Characterizing Algebraic Invariants by Differential Radical Invariants
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1 Characterizing Algebraic Invariants by Differential Radical Invariants Khalil Ghorbal Carnegie Mellon university Joint work with André Platzer ÉNS Paris April 1st, 2014 K. Ghorbal (CMU) Invited Talk 1 Differential Radical Invariants 1 / 29
2 Introduction Context: Hybrid Systems Model Sensing: read data from sensors K. Ghorbal (CMU) Invited Talk 2 Differential Radical Invariants 2 / 29
3 Introduction Context: Hybrid Systems Model Sensing: Control: read data from sensors actuate K. Ghorbal (CMU) Invited Talk 2 Differential Radical Invariants 2 / 29
4 Introduction Context: Hybrid Systems Model Sensing: read data from sensors Control: actuate Plant: evolve K. Ghorbal (CMU) Invited Talk 2 Differential Radical Invariants 2 / 29
5 Introduction Context: Hybrid Systems Model ( Sensing: read data from sensors Control: actuate Plant: evolve ) K. Ghorbal (CMU) Invited Talk 2 Differential Radical Invariants 2 / 29
6 Introduction Context: Hybrid Systems Model Init ( Sensing: read data from sensors Control: actuate Plant: evolve ) Safety K. Ghorbal (CMU) Invited Talk 2 Differential Radical Invariants 2 / 29
7 Introduction Context: Hybrid Systems Model Init [ ( Sensing: read data from sensors Control: actuate Plant: evolve ) ] Safety K. Ghorbal (CMU) Invited Talk 2 Differential Radical Invariants 2 / 29
8 Introduction Context: Hybrid Systems Model Init [ ( Sensing: read data from sensors Control: actuate Plant: Evolve ) ] Safety Evolution Continuous time Ordinary Differential Equations (ODE) K. Ghorbal (CMU) Invited Talk 2 Differential Radical Invariants 2 / 29
9 Introduction Algebraic Differential Equations Example x 1 = x 2 x 3 = x4 2 x 2 = x 1 x 4 = x 3 x 4 Formally dx i (t) dt = ẋ i = p i (x), 1 i n. K. Ghorbal (CMU) Invited Talk 3 Differential Radical Invariants 3 / 29
10 Introduction Algebraic Invariant Expression x 1 = x 2 x 3 = x4 2 x 2 = x 1 x 4 = x 3 x x x1 Solution for x 0 = (1, 0, 0, 1) for t = [0, 1] K. Ghorbal (CMU) Invited Talk 4 Differential Radical Invariants 4 / 29
11 Introduction Algebraic Invariant Expression x 1 = x 2 x 3 = x4 2 x 2 = x 1 x 4 = x 3 x x x1 Solution for x 0 = (1, 0, 0, 1) for t = [0, 2] K. Ghorbal (CMU) Invited Talk 4 Differential Radical Invariants 4 / 29
12 Introduction Algebraic Invariant Expression x 1 = x 2 x 3 = x4 2 x 2 = x 1 x 4 = x 3 x x x1 t, x 1 (t) 2 + x 2 (t) 2 1 = 0 K. Ghorbal (CMU) Invited Talk 4 Differential Radical Invariants 4 / 29
13 Introduction Algebraic Invariant Expression Geometrically O(x 0 ) def = {x(t) t 0} Set of roots of h(x 1, x 2 ) K. Ghorbal (CMU) Invited Talk 5 Differential Radical Invariants 5 / 29
14 Introduction Algebraic Invariant Expression Geometrically O(x 0 ) def = {x(t) t 0} Set of roots of h(x 1, x 2 ) Algebraically h(x 1, x 2 ) = x1 2 + x2 2 1(= 0) K. Ghorbal (CMU) Invited Talk 5 Differential Radical Invariants 5 / 29
15 Introduction Algebraic Invariant Expression Geometrically O(x 0 ) def = {x(t) t 0} Set of roots of h(x 1, x 2 ) Algebraically h(x 1, x 2 ) = x1 2 + x2 2 1(= 0) Algebraic Invariant Expression t, h(x(t)) = 0, for all x(t) solution of the Initial Value Problem. K. Ghorbal (CMU) Invited Talk 5 Differential Radical Invariants 5 / 29
16 Introduction Problems I. Checking the invariance of Algebraic Expression Given p, and x 0 root of h(x) (h(x 0 ) = 0), is h(x) = 0 is an algebraic invariant expression? ( t 0, h(x(t)) = 0) II. Generate Algebraic Invariant Expression Given p, and Init, how to generate algebraic invariant expressions? (h(x) = 0) K. Ghorbal (CMU) Invited Talk 6 Differential Radical Invariants 6 / 29
17 Time Abstraction Outline 1 Introduction 2 Time Abstraction 3 Characterization of Invariant Expressions 4 Checking & Generation 5 Conclusion K. Ghorbal (CMU) Invited Talk 7 Differential Radical Invariants 7 / 29
18 Time Abstraction Variety Embedding of Orbits Zariski Closure Vanishing Ideal: all polynomials that vanish on O(x 0 ) I (O(x 0 )) def = {h R[x] x O(x 0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I Closure: Sound Abstraction V (I ) def = {x R n h I, h(x) = 0} O(x 0 ) V (I (O(x 0 ))) K. Ghorbal (CMU) Invited Talk 8 Differential Radical Invariants 8 / 29
19 Time Abstraction Variety Embedding of Orbits Zariski Closure Vanishing Ideal: all polynomials that vanish on O(x 0 ) I (O(x 0 )) def = {h R[x] x O(x 0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I Closure: Sound Abstraction V (I ) def = {x R n h I, h(x) = 0} O(x 0 ) V (I (O(x 0 ))) K. Ghorbal (CMU) Invited Talk 8 Differential Radical Invariants 8 / 29
20 Time Abstraction Variety Embedding of Orbits Zariski Closure Vanishing Ideal: all polynomials that vanish on O(x 0 ) I (O(x 0 )) def = {h R[x] x O(x 0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I Closure: Sound Abstraction V (I ) def = {x R n h I, h(x) = 0} O(x 0 ) V (I (O(x 0 ))) K. Ghorbal (CMU) Invited Talk 8 Differential Radical Invariants 8 / 29
21 Time Abstraction Variety Embedding of Orbits Zariski Closure Vanishing Ideal: all polynomials that vanish on O(x 0 ) I (O(x 0 )) def = {h R[x] x O(x 0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I Closure: Sound Abstraction V (I ) def = {x R n h I, h(x) = 0} O(x 0 ) V (I (O(x 0 ))) V (I (O(x 0 ))) is the smallest variety that contains O(x 0 ) K. Ghorbal (CMU) Invited Talk 8 Differential Radical Invariants 8 / 29
22 Time Abstraction Variety Embedding O(x 0 ) Orbit α I (O(x 0 )) Vanishing Ideal γ V (I (O(x 0))) Closure O(x 0) Orbit K. Ghorbal (CMU) Invited Talk 9 Differential Radical Invariants 9 / 29
23 Time Abstraction Variety Embedding O(x 0 ) Orbit α I (O(x 0 )) Vanishing Ideal γ V (I (O(x 0))) Closure O(x 0) Orbit K. Ghorbal (CMU) Invited Talk 9 Differential Radical Invariants 9 / 29
24 Time Abstraction Variety Embedding O(x 0 ) Orbit α I (O(x 0 )) Vanishing Ideal γ V (I (O(x 0))) Closure O(x 0) Orbit K. Ghorbal (CMU) Invited Talk 9 Differential Radical Invariants 9 / 29
25 Time Abstraction Variety Embedding O(x 0 ) Orbit α I (O(x 0 )) Vanishing Ideal γ V (I (O(x 0))) Closure O(x 0) Orbit K. Ghorbal (CMU) Invited Talk 9 Differential Radical Invariants 9 / 29
26 Time Abstraction Variety Embedding O(x 0 ) Orbit α I (O(x 0 )) Vanishing Ideal γ V (I (O(x 0))) Closure O(x 0) Orbit Limitation: Zariski Dense Varieties ẋ = x O(x 0 ) = [x 0, [ I = 0 V (I (O(x 0 ))) = R K. Ghorbal (CMU) Invited Talk 9 Differential Radical Invariants 9 / 29
27 Characterization of Invariant Expressions Outline 1 Introduction 2 Time Abstraction 3 Characterization of Invariant Expressions 4 Checking & Generation 5 Conclusion K. Ghorbal (CMU) Invited Talk 10 Differential Radical Invariants 10 / 29
28 Characterization of Invariant Expressions Properties of I (O(x 0 )) (abstract domain) I (O(x 0 )): The set of all polynomials that vanish on O(x 0 ) I (O(x 0 )) is an ideal 0 I (O(x 0 )) if h 1, h 2 I (O(x 0 )), then h 1 + h 2 I (O(x 0 )) if h I (O(x 0 )), then q h I (O(x 0 )), for any polynomial q K. Ghorbal (CMU) Invited Talk 11 Differential Radical Invariants 11 / 29
29 Characterization of Invariant Expressions Properties of I (O(x 0 )) (abstract domain) I (O(x 0 )): The set of all polynomials that vanish on O(x 0 ) I (O(x 0 )) is an ideal 0 I (O(x 0 )) if h 1, h 2 I (O(x 0 )), then h 1 + h 2 I (O(x 0 )) if h I (O(x 0 )), then q h I (O(x 0 )), for any polynomial q I (O(x 0 )) is a Differential Ideal if h I (O(x 0 )), then dh dt I (O(x 0)). K. Ghorbal (CMU) Invited Talk 11 Differential Radical Invariants 11 / 29
30 Characterization of Invariant Expressions Properties of I (O(x 0 )) (abstract domain) I (O(x 0 )): The set of all polynomials that vanish on O(x 0 ) I (O(x 0 )) is an ideal 0 I (O(x 0 )) if h 1, h 2 I (O(x 0 )), then h 1 + h 2 I (O(x 0 )) if h I (O(x 0 )), then q h I (O(x 0 )), for any polynomial q I (O(x 0 )) is a Differential Ideal if h I (O(x 0 )), then dh dt I (O(x 0)). Instead of dh dt, we will use L p(h): The Lie Derivative of h. K. Ghorbal (CMU) Invited Talk 11 Differential Radical Invariants 11 / 29
31 Characterization of Invariant Expressions Lie derivative along a vector field Definition L p (h) def = n i=1 h x i p i (x) R[x] Properties Algebraic differentiation (chain rule) Do not require x(t) (the solution of the ODE) Corresponds to the time derivative when x = x(t) K. Ghorbal (CMU) Invited Talk 12 Differential Radical Invariants 12 / 29
32 Characterization of Invariant Expressions Differential Radical Invariants Theorem h I (O(x 0 )) if and only if g 0,..., g N 1 R[x]: L (N) p (h) = N 1 i=0 g il (i) p (h) N finite L (0) p (h)(x 0 ) = 0,..., L (N 1) p (h)(x 0 ) = 0 K. Ghorbal (CMU) Invited Talk 13 Differential Radical Invariants 13 / 29
33 Characterization of Invariant Expressions Special Cases Invariant Polynomial Functions L p (h) = 0 N = 1 and g 0 = 0 Darboux Polynomials [Darboux 1878, Painlevé, Poincaré... ] L p (h) = g 0 h N = 1 K. Ghorbal (CMU) Invited Talk 14 Differential Radical Invariants 14 / 29
34 Characterization of Invariant Expressions Special Cases Invariant Polynomial Functions L p (h) = 0 N = 1 and g 0 = 0 Darboux Polynomials [Darboux 1878, Painlevé, Poincaré... ] L p (h) = g 0 h N = 1 K. Ghorbal (CMU) Invited Talk 14 Differential Radical Invariants 14 / 29
35 Checking & Generation Outline 1 Introduction 2 Time Abstraction 3 Characterization of Invariant Expressions 4 Checking & Generation 5 Conclusion K. Ghorbal (CMU) Invited Talk 15 Differential Radical Invariants 15 / 29
36 Checking & Generation Checking Invariance of Candidates I. Checking the invariance of Algebraic Expression Given p, and x 0 root of h(x) (h(x 0 ) = 0), is h(x) = 0 is an algebraic invariant expression? ( t 0, h(x(t)) = 0) A Necessary and Sufficient Proof Rule (DRI) h = 0 N 1 i=0 L(i) p (h) = 0 (h = 0) [ẋ = p](h = 0). K. Ghorbal (CMU) Invited Talk 16 Differential Radical Invariants 16 / 29
37 Checking & Generation DRI: Algorithm Data: h, p, x Result: Boolean: True, False. N 1 GB {h} l h symbs Variables[p, h] while true do GB GröbnerBasis[GB, x] LieD LieDerivative[l, p, x] Rem PolynomialRemainder[LieD, GB, x] if Rem = 0 then return True else Reduce QE[ symbs, h = 0 LieD = 0] if Reduce then return False Append[GB, LieD] l LieD N N + 1 K. Ghorbal (CMU) Invited Talk 17 Differential Radical Invariants 17 / 29
38 Checking & Generation Benchmarks Joint work with Andrew Sogokon log10 HtL 1 DI 0 Darboux Lie -1 LieZero LieStar -2 DRI -3 0 K. Ghorbal (CMU) Number of problems solved Invited Talk Differential Radical Invariants 18 / 29
39 Checking & Generation Generation of Invariant Varieties II. Generate Algebraic Invariant Expression Given p, and Init, how to generate algebraic invariant expressions? (h(x) = 0) Theorem S R n is an invariant variety if and only if S is the set of roots of the system L (i) p (h) = 0, for some polynomial h. 0 i N 1 K. Ghorbal (CMU) Invited Talk 19 Differential Radical Invariants 19 / 29
40 Checking & Generation In practice In practice Find h and N, such that: L (N) N 1 p (h) = g i L (i) p (h) i=0 The set of roots of is an invariant variety. 0 i N 1 L (i) p (h) = 0, K. Ghorbal (CMU) Invited Talk 20 Differential Radical Invariants 20 / 29
41 Checking & Generation Matrix Representation: Intuition invariant of degree 1 x 1 = a 1 x 1 + a 2 x 2 h = α 1 x 1 + α 2 x 2 + α 3 x 0 x 2 = b 1 x 1 + b 2 x 2 L p (h) = α 1 (a 1 x 1 + a 2 x 2 ) + α 2 (b 1 x 1 + b 2 x 2 )?β R s.t. L p (h) = βh ( a 1 + β)α 1 + ( b 1 )α 2 = 0 ( a 2 )α 1 + ( b 2 + β)α 2 = 0 (β)α 3 = 0 a 1 + β b 1 0 α 1 a 2 b 2 + β 0 α 2 = β α 3 K. Ghorbal (CMU) Invited Talk 21 Differential Radical Invariants 21 / 29
42 Checking & Generation Matrix Representation Polynomial ( ) n+d d Coefficients (d degree of h) h α = (α 1, α 2,..., α r ) g i β i = (β 1, β 2,..., β si ) Matrix Representation L (N) N 1 p (h) = g i L (i) p (h) M(β)α = 0 i=0 α lies in the Kernel of M(β) def = {α R r M(β)α = 0} K. Ghorbal (CMU) Invited Talk 22 Differential Radical Invariants 22 / 29
43 Checking & Generation Example: n = 2, d = 1, N = 1 invariant of degree 1 x 1 = a 1 x 1 + a 2 x 2 h = α 1 x 1 + α 2 x 2 + α 3 x 0 x 2 = b 1 x 1 + b 2 x 2 L p (h) = α 1 (a 1 x 1 + a 2 x 2 ) + α 2 (b 1 x 1 + b 2 x 2 ) ker(m(0)) = (0, 0, 1) α (0, 0, 1) x 0 M(β) = β(β 2 (a 1 + b 2 )β a 2 b 1 + a 1 b 2 ) K. Ghorbal (CMU) Invited Talk 23 Differential Radical Invariants 23 / 29
44 Checking & Generation Example System x 1 = x 2 x 3 = x4 2 x 2 = x 1 x 4 = x 3 x 4 Differential Radical Invariants h = 1 + x 1 x 4 and L p (h) = x 3 x 2 x 4 and L (2) p (h) = x 2 4 x2 3 1 Orbit Roots of L p (h) Roots of L (2) p (h) K. Ghorbal (CMU) Invited Talk 24 Differential Radical Invariants 24 / 29
45 Checking & Generation Example: cont d Overapproximation of O(x 0 ) K. Ghorbal (CMU) Invited Talk 25 Differential Radical Invariants 25 / 29
46 Checking & Generation Case Study: Longitudinal Dynamics of an Airplane 6th Order Longitudinal Equations u = X g sin(θ) qw m ẇ = Z + g cos(θ) + qu m ẋ = cos(θ)u + sin(θ)w ż = sin(θ)u + cos(θ)w q = M I yy θ = q u : axial velocity w : vertical velocity x : range z : altitude q : pitch rate θ : pitch angle K. Ghorbal (CMU) Invited Talk 26 Differential Radical Invariants 26 / 29
47 Checking & Generation Case Study: Generated Invariants Automatically Generated Invariant Functions ( ) Mz X + gθ + I yy m qw ( ) Mx Z I yy m + qu cos(θ) + q 2 + 2Mθ I yy ( Z cos(θ) + ( X m qw ) m + qu sin(θ) ) sin(θ) K. Ghorbal (CMU) Invited Talk 27 Differential Radical Invariants 27 / 29
48 Conclusion Conclusion Variety embedding of the reachable set (Zariski Closure) Characterizing elements of the vanishing ideal I (O(x 0 )) New Necessary and sufficient proof rule (DRI) Leveraging linear algebra tools to generate algebraic invariant equations K. Ghorbal (CMU) Invited Talk 28 Differential Radical Invariants 28 / 29
49 Conclusion Thank you for attending! K. Ghorbal (CMU) Invited Talk 29 Differential Radical Invariants 29 / 29
50 Conclusion Enforcing Invariants δ def = (a 1 b 2 ) 2 + 4a 2 b 1 0 β {0, 1 2 (a 1 + b 2 + δ), 1 2 (a 1 + b 2 δ)} ( If x 0 a 1 b 2 ± ) ( δ, 2a 2, 0 then α = a 1 b 2 ± ) δ, 2a 2, 0 α is an Eigenvector If a 2 = 0, β {0, a 1, b 2 }... K. Ghorbal (CMU) Invited Talk 30 Differential Radical Invariants 30 / 29
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