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2.3

Sensor Minimisation and Dynamic Observers for Timed

Systems

In the case of static observers, a set of observable events do not change during the execution of a system. This in turn determines the set of unobservable events. Thus, observing an event usually requires some detection mechanism, which brings the design questions of which sensors to use, how many of them, and where to place them. They are often difficult to answer, especially without knowing what these sensors are to be used for.

However, using a fixed set of observable events, observing events is costly in terms of time and energy. Firstly, it consumes time to read and process the infor- mation provided by the sensors. Secondly, it requires power for the operation of the sensors and computation of the data from the sensors. Consequently, it is essential to ensure that the sensors in use are necessary, and provide useful information. It is also important to discard any information given by a sensor that is not needed. In the case of a fixed set of observable events, not all sensors always provide useful information. Also, energy for sensor operation and data computation is sometimes spent for nothing.

The sensor minimisation problem focuses on dynamic observers for fault diagno- sis of timed systems [Brandán-Briones et al., 2008; Cassez, 2010]. In particular, Cassez [2010] studied dynamic observers for a timed automaton. The sensor minimisation problem does not consider the problems of sensor placement, or choosing between different types of sensors. It is reasonable to follow the DES setting where the be- haviours of a plant are known and a model is available as a finite state automaton over Σ∪ {e,f}where Σis a set of observable events of a DES, andeis a single un- observable event to represent invisible actions. f is a special unobservable event that corresponds to a fault. Finally, the aim is to test diagnosability in polynomial time for a given plant and a fixed set of observable events.

2.3.1 Definitions and Examples

This section reviews the definitions of Cassez [2010] for timed word, un-timed word, clock valuation and timed automaton. Firstly, a timed word, denoted as w, is one or more events with time descriptions. For example, 0.4a1.0b2.7c is a timed word. The numbers are the time elapsed between two events. Secondly, Unt(w) repre- sents the un-timed version of w obtained by erasing all durations. For example, Unt(0.4a1.0b2.7c) = abc. Thirdly,clock valuationmeans adding a time constraint to each state and transition in an automaton. Finally, atimed automaton (TA) is a finite

automaton extended with real-valued clocks to specify timing constraints between occurrences of events. A timed automaton TA is a tuple (L,l0,X,ΣT,E,Inv,F,R) where

• Lis a finite set of locations; • l0is the initial location;

• Xis a finite set of clocks; • Σ_T is a finite set of actions;

• E⊆ L×C(X)×ΣT ×2X×Lis a finite set of transitions whereC(X)is a set of conjunctions of constraints of the form x on c with c ∈ Z andon∈ {≤,<,=,>

,≥}; for(l,g,a,r,l0)∈E,

– g is a guard;

– a is an action;

– ris a reset value;

• Inv⊆C(X)Lassociates an invariant with each location; • F⊆L is the final set of locations;

• R⊆ Lis the repeated set of locations.

Figure 2.9: A timed automatonTA

Fig. 2.9 is an example of timed automaton. [x≤3]is an invariant for the statel1.

The transition from l1 tol2 has a guard x≤ 2 and an actiona. Although reset is not

§2.3 Sensor Minimisation and Dynamic Observers for Timed Systems 31

A state of Ais a pair(l,v)∈ L×RX_≥0. R≥0is the set of positive rational numbers.

A run r of A from (l0,v0) is a finite or infinite sequence of alternating delay δ and discrete move a, i.e.

r = (l0,v0) δ0 −→(l0,v0+δ0) a0 −→(l1,v1). . . an−1 −−→(ln,vn) δn −→(ln,vn+δn). . . (2.11)

such that for everyi≥0 :vi+δ |= Inv(li)for 0≤δ≤ δi (2.12)

there is some transition(li,gi,ai,ri,li+1)∈ E (2.13)

such thatv_i+δ_i |= g_i andv_i+1= (vi+δi)[ri] (2.14)

A finite timed wordwis accepted byTAif the trace of a run ofTAends in a final location. A timed automatonTAis deterministic if there is noelabelled transition in TA, and when the same action may occur, the guard must be different. emeans no event is observed, but a new state is entered.

2.3.2 Sensor Minimisation

The sensor minimisation problem has three aspects. First, it requires to decide whether there is a subset of observable events such that faults can be detected by observing only events inΣo. It also needs to find an optimally minimumΣo.

Second, the sensor minimisation problem allows a diagnoser to raise a fault not necessarily immediately after it occurs, but possibly some time later, as long as this time is bounded by some k ∈ N. Time is modelled by counting the moves that a plant makes including observable and unobservable ones.

Third, it needs to address the masking issue. There are cases where two events are observable but not distinguishable. For example,a andbare observable; when a diagnoser knows thataorboccurred, it does not know which of the two. This is not the same as considering a and bto be unobservable, since in that case a diagnoser would not be able to detect the occurrence. Distinguishability of events is captured by the notion of a mask. Amask(M,n)over Σis a total and surjective function:

Then, morphism is defined as:

M∗ : TW∗(Σ)→TW∗({1, 2, 3, . . . ,n})

where TW∗(Σ)is the set of finite timed words overΣ. For example,

ifΣ={a,b,c,d},n=2,M(a) =M(b) =1,M(c) =2,M(d) =e, then M∗(a.b.c.b.d) =1.1.2.1=M∗(a.a.c.a).

2.3.3 Dynamic Observers

A dynamic observer modifies the set of events that it wishes to observe during a system execution. 0 1 2 3 4 5 f a b e b _a e

Figure 2.10:DES model B

Fig. 2.10 show the DES model B. Suppose the target is to detect faults in B. A static diagnoser that observes Σ = {a,b} works. However, no proper subset of Σ can be used to diagnose the faults in B. Thus, the minimum cardinality of the set of observable events for diagnosing B is 2., i.e. a static observer will have to monitor two events during the execution of this DES model. If a mask is used, the minimum cardinality for a mask is 2 as well. Therefore, an observer will have to be receptive to at least two inputs at each point in time to detect a fault in B.

In contrast, it is more efficient to use a dynamic observer, which only switches on sensors when needed. In the DES model B, a diagnoser only switches on the a- sensor at the beginning; once aoccurs, the a-sensor is switched off, and theb-sensor is switched on. Compared to the static diagnoser, a dynamic observer uses half as much energy. In general, switching sensors on and off leads to energy saving.