Observability of Piecewise-Afiine Hybrid Systems Collins and Van Schuppen
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.
Piecewise-Affine hybrid systems - I Definition:
Piecewise-Affine hybrid systems - II Definition (Cont’ed):
Definition 2 - presentation S(q,t)(x0) Xinitq(t) x-(t) q Xq
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
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.
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.
Observability for affine systems Consider the affine system: By derivation we get: Observability matrix Observability vector Output derivative vector
Observability for affine systems - II The observability map Rank( ) = ? Discrete states
Observability for affine systems – IIIDiscrete state Determining the Continuous State
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