# Nonlinear Control and Servo Systems

## FRTN05/FRTN05F Olinjär reglering och servosystem

Syllabus | Schedule |

In real life, most control systems are nonlinear in one aspect or the other (nonlinear dynamics in, e.g., robotics, actuation saturations, sensor nonlinearities, etc). While control designed using linear systems tools may continue to work well in many nonlinear systems, in some situations nonlinear effects should be taken into account in order to get a stable control system. Moreover, the performance and robustness may be significantly improved by considering the nonlinear aspects of the system. In the course Nonlinear Control and Servo Systems (FRTN05) we study ordinary nonlinearities and how to handle those, treat different analysis methods and learn how to do control synthesis for nonlinear systems, both in theory and practise. Brief course information (in Swedish) is available.

### Lecturer

Anders Rantzer, (course responsible) M-building (fifth floor, room 5143) 046 - 222 87 83

**Teaching assistants **

Martin Karlsson, M-building (second floor), 046 - 222 87 60

Kaito Ariu, M-building (second floor), 046 - 222 87 95

### Literature

Recommended textbook is- Glad, T., and Ljung,L., "Reglerteori: Flervariabla och olinjära metoder" (2003, Studentlitteratur, Lund, ISBN 9-14-403003-7) or the English translation "Control Theory: Multivariable and Nonlinear Methods", 2000, Taylor & Francis Ltd, ISBN 0-74-840878-9. Chapter 11-16,18. This book covers MPC and Optimal control which are not covered in the other text book. The first part of this book (Ch. 1-10) covers linear control theory and is useful for the course Multivariable Control - (FRTN10 Flervariabel reglering)

- Khalil, H. K., Nonlinear Systems (3rd ed., 2002, Prentice Hall, ISBN 0-13-122740-8). This is a good textbook on nonlinear control systems, at a bit more advanced level than the course.

Handouts: Lecture notes and extra material will be handed out and posted on this website.

### Lectures and Exercises

Lectures on Mondays are held at 13.15-15.00, other days at 08.15-10.00. Lectures are held in **M:B** on October 30 and November 10, otherwise in **M:E**.

The exercises are held Tuesdays and Wednesdays 15:15-17:00 in **lab A** and **lab B** of the Automatic Control Department, first floor of the M-building. See Exercise schedule and problem set.

*The lecture slides will be handed out during the lectures. Below you find last edition's versions to be replaced when the lecture is given (indicated by red or green bullet). *

Monday 30/10: L1: Introduction. Typical nonlinear problems and phenomena. Models.

Wednesday 1/11: L2: Linearization. Stability. Controllability. Simulation.

Friday 3/11: L3: Phase-plane analysis. Classification of singular points. Stability of periodic solutions.

Monday 6/11: L4: Lyapunov methods.

Wednesday 8/11: L5: Stability theory. Small gain theorem. Circle criteria. Passivity.

Friday 9/11: L6: Describing function analysis.

Monday 13/11: L7: Saturation and antiwindup. Friction.

Wednesday 15/11: L8: Compensation for backlash and quantization.

Reading material on: antiwindup and friction

Monday 20/11: L9: Lyapunov-based design and sliding modes.

Wednesday 22/11: L10: Optimal control: The Maximum Principle, examples.

Lab1: Deadzone-compensation for an air throttle in a car.

Monday 27/11:L11: Optimal control (cont'd).

D. Liberzon, Calculus of variations and optimal control theory: A Coincise Introduction, Princeton University Press, 2012.

Wednesday 29/11: L12: Dynamic programming.

Lab2 : Pendulum swing-up

Monday 4/12: L13: Internal model control. Model predictive control. Nonlinear observers. Gain scheduling.

Wednesday 6/12: L14: Course summary.

Lab 3: Optimal control of pendulum an a cart.

### Exercises and Solutions

Exercise schedule and problem set

### Laboratories

Lab1: Deadzone-compensation for an air throttle in a car (nonlinear compensation and describing function analysis) (Sign-up starts November 13) **If**** all 60 slots have already been booked, please email the lab responsible (gautham@control.lth.se) and the course responsible (rantzer@control.lth.se) immediately and we will find a suitable arrangement. **

Lab2: Pendulum swing-up (Sign-up starts November 20) **If**** all 60 slots have already been booked, please email the lab responsible (martin.karlsson@control.lth.se) and the course responsible (rantzer@control.lth.se) immediately and we will find a suitable arrangement. **

Lab3: Pendulum on a cart. (Sign-up starts November 27) **If**** all 60 slots have already been booked, please email the lab responsible (kaito.ariu@control.lth.se) and the course responsible (rantzer@control.lth.se) immediately and we will find a suitable arrangement. **

### Software

Matlab and Simulink are extensively used in the course. Graphical user interfaces called **pplane** and **dfield** for phase-plane analysis in Matlab are available for educational purposes from http://math.rice.edu/~dfield/. Matlab is available for home computers.

- How to obtain Matlab/Simulink (need StiL or Lucat-account).
- Matlab files (Shift + Mouse Click to download file), pplane8.m and dfield8.m.

Put the file in your working directory, start Matlab and type "pplane8" or "dfield8", respectively.

If you use MATLAB 2014b and onwards, use the following fixed version: pplane8.m and dfield8.m.

This version does not support "Plote while computing", therefore you may turn it off under "Options" to speed up the plotting.

### Old Exams

March 7, 2012

April 5, 2013

Mar 12, 2014

Jan 13, 2016