Introduction to Analysis of Non-linear Systems

Instructor

School of Electrical and Computer Engineering

Purdue University

465 Northwestern Avenue
West Lafayette, IN 47907-2035
Ph: (765) 494-6443
e-mail: zak@purdue.edu

Class Hours

T,Th: 9:00--10:15am in EE 226

Office Hours

M,W,F: 12:30--1:30pm

Prerequisites

Elements of linear algebra, ordinary differential equations, calculus of several variables. In particular: matrix manipulation, linear spaces, quadratic forms, differentiation of real-valued functions of n variables, gradients, the chain rule. Working knowledge of linear dynamic systems.

Textbook

S. H. Żak, Systems and Control, Oxford University Press, New York, 2003

Computer Software

MATLAB (a math-tools program)

Course Objective

“Real world” control problems are non-linear. Design techniques based on the linear system theory have difficulties with accommodating non-linear effects and modeling uncertainties. In this course, we study different approaches to the analysis and design of non-linear and uncertain, dynamical control systems. We then apply these methods to modeling and analysis of biological systems, specifically two endocrine systems are analyzed in order to demonstrate the power of non-linear methods.

Upon completion of the course, the student should understand common non-linear phenomena. The student will become familiar with concepts and tools that are useful in the analysis of non-linear systems and in the design of controllers and observers for such systems. The emphasis will be on design to show how nonlinear system theory fits into practical applications.

Brief Course Description

An introduction to modeling of dynamical control systems. State-plane and numerical methods for solving modeling equations. Linearization techniques. Stability. Controller and observer design for non-linear systems. Variable structure sliding-mode control. Vector field techniques. Introduction to chaos. Decentralized control of non-linear large-scale systems.

Course Outline

• Modeling
• Lagrange equations of motion

• Phase portraits
• The method of isoclines

• Linearization

• Lyapunov’s methods
• Uniform ultimate boundedness
• Lyapunov-like analysis

·         LaSalle’s invariance principle

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o    A generalization of the LaSalle’s principle and its application to the information systems

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·         Linear Matrix Inequalities

• Variable structure sliding mode control---slide presentation
• Conditions for existence of a sliding mode
• Switching surface design and construction of switched feedback gains

·         Potential extra topic

FunWork

INSTRUCTIONS:

·         The assignments must be typed. Recommended package for typing math is LaTeX. A friendly introduction to LaTeX is the book by Jane Hahn, LaTEX for Everyone.

·         Clearly identify the steps you have taken to solve each problem.

·         Make sure to cite completely all sources used.

---Submit only html or pdf file of your MATLAB m-file prepared using the cell mode. Use the publish button in the toolbar to obtain an html file, or go to the workspace and type

to obtain a pdf file. Submit either html or pdf file.

·         Here is a YouTube video about hypothyroidism related to the modeling of this assignment.

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·         Three fun to read papers on modeling endocrine systems:

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·         Some papers on modeling hypothalamus-pituitary-adrenal/thyroid axes:

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·        FunWork #2

·        FunWork #3

·        FunWork #4

·        FunWork #5

·        Final Exam

There will be two midterm exams from the material covered in class. Each midterm will be worth 100 points. The final exam will be worth 200 points. FunWork assignments will be averaged out to be worth 200 points. The course grade will be based on 600 points.

 Cutoffs A, A+ 540--600 A- 500--539 B+ 470--499 B 440--469 B- 410--439 C+ 380--409 C 350--379 C- 320--349 D+ 290--319 D 270--289 D- 250--269 F <250

In order to receive consideration, all requests for re-grades, regardless of type, will have to be submitted within one week of the return of the exam or homework in question.