in the tables.

**Keywords:** bivariate interpolation, spectral collocation, quasi-linearisation, multi-domain approach, non-linear evolution PDEs

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
