GRADUATE COURSE "Analysis and Design of Hybrid Control Systems"


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.


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:
Chapter 1 (sample):

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

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

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

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]


• 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.