slides - LIX
Characterizing Algebraic Invariants by Differential Radical Invariants Khalil Ghorbal Carnegie Mellon university Joint work with Andr´ e Platzer
CEA List April 3rd, 2014
K. Ghorbal (CMU)
Invited Talk
1
Differential Radical Invariants
1 / 30
Introduction
Context: Hybrid Systems Model
Sensing:
read data from sensors
K. Ghorbal (CMU)
Invited Talk
2
Differential Radical Invariants
2 / 30
Introduction
Context: Hybrid Systems Model
Sensing: Control:
read data from sensors actuate
K. Ghorbal (CMU)
Invited Talk
2
Differential Radical Invariants
2 / 30
Introduction
Context: Hybrid Systems Model
Sensing: read data from sensors Control: actuate Plant: evolve
K. Ghorbal (CMU)
Invited Talk
2
Differential Radical Invariants
2 / 30
Introduction
Context: Hybrid Systems Model
( Sensing: read data from sensors Control: actuate Plant: evolve )∗
K. Ghorbal (CMU)
Invited Talk
2
Differential Radical Invariants
2 / 30
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 / 30
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 / 30
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 / 30
Introduction
Algebraic Differential Equations
Example x˙1 = −x2 x˙3 =
x˙2 = x1
x42
x˙4 = x3 x4
Formally dxi (t) = x˙ i = pi (x), 1 ≤ i ≤ n . dt
K. Ghorbal (CMU)
Invited Talk
3
Differential Radical Invariants
3 / 30
Introduction
Algebraic Invariant Expression x˙1 = −x2 x˙3 =
x˙2 = x1
x42
x˙4 = x3 x4
1.0
x2
0.5
0.0
-0.5
-1.0 -1.0
-0.5
0.0 x1
0.5
1.0
Solution for x0 = (1, 0, 0, 1) for t = [0, 1] K. Ghorbal (CMU)
Invited Talk
4
Differential Radical Invariants
4 / 30
Introduction
Algebraic Invariant Expression x˙1 = −x2 x˙3 =
x˙2 = x1
x42
x˙4 = x3 x4
1.0
x2
0.5
0.0
-0.5
-1.0 -1.0
-0.5
0.0 x1
0.5
1.0
Solution for x0 = (1, 0, 0, 1) for t = [0, 2] K. Ghorbal (CMU)
Invited Talk
4
Differential Radical Invariants
4 / 30
Introduction
Algebraic Invariant Expression x˙1 = −x2
x˙2 = x1
x42
x˙3 =
x˙4 = x3 x4
1.0
x2
0.5
0.0
-0.5
-1.0 -1.0
-0.5
0.0 x1
0.5
1.0
∀t, x1 (t)2 + x2 (t)2 − 1 = 0 K. Ghorbal (CMU)
Invited Talk
4
Differential Radical Invariants
4 / 30
Introduction
Algebraic Invariant Expression
Geometrically def
O(x0 ) = {x(t) | t ≥ 0} ∈ Set of roots of h(x1 , x2 )
K. Ghorbal (CMU)
Invited Talk
5
Differential Radical Invariants
5 / 30
Introduction
Algebraic Invariant Expression
Geometrically def
O(x0 ) = {x(t) | t ≥ 0} ∈ Set of roots of h(x1 , x2 ) Algebraically h(x1 , x2 ) = x12 + x22 − 1(= 0)
K. Ghorbal (CMU)
Invited Talk
5
Differential Radical Invariants
5 / 30
Introduction
Algebraic Invariant Expression
Geometrically def
O(x0 ) = {x(t) | t ≥ 0} ∈ Set of roots of h(x1 , x2 ) Algebraically h(x1 , x2 ) = x12 + x22 − 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 / 30
Introduction
Problems
I. Checking the invariance of Algebraic Expression Given p, and x0 root of h(x) (h(x0 ) = 0), is h(x) = 0 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 / 30
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 / 30
Time Abstraction
Variety Embedding of Orbits Zariski Closure
Vanishing Ideal: all polynomials that vanish on O(x0 ) def
I (O(x0 )) = {h ∈ R[x] | ∀x ∈ O(x0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I def
V (I ) = {x ∈ Rn | ∀h ∈ I , h(x) = 0} Closure: Sound Abstraction O(x0 ) ⊆ V (I (O(x0 )))
K. Ghorbal (CMU)
Invited Talk
8
Differential Radical Invariants
8 / 30
Time Abstraction
Variety Embedding of Orbits Zariski Closure
Vanishing Ideal: all polynomials that vanish on O(x0 ) def
I (O(x0 )) = {h ∈ R[x] | ∀x ∈ O(x0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I def
V (I ) = {x ∈ Rn | ∀h ∈ I , h(x) = 0} Closure: Sound Abstraction O(x0 ) ⊆ V (I (O(x0 )))
K. Ghorbal (CMU)
Invited Talk
8
Differential Radical Invariants
8 / 30
Time Abstraction
Variety Embedding of Orbits Zariski Closure
Vanishing Ideal: all polynomials that vanish on O(x0 ) def
I (O(x0 )) = {h ∈ R[x] | ∀x ∈ O(x0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I def
V (I ) = {x ∈ Rn | ∀h ∈ I , h(x) = 0} Closure: Sound Abstraction O(x0 ) ⊆ V (I (O(x0 )))
K. Ghorbal (CMU)
Invited Talk
8
Differential Radical Invariants
8 / 30
Time Abstraction
Variety Embedding of Orbits Zariski Closure
Vanishing Ideal: all polynomials that vanish on O(x0 ) def
I (O(x0 )) = {h ∈ R[x] | ∀x ∈ O(x0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I def
V (I ) = {x ∈ Rn | ∀h ∈ I , h(x) = 0} Closure: Sound Abstraction O(x0 ) ⊆ V (I (O(x0 ))) V (I (O(x0 ))) is the smallest variety that contains O(x0 ) K. Ghorbal (CMU)
Invited Talk
8
Differential Radical Invariants
8 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
K. Ghorbal (CMU)
γ
−→
Invited Talk
V (I (O(x0 ))) Closure
9
⊇
O(x0 ) Orbit
Differential Radical Invariants
9 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
K. Ghorbal (CMU)
γ
−→
Invited Talk
V (I (O(x0 ))) Closure
9
⊇
O(x0 ) Orbit
Differential Radical Invariants
9 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
K. Ghorbal (CMU)
γ
−→
Invited Talk
V (I (O(x0 ))) Closure
9
⊇
O(x0 ) Orbit
Differential Radical Invariants
9 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
K. Ghorbal (CMU)
γ
−→
Invited Talk
V (I (O(x0 ))) Closure
9
⊇
O(x0 ) Orbit
Differential Radical Invariants
9 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
γ
−→
Limitation: Zariski Dense Varieties x˙ = x O(x0 ) = [x0 , ∞[ I = h0i
K. Ghorbal (CMU)
Invited Talk
V (I (O(x0 ))) Closure
⊇
O(x0 ) Orbit
V (I (O(x0 ))) = R
9
Differential Radical Invariants
9 / 30
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 / 30
Characterization of Invariant Expressions
Properties of I (O(x0 )) (abstract domain)
I (O(x0 )): The set of all polynomials that vanish on O(x0 ) I (O(x0 )) is an ideal • 0 ∈ I (O(x0 )) • if h1 , h2 ∈ I (O(x0 )), then h1 + h2 ∈ I (O(x0 )) • if h ∈ I (O(x0 )), then q h ∈ I (O(x0 )), for any polynomial q
K. Ghorbal (CMU)
Invited Talk
11
Differential Radical Invariants
11 / 30
Characterization of Invariant Expressions
Properties of I (O(x0 )) (abstract domain)
I (O(x0 )): The set of all polynomials that vanish on O(x0 ) I (O(x0 )) is an ideal • 0 ∈ I (O(x0 )) • if h1 , h2 ∈ I (O(x0 )), then h1 + h2 ∈ I (O(x0 )) • if h ∈ I (O(x0 )), then q h ∈ I (O(x0 )), for any polynomial q
I (O(x0 )) is a Differential Ideal if h ∈ I (O(x0 )), then
K. Ghorbal (CMU)
dh dt
∈ I (O(x0 )).
Invited Talk
11
Differential Radical Invariants
11 / 30
Characterization of Invariant Expressions
Properties of I (O(x0 )) (abstract domain) I (O(x0 )): The set of all polynomials that vanish on O(x0 ) I (O(x0 )) is an ideal • 0 ∈ I (O(x0 )) • if h1 , h2 ∈ I (O(x0 )), then h1 + h2 ∈ I (O(x0 )) • if h ∈ I (O(x0 )), then q h ∈ I (O(x0 )), for any polynomial q
I (O(x0 )) is a Differential Ideal if h ∈ I (O(x0 )), then Instead of
dh dt ,
dh dt
∈ I (O(x0 )).
we will use Lp (h): The Lie Derivative of h.
K. Ghorbal (CMU)
Invited Talk
11
Differential Radical Invariants
11 / 30
Characterization of Invariant Expressions
Lie derivative along a vector field
Definition n X ∂h Lp (h) = pi (x) ∈ R[x] ∂xi def
i=1
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 / 30
Characterization of Invariant Expressions
Differential Radical Invariants
Theorem h ∈ I (O(x0 )) if and only if (N)
• ∃ g0 , . . . , gN−1 ∈ R[x]: Lp (h) =
PN−1 i=0
(i)
gi Lp (h)
• N finite (0)
(N−1)
• Lp (h)(x0 ) = 0, . . . , Lp
K. Ghorbal (CMU)
(h)(x0 ) = 0
Invited Talk
13
Differential Radical Invariants
13 / 30
Characterization of Invariant Expressions
Special Cases
Invariant Polynomial Functions Lp (h) = 0 ←→ N = 1 and g0 = 0
Darboux Polynomials [Darboux 1878, Painlev´e, Poincar´e . . . ] Lp (h) = g0 h ←→ N = 1
K. Ghorbal (CMU)
Invited Talk
14
Differential Radical Invariants
14 / 30
Characterization of Invariant Expressions
Special Cases
Invariant Polynomial Functions Lp (h) = 0 ←→ N = 1 and g0 = 0
Darboux Polynomials [Darboux 1878, Painlev´e, Poincar´e . . . ] Lp (h) = g0 h ←→ N = 1
K. Ghorbal (CMU)
Invited Talk
14
Differential Radical Invariants
14 / 30
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 / 30
Checking & Generation
Checking Invariance of Candidates
I. Checking the invariance of Algebraic Expression Given p, and x0 root of h(x) (h(x0 ) = 0), is h(x) = 0 is an algebraic invariant expression ? (∀t ≥ 0, h(x(t)) = 0)
A Necessary and Sufficient Proof Rule V (i) h = 0 → N−1 i=0 Lp (h) = 0 (DRI) . (h = 0) → [x˙ = p](h = 0)
K. Ghorbal (CMU)
Invited Talk
16
Differential Radical Invariants
16 / 30
Checking & Generation
DRI: Algorithm Data: h, p, x Result: Boolean: True, False. N ←1 GB ← {h} `←h symbs ← Variables[p, h] while true do GB ← Gr¨obnerBasis[GB, x] LieD ← LieDerivative[`, 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] ` ← LieD N ← N +1
K. Ghorbal (CMU)
Invited Talk
17
Differential Radical Invariants
17 / 30
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)
10
20 30 40 50 Number of problems solved
Invited Talk
18
60
70
Differential Radical Invariants
18 / 30
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 ⊆ Rn is an invariant variety if and only if S is the set of roots of the system ^ (i) Lp (h) = 0, 0≤i≤N−1
for some polynomial h.
K. Ghorbal (CMU)
Invited Talk
19
Differential Radical Invariants
19 / 30
Checking & Generation
In practice
In practice Find h and N, such that: (N)
Lp (h) =
N−1 X
(i)
gi Lp (h)
i=0
: The set of roots of
^
(i)
Lp (h) = 0,
0≤i≤N−1
is an invariant variety.
K. Ghorbal (CMU)
Invited Talk
20
Differential Radical Invariants
20 / 30
Checking & Generation
Matrix Representation: Intuition invariant of degree 1 x˙1 = a1 x1 + a2 x2 x˙2 = b1 x1 + b2 x2
h = α1 x1 + α2 x2 + α3 x0 Lp (h) = α1 (a1 x1 + a2 x2 ) + α2 (b1 x1 + b2 x2 )
∃ ?β ∈ R s.t. Lp (h) = βh
(−a1 + β)α1 + (−b1 )α2 = 0 −a1 + β −b1 0 α1 (−a2 )α1 + (−b2 + β)α2 = 0 ↔ −a2 −b2 + β 0 α2 = 0 (β)α3 =0 0 0 β α3 K. Ghorbal (CMU)
Invited Talk
21
Differential Radical Invariants
21 / 30
Checking & Generation
Matrix Representation Polynomial h gi
↔ ↔ ↔
n+d d
Coefficients (d degree of h) α = (α1 , α2 , . . . , αr ) β i = (β1 , β2 , . . . , βsi )
Matrix Representation (N)
Lp (h) =
N−1 X
(i)
gi Lp (h) ↔ M(β)α = 0
i=0
def
α lies in the Kernel of M(β) = {α ∈ Rr | M(β)α = 0}
K. Ghorbal (CMU)
Invited Talk
22
Differential Radical Invariants
22 / 30
Checking & Generation
Example: n = 2, d = 1, N = 1
invariant of degree 1 x˙1 = a1 x1 + a2 x2 x˙2 = b1 x1 + b2 x2
h = α1 x1 + α2 x2 + α3 x0 Lp (h) = α1 (a1 x1 + a2 x2 ) + α2 (b1 x1 + b2 x2 )
|M(β)| = β(β 2 − (a1 + b2 )β − a2 b1 + a1 b2 ) • ker(M(0)) = h(0, 0, 1)i • α ∈ h(0, 0, 1)i ∩ x0 ⊥
K. Ghorbal (CMU)
Invited Talk
23
Differential Radical Invariants
23 / 30
Checking & Generation
Example System
x˙1 = −x2
x˙2 = x1
x˙3 = x42
x˙4 = x3 x4
Differential Radical Invariants (2)
h = −1 + x1 x4 and Lp (h) = x3 − x2 x4 and Lp (h) = x42 − x32 − 1
Orbit K. Ghorbal (CMU)
Roots of Lp (h) Invited Talk
24
(2)
Roots of Lp (h) Differential Radical Invariants
24 / 30
Checking & Generation
Example: cont’d
Overapproximation of O(x0 ) K. Ghorbal (CMU)
Invited Talk
25
Differential Radical Invariants
25 / 30
Checking & Generation
Case Study: Longitudinal Dynamics of an Airplane 6th Order Longitudinal Equations X − g sin(θ) − qw m Z w˙ = + g cos(θ) + qu m x˙ = cos(θ)u + sin(θ)w u˙ =
u : axial velocity w : vertical velocity x : range
z˙ = − sin(θ)u + cos(θ)w M q˙ = Iyy θ˙ = q
K. Ghorbal (CMU)
Invited Talk
z : altitude q : pitch rate θ : pitch angle
26
Differential Radical Invariants
26 / 30
Checking & Generation
Case Study: Generated Invariants
Automatically Generated Invariant Functions X Z Mz + gθ + − qw cos(θ) + + qu sin(θ) Iyy m m Mx Z X − + qu cos(θ) + − qw sin(θ) Iyy m m 2Mθ − q2 + Iyy
K. Ghorbal (CMU)
Invited Talk
27
Differential Radical Invariants
27 / 30
Conclusion
Conclusion
• Variety embedding of the reachable set (Zariski Closure) • Characterizing elements of the vanishing ideal I (O(x0 )) • 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 / 30
Conclusion
Thank you for attending !
[email protected]
K. Ghorbal (CMU)
Invited Talk
29
Differential Radical Invariants
29 / 30
Conclusion
Enforcing Invariants
def
δ = (a1 − b2 )2 + 4a2 b1 ≥ 0 √ √ • β ∈ {0, 12 (a1 + b2 + δ), 12 (a1 + b2 − δ)} ⊥ √ √ • If x0 ∈ a1 − b2 ± δ, 2a2 , 0 then α = a1 − b2 ± δ, 2a2 , 0 • α is an Eigenvector
If a2 = 0, β ∈ {0, a1 , b2 }
K. Ghorbal (CMU)
...
Invited Talk
30
Differential Radical Invariants
30 / 30
CEA List April 3rd, 2014
K. Ghorbal (CMU)
Invited Talk
1
Differential Radical Invariants
1 / 30
Introduction
Context: Hybrid Systems Model
Sensing:
read data from sensors
K. Ghorbal (CMU)
Invited Talk
2
Differential Radical Invariants
2 / 30
Introduction
Context: Hybrid Systems Model
Sensing: Control:
read data from sensors actuate
K. Ghorbal (CMU)
Invited Talk
2
Differential Radical Invariants
2 / 30
Introduction
Context: Hybrid Systems Model
Sensing: read data from sensors Control: actuate Plant: evolve
K. Ghorbal (CMU)
Invited Talk
2
Differential Radical Invariants
2 / 30
Introduction
Context: Hybrid Systems Model
( Sensing: read data from sensors Control: actuate Plant: evolve )∗
K. Ghorbal (CMU)
Invited Talk
2
Differential Radical Invariants
2 / 30
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 / 30
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 / 30
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 / 30
Introduction
Algebraic Differential Equations
Example x˙1 = −x2 x˙3 =
x˙2 = x1
x42
x˙4 = x3 x4
Formally dxi (t) = x˙ i = pi (x), 1 ≤ i ≤ n . dt
K. Ghorbal (CMU)
Invited Talk
3
Differential Radical Invariants
3 / 30
Introduction
Algebraic Invariant Expression x˙1 = −x2 x˙3 =
x˙2 = x1
x42
x˙4 = x3 x4
1.0
x2
0.5
0.0
-0.5
-1.0 -1.0
-0.5
0.0 x1
0.5
1.0
Solution for x0 = (1, 0, 0, 1) for t = [0, 1] K. Ghorbal (CMU)
Invited Talk
4
Differential Radical Invariants
4 / 30
Introduction
Algebraic Invariant Expression x˙1 = −x2 x˙3 =
x˙2 = x1
x42
x˙4 = x3 x4
1.0
x2
0.5
0.0
-0.5
-1.0 -1.0
-0.5
0.0 x1
0.5
1.0
Solution for x0 = (1, 0, 0, 1) for t = [0, 2] K. Ghorbal (CMU)
Invited Talk
4
Differential Radical Invariants
4 / 30
Introduction
Algebraic Invariant Expression x˙1 = −x2
x˙2 = x1
x42
x˙3 =
x˙4 = x3 x4
1.0
x2
0.5
0.0
-0.5
-1.0 -1.0
-0.5
0.0 x1
0.5
1.0
∀t, x1 (t)2 + x2 (t)2 − 1 = 0 K. Ghorbal (CMU)
Invited Talk
4
Differential Radical Invariants
4 / 30
Introduction
Algebraic Invariant Expression
Geometrically def
O(x0 ) = {x(t) | t ≥ 0} ∈ Set of roots of h(x1 , x2 )
K. Ghorbal (CMU)
Invited Talk
5
Differential Radical Invariants
5 / 30
Introduction
Algebraic Invariant Expression
Geometrically def
O(x0 ) = {x(t) | t ≥ 0} ∈ Set of roots of h(x1 , x2 ) Algebraically h(x1 , x2 ) = x12 + x22 − 1(= 0)
K. Ghorbal (CMU)
Invited Talk
5
Differential Radical Invariants
5 / 30
Introduction
Algebraic Invariant Expression
Geometrically def
O(x0 ) = {x(t) | t ≥ 0} ∈ Set of roots of h(x1 , x2 ) Algebraically h(x1 , x2 ) = x12 + x22 − 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 / 30
Introduction
Problems
I. Checking the invariance of Algebraic Expression Given p, and x0 root of h(x) (h(x0 ) = 0), is h(x) = 0 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 / 30
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 / 30
Time Abstraction
Variety Embedding of Orbits Zariski Closure
Vanishing Ideal: all polynomials that vanish on O(x0 ) def
I (O(x0 )) = {h ∈ R[x] | ∀x ∈ O(x0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I def
V (I ) = {x ∈ Rn | ∀h ∈ I , h(x) = 0} Closure: Sound Abstraction O(x0 ) ⊆ V (I (O(x0 )))
K. Ghorbal (CMU)
Invited Talk
8
Differential Radical Invariants
8 / 30
Time Abstraction
Variety Embedding of Orbits Zariski Closure
Vanishing Ideal: all polynomials that vanish on O(x0 ) def
I (O(x0 )) = {h ∈ R[x] | ∀x ∈ O(x0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I def
V (I ) = {x ∈ Rn | ∀h ∈ I , h(x) = 0} Closure: Sound Abstraction O(x0 ) ⊆ V (I (O(x0 )))
K. Ghorbal (CMU)
Invited Talk
8
Differential Radical Invariants
8 / 30
Time Abstraction
Variety Embedding of Orbits Zariski Closure
Vanishing Ideal: all polynomials that vanish on O(x0 ) def
I (O(x0 )) = {h ∈ R[x] | ∀x ∈ O(x0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I def
V (I ) = {x ∈ Rn | ∀h ∈ I , h(x) = 0} Closure: Sound Abstraction O(x0 ) ⊆ V (I (O(x0 )))
K. Ghorbal (CMU)
Invited Talk
8
Differential Radical Invariants
8 / 30
Time Abstraction
Variety Embedding of Orbits Zariski Closure
Vanishing Ideal: all polynomials that vanish on O(x0 ) def
I (O(x0 )) = {h ∈ R[x] | ∀x ∈ O(x0 ), h(x) = 0} Affine (Real) Variety: common roots of all polynomials in I def
V (I ) = {x ∈ Rn | ∀h ∈ I , h(x) = 0} Closure: Sound Abstraction O(x0 ) ⊆ V (I (O(x0 ))) V (I (O(x0 ))) is the smallest variety that contains O(x0 ) K. Ghorbal (CMU)
Invited Talk
8
Differential Radical Invariants
8 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
K. Ghorbal (CMU)
γ
−→
Invited Talk
V (I (O(x0 ))) Closure
9
⊇
O(x0 ) Orbit
Differential Radical Invariants
9 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
K. Ghorbal (CMU)
γ
−→
Invited Talk
V (I (O(x0 ))) Closure
9
⊇
O(x0 ) Orbit
Differential Radical Invariants
9 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
K. Ghorbal (CMU)
γ
−→
Invited Talk
V (I (O(x0 ))) Closure
9
⊇
O(x0 ) Orbit
Differential Radical Invariants
9 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
K. Ghorbal (CMU)
γ
−→
Invited Talk
V (I (O(x0 ))) Closure
9
⊇
O(x0 ) Orbit
Differential Radical Invariants
9 / 30
Time Abstraction
Variety Embedding
O(x0 ) Orbit
α
−→
I (O(x0 )) Vanishing Ideal
γ
−→
Limitation: Zariski Dense Varieties x˙ = x O(x0 ) = [x0 , ∞[ I = h0i
K. Ghorbal (CMU)
Invited Talk
V (I (O(x0 ))) Closure
⊇
O(x0 ) Orbit
V (I (O(x0 ))) = R
9
Differential Radical Invariants
9 / 30
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 / 30
Characterization of Invariant Expressions
Properties of I (O(x0 )) (abstract domain)
I (O(x0 )): The set of all polynomials that vanish on O(x0 ) I (O(x0 )) is an ideal • 0 ∈ I (O(x0 )) • if h1 , h2 ∈ I (O(x0 )), then h1 + h2 ∈ I (O(x0 )) • if h ∈ I (O(x0 )), then q h ∈ I (O(x0 )), for any polynomial q
K. Ghorbal (CMU)
Invited Talk
11
Differential Radical Invariants
11 / 30
Characterization of Invariant Expressions
Properties of I (O(x0 )) (abstract domain)
I (O(x0 )): The set of all polynomials that vanish on O(x0 ) I (O(x0 )) is an ideal • 0 ∈ I (O(x0 )) • if h1 , h2 ∈ I (O(x0 )), then h1 + h2 ∈ I (O(x0 )) • if h ∈ I (O(x0 )), then q h ∈ I (O(x0 )), for any polynomial q
I (O(x0 )) is a Differential Ideal if h ∈ I (O(x0 )), then
K. Ghorbal (CMU)
dh dt
∈ I (O(x0 )).
Invited Talk
11
Differential Radical Invariants
11 / 30
Characterization of Invariant Expressions
Properties of I (O(x0 )) (abstract domain) I (O(x0 )): The set of all polynomials that vanish on O(x0 ) I (O(x0 )) is an ideal • 0 ∈ I (O(x0 )) • if h1 , h2 ∈ I (O(x0 )), then h1 + h2 ∈ I (O(x0 )) • if h ∈ I (O(x0 )), then q h ∈ I (O(x0 )), for any polynomial q
I (O(x0 )) is a Differential Ideal if h ∈ I (O(x0 )), then Instead of
dh dt ,
dh dt
∈ I (O(x0 )).
we will use Lp (h): The Lie Derivative of h.
K. Ghorbal (CMU)
Invited Talk
11
Differential Radical Invariants
11 / 30
Characterization of Invariant Expressions
Lie derivative along a vector field
Definition n X ∂h Lp (h) = pi (x) ∈ R[x] ∂xi def
i=1
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 / 30
Characterization of Invariant Expressions
Differential Radical Invariants
Theorem h ∈ I (O(x0 )) if and only if (N)
• ∃ g0 , . . . , gN−1 ∈ R[x]: Lp (h) =
PN−1 i=0
(i)
gi Lp (h)
• N finite (0)
(N−1)
• Lp (h)(x0 ) = 0, . . . , Lp
K. Ghorbal (CMU)
(h)(x0 ) = 0
Invited Talk
13
Differential Radical Invariants
13 / 30
Characterization of Invariant Expressions
Special Cases
Invariant Polynomial Functions Lp (h) = 0 ←→ N = 1 and g0 = 0
Darboux Polynomials [Darboux 1878, Painlev´e, Poincar´e . . . ] Lp (h) = g0 h ←→ N = 1
K. Ghorbal (CMU)
Invited Talk
14
Differential Radical Invariants
14 / 30
Characterization of Invariant Expressions
Special Cases
Invariant Polynomial Functions Lp (h) = 0 ←→ N = 1 and g0 = 0
Darboux Polynomials [Darboux 1878, Painlev´e, Poincar´e . . . ] Lp (h) = g0 h ←→ N = 1
K. Ghorbal (CMU)
Invited Talk
14
Differential Radical Invariants
14 / 30
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 / 30
Checking & Generation
Checking Invariance of Candidates
I. Checking the invariance of Algebraic Expression Given p, and x0 root of h(x) (h(x0 ) = 0), is h(x) = 0 is an algebraic invariant expression ? (∀t ≥ 0, h(x(t)) = 0)
A Necessary and Sufficient Proof Rule V (i) h = 0 → N−1 i=0 Lp (h) = 0 (DRI) . (h = 0) → [x˙ = p](h = 0)
K. Ghorbal (CMU)
Invited Talk
16
Differential Radical Invariants
16 / 30
Checking & Generation
DRI: Algorithm Data: h, p, x Result: Boolean: True, False. N ←1 GB ← {h} `←h symbs ← Variables[p, h] while true do GB ← Gr¨obnerBasis[GB, x] LieD ← LieDerivative[`, 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] ` ← LieD N ← N +1
K. Ghorbal (CMU)
Invited Talk
17
Differential Radical Invariants
17 / 30
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)
10
20 30 40 50 Number of problems solved
Invited Talk
18
60
70
Differential Radical Invariants
18 / 30
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 ⊆ Rn is an invariant variety if and only if S is the set of roots of the system ^ (i) Lp (h) = 0, 0≤i≤N−1
for some polynomial h.
K. Ghorbal (CMU)
Invited Talk
19
Differential Radical Invariants
19 / 30
Checking & Generation
In practice
In practice Find h and N, such that: (N)
Lp (h) =
N−1 X
(i)
gi Lp (h)
i=0
: The set of roots of
^
(i)
Lp (h) = 0,
0≤i≤N−1
is an invariant variety.
K. Ghorbal (CMU)
Invited Talk
20
Differential Radical Invariants
20 / 30
Checking & Generation
Matrix Representation: Intuition invariant of degree 1 x˙1 = a1 x1 + a2 x2 x˙2 = b1 x1 + b2 x2
h = α1 x1 + α2 x2 + α3 x0 Lp (h) = α1 (a1 x1 + a2 x2 ) + α2 (b1 x1 + b2 x2 )
∃ ?β ∈ R s.t. Lp (h) = βh
(−a1 + β)α1 + (−b1 )α2 = 0 −a1 + β −b1 0 α1 (−a2 )α1 + (−b2 + β)α2 = 0 ↔ −a2 −b2 + β 0 α2 = 0 (β)α3 =0 0 0 β α3 K. Ghorbal (CMU)
Invited Talk
21
Differential Radical Invariants
21 / 30
Checking & Generation
Matrix Representation Polynomial h gi
↔ ↔ ↔
n+d d
Coefficients (d degree of h) α = (α1 , α2 , . . . , αr ) β i = (β1 , β2 , . . . , βsi )
Matrix Representation (N)
Lp (h) =
N−1 X
(i)
gi Lp (h) ↔ M(β)α = 0
i=0
def
α lies in the Kernel of M(β) = {α ∈ Rr | M(β)α = 0}
K. Ghorbal (CMU)
Invited Talk
22
Differential Radical Invariants
22 / 30
Checking & Generation
Example: n = 2, d = 1, N = 1
invariant of degree 1 x˙1 = a1 x1 + a2 x2 x˙2 = b1 x1 + b2 x2
h = α1 x1 + α2 x2 + α3 x0 Lp (h) = α1 (a1 x1 + a2 x2 ) + α2 (b1 x1 + b2 x2 )
|M(β)| = β(β 2 − (a1 + b2 )β − a2 b1 + a1 b2 ) • ker(M(0)) = h(0, 0, 1)i • α ∈ h(0, 0, 1)i ∩ x0 ⊥
K. Ghorbal (CMU)
Invited Talk
23
Differential Radical Invariants
23 / 30
Checking & Generation
Example System
x˙1 = −x2
x˙2 = x1
x˙3 = x42
x˙4 = x3 x4
Differential Radical Invariants (2)
h = −1 + x1 x4 and Lp (h) = x3 − x2 x4 and Lp (h) = x42 − x32 − 1
Orbit K. Ghorbal (CMU)
Roots of Lp (h) Invited Talk
24
(2)
Roots of Lp (h) Differential Radical Invariants
24 / 30
Checking & Generation
Example: cont’d
Overapproximation of O(x0 ) K. Ghorbal (CMU)
Invited Talk
25
Differential Radical Invariants
25 / 30
Checking & Generation
Case Study: Longitudinal Dynamics of an Airplane 6th Order Longitudinal Equations X − g sin(θ) − qw m Z w˙ = + g cos(θ) + qu m x˙ = cos(θ)u + sin(θ)w u˙ =
u : axial velocity w : vertical velocity x : range
z˙ = − sin(θ)u + cos(θ)w M q˙ = Iyy θ˙ = q
K. Ghorbal (CMU)
Invited Talk
z : altitude q : pitch rate θ : pitch angle
26
Differential Radical Invariants
26 / 30
Checking & Generation
Case Study: Generated Invariants
Automatically Generated Invariant Functions X Z Mz + gθ + − qw cos(θ) + + qu sin(θ) Iyy m m Mx Z X − + qu cos(θ) + − qw sin(θ) Iyy m m 2Mθ − q2 + Iyy
K. Ghorbal (CMU)
Invited Talk
27
Differential Radical Invariants
27 / 30
Conclusion
Conclusion
• Variety embedding of the reachable set (Zariski Closure) • Characterizing elements of the vanishing ideal I (O(x0 )) • 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 / 30
Conclusion
Thank you for attending !
[email protected]
K. Ghorbal (CMU)
Invited Talk
29
Differential Radical Invariants
29 / 30
Conclusion
Enforcing Invariants
def
δ = (a1 − b2 )2 + 4a2 b1 ≥ 0 √ √ • β ∈ {0, 12 (a1 + b2 + δ), 12 (a1 + b2 − δ)} ⊥ √ √ • If x0 ∈ a1 − b2 ± δ, 2a2 , 0 then α = a1 − b2 ± δ, 2a2 , 0 • α is an Eigenvector
If a2 = 0, β ∈ {0, a1 , b2 }
K. Ghorbal (CMU)
...
Invited Talk
30
Differential Radical Invariants
30 / 30
