Robust liveness-enforcing supervisor for Petri nets with unreliable resources based on mixed integer programming

Abstract

Petri nets, as an effective mathematical tool, have been intensively used in modeling and analyzing automated manufacturing systems (AMSs). Many deadlock control policies have been proposed for AMSs, but most of them assume that resources never fail during product processing. However, resource failures may happen in a real world, which may invalidate existing control policies. This paper concentrates on robust liveness-enforcing supervisor design for a system of simple sequential processes with multiple unreliable resources. Recovery subnets model resource failure and recovery, which are added to the holders of unreliable resource places. The proposed method consists of two steps. At the first step, a mixed integer programming (MIP) problem is developed to detect a strict minimal siphon that can be emptied. At the second step, an extended constraint set derived by the complementary set of a siphon is constructed. The siphon is controlled through the extended constraint set by adding a control place. The above two steps are executed in an iterative way until no new empty siphon is found and a robust liveness-enforcing supervisor can be obtained. Examples are used to expose the advantages of the proposed method.

Appendix A

Definition 10

(Liu et al. 2013b) Let $(N_1,M_1)$ and $(N_2,M_2)$ be PNs with $N_1=(P_1,T_1,F_1,W_1)$ and $N_2=(P_2,T_2,F_2,$ $W_2)$, where $P_1\cap P_2=P_C\ne \emptyset$ and $T_1\cap T_2=\emptyset$. (N, M) with $N=(P,T,F,W)$ is said to be the resultant net of composing $(N_1,M_1)$ and $(N_2,M_2)$ via the set of shared places $P_C$ if (1) $P=P_1\cup P_2$, $T=T_1\cup T_2$, $F=F_1\cup F_2$, and $W(x,y)=W_i(x,y)$ if $(x,y)\in F_i$, $i=1,2$; and (2) for all $p\in P_1\setminus P_C$, $M(p)=M_1(p)$, for all $p\in P_2\setminus P_C$, $M(p)=M_2(p)$, and for all $p\in P_C$, $M(p)=max\lbrace M_1(p),M_2(p)\rbrace$. The composition of $N_1$ and $N_2$ is denoted by $N_1\bigotimes N_2$.

Definition 11

(Liu et al. 2020) Let $(N_1,M_1)$ and $(N_2,M_2)$ be two nets with $N_i=(P_i,T_i,F_i,W_i)$, $i=1,2$, satisfying $P_1 \cap P_2=\emptyset$. (N, M) with $N=(P,T,F,W)$ is said to be a synchronous synthesis net resulting from the merge of $(N_1,M_1)$ and $(N_2,M_2)$, denoted by $(N_1,M_1)\Vert (N_2,M_2)$, if (1) $P=P_1\cup P_2$; (2) $T=T_1\cup T_2$; (3) $F=F_1\cup F_2$; (4)$W(f )=W_i(f )$ if $f \in F_i, i = 1, 2$; (5) $M(p)= M_i(p)$ if $p \in P_i, i= 1, 2$.

Definition 12

(Ezpeleta et al. 1995) A system of simple sequential processes with resources ($\hbox {S}^3$PR) is a PN $N = (P_A, \{p^0\}, P_R,T,F)$ defined as the union of a set of nets $N_i= (P_{A_i}, \{p^0_i\}, P_{R_i}; T_i; F_i)$ sharing common places, where the following statements are true:

• (1) $p^0_i$ is called the process idle place of $N_i$. Elements in $P_{A_i}$ and $P_{R_i}$ are called activity and resource places, respectively. A resource place is called a resource for short in case of no confusion.

• (2) $P_{R_i} \ne \emptyset ; P_{A_i}\ne \emptyset ; \{p^0_i\}\notin P_{A_i}; (P_{A_i}\cup \{p^0_i\})\cap P_{R_i} =\emptyset ; \forall p\in P_{A_i}, \forall t\in ^{\bullet }p, \forall t'\in p^{\bullet }, \exists r_p\in P_{R_i}, ^{\bullet }t\cap P_{R_i}=t'^{\bullet }\cap P_{R_i}=\{r_p\};\forall r\in P_{R_i}, ^{\bullet \bullet }r\cap P_{A_i}=r^{\bullet \bullet }\cap P_{A_i}\ne \emptyset$; $\forall r\in P_{R_i}, ^{\bullet }r\cap r^{\bullet }=\emptyset$; $^{\bullet \bullet }\{p^0_i\}\cap P_{R_i}=\emptyset$.

• (3) $N_i$ is a strongly connected state machine, where $\overline{N_i}= (P_{A_i}, \{p^0_i\},P_{R_i};$ $T_i; F_i^\prime )$ is the resulting net after the places in $P_{R_i}$ and related arcs are removed from $N_i$ and $F_i^\prime =F_i\cap \{[(P_{A_i}\cup \{p^0_i\})\times T_i]\cup [T_i\times (P_{A_i}\cup \{p^0_i\})]\}$.

• (4) Every circuit of $\overline{N_i}$ contains place $p^0_i$.

• (5) Any two $N_i^\prime$s are composable when they share a set of common places. Every shared place must be a resource.

• (6) Transitions in $^\bullet (p^0_i)$ and $(p^0_i)^\bullet$ are called the source and sink transitions of an $\hbox {S}^3$PR, respectively.

• (7) $N = (P_A,\{p^0\}, P_R,T,F)=\bigcirc ^n_{i=1} N_i$, where $P_A =\bigcup ^n_{i=1} P_{A_i}; \{p^0\}= \bigcup ^n_{i=1} \{p^0_i\}; P_R= \bigcup ^n_{i=1} P_{R_i}$; and $T =\bigcup ^n_{i=1} T_i$.