**GRADUATE COURSE "Analysis and Design of Hybrid Control Systems"**

**COURSE DESCRIPTION**

Hybrid control systems arise when controlling nonlinear systems with hybrid control algorithms — algorithms that involve logic variables, timers, computer program, and in general, states experiencing jumps at certain events — and also when controlling systems that are themselves hybrid. Recent technological advances allowing for and utilizing the interplay between digital systems with the analog world (e.g., embedded computers, sensor networks, etc.) have in- creased the demand for a theory applicable to the resulting systems, which are of hybrid nature, and for design techniques that may guarantee, through hybrid control, performance, safety, and recovery specifications even in the presence of uncertainty.

This course will present recent advances in the analysis and design of hybrid control systems from a control theory viewpoint. The power of hybrid control for robust stabilization will be displayed in several applications including power systems, robotic networks, underactuated rigid bodies, integrate-and-fire oscillators, neurons, and genetic networks.

**COURSE CONTENT**

*Main references:*

R. Goebel, R. G. Sanfelice and A. R. Teel. Hybrid Dynamical Systems: Modeling, Stability, and Robustness, Princeton University Press, 2012

Publisher's website: http://press.princeton.edu/titles/9759.html

Chapter 1 (sample): http://press.princeton.edu/chapters/s9759.pdf

R. Goebel, R. G. Sanfelice and A. R. Teel. Hybrid Dynamical Systems. IEEE Control Systems Magazine, 2009.

Available from https://hybrid.soe.ucsc.edu/files/preprints/34.pdf

R. G. Sanfelice. Control of Hybrid Dynamical Systems: An Overview of Recent Advances. Wiley, Hybrid Systems with Constraints, 146--177, 2013.

Available from https://hybrid.soe.ucsc.edu/files/preprints/77.pdf*Suggested preliminary reading:* first 5 pages of

R. Goebel, R. G. Sanfelice and A. R. Teel. Hybrid Dynamical Systems. IEEE Control Systems Magazine, 2009.

Available from http://www.u.arizona.edu/~sricardo/Preprints/2009/Goebel-2009_preprint.pdf**Day 1: **

**Modeling hybrid systems**

• Overview of main modeling technique, robustness, and stability results

• Mathematical examples of hybrid systems

Key references: [34], [65], [20], [36].**Concept of solution**

• Introduction to solution concepts to hybrid systems

• Hybrid time domains and hybrid arcs

• Solutions and basic properties

Key references: [34], [65], [40]**Simulation of hybrid systems**

• Introduction to simulation issues

• Hybrid Equations Toolbox

• Examples

Key references: HyEQ Toolbox, [74], [60], [8].

Homework assignment 1 here

**Day 2: **

**Modeling systems as hybrid inclusions**

• Switched systems

• Sample-and-hold control

• Hybrid automata

Key references: [34], [65]

**Asymptotic stability**

• Introduction to stabilization for hybrid systems

• Uniform global pre-asymptotic stability

• Lyapunov functions and sufficient conditions

• Relaxed conditions

• Examples

Key references: [34], [65]

**Well-posed hybrid systems**

• Preliminaries on set-valued maps

• Hybrid basic conditions

Key references: [34], [65]

Homework assignment 2 here

**Day 3: **

**Robustness to measurement noise**

• Generalized solutions

• Regularized hybrid systems and hybrid basic conditions

• Examples

Key references: [29], [34], [65]

**Consequences of hybrid basic conditions**

• Notion of solution revisited

• Basic existence result revisited

• Graphical distance

• Dependence on initial conditions

Key references: [34], [65]

**Invariance principles**

• Weak invariance

• Invariance principle

• Examples

Key references: [18], [34], [65]

**Robust asymptotic stability**

• Local and KL asymptotic stability

• Perturbed hybrid systems

• Converse theorem

Key references: [18], [34], [65]

Homework assignment 3 here

**Day 4: **

**Hybrid Control and Control of Hybrid Systems**

• Hybrid systems with inputs

• Design using control Lyapunov functions

• Minimum norm control

Key references: [85], [75], [103]

**Applications**

• Attitude control

• Control of a power inverter

• Juggling control

Key references: [50], [58], [104], [91], [79], [17]

The final homework assignment is a final report of a team project (two students per team) on a topic or problem related to a hybrid system.