### ControL and Analysis for the Stability of Hybrid and Embedded Systems

Faculty of Electrical Engineering, Mathematics and Computer Science

Tel.: +31 53 489 3438 / 3714

Email: j.w.polderman@utwente.nl / r.langerak@utwente.nl

## Summary

The aim of the project is to develop a theory for stability and control of switched linear systems. A switched linear system is a dynamical system consisting of a finite number of different continuous time linear systems among which the overall system switches. With each continuous time system we associate a discrete state or location. The dynamics of the system has a discrete component that describes the switches from one discrete state to the other and a continuous component corresponding to the dynamics within each discrete state. For that reason switched systems are also referred to as hybrid systems. Hybrid systems arise naturally in the area of embedded systems design, where continuous dynamics of both physical components and the environment interact with logical and discrete processes implemented in software or hardware.

In such cases stability is a crucial property for guaranteeing the system to be correct, dependable, and safe. Stability of switched systems is defined analogously to stability of systems described by ordinary differential equations. Whereas the stability theory for non-switched linear systems is elementary and well-understood, the stability of switched systems is, due to the interaction between the continuous and discrete dynamics, much more involved. In this project we will develop a stability theory for switched linear systems with two branches: one based on transition gain analysis and one based on delayed switching. In both approaches optimization of local Lyapunov functions is used to reduce conservatism. One of the main innovations is that the discrete dynamics and the corresponding interaction between the dynamics in the discrete states will be analyzed using methods from automata theory. Subsequently the stability results will be used for controller synthesis for the case that the continuous time systems in the discrete states are models of linear control systems. Finally, the project aims at the development of a more general and fundamental notion of hybrid Lyapunov function.

**Project duration:** 2007-2011

**Project budget:** 165 k-€

**Number of person/years:** 1 fte / year

**Involved groups: ** Mathematical Systems & Control Theory (MSCT), Formal Methods and Tools (FMT)