Discrete Event Control Kit DECK 1.2013.11 User Manual
Contents
1. Spec H display Building Spec model H Event set if k E alpha gamma mu else E alpha beta gamma mu end H selfloop automaton 1 1 E Uncontrollable events Euc alpha 2 2 n display Supcon 15 tic K supcon H G Euc toc End of code 16 4 Toolbox Functions 4 1 Automaton Create an automaton model automaton object for use by Discrete Event Control Kit DECK Syntax G automaton N TL G automaton N TL Xm Inputs N Number of states TL Transition list Xm Marked states row vector Outputs G Output automaton Description The class automaton has the following properties N Number of states The states must be named using the following convention the state set must be 1 N with 1 used for the initial state TL Transition list TL is an m by 3 matrix where m is the number of transitions Each row of TL repre sents a transition in the form x1 sigma x2 with x1 and x2 being the source and destination states and sigma the corresponding event Xm Marked states row vector G automaton N TL Xm returns an automaton model object G with state set 1 N transition list TL and set of marked states Xm G automaton N TL returns an automaton G with Xm 18 4 2 Automatonchk Verify the validity of an automaton object Syntax flagi automatonchk G flag1 flag2 automatonchk G Inputs G Input automaton Outputs2. distribute the Program or its derivative works These actions are prohibited by law if you do not accept this License Therefore by modifying or distributing the Program or any work based on the Program you indicate your acceptance of this License to do so and all its terms and conditions for copying distributing or modifying the Program or works based on it 6 Each time you redistribute the Program or any work based on the Program the recipient automatically receives a license from the original licensor to copy distribute or modify the Program subject to these terms and conditions You may not impose any further restrictions on the recipients exercise of the rights granted herein You are not responsible for enforcing compliance by third parties to this License 7 If as a consequence of a court judgment or allegation of patent infringement or for any other reason not limited to patent issues conditions are imposed on you whether by court order agreement or otherwise that contradict the conditions of this License they do not excuse you from the conditions of this License If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations then as a consequence you may not distribute the Program at all For example if a patent license would not permit royalty free redistribution of the Program by all those who receive copies directly or indirectly through you then
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4. the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program 39 If any portion of this section is held invalid or unenforceable under any particular circumstance the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims this section has the sole purpose of protecting the integrity of the free software distribution system which is implemented by public license practices Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system it is up to the author donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License 8 If the distribution and or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries so that distribution is permitted only in or among countries not thus excluded In such case this Li
5. Gt Xrc trim G returns the states of G that are reachable and coreachable in Xrc 32 A Installation Simply unzip the downloaded file If you wish to keep your data files and scripts in a separate directory folder and run DECK from that directory then on the HOME tab of MATLAB Desktop in the ENVIRONMENT section click on Set path to add the directory of DECK files to MATLAB s search path Alternatively this can be done using the command path 33 B Release Notes DECK 1 2013 11 e New functions deterministic isnondet and project e Bug fixes The functions reachable supcon and trim in the previous version of DECK resulted in MATLAB error message for some single state automata that had two or more transitions The problem is fixed in this version 34 C GNU General Public License v 2 GNU GENERAL PUBLIC LICENSE Version 2 June 1991 Copyright C 1989 1991 Free Software Foundation Inc 51 Franklin Street Fifth Floor Boston MA 02110 1301 USA Everyone is permitted to copy and distribute verbatim copies of this license document but changing it is not allowed Preamble The licenses for most software are designed to take away your freedom to share and change it By contrast the GNU General Public License is intended to guarantee your freedom to share and change free software to make sure the software is free for all its users This General Public License applies to most of the Free Software Fo
6. W M Wonham Supervisory Control of Discrete Event Systems Systems Control Group Edward S Rogers Sr Dept of Electrical and Computer Engineering University of Toronto Canada 2013 available at http www control utoronto ca DES 44
7. cr ge cae e ave Gor ee peed Ge ae Ge Ge ae a ee a a 4 10 Reachable mia roer 4 ee Wad and e eee Pee ewe De a IEA 4 12 SUPCON s hg dhe Oe doe oe 2 he See he oe as ee ee ee s AIG OVC 424k home Chews Oh eae 6a AAA e AA IA era ea ee Ges Gs eee TG O Ge me EOE Ge Eee A Installation B Release Notes C GNU General Public License v 2 References O 17 18 19 20 21 22 23 24 25 26 21 28 29 30 32 33 34 35 44 1 Introduction Discrete Event Control Kit DECK is a toolbox set of M file functions written in the programming language of MATLAB 2 for the analysis and design of supervisory control systems based on the Ramadge Wonham RW theory of supervisory control of discrete event systems DES 3 The current version of DECK supports the case of supervision under full event observation Future versions will extend the support to other cases For information about supervisory control the reader is referred to 5 1 DECK has been developed in the familiar environment of MATLAB as an educational tool for a graduate course at Concordia University on the supervisory control of discrete event systems Furthermore DECK offers a set of functions that along with the matrix and set operations of MATLAB provide a convenient setup for implementing new algorithms and applying them to useful and interesting test cases The rest of this user manual is organized as follows Section 2 explains how discrete event models are bu
8. of FSco is empty and hence L FACT Em SPEC FSco Xm ans Empty matrix 1 by 0 11 Figure 7 Dining Philosophers Philosopher P Finally to see if the reachable automaton FACTuSUP is nonblocking the trim procedure is used FACTuSUPt trim FACTuSUP FACTuSUPt automaton Properties N 12 TL 25x23 double Xm 1 3 Methods FACTuSUP has the same number of states as FACTuSUPt and hence it is nonblocking 3 2 Dining Philosophers In this example an extension of the problem of Dining Philosophers which was proposed in WODES 2008 as a benchmark problem for supervisory control software tools is exam ined In this problem n philosophers P P n gt 2 are seated around a table and n forks F F placed on the table in the following order F1 P1 F2 P2 En Pn The automaton modeling philosopher P is shown in Fig 7 Philosopher P takes the fork on his left F event a and executes k 1 k gt 1 intermediate events 3 till it reaches state k 1 Then it takes the fork on his right that is F when 1 lt i lt n 1 and F when i n and enters eating state k 2 event y Finally P returns the forks event u and goes to its initial idle state The automata for forks are shown in Fig 8 The events philosopher takes the left fork 0 are assumed uncontrollable when 7 is even The rest of events are controllable The objective is to design a maximally permiss
9. that BUFSUP is not admissible Uncontrollable event disablement would occur when the plant and supervisor are in states given by the rows of S The corresponding disabled uncontrollable events are provided in E For instance one such disablement occurs when FACT and BUFSPEC are in states 2 and 2 To determine the states of MACH1 and MACH2 when FACT is in state 2 we use FACT_States 2 ans 2 1 Thus MACH1 is in state 2 and MACH2 in state 1 In other words when the buffer is full MACH1 is in state 2 W and potentially ready to deposit another workpiece in the buffer BUPSUP attempts to disable the uncontrollable event 3 event 10 In order to correct this issue we remove the a selfloop in state 2 of BUFSUP after the occurrence of 3 workpiece deposit in buffer and effectively disable controllable event a when the buffer is full The resulting supervisor BUFSUPrev is shown in Fig 6 We can see that this supervisor is addmissible 10 Licrev Srev Erev controllable BUFSUPrev FACT Euc 2crev Srev 7 Erev U To verify that the plant under supervision is nonblocking and satisfies the design spec first it must be built FACTuSUP product FACT BUFSUPrev FACTuSUP automaton Properties N 12 TL 2523 double Xm 1 3 Methods Next the automaton FSco is constructed which marks Lm FACT A Lm SPEC SPECco complement BUFSPEC FSco product FACTuSUP SPECco We note that the marked state set
10. Discrete Event Control Kit DECK 1 2013 11 User Manual Shahin Hashtrudi Zad Department of Electrical and Computer Engineering Concordia University November 2013 Copyright by Shahin Hashtrudi Zad Concordia University 2013 Abstract Discrete Event Control Kit DECK is a toolbox a set of functions written in the programming language of MATLAB 2 for the analysis and design of supervisory control systems based on discrete event models This software has been developed by Shahin Hashtrudi Zad and two of his graduate students Shauheen Zahirazami and Farzam Boroomand It has been developed and tested on MATLAB release R2012b DECK is provided under the terms of the GNU General Public License version 2 as published by the Free Software Foundation The text of the license appears in Appendix C ii Contents 1 Introduction 2 Building Models and Solving Problems 3 Examples Gel omal Factory ss esena BAL ros ek ee e a RRR BS 3 2 Dining Philosophers s s 2420 era d g dee RG be 2 hae OOS Ee Bd Oo eS 4 Toolbox Functions A Automaton lt lt cesar votar be ew oe we e BE Boel we 4 42 A utomatonchk e se be ee ee we eo ee ewe ww ee daras 43 Complement sed 40 46 6 e rs ee a a a 4 4 Controllable 00 aona a ea a d ea e a aa 4 5 Deterministic sasaa sre darda ee 4 6 Isnond t 4 sce iw Bae eee DEE KE SS RR OR a Ast PROGUCU am ere nee ao a OG ee Ge Se ee A PrOjJeChe a2 246 soe ad Bee ee A be ALO STRGACIS aaa
11. E Gco complement G Ea returns an automaton Gco with Lig Ges E Lm G TL Giza E e where E EU Ea The event set of the input automaton E and E must be disjoint 20 4 4 Controllable Determine if a language is controllable Syntax isctrb controllable K G Euc Lisctrb States controllable K G Euc Lisctrb States Events controllable K G Euc Inputs K Automaton representing the test language G Plant Automaton Euc Uncontrollable events vector Outputs isctrb Test result States List of states where disablement of uncontrollable events occurs Events List of disabled uncontrollable events cell array Description Let Lm and L denote marked behavior marked language and closed behavior gener ated language isctrb controllable K G Euc returns isctrb 1 if and only if L K is controllable with respect to L G and Eue Otherwise it returns isctrb 0 When K is trim controllable K G Euc returns 1 if and only if Lm K is controllable with respect to L G and Eye Lisctrb States controllable K G Euc returns a two column matrix States Each row of States xK xG is a state of product K G where disablement of uncontrollable events if any occurs i e the test fails If L K is controllable isctrb 1 then States Lisctrb States Events controllable K G Euc returns the list of disabled uncon trollable events if any in the cell array Events The i th cell of Events Events i is a row vector
12. Small Factory proposed supervisor BUFSUP 10 23 2 11 23 4 12 23 3 FACTSUP Xm ans 1 3 In the second part of this problem let us consider Small Factory again with the objective of finding a nonblcking supervisor enforcing design specification S1 regarding buffer overflow and underflow First suppose the automaton in Fig 5 has been proposed for supervisor Note that this automaton is the same as the spec automaton BUFSPEC The following script first builds the plant model which contains 9 states This time in addition to FACT we have decided to get the information of the states of the components MACH1 and MACH2 in the 9 x 2 matrix FACT_States Next the proposed suprevisor BUFSUP is built and the controllable procedure is used to determine if BUFSUP is admissible i e the closed behavior L BUFSUP is controllable MACH1 automaton 3 1 11 2 2 10 1 2 12 3 3 13 1 1 MACH2 automaton 3 1 21 2 2 20 1 2 22 3 3 FACT FACT_States sync MACH1 MACH2 2 BUFSPEC automaton 2 1 10 2 2 21 1 1 2 BUFSPEC selfloop BUFSPEC 11 12 13 20 22 23 BUFSUP automaton 2 1 10 2 2 21 1 1 2 BUFSUP selfloop BUFSUP 11 12 13 20 22 23 Euc 10 12 20 22 Lic S E controllable BUFSUP FACT Euc The results are returned in ic S E Q Br BUFSUPrev SS o selfloop A1 41 2 Az H2 SOO a2 Figure 6 Small Factory revised supervisor ic 0 S 2 2 2 5 2 8 E 10 10 10 It is observed
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14. actory design specifications States I W and D are named 1 2 and 3 The following script design a minimally restrictive supervisor N1 3 TLi 1 11 2 2 10 1 2 12 3 3 13 1 Xmi 1 MACH1 automaton N1 TL1 Xm1 N2 3 TL2 1 21 2 2 20 1 2 22 3 3 23 1 Xm2 1 MACH2 automaton N2 TL2 Xm2 FACT sync MACH1 MACH2 BUFSPEC automaton 2 1 10 2 2 21 1 1 2 BUFSPEC selfloop BUFSPEC 11 13 14 20 22 23 BRSPEC automaton 2 1 13 1 1 22 2 2 23 1 1 2 BRSPEC selfloop BRSPEC 10 11 13 20 21 SPEC product BUFSPEC BRSPEC Euc 10 12 20 22 FACTSUP supcon SPEC FACT Euc First two automata objects MACH1 and MACH2 are created Next the plant model FACT is obtained using the sync procedure After automata objects for BUFSPEC and BRSPEC are built the automaton object SPEC is found using the product command Finally the set of uncontrollable events are defined and the supervisor in the form of an automaton FACTSUP is obtained using supcon FACTSUP has 12 states and 24 transitions FACTSUP FACTSUP automaton Properties N 12 TL 24x23 double Xm 1 3 Methods The transition list and marked states can be examined as follows FACTSUP TL ans 11 10 12 21 13 11 20 22 10 12 20 22 11 23 13 20 22 20 22 10 12 hata VVUOR xO0kooovovoo2R2O0k AX UD BA kk SSCoO OA AA RYRAAHAANHAGTK MY Re kk Be BUFSUP 0 selfloop a1 A1 1 82 A2 H2 Figure 5
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17. containing the uncontrollable events disabled in xKi xGi the i th row of States Tip The controllable function can be used to generate the list of all event disable ments in a DES plant G by a supervisor K if the input argument Euc the set of uncontrollable events is replaced with the entire event set E that is using Lisctrb States Events controllable K G E 21 4 5 Deterministic Convert nondeterministic automaton to deterministic automaton Syntax Gdet deterministic G Gdet States deterministic G Inputs G Input automaton deterministic or nondeterministic Outputs Gdet Output automaton deterministic States State set of output automaton cell array Description Gdet deterministic G returns a deterministic automaton Gdet that has the same marked and closed behavior as G Lm Gact Lm G L Gaet L G Here Lm and L denote marked behavior marked language and closed behavior gener ated language If G is deterministic then Gdet G Deterministic uses the subset construction to build Gdet Gdet States deterministic G returns a cell array States containing informa tion about the states of Gdet The i the cell of States States i is a row vector containing the subset of states of the input automaton G associated with state i of Gdet 22 4 6 Isnondet Determine whether automaton is nondeterministic Syntax flag isnondet G Inputs G Input automaton Outputs flag logical 1 true
18. f its input automaton and selfloop adjoins selfloops to each state of its input automaton i e performs inverse projection The function isnondet determines if an automaton is nondeterministic and deterministic converts nondeterministic automata to deterministic automata The functions product and sync perform the parallel product also known as meet and the synchronous product of an arbitrary number of automata G1 Gn The product and sync functions can also find the parallel and synchronous products of an array of au tomata Furthermore for every state of the resulting automaton these functions return the information about the state of each of the constituent automata G1 Gn Finally the function supcon can be used to find the supremal controllable sublanguage and design minimally restrictive supervisor The function controllable can be used to see if a language is controllable and in cases where controllability test fails to obtain information about the circumstances under which the property has failed More details about each function is provided in Sec 4 Toolbox Functions In the following section two illustrative examples will be discussed 3 Examples 3 1 Small Factory This example has two parts First it illustrates supervisor design using the Small Factory problem Example 3 4 4 5 Next it explains how it can be verified whether the plant small factory under the supervision of a given supervisor meet
19. flagi Automaton validity flag part 1 flag2 Automaton validity flag part 2 Description flag1 flag2 automatonchk G verifies the validity of the automaton object G and returns the result in flag1 In cases where the automaton is not valid the invalid property is identified in flag2 The various cases are explained in the following table flagi flag2 Description N has wrong size is not a scalar TL has wrong size Xm has wrong size N is not an integer TL contains entry that is not an integer Xm contains entry that is not an integer N is negative G G TL has out of range entry G Xm has out of range entry The valid range for states is 1 G N and for events nonnegative integers G TL has repeated rows G Xm has repeated entries Automaton is valid Automatonchk examines the above list of cases from the top Once a case is identified the function returns with the corresponding flags 19 4 3 Complement Complement of a deterministic automaton Syntax Gco complement G Gco complement G Ea Inputs G Input deterministic automaton Ea List of events vector Outputs Gco Output deterministic automaton Description Let E denote the event set of the input automaton G Gco complement G returns an automaton Gco with Lin Geo E Lm G IAG ea E where Lm and L denote marked behavior marked language and closed behavior gen erated language and E is the Kleene closure of
20. ilt in DECK and provides an overview of the analysis and design functions that are available in DECK The application of these functions to two examples is covered in Section 3 The second example is a well known benchmark problem for which performance results specifically execution times are provided Section 4 contains the description of the functions Appendix A contains a note on installation and Appendix B provides the release notes Appendix C contains the text of the GNU General Public License version 2 2 Building Models and Solving Problems In the RW supervisory control theory 3 it is assumed that the plant and the design speci fications can be modeled as finite state automata A finite state automaton is a five tuple G X Xi n Lo Aug where X is the finite state set the event set xy the initial state Xm C X the set of marked states and 7 X x X gt X is the transition partial function An example is given in Fig 1 with X 1 2 3 2 11 12 zo 1 and Xm 2 3 The marked states are Figure 1 Automaton G shown with thick circles In DECK an automaton is characterized by 1 N the number of states 2 TL the list of transitions and 3 Xm the set of marked states N is a nonnegative N gt 0 integer For an automaton with N gt 1 the state set is 1 N with 1 used for the initial state For an empty automaton N 0 The event labels are nonnegative integers Zero 0 is reserved for cl
21. ive nonblocking supervisor The script for solving the problem is provided at the end of this section Note that the automata for philosophers and forks are put together to form an array of automata Gs The commands tic and toc measure the execution time of supcon The execution times along 12 a1 Yn a A H1 Hn Yi 1 Qi gt O _ Hi 1 Hi Figure 8 Dining Philosophers Fork F with the number of states of the supervisor are provided for various values of n and k in Table 1 Remark Note that the philosopher and fork automata are placed in array Gs in the order Fi P1 Fn Pn The order affects the execution time and space requirement of the sync operation despite the fact that the order of arguments in sync function does not change the final result except perhaps for the name of the states For instance for n 10 and k 1 sync P1 Pn F1 Fn takes 24 times longer to run compared with sync F1 P1 Fn Pn For an explanation refer to the description of the sync function in Sec 4 AE CI A CI o E A 1 lt 0 1 S 1 0 5 1 7 185 1476 T sec Ei E a 1138 an P N 1 a 8 1240 T sec 2795 sec a EAT 1 3 325 sec TT 6 lt 0 1 313 sec mak SRE ECE EEE Te Table 1 Dining Philosophers Execution time of supcon T sec and the number of states of the resulting automaton N for various values of philosopher number n and intermediate states k Execution times are measured on a PC wi
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24. ock tick In supervisory control the event set is partitioned into controllable and uncontrollable event sets The controllability status of an event depends on the extent and scope of a supervisor s control over the plant and hence on both the plant and the supervisor This issue becomes important for instance in decentralized supervisory control Therefore in DECK when an automaton is defined the controllable uncontrollable status of events are not specified This status is declared only when a function related to supervisory control i e supcon and controllable is used TL transition list is a matrix with three columns Each row represents a transition in the form x1 e x2 where x1 e and x2 are the source state event and target state The event set X is implicitly taken to be the event labels that appear in the second column of the transition list TL The default type for numerical variables in MATLAB is double precision floating point To prevent roundoff errors in storing integer labels the event labels should be no larger than 9 x 10 Xm is a row vector containing the marked states The function automaton builds automaton objects For example the following statements build an automaton object G corresponding to the automaton G in Fig 1 N 3 TL 1 11 2 2 12 3 3 11 1 Xm 2 3 G automaton N TL Xm Alternatively the following compact form may be used G automaton 3 1 11 2 2 12 3 3 11 1 2 3 The vect
25. on G Ewo the events to be erased unobservable events and P E gt E Ewo the natural projection onto E Ewo Furthermore let Lm and L denote marked behavior marked language and closed behavior generated language Go project G Euo returns a deterministic automaton Go which marks P L G and generates P L G Lm Go P Lm G and L G P L G Go may be regarded as an observer automaton Go States project G Euo returns a cell array States containing information about the states of Go The i the cell of States States i is a row vector containing the state estimate for the input automaton G when the observer Go is in state i 25 4 9 Reach Find the reachable states of transition graph Syntax Xr reach TL S Inputs TL Transition list S Source states vector Outputs Xr States reachable from S row vector Description Xr reach TL S returns the states of the automaton transition graph that are reachable from the set of source states S using the breadth first search algorithm The reachable states appear in Xr in the order they are discovered in the breadth first search 26 4 10 Reachable Find reachable subautomaton Syntax Gr reachable G Gr Xr reachable G Inputs G Input automaton Outputs Gr Reachable subautomaton Xr Reachable states of G row vector Description Gr reachable G returns the subautomaton of G that is reachable from the initial state of G The s
26. or 0 false Description Returns logical 1 true if the input automaton G is nondeteterminstic Otherwise it returns logical 0 false 23 4 7 Product Product of automata Syntax G product G1 Gn G States product G1 Gn G product Ga G States product Ga Inputs Gi Input automaton i i 1 n Ga Cell array containing input automata Outputs G Output automaton States State set of output automaton Description G product G1 Gn returns the product of G1 Gn n gt 2 If L and L denote marked behavior marked language and closed behavior generated language then Le G L G N O LmlGn LG LG L G G States product G1 Gn returns an N x n matrix States where N is the number of states of G Let xi1 xin be the ith row of States Then xil xin are the states of G1 Gn when G is in state i Product can be used with arrays of automata Let Ga denote a cell array containing automata Gal Gan n gt 2 product Ga returns the product of Gal Gan 24 4 8 Project Find a deterministic automaton to represent the projection of a language Syntax Go project G Euo Go States project G Euo Inputs G Input automaton deterministic or nondeterministic Euo Events to be erased vector Outputs Go Output automaton deterministic States State set of output automaton cell array Description Let E be the event set of the input automat
27. or Xm marked states is optional and if omitted in the automaton statement the default value of empty Xm will be taken Once the automaton object G is created typing G returns a list of its properties namely the number of states transition list and marked states An automaton object does not have any methods G G automaton Properties N 3 TL 323 double Xm 2 3 Methods Note that thoroughout this manual MATLAB DECK responses are shown in italics The number of states transition list and marked states of an automaton object G can be accessed through G N G TL and G Xm For example G TL ans 1 11 2 2 12 3 3 11 1 The function automatonchk can be used to verify that an automaton object conforms to the conventions of DECK for naming states and event labels This function is useful for checking DECK models that are created for the first time or imported from an input file It can also help with debugging new functions developed in the DECK environment After the desired automata are built they can be analyzed and manipulated using the available functions in DECK The reach function can be used to perform reachability anal ysis on the transition graphs of automata The function reachable and trim return the reachable and trim subautomata while complement finds the complement of an automaton The project function returns a deterministic automaton that represents the natural pro jection of the marked and closed behaviors o
28. ronous product of Gal Gan 30 Remark The execution time and space requirement of the sync operation depends on the order of the input arguments G1 Gn This can be explained as follows In DECK G sync G1 G2 G3 G4 for instance is evaluated in the following steps First sync G1 G2 is calculated Let us call the result G12 Next G123 sync G12 G3 and fi nally G sync G123 G4 are found If automata that have common events are listed next to eachother then the intermediate automata resulting from successive application of sync in our example G12 G123 are likely to have fewer states because of the interactions among the neighboring automata on the list of input arguments Therefore this results in smaller intermediate automata hence less space requirement and faster execution time Cover sely if automata with no common events appear first on the list of sync the intermediate automata in sync computations could become very large resulting in prolonged execution times 31 4 14 Trim Find the reachable and coreachable subautomaton Syntax Gt trim G Gt Xrc trim G Inputs G Input automaton Outputs Gt Trim subautomaton Xrc States of G that are reachable and coreachable row vector Description Gt trim G returns the trim subautomaton of G containing only those states of G that are both reachable and coreachable The states of Gt are renamed in the order they are discovered in a breadth first search
29. rts of the General Public License Of course the commands you use may be called something other than show w and show c they could even be mouse clicks or menu items whatever suits your program You should also get your employer if you work as a programmer or your school if any to sign a copyright disclaimer for the program if necessary Here is a sample alter the names Yoyodyne Inc hereby disclaims all copyright interest in the program Gnomovision which makes passes at compilers written by James Hacker lt signature of Ty Coon gt 1 April 1989 Ty Coon President of Vice This General Public License does not permit incorporating your program into 42 proprietary programs If your program is a subroutine library you may consider it more useful to permit linking proprietary applications with the library If this is what you want to do use the GNU Lesser General Public License instead of this License 43 References 1 C G Cassandras and S Lafortune Introduction to Discrete Event Systems Springer 2008 Mathworks Inc MATLAB http www mathworks com help matlab index html P J Ramadge and W M Wonham Supervisory control of a class of discrete event processes SIAM J Control Optim vol 25 no 1 pp 206 230 1987 W M Wonham and P J Ramadge On the supremal controllable sublanguage of a given language SIAM J Control Optim vol 25 no 3 pp 635 659 1987
30. s the design specifications The Small Factory Fig 2 consists of two machines MACH1 and MACH2 and a buffer 01 Bi a2 Ba MACHI1 BUF MACH2 Figure 2 Small Factory with a capacity of one The automaton model of MACHi is depicted in Fig 3 Starting MACHi Figure 3 MACHi i 1 2 from the initial state I idle MACH1 takes a piece a1 from an infinite input bin and enters state W work Once the work of MACH1 is done it deposits the workpiece in the buffer 8 If MACH1 breaks down and enters state D down the workpiece will be discarded but the machine can be repaired 41 MACH2 takes workpieces from the buffer and puts the finished workpieces in an output bin The objective is to design a supervisor to have a nonblocking system i e plant under supervision that meets the following design specifications S1 The buffer must not oveflow or underflow S2 In case both machines fail MACH2 must be repaired first The above two specifications can be captured using the following automata 5 For the purpose of control the events a1 a2 uy and fz are controllable and the rest of events are uncontrollable To design supervisor using DECK we use the following codes for events which are the same as those in 5 a 11 B1 10 Ay 1 12 ra a ag 21 Ba 20 Ag 22 fa 23 BUFSPEC O selfloop a1 A1 H1 Ba Az H2 pi de BRSPEC selfloop a1 81 A1 02 b2 Sis a H2 Figure 4 Small F
31. t to control the distribution of derivative or collective works based on the Program In addition mere aggregation of another work not based on the Program with the Program or with a work based on the Program on a volume of a storage or distribution medium does not bring the other work under the scope of this License 3 You may copy and distribute the Program or a work based on it under Section 2 in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following a Accompany it with the complete corresponding machine readable source code which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange or b Accompany it with a written offer valid for at least three years to give any third party for a charge no more than your cost of physically performing source distribution a complete machine readable copy of the corresponding source code to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange or c Accompany it with the information you received as to the offer to distribute corresponding source code This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer in accord with Subsection b above The source code for a work means the preferred form of the work for
32. tates of Gr are renamed in the order they are discovered in a breadth first search Gr Xr reachable G returns the reachable states of G in Xr 27 4 11 Selfloop Add selfloops to automaton Syntax Gs selfloop G Es Inputs G Input automaton Es List of events vector Outputs Gs Output automaton Description Adds selfloop transitions x e x to the transition list of the input automaton G for all states x of G and all events e in the event list Es The event set of G and Es must be disjoint 28 4 12 Supcon Supremal Controllable Sublanguage Syntax K supcon H G Euc Inputs H Specification deterministic automaton G Plant deterministic automaton Euc Uncontrollable events vector Outputs K Trim deterministic automaton marking supremal controllable sublangage Description Let Lm and L denote marked behavior marked language and closed behavior generated language Supcon calculates the supremal sublanguage of Lm H N Lm G that is control lable with respect to L G and E The result is returned in the trim automaton K which marks the supremal controllable sublanguage The calculations are based on the algorithm introduced in W M Wonham and P J Ramadge On the supremal controllable sublanguage of a given language STAM J Control and Optimization Vol 25 No 3 May 1987 29 4 13 Sync Synchronous product of automata Syntax G sync G1 Gn G States sync G1 Gn G Sta
33. tes Blocked_events sync G1 Gn G sync Ga G States sync Ga G States Blocked_events sync Ga Inputs Gi Input automaton i i 1 n Ga Cell array containing input automata Outputs G Output automaton States State set of output automaton Blocked_events Events blocked absent in output automaton row vector Description G sync G1 G2 returns the synchronous product of G1 and G2 Let E and E be the event sets of G4 and Gs If Lm and L denote marked behavior marked language and closed behavior generated language then Lm G L G Lm G1 Lm G2 L G1 L G2 Here L La is the synchronous product of languages L and La defined according to Lil Lo Pp Li O Py L2 where P resp P2 is the natural project of E U E2 onto Ej resp Ez G sync G1 Gn returns the synchronous product of G1 Gn n gt 2 G States sync G1 Gn returns the N xn matrix States where N is the number of states of G Let xil xin be the i th row of States Then xii xin are the states of G1 Gn when G is in state i G States Blocked_events sync G1 Gn returns the row vector Blocked_events containing the events that are in the transition list of at least one of the input automata Gi and absent in the transition list of the output automaton G Sync can be used with arrays of automata Let Ga denote a cell array containing automata Gal Gan n gt 2 sync Ga returns the synch
34. th Intel Core i5 2500 3 10 GHz 8GB RAM running 64 bit Windows 7 and MATLAB R2011b 64 bit Execution times longer than one hour have not been measured 2 lt 0 1 lt 0 1 lt 0 0 0 1 T a an 2 1 13 Script for solving the problem of Dining Philosophers Solves the Dining Philosophers benchmark problem of WODES 2008 display Dining Philosophers Number of philosophers and forks n gt 2 n 9 fprintf Number of philosophers n d n n Number of intermediate states k gt 1 k 1 fprintf Number of intermediate states k d n K Plant model G display Building Plant model G Event coding For philosopher Pi alpha i 10 i 1 beta i 10 i 2 gamma i 10 i 3 mu i 10 i 4 alpha 10 1 n 1 beta 10 1 n 2 gamma 10 1 n 3 mu 10 1 n 4 Gs is an array of the automata of Philosophers and Forks in the order F1 Pi F2 P2 Fn Pn for i 1 2x n Gs i automaton 0 end h Philosophers ie for i 1 n 14 TLi 1 alpha i 2 for j 2 k TLi TLi j beta i j 1 end TLi TLi k 1 gamma i k 2 k 2 mu i 1 Gs12xij automaton kx 2 TLi 1 end Forks The first fork Gs i automaton 2 1 alpha 1 2 1 gamma n 2 2 mu 1 1 2 mu n 1 1 de The other forks for i 2 n Gs 2 i 1 automaton 2 1 gamma i 1 2 1 alpha i 2 2 mu i 1 1 2 mui 1 1 end G sync Gs
35. tion are not covered by this License they are outside its scope The act of running the Program is not restricted and the output from the Program is covered only if its contents constitute a work based on the Program independent of having been made by running the Program Whether that is true depends on what the Program does 1 You may copy and distribute verbatim copies of the Program s source code as you receive it in any medium provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty keep intact all the notices that refer to this License and to the absence of any warranty 36 and give any other recipients of the Program a copy of this License along with the Program You may charge a fee for the physical act of transferring a copy and you may at your option offer warranty protection in exchange for a fee 2 You may modify your copy or copies of the Program or any portion of it thus forming a work based on the Program and copy and distribute such modifications or work under the terms of Section 1 above provided that you also meet all of these conditions a You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change b You must cause any work that you distribute or publish that in whole or in part contains or is derived from the Program or any part thereof to be licensed as a whole at
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