AME 549

AME 549: Hybrid Control Systems

Meetings Wednesday and Friday at 4:30 to 5:45pm, AME S336 
Course website at D2L
Office hours: Wednesday 6-7pm AME S336, Friday 2-3pm AME N619

Driven by recent technological advances and user specifications, most systems of today combine digital and analog devices, humans interacting with embedded computers, software distributed through networks, etc. As a result, they have state variables evolving both continuously and discontinuously due to features such as events, logic transitions, and impacts; they rely on algorithms implemented in embedded computers, which are interfaced with the plants through analog/digital and digital/analog devices; and they involve sensing and actuation through networks using communication protocols. Due to the presence of two types of dynamics, continuous and discrete, these systems are called hybrid control systems.

AME549 provides an introduction to hybrid control systems. After a short review of several mathematical concepts, a general modeling framework is introduced and exercised in several applications. A definition of solutions (or trajectories) to these systems is introduced next and their structural properties are investigated. A phenomenon unique to hybrid systems, called Zeno behavior, is introduced and discussed. Definitions of stability and convergence are presented next. Sufficient conditions for convergence to and asymptotic stability of equilibrium sets are given first for the linear case, which consist of invariance and eigenvalue conditions, respectively, and then for the nonlinear case, which are Lyapunov based. A characterization of the robustness properties induced by asymptotic stability follow and is illustrated with several applications. The last topic of the course is hybrid control design, where the tools for modeling and asserting convergence and stability for these systems are applied. Throughout the course, the students will be guided on methods for simulation and hardward implementation and encouraged to apply them to several applications.

TOPICS: This course will cover:
Modeling; Definition of solutions; Zeno behavior; Equilibrium sets; Stability and convergence; Nominal robustness; Hybrid control design; Numerical simulation. The content will be mainly theoretical. Applications will be on modeling, analysis, and control of unmanned aerial vehicles, robotic manipulators, hard-disk drive, mechanical systems with impacts, impulsively-coupled oscillators, cellular networks, and others.

• Instructor notes (to be available at D2L).
• Goebel, R.; Sanfelice, R.G.; Teel; A.R., Hybrid Dynamical Systems, IEEE Control Systems Magazine, pp. 28-93, April 2009.
• Franklin, G.; Powell, J. & Emami-Naeini, A. Feedback Control of Dynamic Systems 4th. Edition Prentice Hall, 2002.
• Chi-Tsong Chen, Linear System Theory and Design, Oxford University Press US, 1998
• H. Khalil, Nonlinear Systems, Prentice Hall, 2002.

PREREQUISITES and COMPUTER SKILLS: The course is self contained. Students are expected to have basic background on differential equations, calculus, and linear algebra. Knowledge of Matlab/Simulink will be useful.

EMAIL: My email will be the preferred way to communicate. Please check your email frequently for announcements. An immediate response is not guaranteed, but you should expect to get one within 24hs when your email arrives to the instructor’s email inbox Monday to Friday before class time.

HOMEWORK: Will account for 15% of final grade (between 2 and 4). Homework will be due at the beginning of class with no exception.

EXAMS: There will be one written take home midterm exam and one written final comprehensive exam. There will be no make-ups for any exam. If you are unable to take a scheduled exam due to health reasons, you must notify the Instructor prior to the beginning of the exam. If you will be absent due to a death (or life-threatening illness) in your family, similar advance notification and subsequent documentation will be required. Students absent from exams for one of the above reasons will be assigned a grade reflecting performance on another examination. Students missing exams under conditions not discussed above will normally be awarded a zero.

FINAL PROJECT: A project incorporating material covered in the course is required. The project is to be proposed by a team of 3-5 students and should consist of symbolic, simulation, or experimental analysis/design. Project topics will be proposed during the lectures. The teams are encouraged to propose a topic of their own interest on the subject. Tentative schedule:

October 19, 2012, in class: Project proposal (5 minutes oral presentation and 1 page team proposal due)
November 16, 2012, in class: Progress report (5 minutes oral presentation and 3-5 pages team progress report due)
November 28 and November 30, 2012, in class: Final Report (10 minutes oral presentation)
December 14, 2012, in class: Final report for project due.

If you already have project ideas, please contact me.

NEW PROJECT for FALL 2011Modeling and Global Stabilization of the Pan/Tilt Motion Control System

This purpose of this project is to develop a rigid body model of the fully actuated pan/tilt device shown in Figure 1 and to design a global stabilizer of a specific orientation using hybrid control techniques. To derive the rigid model, a 6 DOF model tailored to the Pan/Tilt Motion Control System is to be developed. A practical use of the results consist of distributed wind and solar energy farms in which the Pan/Tilt Motion Control System will efficiently capture energy from the environment by autonomously adjusting the angle of the Pan/Tilt Motion Control System according to the wind/sun properties. Slides introducing the Pan/Tilt Motion Control System are available here.

Student reports are available upon request.

The development of the Pan/Tilt Motion Control System and related curriculum material has been funded by Mathworks.

Pan/Tilt Motion Control System website:


GRADING: The course grades will be posted at d2l and determined using the following percentages:

Homework 15%
Midterm Exam 20%
Final Exam 20%
Final Project All Parts 45%

(20% oral presentations, 25% written reports)

The “break points” dividing letter grades will be determined by the Instructor at the end of the semester, based on the overall performance of the class and other relevant factors. Class participation will be taken into consideration when determining boundary cases.

Grading exam papers is a difficult task, and errors or misjudgments occasionally occur. Any student who feels that his or her paper has not been graded properly may request that the paper be re-graded. However, all such requests must be made to the TA no later than one week after the assignment has been returned. The complete paper will be reexamined, and the student’s grade may change in either direction.


Date Topics Events
Week 1

L1: Introduction to hybrid systems

L2: Summary of required linear systems concepts
Week 2

L3: Summary of required nonlinear systems concepts

L4: Modeling of hybid systems
HW1 posted
Week 3

L5: Modeling of hybrid systems and examples

L6: Solution concepts
HW1 due
Week 4

L7: Basic properties of solutions

L8: Types of solutions
Week 5

No class

No class
HW2 posted
Week 6

L9: Uniqueness of solutions

L10: Asymptotic stability
HW2 due
Week 7

L11: Asymptotic stability

L12: Asymptotic stability
HW3 posted
Week 8

L13: Examples

L14: Examples
Week 9

L15: Proof of Lyapunov theorem

HW3 due
Week 10

L16: Special cases of Lyapunov theorem

L17: Special cases of Lyapunov theorem
Week 11

L18: Perturbed hybrid systems

L19: Generalized solutions
Week 12

L20: Measurement noise in feedback control

L21: Well-posed hybrid systems
HW4 posted

11/09: Prog. Report due
Week 13

L22: Properties of well-posed hybrid systems

L22: Asymptotic stability for well-posed hybrid systems
Week 14

L23: Robustness of asymptotic stability

L24: Invariance principles
HW4 due
Week 15

Project Report Oral Presentations (all week)  
Week 16

12/07 Final Exam  
Finals week

12/14: Final Report due