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 ofa finite state automaton and a family of finite-dimensional affine systems onpolytopes.

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:

Piecewise-Affine hybrid systems - IDefinition:

Piecewise-Affine hybrid systems - II

Definition (Cont’ed):

• •

•

Piecewise-Affine hybrid systems - III

Definition 2 - presentation

q

Xinitq(t)

Xq

x-(t)

S(q,t)(x0)

The systemAssumption: (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) 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.

Observability of PATHAn 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 (ti) of event times.

The timed event sequence of 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

Conditions for observability of PATH

Single-Event observability

Examples

Relevant referencesBalluchi, 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 Decisionand Control, Maui, Hawaii, USA (2003)

Bemporad, A., Ferrari-Trecate, G., Morari, M.: Observability and controllabilityof 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