**GRADUATE COURSE "Hybrid feedback control systems: Analysis and design"**

**COURSE DESCRIPTION**

Hybrid control systems arise from 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 from controlling systems that are themselves hybrid, such as computer networks, power systems, and nonsmooth mechanical systems. 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.

This course presents recent advances in the analysis and design of hybrid control systems from a control theory viewpoint. The power of hybrid feedback 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.

This course is a natural follow-up to the introductory course on hybrid dynamical systems, to be taught in the preceding week by R. Goebel as module M20. Similar introductory courses have been offered through EECI and HYCON previously by A.R. Teel in 2010 and R.G. Sanfelice in 2011, 2013, and 2014.

**GRADING SCHEME**

• Homework will be assigned at the end of the day.

• In-class activities will be proposed during some of the lectures.

• A final project is required for validation of the course. A short presentation outlining the ideas is required (Friday morning). A final report in pdf using IEEE conference format is due on June 5 midnight PST (e-mail submission)

**COURSE CONTENT**

Some of the files are password protected (pass is the same as the WiFi password for the course)

**PART 1: Introduction to hybrid control systems**

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

• Mathematical examples of hybrid systems

• Motivation to hybrid feedback control

References: Slides, [1], [2], [3]

**PART 2: Modeling hybrid control systems**

• Modeling of hybrid plant and hybrid controller

• Basic notions and properties

• Examples

Assignments: Homework 1

References: [1], [4], [5], [6]

**PART 3: Supervisory hybrid control**

• General control architecture

• Uniting control (state and output feedback)

• Throw-and-catch control

• Examples

• Other control methods

Assignments: Homework 2

References: [7], [8], [9], [10], [11]

**PART 4: CLF-based control**

• Control Lyapunov function concept

• Existence of continuous state feedbacks

• Minimum pointwise norm control

• Examples

**PART 5: Passivity-based control**

• General, Flow, and Jump passivity

• Link to Stability and Attractivity

• Synthesis of stabilizing feedback law

• Examples

**PART 6: Invariance-based control**

• Invariance notions

• Sufficient conditions

• Design of feedback law inducing invariance

• Examples

**PART 7: Stability of interconnections of hybrid systems**

• Feedback and series interconnection

• I/O stability

• Small gain theorem

• Examples

Assignments: Homework 3

References: [5], [4]

*Main references:*

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/sites/default/files/preprints/77.pdf

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/sites/default/files/preprints/34.pdf

### References

- [77] Control of Hybrid Dynamical Systems: An Overview of Recent Advances, , Hybrid Systems with Constraints, April, Issue NULL, p.146-177, (2013)
- [65] Hybrid Dynamical Systems: Modeling, Stability, and Robustness, , New Jersey, (2012)
- [34] Hybrid dynamical systems, , IEEE Control Systems Magazine, April, Volume 29, Issue NULL, Number 2, p.28-93, (2009)
- [96] Input-Output-to-State Stability Tools for Hybrid Systems and Their Interconnections, , IEEE Transactions on Automatic Control, May, Volume 59, Number 5, p.1360-1366, (2014)
- [52] Interconnections of Hybrid Systems: Some Challenges and Recent Results, , Journal of Nonlinear Systems and Applications, Volume 2, Number 1-2, p.111–121, (2011)
- [44] Results on Input-to-Output and Input-Output-to-State Stability for Hybrid Systems and their Interconnections, , Proc. 49th IEEE Conference on Decision and Control, Issue NULL, p.2396–2401, (2010)
- [70] Robust Supervisory Control for Uniting Two Output-Feedback Hybrid Controllers with Different Objectives, , Automatica, July, Volume 49, Issue NULL, Number 7, p.1958–1969, (2013)
- [33] Supervising a family of hybrid controllers for robust global asymptotic stabilization, , Proc. 47th IEEE Conference on Decision and Control, Issue NULL, p.4700–4705, (2008)
- [11] A ``throw-and-catch" hybrid control strategy for robust global stabilization of nonlinear systems, , Proc. 26th American Control Conference, Issue NULL, p.3470–3475, (2007)
- [25] Robust global swing-up of the pendubot via hybrid control, , Proc. 27th American Control Conference, Issue NULL, p.1424–1429, (2008)
- [97] An Observer with Measurement-triggered Jumps for Linear Systems with Known Input, , Proceedings of the 19th IFAC World Congress, Issue NULL, p.140--145, (2014)
- [75] On the existence of control {L}yapunov functions and state-feedback laws for hybrid systems, , IEEE Transactions on Automatic Control, December, Volume 58, Issue NULL, Number 12, p.3242–3248, (2013)
- [85] Pointwise Minimum-norm Control Laws for Hybrid Systems, , Proceedings of the IEEE Conference on Decision and Control, Issue NULL, p.2665–2670, (2013)
- [69] Passivity-based Control for Hybrid Systems with Applications to Mechanical Systems Exhibiting Impacts, , Automatica, May, Volume 49, Number 5, p.1104–1116, (2013)
- [92] Sufficient Conditions for Passivity and Stability of Interconnections of Hybrid Systems using Sums of Storage Functions, , Proceedings of the 2014 American Control Conference, Issue NULL, p.1432-1437, (2014)
- [18] Invariance principles for hybrid systems with connections to detectability and asymptotic stability, , IEEE Transactions on Automatic Control, Volume 52, Issue NULL, Number 12, p.2282–2297, (2007)
- [91] A Robust Hybrid Control Algorithm for a Single-Phase DC/AC Inverter with Variable Input Voltage, , Proceedings of the 2014 American Control Conference, Issue NULL, p.1420-1425, (2014)