2. Page 1 Aalborg University Control in Discrete Event Systems

3. Page 2 Aalborg University State Dynamics X n+1 =F(x n,u n ), y n =G(x n,u n ) M=(X,E, ,x 0,F) S(s)={a,b} {u n }{y n } {x n } a b c x0x0 s

4. Page 3 Aalborg University Control – Automation vs. CS views u n =C(s n ) {x n } s n ={y j | n < j} u n 2 S(s n ) {x n } S(s n ) s n ={y j | n < j} C(s n ) Autumation CS

5. Page 4 Aalborg University Linguistic Specification L(M)={s=a 0,..a n | (x i,a i,x i+1 ) 2  for i=0,,n} L m (M)={s=a 0,..a n | (x i,a i,x i+1 ) 2  and x i+1 2 F for i=0,,n} x0x0 a a a b aaab

6. Page 5 Aalborg University Linguistic control S limits the output language to L(S/M) K µ E* specifies desired behaviour of the closed loop Design S so that L(S/M)=K u n 2 S(s n ) M S s n ={y j | n < j}

7. Page 6 Aalborg University Regular languages A regular language iff. generated by a FSA Not all languages are regular L={a n b n } is not regular Assume there is a FSA (M) with N states generating L a N b N ->x 0,x 1,..,x 2N+1 x 0,x 1,..,x 2N+1 = x 0,x 1,..,x i-1, x,x i+1,.., x j-1,x, x j+1,.., x_{2N+1} M accepts a 0,a 1,..,a i-1, a i,a i+1,.., a j-1,a j, a j+1,.., a_{2N+1} M also accepts a 0,a 1,..,a i-1, a i,a i+1,.., a j-1,a i+1,.., a j-1,a j, a j+1,.., a_{2N+1} M also accepts either a N+m b N or a N b m or a n b N or a N b N+m

8. Page 7 Aalborg University Regular language control Assume K is regular S is an FSA generating K Closed loop is S||M (parallel composition) Examples 2DP problem K={(ab) * U (ab) * a} M S a b a b c x0x0 x0x0

9. Page 8 Aalborg University Some examples State avoidance (safety) remove undesired states from M - > S Desired sequence of labels abc Avoid sequence of labels aaab abc aaab a b b

10. Page 9 Aalborg University Uncontrollable Events E = E c U E uc E c : controllable events E uc : uncontrollable events Allowed events: S(s) U E uc Uncontrollable events cannot be avoided E uc ={c} a b c x0x0 s

11. Page 10 Aalborg University Controllable Languages A language K is controllable w.r.t. M if S exists so that L(S/M)=K K : prefix closure of K K is K together with all its prefixes K = K U {s| 9 t so that st 2 K} K is prefix closed iff K=K L(M) is prefix closed by definition

12. Page 11 Aalborg University Controllability Controllability theorem K is controllable w.r.t. M iff. K E uc Å L(M) µ K Explanation: s is the string generated until now, proceeding with a 2 E uc within M should always have a continuation in K

13. Page 12 Aalborg University Example a b c d c e x0x0 L(M)={(ab) * U (ab) * a U (ab) * ac * U (ab) * acc * de * } K=(ab) * acc * (prefix closed?, controllable ?) K’=(ab)*a (prefix closed?, controllable ?) K = (ab) * U (ab) * a U K K’ = (ab) * U K’ K E uc Å L(M)=(ab)*acc*d ( K K’ E uc Å L(M)=; µ K’

14. Page 13 Aalborg University Basic DES control problems Safety: L(S/M) µ L a (if L a is not controllable we want L(S/G) to be as large as possible within L a ) Liveness: L r µ L(S/M) (if L r is not controllable we want L(S/M) to be as small as possible) Tolerance: L(S/M) µ L tol, where L(S/M) is also the controllable language achieving as much as possible of L des, while being smallest possible amongst those realizing equally much.

15. Page 14 Aalborg University Some abstract stuff K is a set of prefix closed controllable languages L=U {K \in K} L is prefix closed and controllable L’=Å {K \in K} L’ is prefix closed and controllable C in (K) is the set of all prefix closed controllable sublanguages of K C out (K) is the set of all prefix closed controllable languages including K K C =U {L 2 C in (K)} K C =Å {L 2 C out (K)}

16. Page 15 Aalborg University Solutions to basic control problems L a C solves the safety problem, i.e. L(S/M)= L a C L rC solves the liveliness problem, L(S/M)= L rC (L C tol Å L des ) C solves the tolerance problem If K is regular then K C is regular

17. Page 16 Aalborg University Closed form solutions K C =K \ [(M\K)/E uc * ]E uc * K C =KE uc * Å M