Metodolog´ıa del Desarrollo

This section develops an algorithm to synthesize an optimal liveness-enforcing su-pervisor M^∗and the controlled system(N^m, M₀^m) for an S³PR(N, M0) when such a supervisor exists. With the aid of the MIP-based deadlock detection method [7], this synthesis algorithm enjoys high computational efficiency. It generates an optimal controlled system with liveness represented by(N^m, M₀^m) or reports “Undecided”.

The latter means that the algorithm cannot sufficiently decide whether there exists an M^∗for a plant model(N, M0).

Algorithm 6.1Synthesis of an M^∗ Input: an S³PR plant model(N, M0) Output:(N^m, M₀^m) or “Undecided”

6.4 Synthesis of Optimal Supervisors 179 N₁:= N

M₁:= M0

L:= |PA| + |P⁰| + |PR| ΠE:= /0

ΠD:= /0 Π:= /0 P_V := /0 flag a := 1 flag b := 1

while (G^MIP(M1) < L and flag b = 1) do find a maximal unmarked siphon Smax

ΠS_max:= {S|S is a minimal siphon derived from S^maxand∀p ∈ S, p ∈ P^V} find a siphon S fromΠS_max\Π such that∀S^′∈ΠS_max,|S^′R| ≥ |SR|

Π:=Π_{∪ {S}}

if S is elementary with respect toΠEthen ΠE:=ΠE∪ {S}

add monitor VSfor S by Proposition 6.3 withξS= 1 P_V := PV∪ {VS}

L:= L + 1

the augmented net is denoted by(N^m, M₀^m) N₁:= N^m

M₁:= M₀^m else

Π_D:=Π_{D∪ {S}}

add monitor VSfor S by Proposition 6.3 withξ_S= 1 ifη_Sis optimally controlled then

PV:= PV∪ {V^S} L:= L + 1

the augmented net is denoted by(N^m, M₀^m) N₁:= N^m

M₁:= M₀^m else

flag a := 0 flag b := 0 end if end if end while

if (flag a= 1) then N^m:= N1

M₀^m:= M1

output(N^m, M₀^m) else

output “Undecided”

end if

This algorithm can synthesize an M^∗for a plant S³PR model(N, M0) if some conditions are satisfied. Note that if there is no emptiable siphon in the plant,(N, M0) itself can be considered to be optimally controlled already. An intuitive but naive algorithm to synthesize such a supervisor is first to compute the set of strict minimal siphonsΠin(N, M0). Then, a monitor VSis added for each siphon S inΠwithξS= 1. For each dependent siphon, if its T -vector is optimally controlled, the intuitive algorithm can output an M^∗. Note that the number of additional monitors exactly equals that of strict minimal siphons in(N, M0), i.e., |PV| = |Π|, which is in theory exponential with respect to the size of N.

The naive algorithm suffers from two problems: computational complexity and structure complexity. For example, as shown in [16], it takes more than 6 hours to compute all 169 strict minimal siphons in an S³PR with 68 places and 54 transitions by using INA [27]. However, it takes 178 CPU seconds only for the MIP-based deadlock detection method to find 24 strict minimal siphons among the 169, where Lindo [19] is used to solve the MIP problems. The high efficiency of the MIP-based method is also fully shown in [7] by case studies.

The structural complexity of a supervisor is usually measured by the number of monitors. The intuitive algorithm adds exactly|Π| monitors. However, the potential redundancy of some monitors motivates us to develop a method to systematically remove them.

Algorithm 6.2Structural reduction of M^∗ Input:(N^m, M₀^m), an optimal controlled system

Output(N^Rm, M₀^Rm), a structurally reduced controlled system suppose that PV = {VSi|i ∈ Nn} is the set of monitors

L:= |PA| + |P⁰| + |PR| + |PV| N1:= N^m

M₁:= M₀^m i:= 1

while (i≤ n) do

remove VS_i and its related arcs from(N1, M1) denote the resultant net as G= (N2, M2) if G^MIP(M2) = L − 1 then

L:= L − 1 N₁:= N2

M₁:= M2

i:= i + 1 P_V := PV\{VSi} else

i:= i + 1 end if end while N^Rm:= N1

M₀^Rm:= M1

output(N^Rm, M₀^RM)

6.4 Synthesis of Optimal Supervisors 181 Algorithm 6.2 can sufficiently distinguish monitors whose removal does not change the liveness property of the optimal controlled system computed by Algo-rithm 6.1. It depends on the order in which monitors are taken into consideration.

Furthermore it requires that all monitors are generated first. Note that a redundancy test algorithm for monitors is also proposed in [30], which needs the complete state enumeration.

To illustrate the supervisor synthesis and reduction algorithms, the net in Fig.

6.5(a) is taken as an example.

Example 6.7.Applying the MIP-based deadlock detection method to the net shown in Fig. 6.5(a), we can first obtain a maximal unmarked siphon Smax= {p5, p6, p9, p12

− p14} from which a strict minimal siphon S1= {p5, p9, p12, p13} with M0(S1) = 3 is derived. It is easy to see that[S1] = {p3, p4}. By Proposition 6.3, a monitor VS₁is added such that VS1+ p3+ p4is a P-invariant of the resultant net that is denoted by (N^α, M₀^α) withξ_S1 = 1 and M₀^α(VS1) = 2, as shown in Fig. 6.5(b).

The MIP-based deadlock detection method is applied to(N^α, M₀^α) and we can obtain a maximal unmarked siphon Smax= {p4, p6, p9, p12− p14, VS1}. From Smax, we can derive a minimal siphon S2= {p4, p6, p13, p14} with [S2] = {p5, p8}. Since η_S1 andη_S2are linearly independent, we add monitor VS2. The resultant net system, as shown in Fig. 6.5(b), is denoted by(N^α, M₀^α), where M₀^α(VS₂) = 2.

The same method is applied to(N^α, M₀^α). A maximal unmarked siphon Smax= {p4, p6, p9, p12− p14, VS₁, VS₂} is detected, from which a strict minimal siphon S3= {p6, p9, p12− p14} is found. The factηS₃=ηS₁+ηS₂indicates that S3is a strongly dependent siphon. A monitor VS₃is added withξS₃= 1. The resultant net is denoted by(N^α, M₀^α). Clearly, we have M₀^α(VS₃) = 3. Since M0(S^R_1,2) = M0(p13) = 2 > 1, ηS₃ =ηS₁+ηS₂ is optimally controlled. When the MIP-based deadlock detection method is applied to G= (N^α, M₀^α) that has three monitors, we have G^MIP(M₀^α) = 18, implying that no unmarked siphon can be derived. As a result, the net shown in Fig. 6.5(b) is an optimal controlled system for the plant model in Fig. 6.5(a).

For this example, Algorithm 6.2 indicates that the removal of any monitor de-teriorates the liveness of the controlled system. Hence, we have the monitor set P_V = {VS1,VS2,VS3}. ByΠ _{= {S1}, S2, S3}, we conclude |PV| = |Π_|.

Remark 6.2.A hybrid policy that combines deadlock prevention and avoidance is developed in [1, 4], and [6], respectively. It consists of two stages. First, a monitor V_S is added by Proposition 6.3 for each strict minimal siphon S with ξS= 1 on condition that all strict minimal siphons in a plant net model(N, M0) are known.

Then, the exertion of an online deadlock avoidance policy depends on whether the net(N^α, M₀^α) resulting from the first stage contains deadlocks, which is verified via computing the existence of emptiable siphons in(N^α, M₀^α).

A two-stage deadlock prevention policy is developed in [13]. The first stage adds monitors to plant model(N, M0) such that no siphons in it can be unmarked, leading to an augmented net(N^α, M₀^α). The second stage, control-induced siphon control, becomes necessary via adding monitors if(N^α, M₀^α) contains deadlocks, which is verified by solving an MIP problem.

The deadlock control strategy presented in [5] is also of two stages. The first is identical to those proposed in [1, 4], and [6], leading to an augmented net(N^α, M₀^α).

The second stage aims to eliminate deadlocks in(N^α, M₀^α) by properly modifying the initial markings of the monitors added in the first stage. Likewise, the second stage is put into execution provided that (N^α, M₀^α) contains deadlocks, which is verified by finding the emptiable siphons in it.

The two-stage policies mentioned above suffer from the siphon computation problem in the augmented net(N^α, M₀^α) resulting from the first stage. As is known, the number of siphons in a net is exponential with respect to its size. Since each strict minimal siphon needs a monitor to add to prevent it from being emptied or insufficiently marked, in theory, the size of (number of the monitors in)(N^α, M₀^α) is exponential with respect to the size of(N, M0). Therefore, siphon computation in (N^α, M₀^α) is time-consuming or impossible by the above methods in the case of a large-sized plant model.

However, the results in this chapter indicate that onceΠE=Π in an S³PR plant model is true, the first stage of these two-stage policies can result in an M^∗, leading to the fact that the second stage is no longer needed.