**GRADUATE COURSE ON ROBUST 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 increased 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. In the workshop, we will present recent advances in the theory and design of hybrid control systems, with focus on robustness properties.

In this short course, we will present a general modeling framework for hybrid systems and relevant modern mathematical tools. Next, we will introduce asymptotic stability and its robustness, and describe systematic tools like Lyapunov functions and invariance principles. The power of hybrid control for (robust) stabilization of general nonlinear systems will be displayed in applications including genetic networks, smart grid systems, groups of neurons, control of robotic manipulators, autonomous vehicles, and juggling systems.

**COURSE CONTENT**

The course consists of five modules described below. Each module includes time for questions and informal discussions.

*Main reference:*

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

*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**PART 1: Modeling hybrid systems**

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

• Mathematical examples of hybrid systems

• Applications:

o Smart grids

o Sensor networks

o Coordination to UAVs

o Genetic networks

o Spiking neurons

o Juggling systems

Key references: [34], [20], [36].

Assignments: Homework 1**PART 2: Concept of solution**

• Introduction to solution concepts to hybrid systems

• Hybrid time domains and hybrid arcs

• Solutions and basic properties

Key references: [34], (2), [40], (1).

Assignments: Homework 2**PART 3: Simulation of hybrid systems**

• Introduction to simulation issues

• Hybrid Equations Toolbox

• Examples

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

**PART 4: Introduction to asymptotic stability**

• Introduction to stabilization for hybrid systems

• Well-‐posed hybrid systems

• Lyapunov functions and sufficient conditions

• Applications:

o Conversion in smart grids

o Stability of genetic networks

o Synchronization and desynchronization in spiking neurons

Key references: (6), (3), [18].

Assignments: Homework 3

**PART 5: Applications of hybrid control**

• Advantages of hybrid control

• Combination of multiple controllers

• Trajectory tracking control

• Discussion on open problems and future directions

Key references: [7], [33], [11], [17], [51].

Assignments: Final project

**REFERENCES:**

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

[*]: https://hybrid.soe.ucsc.edu/biblio

(1): https://hybrid.soe.ucsc.edu/software

(2): R. Goebel and A. R. Teel, “Solutions to Hybrid Inclusions Via Set and Graphical Convergence with Stability Theory Applications,” Automatica, 2006, 42, 573-587.

(3): R. Goebel, R. G. Sanfelice, and A. R. Teel, “Hybrid Dynamical Systems: Modeling, Stability, and Robustness,” Accepted for Pub. in Princeton University Press, 2010.

(4): H. Hermes, “Discontinuous Vector Fields and Feedback Control,” Differential Equations and Dynamical Systems, Academic Press, 1967, 155-165.

(5): O. Hàjek, “Discontinuous Differential Equations I,” Journal of Differential Equations, 1979, 32, 149-170.

(6): R. Rockafellar, R. J.-B. Wets, “Variational Analysis,” Springer, 1998.