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

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

Available fromhttp://www.u.arizona.edu/~sricardo/Preprints/2013/Sanfelice.13.Wiley_preprint.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 Spiking neurons

o Juggling systems

o Genetic networks

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: Asymptotic stability and Invariance**

• Introduction to stabilization for hybrid systems

• Well-‐posed hybrid systems

• Lyapunov functions and sufficient conditions

• Well posedness

• Invariance principles

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

Assignments: Homework 3**PART 5: Robustness**

• Effect of small noise

• Generalized solutions

• Perturbed hybrid systems

• Robustness of stability

Key references: (4), (5), [29]. **PART 6: Hybrid Control**

• Control Lyapunov functions

• Existence of stabilizing state-feedback laws

• Minimum norm control

• Passivity based control

• Tracking control

Key references: [51], [55], [56], [67], [69], [75], [85], [92]. **PART 7: Applications**

• Juggling systems

• Neuron control

• Genetic networks

• Power conversion

Key references: [17], [79], [80], [82], [83], [89], [91], [93].

**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

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