## Nonlinear Systems Khalil Homework Solutions

## Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

## Course Description

This course studies state-of-the-art methods for modeling, analysis, and design of nonlinear dynamical systems with applications in control. Topics include:

- Nonlinear Behavior
- Mathematical Language for Modeling Nonlinear Behavior
- Discrete Time State Space Equations
- Differential Equations on Manifolds
- Input/Output Models
- Finite State Automata and Hybrid Systems

- Linearization
- Linearization Around a Trajectory
- Singular Perturbations
- Harmonic Balance
- Model Reduction
- Feedback Linearization

- System Invariants
- Storage Functions and Lyapunov Functions
- Implicitly Defined Storage Functions
- Search for Lyapunov Functions

- Local Behavior of Differential Equations
- Local Stability
- Center Manifold Theorems
- Bifurcations

- Controllability of Nonlinear Differential Equations
- Frobenius Theorem
- Existence of Feedback Linearization
- Local Controllability of Nonlinear Systems

- Nonlinear Feedback Design Techniques
- Control Lyapunov Functions
- Feedback Linearization: Backstepping, Dynamic Inversion, etc.
- Adaptive Control
- Invariant Probability Density Functions
- Optimal Control and Dynamic Programming

Prerequisite: 6.241 or an equivalent course.

## Information Resources and Literature

This year, there will be no required textbook. All necessary information will be supplied in the lecture notes.

The books *Nonlinear Systems* by Hassan K. Khalil, published by Prentice Hall, and the more advanced *Nonlinear Systems: Analysis, Stability, and Control* by Shankar Sastry, published by Springer, can both serve as basic references on Nonlinear Systems Theory, frequently covering the topics skipped in the lectures.

## Instructor

Prof. Alexandre Megretski

## Class Schedule

## Homework

Homework assignments are usually given on Wednesdays. Homework papers are to be submitted during the lecture hours on the following Wednesday. The homework will be corrected, graded, and returned as soon as possible. Solutions to the homework will be distributed when the corrected homework is returned.

Team work on home assignments is strictly encouraged, as far as generating ideas and arriving at the best possible solution is concerned. However, you have to write your own solution texts (and your own code, when needed).

## MATLAB®

MATLAB®, the "language of technical computing'', will be used in some assignments. We will need Simulink®, Control Systems, and LMI Control Toolboxes. You may wish to consult its online help for general information and for specific commands for simulating and analysing systems.

## Examinations

There will be two take-home quizzes, to be completed and returned within 24 hours, but no final exam. The quizzes will cover the theory of 6.243J (divided as equally as possible). The questions will be based on the ideas used in the problem set solutions made available at least a week before the test. No homework will be given on the last Wednesdays before the quizzes. No cooperation is allowed on take-home quizzes.

## Grading

The letter grade will be determined at the end of the semester from a numerical grade N, obtained from the formula

N=0.5*H+0.25*Q1+0.25*Q2

where H is the average homework grade, and Q1, Q2 are quiz grades (H, Q1, Q2 are numbers between 0 and 100). From the distribution of N for the entire class, boundaries will be chosen to define letter grades. For students near the boundaries, other factors may be taken into account to determine the letter grade, such as effort, classroom activity, etc.

## EE 8215 – Nonlinear Systems

Mihailo Jovanovic, University of Minnesota, Spring 2013

## Course description

Introduction. Examples of nonlinear systems. State-space models. Equilibrium points. Linearization. Range of nonlinear phenomena: finite escape time, multiple isolated equilibria, limit cycles, chaos. Bifurcations. Phase portraits. Bendixson and Poincare-Bendixson criteria. Mathematical background: existence and uniqueness of solutions, continuous dependence on initial conditions and parameters, normed linear spaces, comparison principle, Bellman-Gronwall Lemma. Lyapunov stability. Lyapunov's direct method. Lyapunov functions. LaSalle's invariance principle. Estimating region of attraction. Center manifold theory. Stability of time-varying systems. Input-output and input-to-state stability. Small gain theorem. Passivity. Circle and Popov criteria for absolute stability. Perturbation theory and averaging. Singular perturbations. Feedback and input-output linearization. Zero dynamics. Backstepping design. Control Lyapunov functions.

Class schedule

## Instructor and Teaching Assistant

**Instructor**

Mihailo Jovanovic

Office: Keller Hall 5-157

Office hours: Tu 3:45pm - 4:45pm (or by appointment)

**Teaching Assistant**

Xiaofan Wu

Office: Keller Hall 2-276

Regular office hours: Wednesday, 4:45pm - 5:45pm

Extra office hours: Monday, 4:45pm - 5:45pm (**only when HW is due Tuesday**)

## Textbook and software

**Textbook**

Hassan K. Khalil**Nonlinear Systems**

Prentice Hall, Third Edition, ISBN 0-13-067389-7

**Software**

Homework sets will make a use of Matlab and Simulink

## Grading policy

Homework (40%)

Midterm exam (30%)

Final exam or Project (30%)

**Homework policy**

Homework is intended as a vehicle for learning, not as a test. Moderate collaboration with your classmates is allowed. However, I urge you to invest enough time alone to understand each homework problem, and independently write the solutions that you turn in. Homework is generally handed out every Thursday, and it is due at the beginning of the class a week later. Late homework will not be accepted. Start early!

**Tentative exam schedule**

Midterm: Feb 26

Final exam or Project presentation: during exam week

## Prerequisites

Even though I plan to cover everything from scratch, the students would benefit from a solid background in linear systems (EE 5231 or an equivalent course). Those interested should contact the instructor.

## Acknowledgment

I would like to thank Prof. Murat Arcak for sharing with me the material that he developed for his Nonlinear Systems Course (EE 222) at UC Berkeley.

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