1 / 17

Observability of Piecewise-Afiine Hybrid Systems

Observability of Piecewise-Afiine Hybrid Systems. Collins and Van Schuppen. Overview. 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. The Goal:.

danton
Download Presentation

Observability of Piecewise-Afiine Hybrid Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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. The Goal: 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.

  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 S(q,t)(x0) Xinitq(t) x-(t) q Xq

  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 [t0, t1) is the functional : X × U[t0,t1)Y [t0,t1) assigning to each initial state x0∈ 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(t0) = x0. A system is (initial-state) observableif the initial state can be determined from the output function y(t) ∈ Y [t0,t1), and final-state observableif 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 qif it is detectable at all points in the guard set The event-time sequenceof a trajectory is the sequence (ti) of event times. The timed event sequenceof a trajectory is the sequence of pairs (ei, ti) 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 – IIIDiscrete 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

More Related