Failure Diagnosis of Discrete Event Systems: A Temporal Logic Approach

Transcription

1 Failre Diagnosis of Discrete Event Systems: A Temporal Logic Approach Shengbing Jiang Electrical & Controls Integration Lab General Motors R&D 1

2 Otline Introdction Notion of Diagnosability in Temporal Logic Setting Algorithm for Diagnosis Example Conclsion 2

3 DES: Introdction Discrete states: driven by randomly occrring events Events: discrete qalitative changes arrival of part in a manfactring system loss of message packet in a commnication network termination of a program in an operating system exection of operation in database system arrival of sensor packet in embedded control system Examples of discrete event systems: compter and commnication networks robotics and manfactring systems compter programs atomated traffic systems 3

4 DES: Example States change in response to randomly occrring events piecewise-constant state trajectory floor 1 state floor 0 moving top on +on bot t t t t time typical state trajectory of an elevator State trajectory a seqence of triples: (x 0, σ 1, t 1 )(x 1, σ 2, t 2 )... Untimed trajectory (for ntimed specification): (x 0, σ 1 )(x 1, σ 1 )... Under determinism this is eqivalent to: x 0 and σ 1 σ 2... Collection of all event traces, langage, L = pr(l) Collection of final traces, marked langage, L m L Langage model, (L, L m ), also modeled as atomaton: G := (X,Σ, α, x 0, X m ); (L(G), L m (G)) langage model 4

5 Failre Diagnosis of DESs Failre: deviation from normal or reqired behavior occrrence of a failre event visiting a failed state reaching a deadlock or livelock Failre Diagnosis: detecting and identifying failres plant model M (e) diagnosability test diagnoser failre / no failre falt specification model e: event generated; M(e): event observed/sensored M(e) = ǫ: no sensor for the event e M(e 1 ) = M(e 2 ) ǫ: e.g. motion sensor, detect the movement of a part, not the moving direction and the part type Diagnoser: observes the seqence of generated events, and determines (possibly with a delay that is bonded) whether or not a failre occrred 5

6 Prior Reslts: Formal langage / atomaton falt specification Falt spec.: only safety properties Or Contribtion: A temporal logic approach for failre diagnosis of DESs Temporal logic has a syntax similar to natral langage and has a formal semantics Falt spec.: both safety and liveness properties 6

7 Linear-time Temporal Logic (LTL): Introdction Describing properties of seqence of proposition set traces Notations AP: set of atomic propositions Σ AP = 2 AP : power set of AP Σ AP : set of all finite proposition set traces Σ ω AP : set of all infinite proposition set traces Temporal operators & their interpretations X : next, U : ntil, F : ftre or eventally, Ff=TreUf G : globally or always, Gf=~F~f B : before, fbg=~(~fug) Xf f fug Ff f f f f g f Gf f f f f f fbg f g Syntax of LTL formlae P1 p AP p is a LTL formla. P2 f 1 and f 2 are LTL formlae so are f 1, f 1 f 2, and f 1 f 2. P3 f 1 and f 2 are LTL formlae so are Xf 1, f 1 Uf 2, Ff 1, Gf 1, and f 1 Bf 2. 7

8 Semantics: proposition-trace π = (L 0, L 1, ) Σ ω AP π i = (L i, ), i 0 1. p AP, π = p p L π = f 1 π = f π = f 1 f 2 π = f 1 or π = f π = f 1 f 2 π = f 1 and π = f π = Xf 1 π 1 = f π = f 1 Uf 2 k 0, π k = f 2 and j {0,1,, k 1}, π j = f π = Ff 1 k 0, π k = f π = Gf 1 k 0, π k = f π = f 1 Bf 2 k 0 with π k = f 2, j {0,1,, k 1}, π j = f 1. Examples of LTL formlae G R 1 : an invariance (a type of safety) property. G((message sent) F(message received)) : a recrrence (a type of liveness) property. FGp : a stability (a type of liveness) property. 8

9 Notion of Diagnosability in Temporal Logic Setting System Model: P = (X,Σ, R, X 0, AP, L) X, a finite set of states; Σ, a finite set of event labels; R : X Σ {ǫ} X, a transition relation, x X, σ Σ {ǫ}, x X, (x, σ, x ) R (P is nondeterministic & nonterminating); X 0 X, a set of initial states; AP, a finite set of atomic proposition symbols; L : X 2 AP, a labelling fnction. M : Σ {ǫ} {ǫ}, an observation mask. Falt specification: LTL formla f. π = (x 0, x 1, ), π AP = (L(x 0 ), L(x 1 ), ): π = f if π AP = f 9

10 Falty state-trace An infinite state-trace π is falty if π = f. Remark captres both safety and liveness failres Indicator A finite state-trace π is an indicator if all its infinite extensions in P are falty. Remark: can only detect indicators throgh observation of finite length event-traces Pre-diagnosability P is pre-diagnosable w.r.t. f if every falty state-trace in P possesses an indicator as its prefix. Remark Needed for detecting all failres throgh observation of finite length event-traces Example b d x a c 0 x 1 p 1 p 1 x 2 p 2 f = GFp 2 : not pre-diagnosable; no indicator for x 0 x ω 1 f = GFp 1 : pre-diagnosable 10

11 Diagnosability: single specification P is diagnosable w.r.t. M and f if P is pre-diagnosable and Example Exists a detection delay bond n sch that For all indicator trace π 0 For all extension-sffix π 1 of π 0, π 1 n For all π indistingishable from π 0 π 1 It holds that π is an indicator trace p 2 x b 2 c 1 1 p 1 p 1 a a x 0 x 1 p 1, p 2 b x a 2 3 x 4 c 2 p 1 M(b 1 ) = M(b 2 ) = b; M(c 1 ) = M(c 2 ) = c f = GFp 2 : diagnosable (no falty trace looks like a nonfalty trace) f = Gp 1 : pre-diagnosable bt not diagnosable Diagnosability: mltiple specifications P is diagnosable w.r.t. M and {f i, i = 1,2,, m} if P is diagnosable w.r.t. M and each f i, i = 1,2,, m. Remark: Sffices to stdy the case of only one falt specification 11

12 Algorithm for Diagnosis with LTL Specifications Problem of failre Diagnosis Given P, M, f: Test the diagnosability of P w.r.t. M and f; If P is diagnosable, then constrct a diagnoser for P. 12

13 Algorithm 1: diagnosis for single falt specification 1. Constrct a tablea T f for f that contains all infinite proposition-traces satisfying f, and let {F i,1 i r} denote the generalized Büchi acceptance condition of f. 2. Test the pre-diagnosability of P. Constrct T 1 = T f AP P that generates traces that are accepted by P and are limits of traces satisfying f. Check whether every infinite state-trace generated by T 1 satisfies f, i.e., whether every infinite statetrace generated by T 1 visits each F i infinitely often: T 1 = r i=1 GFF i, a LTL model checking problem. NO P is not pre-diagnosable. 3. Test the diagnosability of P. Constrct T 2 = M 1 M(T 1 ) Σ P that accepts finite traces of P indistingishable from finite traces of T 1, i.e., prefixes of non-falty traces. Check whether every infinite trace in T 2 satisfies f: T 2 = f, a LTL model checking problem. NO P is not diagnosable. 4. Otpt M(T 1 ) as the diagnoser D. Complexity: O(2 f X 4 ); size of D: O(2 f X ). 13

14 Illstrative Example: Mose in a Maze 0 cat 1 5 mose food : observable : nobservable Spec 1 Never visit room 1 (an invariance, a type of safety, property). G R 1 : Globally (always) not in Room 1 Spec 2 Visit room 2 for food infinitely often (a recrrence, a type of liveness, property). GFR 2 : Globally (always) in ftre in Room 2 14

15 System model o x 1 3 x 0 x 1 o o3 2 R 1 x 4 R 2 x 2 o 4 x 5 M() = ǫ, M(o i ) = o i for 0 i 4; AP = {R 1, R 2 }; L(x i ) = for i {1,2}, L(x 1 ) = {R 1 }, L(x 2 ) = {R 2 }. Specifications: f 1 = G R 1, f 2 = GFR 2. 15

16 Tablea T fi, i = 1,2. F 1 F 1 s 1 t 1 t 2 R 1 R 2 R 2 (a) (b) Checking pre-diagnosability: T f i 1 = T f i AP P, i = 1,2; T f i 1 = GFF 1? i = 1,2. F 1 o F x 1 x0 1 3 x 0 x 1 R 1 o 2 o 3 F 1 o 2 o 3 (a) x 5 x 2 o 4, R 2 2 x 4 x 2 F 1 F F 1, R 1 o 4 (b) x 5 P is pre-diagnosable w.r.t. f 1 and f 2 respectively. 16

17 Checking diagnosability: T f i 2 = M 1 M(T f i 1 ) ΣP, i = 1,2; T f 1 2 = G R 1? T f 2 2 = GFR 2? q 0 q 1 R 1 q 1 o 1 q 0 o 2 o 3 o 2 o 3 R q 2 2 o 4 q 3 q 2 q 3 q 4 R 2 o 4 o 4 q 4 (a) (b) P is diagnosable w.r.t. f 1 and f 2 respectively. 17

18 Diagnoser D = {M(T f 1 1 ), M(T f 2 1 )} x 3 o 1 x 0 x 0 o 2 o 3 o 4 o 2 o 3 x 2 x 2 o 4 o 4 (a) (b) o x 1 3 x 0 x 1 o o3 2 R 1 x 4 R 2 x 2 o 4 x 5 Specifications: f 1 = G R 1, f 2 = GFR 2. 18

19 Conclsion A framework for failre diagnosis in LTL setting Notions of indicator, pre-diagnosability, and diagnosability Algorithms for checking pre-diagnosability & diagnosability in proposed framework Constrction of diagnoser for on-line diagnosis in proposed framework Complexity analysis (polynomial in the plant size, exponential in the formla length) 19