Discrete Event Systems. Lecture 14: Discrete Control. Continuous System. Discrete Event System. Discrete Control Systems.

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1 Lecure 14: Discree Conrol Discree Even Sysems [Chaper: Sequenial Conrol + These Slides] Discree Even Sysems Sae Machine-Based Formalisms Saechars Grafce Laboraory 2 Peri Nes Implemenaion No covered in he lecure. Homework. 1 Definiion: A Discree Even Sysem (DES) is a discree-sae, even-driven sysem, ha is is sae evoluion depends enirely on he occurrence of asynchronous discree evens over ime. Someimes he name Discree Even Dynamic Sysem (DEDS) is used o emphasize he dynamic naure of DES. 2 DES: Coninuous Sysem 1. The sae space is a discree se. 2. The sae ransiion mechanism is even-driven. 3. The evens need no o be synchronized by, e.g., a clock. Coninuous sysems: 1. Coninuous-sae sysems (real-valued variables) 2. The sae-ransiion mechanism is ime-driven. Coninuous discree-ime sysems x(k+1) =Ax(k)+Bu(k) can be viewed as an even-driven sysem synchronized by clock evens. Sae rajecory is he soluion of a differenial equaion ẋ() = f (x(), u(), ) x() X = R 3 4 Discree Even Sysem Sae rajecory (sample pah) is piecewise consan funcion ha jumps from one value o anoher when an even occurs. x() s 6 s 5 s 4 s 3 s 2 s 1 X =(s 1, s 2, s 3, s 4, s 5, s 6 ) Discree Conrol Sysems All processes conain discree elemens: coninuous processes wih discree sensors and/or acuaors discree processes manufacuring lines, elevaors, raffic sysems,... mode changes manual/auo, sarup/shudown producion (grade) changes alarm and even handling e 1 e 2 e 3 e 4 e 5 e 6 e 7 5 6

2 Discree Logic Basic Elemens Boolean (binary) signals 0, 1, false, rue, a,ā expressions a Larger in volume han coninuous conrol ery lile heoreical suppor a or b (a + b) a and b (a b) Truh values Truh value ables verificaion, synhesis formal mehods beginning o emerge sill no widespread in indusry evens Boolean algebra a ^a 7 a 8 Combinaorial nes oupus = f(inpus) inerlocks, "förreglingar" Sequence nes newsae = f(sae,inpus) oupus = g(sae,inpus) sae machines auomaa Logic Nes Asynchronous nes or synchronous (clocked) nes Discree Even Sysems Sae Machine-Based Formalisms Saechars Grafce Laboraory 2 Peri Nes Implemenaion No covered in he lecure. Homework. Inpus Sae Logic ne Oupus New sae Delay 9 10 Sae Machines Moore Machine Formal properies analysis possible in cerain cases In a Using sae machines is ofen a good way o srucure code. Sae 0 In a Sae 1 Sysemaic ways o wrie auomaa code ofen no augh in programming courses. Ou a In b In b Ou b In c Sae 2 Ou c In a Sae ransiions in response o inpu evens 11 Oupu evens (acions) associaed wih saes 12

3 Mealy Machine Sae Machine Exensions In b Ou b In a Ou a Ordinary sae machines lack srucure Sae 0 Sae 1 Exensions needed o make hem pracically useful In a Ou b In b Ou a In c Ou c hierarchy concurrency hisory (memory) Sae 2 In c Ou c Oupu evens (acions) associaed wih inpu evens Discree Even Sysems Sae Machine-Based Formalisms Saechars Grafce Laboraory 2 Peri Nes Implemenaion No covered in he lecure. Homework. D. Harel, 1987 Saechars = sae machine hierarchy concurrency hisory Saechars Saechar Synax XOR Supersae AND Supersaes: Saechars Concurrency Condiion "guard" D A c (P) Inpu even Oupu even Sae a / e b B Y A B a b (in G) C D G E c g F d a C d Yisheorhogonal produc of A and D When in sae (B,F) and even a occurs, he sysem ransfers simulaneously o (C,G)

4 Hisory Arrows Synax Alarm Off a H Inerfaces for AND Supersaes: J H ν δ η(in B) A D d d B E On F C G H On even a he las visied sae wihin D becomes acive. ω α θ β ɛ K L δ exi from J (B, E) α exi from K (C, F) ν exi from J (B, F) β exi from L (C,mos recenly visied sae in D) ω exi from (B, G) K η exi from (B, F) H θ exi from (C, D) K ɛ exi from (A, D) L Saechar Tools Saechars popular for modeling, simulaion, and code generaion Used o represen sae-ransiion diagrams in UML ools (Raional/Rose, Rhapsody,...) Saeflow for Malab/Simulink Saechar Semanics Unforunaely, Harel only gave an informal definiion of he semanics As a resuls a number of compeing semanics were defined. In 1996, Harel presened his semanics (he Saemae semanics) of Saechar and compared wih 11 oher semanics. The lack of a single semanics is sill he major problem wih Saechars Each ool vendor defines his own. Discree Even Sysems Sae Machine-Based Formalisms Saechars Grafce Laboraory 2 Peri Nes Implemenaion No covered in he lecure. Homework

5 Grafce Exended sae machine formalism for implemenaion of sequence conrol Indusrial name: Sequenial Funcion Chars (SFC) Defined in France in 1977 as a formal specificaion and realizaion mehod for logical conrollers Par of IEC (indusry sandard for PLC conrollers) Seps: acive or inacive Transiions ("övergång"): Basic elemens.x = 1 when acive.t = number of ime unis since he sep las became acive Iniial sep condiion rue and/or even occurred + previous sep acive Alernaive pahs: branches Conrol srucures muually exclusive Parallel pahs: spli a a join (synchronizaion) repeiion Illegal Grafce Legal Grafce Semanics The iniial sep(s) is acive when he funcion char is iniiaed. 2. A ransiion is fireable if: all seps preceding he he ransiion are acive (enabled). he recepiviy (ransiion condiion and/or even) of he ransiion is rue A fireable ransiion mus be fired. 3. All he seps preceding he ransiion are deacivaed and all he seps following he ransiion are acivaed when a ransiion is fired 4. All fireable ransiions are fired simulaneously 5. When a sep mus be boh deacivaed and acivaed i remains acivaed wihou inerrup 30

6 Unreachable grafces a = 1 or 0 a = 0 a) No enabled b) Enabled bu no firable a = 1 c) Firable d) Afer he change from c) Unsafe grafces Acions Acion block Acion ypes: sandard acion (level acion) S2 S sored acion (impulse acion) logical assignmen Sandard acion S = 1 S2 1 S = 0 Unsable siuaion (sored acions performed) Condiional acion C condiion cond. S = 1 Sored acion 1 1 S2 S2 S =

7 Time limied acion L 8 s. 8 Macro Seps: Hierarchy Time delayed acion D 5s. 5 S2 S21 S22 S3 S Grafce/SFC and IEC-1131 Ediors A large number of graphical IEC ediors are available. Generaes PLC code or C-code. Discree Even Sysems Sae Machine-Based Formalisms Saechars Grafce Laboraory 2 Peri Nes Implemenaion No covered in he lecure. Homework Laboraory 2 JGrafchar Sequenial Conrol bead sorer process JGrafchar - Lund Universiy Grafce/SFC graphical edior Grafce/SFC run-ime sysem 41 42

8 Bead Sorer process Black bead comparmen Process Solenoid 1 Solenoid 2 Solenoid 3 Bead Sensor Colour Sensor Yellow bead comparmen Solenoid 4 Discree Even Sysems Sae Machine-Based Formalisms Saechars Grafce Laboraory 2 Peri Nes Implemenaion No covered in he lecure. Homework. SORTER Peri Nes C.A Peri, TU Darmsad, 1962 A mahemaical and graphical modeling mehod. Describe sysems ha are: concurren asynchronous or synchronous disribued nondeerminisic or deerminisic Peri Nes Can be used a all sages of sysem developmen: modeling analysis simulaion/visualizaion ( playing he oken game ) synhesis implemenaion (Grafce) Applicaion areas communicaion proocols disribued sysems disribued daabase sysems flexible manufacuring sysems logical conroller design muliprocessor memory sysems daaflow compuing sysems faul oleran sysems... Inroducion A Peri ne is a direced biparie graph consising of places P and ransiions T. Places are represened by circles. Transiions are represened by bars (or recangles) Places and ransiions are conneced by arcs. In a marked Peri ne each place conains a cardinal (zero or posiive ineger) number of okens of marks

9 P1 Firing rules T1 T6 T3 P3 T2 P2 T4 P4 1. A ransiion is enabled if each inpu place conains a leas one oken. 2. An enabled ransiion may or may no fire. 3. Firing an enabled ransiion means removing one oken from each inpu place of and adding one oken o each oupu place of. P5 P6 The firing of a ransiion has zero duraion. T5 P7 49 The firing of a sink ransiion (only inpu places) only consumes okens. The firing of a source ransiion (only oupu places) only produces okens. 50 T1 P1 Typical inerpreaions of places and ransiions: P2 Inpu Places Transiion Oupu Places T6 T2 Precondiions Inpu daa Even Compuaion sep Poscondiions Oupu daa P3 P4 Inpu signals Signal processor Oupu signals T3 T4 Resources needed Condiions Task or job Clause in logic Resources needed Conclusions P5 P6 Buffers Processor Buffers T5 P Generalized Peri Nes P1 T1 P3 2 Firing rules: 1. A ransiion is enabled if each inpu place p of conains a leas w(p,) okens 2. Firing a ransiion means removing w(p,) okens from each inpu place p and adding w(,q) okens o each oupu place q. P Peri Ne arians Timed Peri Nes: Times associaed wih ransiions or places High-Level Peri Nes: Tokens are srucured daa ypes (objecs) Coninuous & Hybrid Peri Nes: The markings are real numbers insead of inegers Mixed coninuous/discree sysems 54

10 Analysis Analysis Properies: Live: No ransiions can become unfireable. Deadlock-free: Transiions can always be fired Bounded: Finie number of okens... Analysis mehods: Reachabiliy mehods exhausive enumeraion of all possible markings Linear algebra mehods describe he dynamic behaviour as marix equaions Reducion mehods ransformaion rules ha reduce he ne o a simpler ne while preserving he properies of ineres The classical real-ime problems Muual Exclusion Dijksra s classical problems Process A Process B muual exclusion problem producer-consumer problem readers-wriers problem dining philosophers problem All can be modeled by Peri Nes. Execuing ouside criical secion Waiing for criical secion Execuing inside criical secion Muex semaphore Waiing for criical secion Execuing inside criical secion Execuing ouside criical secion Producer-Consumer Bounded buffer: Producer processes Consumer processes Unbounded buffer: Producer processes Consumer processes Buffer Full places Read Buffer Read Wrie Free places Wrie 59 60

11 Readers-Wriers Dining Philosophers Wriers processes Readers processes Ready o wrie 3 Ready o read Wriing Reading 3 Access Conrol Lef fork Righ fork Thinks Eas Thinks Fork Picks lef fork Fork Picks righ fork Eas Drops lef fork Drops righ fork Philosopher Discree Even Sysems Sae Machine-Based Formalisms Saechars Grafce Laboraory 2 Peri Nes Implemenaion No covered in he lecure. Homework. Coding Sae Machines Using sae machines is ofen a good way o srucure code. Sysemaic ways o wrie auomaa code ofen no augh in programming courses. Issues: acive or passive objec Mealy vs Moore machines saes wih imeou evens saes wih periodic aciviies Ofen convenien o implemen sae machines as periodic processes wih a period ha is deermined by he shores ime required when making a sae ransiion

12 Example: Passive sae machine The sae machine is implemened as a synchronized objec public class PassiveMealyMachine { privae saic final in STATE0 = 0; privae saic final in STATE1 = 1; privae saic final in STATE2 = 2; privae saic final in INA = 0; privae saic final in INB = 1; privae saic final in INC = 2; privae saic final in OUTA = 0; privae saic final in OUTB = 1; privae saic final in OUTC = 2; privae in sae; PassiveMealyMachine() { sae = STATE0; privae void generaeeven(in oueven) { // Do somehing 67 public synchronized void inpueven(in even) { swich (sae) { case STATE0 : swich (even) { case INA : generaeeven(outa); sae = STATE1; break; case INB : generaeeven(outb); break; defaul : break; ; break; case STATE1 : swich (even) { case INC : generaeeven(outc); sae = STATE2; break; defaul : break; ; break; case STATE2 : swich (even) { case INA : generaeeven(outb); sae = STATE0; break; case INC : generaeeven(outc); break; defaul : break; ; break; 68 Acive Sae Machines Example: Acive sae machine 1 The sae machine could also be implemened as an acive objec (hread) The hread objec would ypically conain an even-buffer (e.g., an RTEvenBuffer). The run mehod would consis of an infinie loop ha wais for an incoming even (RTEven) and swiches sae depending on he even. An aciviy is an acion ha is execued periodically while a sae is acive. More naural o implemen he sae machine as a hread public class AciveMachine1 exends Thread { privae saic final in STATE0 = 0; privae saic final in STATE1 = 1; privae saic final in STATE2 = 2; privae in sae; AciveMachine1() { sae = STATE0; privae boolean cond0() { // Reurns rue if condiion 0 is rue privae boolean cond1() { privae boolean cond2() { privae void acion0() { // Execues acion 0 privae void acion1() { privae void acion2() { 71 public void run() { long = Sysem.currenTimeMillis(); long duraion; while (rue) { swich (sae) { case STATE0 : { acion0(); = + 20; duraion = - Sysem.currenTimeMillis(); if (duraion > 0) { ry { sleep(duraion); cach (InerrupedExcepion e) { if (cond0()) {sae = STATE1; break; case STATE1 : { // Similar as for STATE0. Execues acion1, wais for 50 ms, checks // cond1 and hen changes o STATE2 ; break; case STATE2 : { // Similar as for STATE0. Execues acion2, wais for 10 ms, checks // cond2 and hen changes o STATE0 ; break; 72

13 Commens Example: Acive sae machine 2 Condiions esed a a frequency deermined by he aciviy frequencies of he differen saes. sleep() spread ou in he code The hread runs a a consan (high) base frequency. Aciviy frequencies muliples of he base frequency. Condiions esed a he base frequency public void run() { long = Sysem.currenTimeMillis(); long duraion; in couner = 0; while (rue) { couner++; swich (sae) { case STATE0 : { if (couner == 4) { couner = 0; acion0(); if (cond0()) { couner = 0; sae = STATE1; ; break; case STATE1 : { // Similar as for STATE0. Execues acion1 if couner == 10. Changes o STATE2 if cond ; break; case STATE2 : { // Similar as for STATE0. Execues acion2 if couner == 12. Changes o STATE0 if cond ; break; = + 5; // Base sampling ime duraion = - Sysem.currenTimeMillis(); if (duraion > 0) { ry { sleep(duraion); 75 cach (InerrupedExcepion e) { Commens Polled ime handling Complicaed handling of couner Condiions esed a a high rae 76