Formal Verification and Automated Generation of Invariant Sets

Transcription

1 Formal Verification and Automated Generation of Invariant Sets Khalil Ghorbal Carnegie Mellon University Joint work with Andrew Sogokon and André Platzer Toulouse, France June, 2015 K. Ghorbal (CMU, Pittsburgh) 1 VORACE Workshop 1 / 44

2 Cyber-Physical Systems K. Ghorbal (CMU, Pittsburgh) 2 VORACE Workshop 2 / 44

3 Hybrid Systems Init [ ( Sensing: Control: Plant: ) ] Safety Evolution read data from sensors actuate evolve Continuous time Ordinary Differential Equations (ODE) K. Ghorbal (CMU, Pittsburgh) 3 VORACE Workshop 3 / 44

4 Qualitative Analysis of Dynamical Systems ( x 1, ẋ 2 ) = (x 1 x1 3 x 2 x 1 x2 2, x 1 + x 2 x1 2 x 2 x2 3 ) Algebraic - Invariant Equation The solution for x 0 = (1, 0) respects x 1 (t) 2 + x 2 (t) 2 1 = 0 t K. Ghorbal (CMU, Pittsburgh) 4 VORACE Workshop 4 / 44

5 Why Are Invariants Important? Numerical Integration & Qualitative Analysis More precise numerical integration (Geometrical Integration) Better understanding of the dynamics without solving the problem (some invariants represent conserved quantities like momentum or energy) Formal Verification Formal verification for dynamical and hybrid systems Static Analysis (as templates to statically analyze an implementation) Safety, Reachability, Stability K. Ghorbal (CMU, Pittsburgh) 5 VORACE Workshop 5 / 44

6 Problem I. Checking Invariance of Algebraic Equations Given ẋ = ( 2x 2, 2x 1 3x 2 1 ), p(x 0) = 0, is p(x(t)) = 0 for all t? p(x 1, x 2 ) = x x3 1 x2 2 p(x 1, x 2 ) = x 1 x 2 2 K. Ghorbal (CMU, Pittsburgh) 6 VORACE Workshop 6 / 44

7 Problem I. Checking Invariance of Algebraic Equations Given ẋ = ( 2x 2, 2x 1 3x 2 1 ), p(x 0) = 0, is p(x(t)) = 0 for all t? p(x 1, x 2 ) = x x3 1 x2 2 p(x 1, x 2 ) = x 1 x 2 2 K. Ghorbal (CMU, Pittsburgh) 6 VORACE Workshop 6 / 44

8 Problem II. Generate Algebraic Invariant Equations Given ẋ = ( x 1 + 2x 2 1 x 2, x 2 ), how to generate p such that p(x(t)) = 0? K. Ghorbal (CMU, Pittsburgh) 7 VORACE Workshop 7 / 44

9 Problem II. Generate Algebraic Invariant Equations Given ẋ = ( x 1 + 2x 2 1 x 2, x 2 ), how to generate p such that p(x(t)) = 0? p (x1 (0),x 2 (0))(x 1, x 2 ) = (x 2 (0) x 1 (0)x 2 (0) 2 )x 1 x 1 (0)(x 2 x 1 x 2 2 ) = 0 K. Ghorbal (CMU, Pittsburgh) 7 VORACE Workshop 7 / 44

10 Problem II. Generate Algebraic Invariant Equations Given ẋ = ( x 1 + 2x 2 1 x 2, x 2 ), how to generate p such that p(x(t)) = 0? p (x1 (0),x 2 (0))(x 1, x 2 ) = (x 2 (0) x 1 (0)x 2 (0) 2 )x 1 x 1 (0)(x 2 x 1 x 2 2 ) = 0 x 1 x 2 x 1 x 2 2 is an invariant rational function. K. Ghorbal (CMU, Pittsburgh) 7 VORACE Workshop 7 / 44

11 Definitions Gradient p := ( p x 1,..., p x n ) Lie Derivation D f (p) := dp(x(t)) dt = p, f (ẋ = f) Singular Locus SL(p) := {x R n p = 0} = { x R n p = 0 p } = 0 x 1 x n x V R (p) (p(x) = 0) is singular if x SL(p), regular otherwise. K. Ghorbal (CMU, Pittsburgh) 8 VORACE Workshop 8 / 44

12 Outline 1 Context 2 The Checking Problem 3 The Generation Problem 4 Future Avenues K. Ghorbal (CMU, Pittsburgh) 8 VORACE Workshop 8 / 44

13 Notation for S R n is invariant for f S [ẋ = f]s The set S is an invariant set for f Starting with x 0 s.t x 0 S: for all t > 0, x(t) solution of the IVP (ẋ = f, x(0) = x 0 ) is in S N.B. Treating ẋ = f as a program, one can think of the top formula as representing the Hoare triple {S} ẋ = f {S}. K. Ghorbal (CMU, Pittsburgh) 9 VORACE Workshop 9 / 44

14 Notation for S R n is invariant for f S [ẋ = f]s The set S is an invariant set for f Starting with x 0 s.t x 0 S: for all t > 0, x(t) solution of the IVP (ẋ = f, x(0) = x 0 ) is in S N.B. Treating ẋ = f as a program, one can think of the top formula as representing the Hoare triple {S} ẋ = f {S}. K. Ghorbal (CMU, Pittsburgh) 9 VORACE Workshop 9 / 44

15 Notation for S R n is invariant for f S [ẋ = f]s The set S is an invariant set for f Starting with x 0 s.t x 0 S: for all t > 0, x(t) solution of the IVP (ẋ = f, x(0) = x 0 ) is in S N.B. Treating ẋ = f as a program, one can think of the top formula as representing the Hoare triple {S} ẋ = f {S}. K. Ghorbal (CMU, Pittsburgh) 9 VORACE Workshop 9 / 44

16 Lie s Criterion [Platzer, ITP 2012] Necessary and sufficient for smooth invariant manifolds (Lie, 1893). (Lie) p = 0 (D f(p) = 0 p 0) (p = 0) [ẋ = f] (p = 0) x 2 x 2 x 1 p = 0 non-smooth x 1 p = 0 smooth K. Ghorbal (CMU, Pittsburgh) 10 VORACE Workshop 10 / 44

17 Extensions of Lie Contribution Handling certain singularities (points where p = 0) No flow in the problem variables at singularities on the variety (Lie ) p = 0 ( D f (p) = 0 ( p = 0 f = 0 )) (p = 0) [ẋ = f] (p = 0) Flow at singularities on the variety is directed into the variety (Lie ) p = 0 ( D f (p) = 0 ( p = 0 p(x + λf) = 0) ) (p = 0) [ẋ = f] (p = 0). K. Ghorbal (CMU, Pittsburgh) 11 VORACE Workshop 11 / 44

18 Extensions of Lie Contribution Handling certain singularities (points where p = 0) No flow in the problem variables at singularities on the variety (Lie ) p = 0 ( D f (p) = 0 ( p = 0 f = 0 )) (p = 0) [ẋ = f] (p = 0) Flow at singularities on the variety is directed into the variety (Lie ) p = 0 ( D f (p) = 0 ( p = 0 p(x + λf) = 0) ) (p = 0) [ẋ = f] (p = 0). K. Ghorbal (CMU, Pittsburgh) 11 VORACE Workshop 11 / 44

19 Extensions of Lie Contribution Handling certain singularities (points where p = 0) No flow in the problem variables at singularities on the variety (Lie ) p = 0 ( D f (p) = 0 ( p = 0 f = 0 )) (p = 0) [ẋ = f] (p = 0) Flow at singularities on the variety is directed into the variety (Lie ) p = 0 ( D f (p) = 0 ( p = 0 p(x + λf) = 0) ) (p = 0) [ẋ = f] (p = 0). K. Ghorbal (CMU, Pittsburgh) 11 VORACE Workshop 11 / 44

20 Extensions of Lie: Lie Handling certain singularities (points where p = 0) Contribution x 2 x 1 Lie Lie Lie K. Ghorbal (CMU, Pittsburgh) 12 VORACE Workshop 12 / 44

21 Extensions of Lie: Lie Contribution Handling certain singularities (points where p = 0) Lie Lie Lie K. Ghorbal (CMU, Pittsburgh) 13 VORACE Workshop 13 / 44

22 First Integral (FI) [Platzer, J. Log. Comput. 2010] Necessary and sufficient for conserved quantities (integrals of motion). D f (p) = 0 (FI) (p = 0) [ẋ = f] (p = 0) x 2 x 2 x 1 p conserved x 1 p not conserved K. Ghorbal (CMU, Pittsburgh) 14 VORACE Workshop 14 / 44

23 Extensions of FI [Sankaranarayanan et al., FMSD 2008] Continuous consecutions (C-c) and polynomial consecutions (P-c) are Darboux polynomials (Darboux, 1878). λ R, D f (p) = λp (C-c) (p = 0) [ẋ = f] (p = 0), (P-c) λ R[x], D f(p) = λp (p = 0) [ẋ = f] (p = 0). K. Ghorbal (CMU, Pittsburgh) 15 VORACE Workshop 15 / 44

24 Extensions of FI [Sankaranarayanan et al., FMSD 2008] ( ) f = (3 x1 2 4, x2 2 + x 1x 2 + 3), p = x x 1x x x 1x 2 + x , D f (p) = (6x 1 4x 2) p }{{} λ x 2 x 1 FI C-c P-c K. Ghorbal (CMU, Pittsburgh) 16 VORACE Workshop 16 / 44

25 Differential Radical Invariants (DRI) [G. et al., TACAS 2014, SAS 2014] Necessary and sufficient for invariant varieties. (DRI) p = 0 N 1 i=0 D(i) f (p) = 0 (p = 0) [ẋ = f] (p = 0) x 2 x 2 x 2 x 1 x 1 x 1 K. Ghorbal (CMU, Pittsburgh) 17 VORACE Workshop 17 / 44

26 Order Relation A B (R A ) (R B ) S A : T A [ẋ = f]s A : T A S B : T B [ẋ = f]s B : T B Partial Order R A R B if and only if A = B and T A is a subtype of T B. R A R B (R A R B and R A R B ) Equivalence. R A R B (R A R B and R A R B ) Strict increase of deductive power K. Ghorbal (CMU, Pittsburgh) 18 VORACE Workshop 18 / 44

27 Algebraic Sets Deductive Hierarchy DRI Lie Lie Lie P-c C-c FI R A R B means everything proved by R A can be proved by R B. no link means R A and R B are incomparable. K. Ghorbal (CMU, Pittsburgh) 19 VORACE Workshop 19 / 44

28 Algebraic Sets Deductive Hierarchy DRI Lie P-c Lie C-c Lie-based Lie FI Darboux-based R A R B means everything proved by R A can be proved by R B. no link means R A and R B are incomparable. K. Ghorbal (CMU, Pittsburgh) 19 VORACE Workshop 19 / 44

29 Square-free Reduction Square-free reduction of a polynomial h = h α 1 1 hα 2 2 hα k k Geometrically V R (h) R V R (SF(h)). SF automated pre-processing step in computer algebra systems Is it a good idea to apply SF for invariance checking? K. Ghorbal (CMU, Pittsburgh) 20 VORACE Workshop 20 / 44

30 Square-free Reduction Square-free reduction of a polynomial SF(h) = h α 1 1 h α 2 2 h α k k Geometrically V R (h) R V R (SF(h)). SF automated pre-processing step in computer algebra systems Is it a good idea to apply SF for invariance checking? K. Ghorbal (CMU, Pittsburgh) 20 VORACE Workshop 20 / 44

31 Square-free Reduction DRI Lie Lie Lie P-c C-c FI K. Ghorbal (CMU, Pittsburgh) 21 VORACE Workshop 21 / 44

32 Square-free Reduction DRI Lie Lie Lie P-c C-c FI SF C-c SF FI K. Ghorbal (CMU, Pittsburgh) 21 VORACE Workshop 21 / 44

33 Square-free Reduction DRI Lie Lie Lie P-c C-c FI SF P-c SF C-c SF FI K. Ghorbal (CMU, Pittsburgh) 21 VORACE Workshop 21 / 44

34 Square-free Reduction DRI SF Lie Lie P-c SF P-c SF Lie Lie C-c SF C-c SF Lie Lie FI SF FI K. Ghorbal (CMU, Pittsburgh) 21 VORACE Workshop 21 / 44

35 Square-free Reduction SF Lie Lie DRI SF DRI P-c SF P-c SF Lie Lie C-c SF C-c SF Lie Lie FI SF FI K. Ghorbal (CMU, Pittsburgh) 21 VORACE Workshop 21 / 44

36 Semi-algebraic Sets So far we have only seen algebraic sets: p = 0 For i p i = 0, we can rewrite using i p2 i = 0 We can do better [SAS 14] What about semi-algebraic sets: Sets of the form p 0 Closed Sets Arbitrary Sets: boolean formulae with polynomial equalities and inequalities K. Ghorbal (CMU, Pittsburgh) 22 VORACE Workshop 22 / 44

37 The Nagumo Theorem Sub-tangent Vector A vector v R n is sub-tangential to a set S R n at x S if lim inf λ 0 + dist (S, x + λv) λ = 0. (dist(s, x) inf y S x y 1) Contingent Cone K x (S): the set of all sub-tangent vectors to a set S at x S. Nagumo s Theorem A closed set S R n is positively invariant under the flow of the system if and only if f(x) K x (S) for all x bdr(s) (boundary of S). K. Ghorbal (CMU, Pittsburgh) 23 VORACE Workshop 23 / 44

38 Contingent Cone of Boolean Formulae K x(s 1) K x(s 2) K x(s 1 S 2) x 2 x 2 x 1 x 2 + x 2 1 = 0 x 2 x 2 1 = 0 ẋ 1 = 1, ẋ 2 = 0 x 1 K. Ghorbal (CMU, Pittsburgh) 24 VORACE Workshop 24 / 44

39 Barrier Certificates Prajna s Thesis (Non-strict) Barrier Certificate ( x R n, D f (p) 0) p 0 is a positive invariant Unsound Barrier Certificate ( x s.t. p(x) = 0, D f (p) 0) p 0 is a positive invariant Strict Barrier Certificate ( x s.t. p(x) = 0, D f (p) < 0) p 0 is a positive invariant K. Ghorbal (CMU, Pittsburgh) 25 VORACE Workshop 25 / 44

40 Differential Invariants (DI) Differential Invariants (DI) Conservative Lifting of Barrier Certificates to Boolean Connectives D(S) ḟ x (DI) S [ẋ = f] S, D(S) ḟ x : substitute each ẋ i in D(S) with f i (x) D(r) = 0 for numbers, D(x) = ẋ for variables, D(a + b) = D(a) + D(b), D(a b) = D(a) b + a D(b), D(a b) D(a) D(b), accordingly for, >, <. (BC) D(S 1 S 2 ) D(S 1 ) D(S 2 ), D(S 1 S 2 ) D(S 1 ) D(S 2 ), ( here is important for soundness) K. Ghorbal (CMU, Pittsburgh) 26 VORACE Workshop 26 / 44

41 Non-smooth Strict Barrier Certificate (NSSBC) Apply Strict Barrier Certificate on Active Boundaries (NSSBC) ( min i=1,...,k ( k i=1 ) ( max p ij = 0 D f min j=1,...,m(i) i=1,...,k ) ( m(i) j=1 p k ij 0 [ẋ = f] i=1 max j=1,...,m(i) p ij m(i) j=1 p ij 0 ) < 0 ). Check Invariance of p 1 0 p 2 0 NSSBC (p 1 < p 2 p 1 = 0 D f (p 1 ) < 0) (p 1 > p 2 p 2 = 0 D f (p 2 ) < 0) (p 1 = p 2 = 0 D f (p 1 ) < 0 D f (p 2 ) < 0) DI D f (p 1 ) 0 D f (p 2 ) 0 K. Ghorbal (CMU, Pittsburgh) 27 VORACE Workshop 27 / 44

42 Liu, Zhan, Zhao (LZZ) Characterization [EMSOFT 2011] Definitions In f (S) {x R n ɛ > 0. t (0, ɛ). x(t) S}, In ( f ) (S) {x R n ɛ > 0. t (0, ɛ). x( t) S}, Characterization An arbitrary set S R n is a positive invariant for f if and only if S In f (S) and S c In ( f ) (S c ) or equivalently In ( f ) (S) S In f (S) In f (S) can be computed using high-order Lie derivatives! K. Ghorbal (CMU, Pittsburgh) 28 VORACE Workshop 28 / 44

43 Overview Proof Rules for Checking Invariance of Semi-algebraic Sets (Closed Semi-algebraic Sets) Nagumo LZZ (Semi-algebraic Sets) NSSBC DRI Lie Lie (Algebraic Sets) DI P-c C-c Lie (Lie) FI (Darboux) K. Ghorbal (CMU, Pittsburgh) 29 VORACE Workshop 29 / 44

44 Experimental Performance: Algebraic Sets 10 Time ( s) per problem Number of problems solved K. Ghorbal (CMU, Pittsburgh) 30 VORACE Workshop 30 / 44

45 Experimental Performance: Semi-algebraic Sets Time ( s) per problem Number of problems solved LZZ vs DI Time ( s) per problem Number of problems solved LZZ vs NSSBC LZZ DI NSSBC K. Ghorbal (CMU, Pittsburgh) 31 VORACE Workshop 31 / 44

46 Outline 1 Context 2 The Checking Problem 3 The Generation Problem 4 Future Avenues K. Ghorbal (CMU, Pittsburgh) 31 VORACE Workshop 31 / 44

47 Overview Template-Based Fix a template: generic polynomials with symbolic coefficients Use sufficient conditions to derive constraints over the coefficients Solve the system In practice: templates of the form p = 0 (algebraic sets) with Darboux condition (D f (p) p ) work for n 10 and d 3. templates of the form p 0 relying on Barrier Certificates (SOSTools) Discrete Abstraction Discrete the space with the signs of a given set of polynomials Relies on sufficient conditions to remove spurious transitions Issue: How to populate and eventually refine the initial set of polynomials K. Ghorbal (CMU, Pittsburgh) 32 VORACE Workshop 32 / 44

48 Matrix Representation: Intuition Suppose we have a 2-dimensional ODE (ẋ 1, ẋ 2 ) = (x 1, x 2 ) K. Ghorbal (CMU, Pittsburgh) 33 VORACE Workshop 33 / 44

49 Matrix Representation: Intuition Suppose we have a 2-dimensional ODE (ẋ 1, ẋ 2 ) = (x 1, x 2 ) 1 Start with parametric h of degree 1: h = α 1 x 1 + α 2 x 2 + α 3 K. Ghorbal (CMU, Pittsburgh) 33 VORACE Workshop 33 / 44

50 Matrix Representation: Intuition Suppose we have a 2-dimensional ODE (ẋ 1, ẋ 2 ) = (x 1, x 2 ) 1 Start with parametric h of degree 1: h = α 1 x 1 + α 2 x 2 + α 3 2 Start with N = 1 K. Ghorbal (CMU, Pittsburgh) 33 VORACE Workshop 33 / 44

51 Matrix Representation: Intuition Suppose we have a 2-dimensional ODE (ẋ 1, ẋ 2 ) = (x 1, x 2 ) 1 Start with parametric h of degree 1: h = α 1 x 1 + α 2 x 2 + α 3 2 Start with N = 1 3 Find β R such that: D f (h) = βh K. Ghorbal (CMU, Pittsburgh) 33 VORACE Workshop 33 / 44

52 Matrix Representation: Intuition Suppose we have a 2-dimensional ODE (ẋ 1, ẋ 2 ) = (x 1, x 2 ) 1 Start with parametric h of degree 1: h = α 1 x 1 + α 2 x 2 + α 3 2 Start with N = 1 3 Find β R such that: D f (h) = βh D f (h) = α 1 x 1 + α 2 x 2 = β(α 1 x 1 + α 2 x 2 + α 3 ) K. Ghorbal (CMU, Pittsburgh) 33 VORACE Workshop 33 / 44

53 Matrix Representation: Intuition Suppose we have a 2-dimensional ODE (ẋ 1, ẋ 2 ) = (x 1, x 2 ) 1 Start with parametric h of degree 1: h = α 1 x 1 + α 2 x 2 + α 3 2 Start with N = 1 3 Find β R such that: D f (h) = βh D f (h) = α 1 x 1 + α 2 x 2 = β(α 1 x 1 + α 2 x 2 + α 3 ) ( 1 + β)α 1 = 0 ( 1 + β)α 2 = 0 (β)α 3 = β 0 0 α β 0 α 2 = β α 3 K. Ghorbal (CMU, Pittsburgh) 33 VORACE Workshop 33 / 44

54 Matrix Representation: Intuition Suppose we have a 2-dimensional ODE (ẋ 1, ẋ 2 ) = (x 1, x 2 ) 1 Start with parametric h of degree 1: h = α 1 x 1 + α 2 x 2 + α 3 2 Start with N = 1 3 Find β R such that: D f (h) = βh D f (h) = α 1 x 1 + α 2 x 2 = β(α 1 x 1 + α 2 x 2 + α 3 ) ( 1 + β)α 1 = 0 ( 1 + β)α 2 = 0 (β)α 3 = β 0 0 α β 0 α 2 = β α 3 Study the null space (kernel) of M(β) K. Ghorbal (CMU, Pittsburgh) 33 VORACE Workshop 33 / 44

55 Symbolic Linear Algebra ( 1 + β)α 1 = 0 ( 1 + β)α 2 = 0 (β)α 3 = 0 h = α 1 x 1 + α 2 x 2 + α β β β Study the null space (kernel) of M(β) α 1 α 2 α 3 = 0 Max dim of ker of M(β) more freedom for α = (α 1, α 2, α 3 ) Increases the chances of finding first integrals Dually, minimize the rank of M(β) K. Ghorbal (CMU, Pittsburgh) 34 VORACE Workshop 34 / 44

56 Symbolic Linear Algebra ( 1 + β)α 1 = 0 ( 1 + β)α 2 = 0 (β)α 3 = 0 h = α 1 x 1 + α 2 x 2 + α β β β Study the null space (kernel) of M(β) α 1 α 2 α 3 = 0 Max dim of ker of M(β) more freedom for α = (α 1, α 2, α 3 ) Increases the chances of finding first integrals Dually, minimize the rank of M(β) K. Ghorbal (CMU, Pittsburgh) 34 VORACE Workshop 34 / 44

57 Symbolic Linear Algebra ( 1 + β)α 1 = 0 ( 1 + β)α 2 = 0 (β)α 3 = 0 h = α 1 x 1 + α 2 x 2 + α β β β Study the null space (kernel) of M(β) α 1 α 2 α 3 = 0 Max dim of ker of M(β) more freedom for α = (α 1, α 2, α 3 ) Increases the chances of finding first integrals Dually, minimize the rank of M(β) K. Ghorbal (CMU, Pittsburgh) 34 VORACE Workshop 34 / 44

58 Symbolic Linear Algebra ( 1 + β)α 1 = 0 ( 1 + β)α 2 = 0 (β)α 3 = 0 h = α 1 x 1 + α 2 x 2 + α β β β Study the null space (kernel) of M(β) α 1 α 2 α 3 = 0 Max dim of ker of M(β) more freedom for α = (α 1, α 2, α 3 ) Increases the chances of finding first integrals Dually, minimize the rank of M(β) NP-hard [Buss et al. 1999] K. Ghorbal (CMU, Pittsburgh) 34 VORACE Workshop 34 / 44

59 Symbolic Linear Algebra ( 1 + β)α 1 = 0 ( 1 + β)α 2 = 0 (β)α 3 = 0 h = α 1 x 1 + α 2 x 2 + α β β β Study the null space (kernel) of M(β) α 1 α 2 α 3 = 0 Max dim of ker of M(β) more freedom for α = (α 1, α 2, α 3 ) Increases the chances of finding first integrals Dually, minimize the rank of M(β) NP-hard [Buss et al. 1999] h = x 2 (0)x 1 x 1 (0)x 2 K. Ghorbal (CMU, Pittsburgh) 34 VORACE Workshop 34 / 44

60 Discrete Abstraction Discretization Transition system K. Ghorbal (CMU, Pittsburgh) 35 VORACE Workshop 35 / 44

61 Outline 1 Context 2 The Checking Problem 3 The Generation Problem 4 Future Avenues K. Ghorbal (CMU, Pittsburgh) 35 VORACE Workshop 35 / 44

62 Challenges K. Ghorbal (CMU, Pittsburgh) 36 VORACE Workshop 36 / 44

63 Differential-Algebraic Equations (DAE) Verification and Faithful Simulation for cyber-physical systems with DAE K. Ghorbal (CMU, Pittsburgh) 37 VORACE Workshop 37 / 44

64 Differential-Algebraic Equations (DAE) Verification and Faithful Simulation for cyber-physical systems with DAE 0 = F(x, ẋ, t) { ẋ = f(y, x) 0 = g(y, x) Compositional design Tools: Dymola (Dassault Systèmes), Modelica K. Ghorbal (CMU, Pittsburgh) 37 VORACE Workshop 37 / 44

65 V6 Engine Simple Combustion Model (Modelica) Engine Cylinder Simulation K. Ghorbal (CMU, Pittsburgh) 38 VORACE Workshop 38 / 44

66 Combining Static Analysis and Symbolic Computation Invariants Generation for DAE Extends previous work (algebraic approach) Semi-explicit form of index 1 (specific class of DAE) Hybrid Simulation is Hard Well-founded operational semantics (compilation, index reduction) Preserve composionality in presence of mode changes Proper handling of zero-crossing (detection and consistent initialization) Cascades of zero-crossings (sliding modes) Investigate the use of static analysis and symbolic computation K. Ghorbal (CMU, Pittsburgh) 39 VORACE Workshop 39 / 44

67 Combining Static Analysis and Symbolic Computation Invariants Generation for DAE Extends previous work (algebraic approach) Semi-explicit form of index 1 (specific class of DAE) Hybrid Simulation is Hard Well-founded operational semantics (compilation, index reduction) Preserve composionality in presence of mode changes Proper handling of zero-crossing (detection and consistent initialization) Cascades of zero-crossings (sliding modes) Investigate the use of static analysis and symbolic computation K. Ghorbal (CMU, Pittsburgh) 39 VORACE Workshop 39 / 44

68 Combining Static Analysis and Symbolic Computation Invariants Generation for DAE Extends previous work (algebraic approach) Semi-explicit form of index 1 (specific class of DAE) Hybrid Simulation is Hard Well-founded operational semantics (compilation, index reduction) Preserve composionality in presence of mode changes Proper handling of zero-crossing (detection and consistent initialization) Cascades of zero-crossings (sliding modes) Investigate the use of static analysis and symbolic computation K. Ghorbal (CMU, Pittsburgh) 39 VORACE Workshop 39 / 44

69 Thanks for your attention! K. Ghorbal (CMU, Pittsburgh) 40 VORACE Workshop 40 / 44

70 Smooth invariant manifolds (Lie vs DRI) Lie and DRI decide invariance for smooth invariant manifolds. DRI Lie P-c Lie C-c Time ( s) per problem Lie FI Number of problems solved K. Ghorbal (CMU, Pittsburgh) 41 VORACE Workshop 41 / 44

71 Functional invariants (DI vs DRI) DI = and DRI decide invariance of varieties of conserved quantities. DRI Lie P-c Lie C-c Time ( s) per problem Lie FI Number of problems solved K. Ghorbal (CMU, Pittsburgh) 42 VORACE Workshop 42 / 44

72 Singularities at Equilibria (Lie, Lie & Lie vs DRI) Lie, Lie and DRI decide invariance for varieties of with singularities that are equilibrium points. DRI Lie P-c Lie C-c Time ( s) per problem Lie FI Number of problems solved K. Ghorbal (CMU, Pittsburgh) 43 VORACE Workshop 43 / 44

73 Min Max (h 1 > max(h 2,..., h m ) D f (h 1 ) < 0) (h 1 < max(h 2,..., h m ) D f (max(h 2,..., h m )) < 0) (h 1 = max(h 2,..., h m ) D f (h 1 ) < 0 D f (max(h 2,..., h m )) < 0) (g 1 < min(g 2,..., g m ) D f (g 1 ) < 0) (g 1 > min(g 2,..., g m ) D f (min(g 2,..., g m )) < 0) (g 1 = min(g 2,..., g m ) D f (g 1 ) < 0 D f (min(g 2,..., g m )) < 0). K. Ghorbal (CMU, Pittsburgh) 44 VORACE Workshop 44 / 44