# NONLINEAR CONTROL 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.

- Skil to simulate nonlinear systems on a computer.

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

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 feedback nonlinear control.

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

- Skill to analyze nonlinear systems.

- Skill to design and simulate nonlinear control systems in the scalar case with the aid of a computer.

## 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. [10 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. [12 hours]

3) Nonautonomous systems. Stability of state motions. Comparison K and KL 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. [6 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. [14 hours]

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.

2) Autonomous systems. Stability theory: the direct method of Lyapunov, Lyapunov functions and the variable gradient method. 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.

3) Nonautonomous systems. Stability of state motions. Comparison K and KL 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.

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.

## Recommended readings

Didactical materials and the pdf slides of the lessons can be downloaded from the course site http://elly.dia.unipr.it/2017/course/view.php?id=274. 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.

- Pdf slides of the lessons on the web site of the course.

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, 2013.

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

## Teaching methods

Classroom lessons with the alternate use of slides and explanations at the blackboard. Exercises of analysis and synthesis with the aid of the MATLAB software in the Informatics Laboratory. The slides used to support lessons are uploaded weekly on the course website. They are the main didactical material for the lessons.

Classroom sessions with alternate use of slides and explanations at the blackboard. Exercises of analysis and synthesis with the aid of MATLAB software.

## 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 is done, 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 vote is expressed on the 0-30 scale and is obtained as a weighted average of the scores in the written and oral tests.

Written examination and subsequent oral examination.