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As already said in Section 3.4, by sampling the continuous-time CPN system with a sampling period Θ, we obtain the discrete-time TCPN ([64]) given by:

m_k+1= mk+ Θ · C · wk

0 ≤ wk≤ fk (4.3)

Here mkand wk = fk− ukare the marking and controlled flow at sampling instant k, i.e., at τ = k · Θ, while fk and uk are the uncontrolled flow and control input.

It is proved in [64] that if the sampling period satisfies (4.4), the interior reach- ability spaces of discrete-time and continuous-time CPN systems are the same.

∀p ∈ P : X

tj∈p•

λj· Θ < 1 (4.4)

In the sequel, we assume that the sampling period Θ is small enough to satisfy (4.4). We first develop the ON/OFF controller based on the discrete-time model,

4.2. Minimum-time controller for Choice-Free nets

In a CF net system, if two transitions t1 and t2 are enabled at the same time, the order of firing is not important (i.e., both sequence t1t2 and t2t1 are fireable). Based on this observation, if there exists a transition that has not fired with the maximal amount at one moment, certain amount of its firings may be moved ahead in order to reach this maximal quantity.

Example 4.2.4. Let us consider the trivial CF net system in Fig. 4.2 and as- sume mf = [0.2 0.5 0.3]T, the minimal firing count vector for reaching the fi- nal state is σ = [0.8 0.3 0]T. Following this vector, one firing sequence may be σ1 = t1(0.5)t2(0.3)t1(0.3). It can be observed that t1 is 1-enabled under m0, and the required amount that t1 should fire is 0.8. Therefore, we can fire t1 more than 0.5 in the beginning. In particular, the final marking is also reached by the firing sequence σ2 = t1(0.8)t2(0.3), for example.

p3 t1 p2 t2 t3 p1

Figure 4.2: A trivial CF net system with m0 = [1 0 0].

The strategy of the ON/OFF controller is quite simple: every transition fires as fast as possible at any moment until the required firing count σ[tj] (mf = m0+C ·σ) is reached. The control input of tj at kth sampling period is:

uk[tj] =                                  0 if Θ · k−1 P i=0 wi[tj] + Θ · fk[tj] ≤ σ[tj] (a) fk[tj] if Θ · k−1 P i=0 wi[tj] = σ[tj] (b) fk[tj] − σ[tj]−Θ· k−1_P i=0 wi[tj] Θ if Θ ·k−1P i=0 wi[tj] < σ[tj] and (c) Θ ·k−1P i=0 wi[tj] + Θ · f [tj] > σ[tj] (4.5) where at k = 0, Θ ·k−1P i=0

wi[tj] = 0. Remember wk = fk− uk, (a) says that before reaching the required total firing count σ[tj], we simply let transition tj to fire free (ON), i.e. uk[tj] = 0; (b) means once σ[tj] is reached, the transition is completely stopped (OFF), i.e. uk[tj] = fk[tj]; (c) represents the last firing of tj. Algorithm 1 synthesizes the ON/OFF controller, in which w0, w1, w2, . . . is the sequence of control inputs at the time instants.

Lemma 4.2.5. Let hN , λ, m0i be a discrete-time continuous CF net system and mf > 0 be a reachable final state. Among all the controllers that drive the system to

Algorithm 1 ON/OFF controller Input: hN , λ, m0i, mf, σ, Θ Output: w0, w1, w2, . . . 1: k = 0 2: while Θ · k−1 P i=0 wi≤ σ do

3: Solve the f ollowing LP P :

max 1T · wk s.t. mk+1 = mk+ Θ · C · wk 0 ≤ Θ · wk ≤ σ − Θ · k−1 P i=0 wi wk[tj] ≤ λj· enab(tj, mk), ∀tj ∈ T (4.6) 4: Apply w_k: m_k+1 = m_k+ Θ · C · w_k 5: k := k + 1 6: end while 7: return w₀, w₁, w₂, . . .

mf by firing σ, i.e., mf = m0+ C · σ, the ON/OFF controller costs the minimum- time.

Proof: Assume an arbitrary non ON/OFF controller G. Hence, at a sampling

period k there exists a transition tj that is not sufficiently fired, i.e., not fired as much as possible. In other words, tj has to fire later in a sampling period l, l > k. Let us assume, without loss of generality, that tj does not fire between the kth and the lth_{sampling periods. It is always possible to “move” some amounts of its firings} from the lth _{sampling period to the k}th _{one until t}

j becomes sufficiently fired in k. According to the persistency property of CF nets, this move is not reducing the enabling degree of the other transitions. Iterating the procedure, all transitions can be sufficiently fired in all sampling periods and the obtained controller is an ON/OFF one. Obviously, the number of discrete-time periods required to reach the final marking after moving firings from a sampling period l to another one k with k ≤ l is at least the same. Hence the number of sampling steps of the ON/OFF controller is not more than the one of controller G, i.e., the ON/OFF controller costs the minimum time.

Lemma 4.2.5 only holds for CF nets. For a net system that is not CF, the ON/OFF controller may be not a minimum-time controller for a given σ, and in the worst case, the final state may not be reached, i.e., the stability is not guaranteed (see Ex. 4.3.1 for an example, in which the (reachable) final state cannot be reached

by applying the ON/OFF controller using the given σ.)

4.2. Minimum-time controller for Choice-Free nets

and σ′ be firing count vectors, such that σ ≤ σ′ and they both drive the system to mf > 0, i.e., mf = m0 + C · σ and mf = m0 + C · σ′. By using the ON/OFF controller, firing σ′ _{costs at least the time of firing σ.}

Proof: If firing σ′ _{costs less time than firing σ, there must exist at least a} transition tj, such that firing σ′[tj] costs less time than firing σ[tj]. We will prove that it is not possible.

Since by firing σ′ _{and σ the same final state is reached, σ}′_[t

j] ≥ σ[tj] implies that in the case of firing σ′, more tokens should be put into each of its input place pi ∈ •tj (with the quantity of P re[pi, tj] · (σ′[tj] − σ[tj])), and later they are all moved out (by firing tj). Remember that tj is the unique output transition of pi (the net is CF), so the time spent for moving out this quantity of tokens depends only on tj; and since this quantity of tokens have to be moved out, they do not contribute to the firing of σ[tj] (they do not make σ[tj] firing faster). Therefore, firing σ′[tj] cannot cost less time than firing σ[tj].

Lemma 4.2.6 holds also only for CF nets. Let us consider the following simple example:

Example 4.2.7. Assume that in the non-CF net system in Fig. 4.3 the firing rate vector is λ = [1 0.01 1 1 1]T, the sampling period is Θ = 0.1, and we want to reach a final state mf = [0 0.5 0.5 4]T. mf can be reached, for example, by using σ = [0 0.5 0 0 0]T _{or σ}′ _{= [0 0.5 0 4 4]}T_{. Although σ ≤ σ}′_{, we can verify that if we} apply the ON/OFF controller using σ the final state is reached in 692 time steps; if we we apply the ON/OFF controller using σ′ the final state is reached in only 673 time steps. In the case of σ, only t2 fires, so the time used for the firing of σ[t2] determines the time to reach the final state. In the case of σ′_{, transitions t}

4 and t5 also fire (an additional T-semiflow [0 0 0 1 1]T _{is fired). By firing t}

4 we can increase the marking of p2 (at some time instants, later this increased quantity of marking is moved back to p4 by firing t5), so t2 fires faster; on the other hand, since the firing rate of t2 is much smaller than the others, the firing of σ′[t2] still determines the total time to reach mf. Therefore, the final state is reached faster in the case of firing σ′.

On the other hand, even for CF nets, given σ ≤ σ′_{, m}

f = m0 + C · σ and

mf′ = m0 + C · σ′, if mf 6= mf′, by applying the ON/OFF controller, reaching mf′ with σ′ may be faster than reaching mf with σ.

Example 4.2.8. Let us consider again the trivial MG (a subclass of CF) in Fig. 4.2 of Ex. 4.2.4. Now assume that m0 = [1 1 0]T; λ = [10 1 1]T; and sampling period is Θ = 0.01. Given σ = [0 0.5 0]T _{and σ}′ _{= [0.5 0.5 0]}T _{(σ ≤ σ}′_{), by using} the ON/OFF controller, σ is fired in 69 time steps ([1 0.5 0.5]T _{is reached) and σ}′ is fired in only 42 time steps ([0.5 1 0.5]T _{is reached).}

Proposition 4.2.9. Let hN , λ, m0i be a consistent and strongly connected discrete- time continuous CF net system and σ ≥ 0 be a firing count vector driving the system

p

t

t

p

p

t

t

t

p

Figure 4.3: A simple non-CF net (a state machine): by using the ON/OFF controller, firing σ may cost more time than firing σ′ ≥ σ

to mf > 0, i.e., mf = m0+ C · σ. The ON/OFF controller is a minimum-time controller driving the system to mf if σ is the minimal firing count vector.

Proof: In consistent and strongly connected CF nets, there exist a unique minimal firing count vector and a unique minimal T-semiflow x > 0 [93], hence for any other firing count vector σ′ 6= σ that drives the system to mf we must have σ′ = σ + α · x > σ, α > 0. According to Lemma 4.2.5 and 4.2.6, the results can be derived straightforwardly.

Remark 4.2.10. When the final state mf has been reached by applying the ON/OFF controller, all the transitions are stopped. If mf is an equilibrium point, it can be maintained by using an appropriated control uk, such that C · wk= 0.