1. Observability of Piecewise- Afiine Hybrid Systems Collins and Van Schuppen

2. Overview

3. Definition and goal A piecewise-affine hybrid system (PAHS) can be considered as a product of a finite state automaton and a family of finite-dimensional affine systems on polytopes. Discuss the observability conditions (necessary and sufficient conditions) for a restricted class of hybrid systems called jump-linear systems.The focus is on discontinuous jumps in the systems state, and switches induced by guard conditions. The Goal:

4. Piecewise-Affine hybrid systems - I Definition:

5. Piecewise-Affine hybrid systems - II Definition (Cont’ed):

6. Piecewise-Affine hybrid systems - III

7. Definition 2 - presentation q X init q(t) XqXq x - (t) S (q,t) (x 0 )

8. The system Assumption: (non-blocking) every trajectory can be continued for infinite time, Assumption: (non-Zenoness) only finitely many events occur on any finite time interval. Considered systems belong to the class of PAHS without inputs. Where

9. Observability The state-output map of a deterministic system on the time interval [t 0, t 1 ) is the functional : X × U [t0,t1)  Y [t0,t1) assigning to each initial state x 0 ∈ X and each admissible input function u(t) the output function y(t) for the trajectory x(t) giving the response of the system to the input function u(t) with x(t 0 ) = x 0. A system is (initial-state) observable if the initial state can be determined from the output function y(t) ∈ Y [t0,t1), and final-state observable if the final state can be determined from the output function.

10. Observability of PATH An event s detectable at a point x if it produces a measurable change in output, otherwise it is undetectable at x. An event is detectable in a state q if it is detectable at all points in the guard set The event-time sequence of a trajectory is the sequence (t i ) of event times. The timed event sequence of a trajectory is the sequence of pairs (e i, t i ) of events and event times.

11. Observability for affine systems Consider the affine system: By derivation we get: Observability matrix Observability vector Output derivative vector

12. Observability for affine systems - II The observability map Rank( ) = ? Discrete states

13. Observability for affine systems – III Discrete state Determining the Continuous State

14. Conditions for observability of PATH

15. Single-Event observability

16. Examples

17. Relevant references Balluchi, A., Benvenuti, L., Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L.: Design of observers for hybrid systems. In Tomlin, C.J., Greenstreet, M.R., eds.: Hybrid Systems: Computation and Control. Volume 2289 of Lecture Notes in Computer Science. Springer-Verlag, Berlin Heidelberg New York (2002) 76–89 Balluchi, A., Benvenuti, L., Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L.: Observability for hybrid systems. In: Proc. 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA (2003) Bemporad, A., Ferrari-Trecate, G., Morari, M.: Observability and controllability of piecewise a.ne and hybrid systems. IEEE Trans. Automatic Control 45 (2000) 1864–1876 Vidal, R., Chiuso, A., Soatto, S., Sastry, S.: Observability of linear hybrid systems. In Maler, O., Pnueli, A., eds.: Hybrid Systems: Computation and Control (Prague). Number 2623 in Lecture Notes in Computer Science, Springer (2003) 527–539 ¨Ozveren, C., Willsky, A.: Observability of discrete event systems. IEEE Trans. Automatic Control 35 (1990) 797–806