Fundamentos de Derecho

Grastien et al. [2007] studied modelling and solving diagnosis problems of DES using satisfiability (SAT). Diagnosis of a DES is translated to a propositional SAT problem in order to be solved by the state-of-the-art SAT algorithms. The results show that SAT algorithms are able to efficiently explore the search space. The related work is using SAT in the context of planning [Kautz and Selman, 1992; Rintanen, 2009].

In SAT, a literal is a state variable or its negation. The set of all literals is L =

A∪ {¬a| a∈ A}. The languageLover Aare all formulae that can be formed from A, or using the connectives ∨ and ¬. The standard definitions for connectives are adapted:

• φ∧ψ≡ ¬(¬φ∨ ¬ψ) • φ→ψ≡ ¬φ∨ψ

§2.1 Existing Work of On-line Diagnosis of DES 15

In the approach of diagnosis using SAT, the states of a DES model are represented by the assignment of a finite set of two-valued Boolean variables in the finite domains.

ΦSD is a formula for the system description SD = hA,Σu,Σo,δ,s0i. A SAT model

consists of:

• a set of state variables A,

• a set of unobservable eventsΣu,

• a set of observable eventsΣo,

• δ ⊆Σo∪Σu→2L×2

L

, which assigns each event a set ofevent instanceshφ,ci, • an initial states0.

A state s : A → {0, 1} is a total function from the state variables to the constants 0 and 1 where 0 means f alse, and 1 means true. Grastien et al. [2007] used an event instanceto describe an eventeas a pairhφ,ci ∈δ(e)meaning thateis associated with changes, denoted asc, in states that satisfy the condition of φ. When etakes place in s, one of the pairshφ,ci ∈δ(e)satisfyings|= φis chosen, and the effect of the event is that the literals in cbecome true. The propositional variables with superscript of time stept are the following:

• at for alla∈ Awhere t∈ {0, 1, 2, . . . ,n}

• et for alle∈Σ_u∪Σ_o wheret ∈ {0, 1, 2, . . . ,n−1}

• ωt for alle∈ Σu∪Σo whereω ∈δ(e)∧t∈ {0, 1, 2, . . . ,n−1}

The successor states0 of a states is denoted assucc(s,c). An event sequencee0,e1,e2,

. . . ,en−1 leads to states s0,s1,s2, . . . ,sn such that ∀i ∈ {0, 1, 2, . . . ,n−1} , ∃hφ,ci ∈

δ(ei)|si |=φ∧si+1 =succ(si,c). However, a statesiand an eventei may not uniquely

determine the successor statesi+1. When an event occurs, the event must be possible

in the current state, i.e.

for everyω= hφ,ci ∈δ(e),ωt→φt. The effects of the transitions are:

for everyω =hφ,ci ∈δ(e),ωt → ^

l∈c

lt+1.

For a diagnostic task, observationsOBS is a set of pairshe,tiwhere e ∈ Σ_o is an observable event, andt ∈N+ is a positive integer. Hence, observations are regarded

as a collection of timed observable events. A sequence of states and events is called a trajectory, which models the behaviour of a system. The diagnosis label of a trajectory says a trajectory is faulty if it contains any faulty event. Otherwise, a trajectory is nominal. A system status isambiguouswhen nominal and faulty behaviours cannot be distinguished from a sequence of observations. The assumption is that it is sufficient to find any one nominal behaviour that is consistent with the observations, and the diagnostic output is that the system status is nominal. Formally, a system is nominal if there exists a sequence E = E0,E1,E2, . . . ,En−1 of events on the system model

consistent with OBS such that E is nominal. Furthermore, Grastien et al. [2007] proposed a theorem that the solutions to a SAT problem ΦSD ∧ΦOBS represent the

set of traces on SD consistent with the observations OBS and ending with the last observation ofOBS.

In summary, diagnosis using SAT has advantages and disadvantages. Firstly, the formulation of a SAT problem enables a SAT solver to run efficiently. Diagnosis using SAT takes the advantage that once a trace has been computed, it can be reused for different diagnostic queries. Therefore, SAT diagnosis is beneficial when utilising the set of all computed traces. Secondly, Section 2.1.1 has reviewed that the construction of a Sampath et al. diagnoser may be extremely expensive because the size of a Sampath et al. diagnoser is exponential to the number of states in the system. The SAT approach has the advantage that there is no such direct dependency between run time and the number of states [Grastien et al., 2007].

On the other hand, diagnosis using SAT has a potential problem that the run time in the worst case grows exponentially to the length of the observation sequence. Another disadvantage is that the complexity of a SAT problem is exponential to the number of propositional variables, which is linear to the number of state variables, and linear to the number of time steps [Grastien et al., 2007].

In conclusion, diagnosis using SAT is still a difficult problem when the number of observations increases. Also, it is an open question which SAT solver is the best for a particular diagnosis problem. It has been recommended that incremental diagnosis and temporal windows have advantages to deal with this problem such that the diagnosis of a temporal window is ensured to be consistent with the next one. Notice that Section 2.1.5 will review the existing work on incremental diagnosis. Then, this work will propose time-window-based diagnostic algorithms in Chapter 4 and 5.