Graduate course Robust 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 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 workshop, 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 control of robotic manipulators, autonomous vehicles, and juggling systems.

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

Final assignment: a project report will have to be submitted after the course only if you are interested or you need a grade. The topic of your report will be discussed during the informal discussions within each module. If time allows, a short presentation of each project will be scheduled for the last day. The final project report (in pdf format) will be due by email to the instructor no later than May 31st 2013, AOE.

Module 1a: Introduction and Modeling

• Overview of main modeling technique, robustness, and stability results • Examples of hybrid systems in proposed modeling framework

Additional suggested references: [34], [20], [36].

Module 1b: Concept of solution

• Introduction to solution concepts to hybrid systems • Hybrid time domains and hybrid arcs • Solutions and basic properties

Additional suggested references: [34], (2), [40], (3), (1), [74].

Brief Tutorial: A Matlab/Simulink implementation for simulation of hybrid systems

Suggested exercise for module 1: Think of an example of a hybrid system not discussed in class today, write it in the proposed modeling framework, and simulate it in the Matlab/Simulink implementation in (1), [74].

Module 2: Uniform asymptotic stability

• Introduction to stabilization for hybrid systems • Lyapunov functions and sufficient conditions • Equivalent characterizations

Module 3: Perturbations and generalized solutions

• Introduction to generalized solutions • Definitions of Hermes and Krasovskii solutions to hybrid systems • Hybrid system results

Additional suggested references for modules 2 and 3: [27], (3), (4), [29].

Suggested exercise for modules 2 and 3: For your example of a hybrid system, propose a set to be stabilized and attempt a Lyapunov function candidate. Study the existence of solutions, with and without perturbations.

Module 4: Introduction to well-posed hybrid systems

• Well-posed hybrid systems • Consequences

Module 5: Asymptotic stability and invariance for well-posed hybrid systems

• Asymptotic stability for well posed hybrid systems • Invariance of sets and Omega limit sets • Invariance principles

Additional suggested references: (5), [18].

Suggested exercise for modules 5 and 6: For your example of a hybrid system, check if your initial model corresponds to a well-posed hybrid system. If not, study the effect on the solution set when it is modified to be well posed.

Module 6a: Applications of asymptotic stability to hybrid control

• Advantages of hybrid control • Selected hybrid feedback control solutions: o Hybrid control to overcome topological obstructions o Combination of multiple controllers o Trajectory tracking control • Discussion of other control solutions

Additional suggested references: [7], [33], [11], [17], [51].

Module 6b: Summary of recent results and future directions

• Control of hybrid systems via CLFs • Existence of feedback control laws • Interconnections • …

Additional suggested references: [44], [51], [55], [56], [67], [69], [70], [75].


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

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

(1): Download software from 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): H. Hermes, “Discontinuous Vector Fields and Feedback Control,” Differential Equations and Dynamical Systems, Academic Press, 1967, 155-165.
(4): O. Hàjek, “Discontinuous Differential Equations I,” Journal of Differential Equations, 1979, 32, 149-170.
(5): R. Rockafellar, R. J.-B. Wets, “Variational Analysis,” Springer, 1998.