# NONLINEAR SYSTEMS

## Learning outcomes of the course unit

The aims of the course in relation to knowledge and understanding are:

- Understanding of the phenomena of nonlinear dynamical systems: multiple equilibria, stability/instability, limit cycles.

- Knowledge of the stability theory and its extensions.

- Knowledge of the main methods of nonlinear control.

In relation to the capability of applying knowledge and understanding, the aims are:

- Skill to analyze nonlinear systems.

- Skill to simulate nonlinear systems on a computer with MATLAB and Simulink.

- Skill to design nonlinear control systems in the scalar case.

## Course contents summary

1) Introduction. Nonlinear phenomena and mathematical models, applications to automation and robotics. Examples. Nonlinear state models: existence and uniqueness of solutions. Second-order dynamical systems: qualitative behavior of linear systems, phase diagrams, multiple equilibria, limit cycles. Useful mathematics for nonlinear systems. Modeling examples: kinematics of wheeled vehicles, magnetic levitators, overhead cranes. [11 hours]

2) Autonomous systems. Stability theory: the direct method of Lyapunov, Lyapunov functions and the variable gradient method. The region of attraction of an equilibrium state. Global asymptotic stability: Barbashin-Krasovkii theorem. Instability: Chetaev theorem. The algebraic Lyapunov equation and the indirect method. The invariance principle: LaSalle theorem. Stability and attractiveness of state sets. Limit cycles in feedback systems: method of the describing function. [14 hours]

3) Nonautonomous systems. Stability of state motions. Class K and class KL (comparison) functions. The direct method for the uniform asymptotic stability. Input-to-state stability. The direct and indirect method for exponential stability. Converse theorems in stability theory. [7 hours]

4) Nonlinear control. The stabilization problem. State-input feedback methods: control Lyapunov functions, integrator backstepping. Relative degree and normal form of a scalar affine control system. Input-output linearization by state-input feedback (feedback linearization). Zero dynamics and minimum-phase systems. Application to stabilization. Regulation of nonlinear scalar systems: integral control. Input-output inversion-based feedforward control. Stable inversion for nonminimum-phase systems: closed-form solutions for linear systems and iterative method for nonlinear systems. Feedforward-feedback control schemes. [16 hours]

## Recommended readings

Didactical materials and the pdf slides of the lessons can be downloaded from the course website. The following books contain suggested further readings:

1) H.K. Khalil, Nonlinear Control, Pearson Education Limited (Global Edition Paperback), 2014.

2) H.J. Marquez, Nonlinear control systems: analysis and design, Wiley, 2003.

3) J.-J. E. Slotine, W. Li, Applied Nonlinear Control. Prentice-Hall, 1991.

4) A. Isidori, Nonlinear Control Systems, third edition, Springer, 1995.

5) H.K. Khalil, Nonlinear Systems, third edition, Prentice-Hall, 2002.

## Teaching methods

Lessons with theory illustrated by examples. Exercitations with problem solving on all teaching topics. Analysis and synthesis exercises solved with the aid of the MATLAB software. Simulations with MATLAB and Simulink.

The slides used to support the lessons and exercitations are available on the online teaching site (Elly) and constitute the main teaching material of the course.

## Assessment methods and criteria

Assessment of learning is carried out by a written test followed a week later by an oral exam.

To enroll in the written tests, it is mandatory the registration on the ESSE3 website of the University. During the written test, it is not permitted to read notes, manuals, books, etc. Some parts of the written test require the use of a basic scientific calculator.

Two-three days after the written test, the results are published on the course website. To proceed to the oral exam, the score in the written test must be 16 points or greater on a scale from 0 to 30. The final grade is obtained as a weighted average of the scores gained in the written test and oral exam.