Parsimonious, Simulation Based Verification Of Linear...
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Parsimonious, Simulation Based Verification Of Linear Systems
Parasara Sridhar Duggirala Mahesh Viswanathan
Safety Verification: Motivation
Adaptive cruise control
Program controller such that v → vf while having 𝑠 > 𝑙𝑖𝑚𝑖𝑡.
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𝒔
𝐯𝐟𝐯
Safety Verification: Motivation
Adaptive cruise control
Program controller such that v → vf while having 𝑠 > 𝑙𝑖𝑚𝑖𝑡.
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𝒔
𝐯𝐟𝐯
ODE model.ሶs = vf − v;ሶv = −kav + u;
Controller designu = c1v + c2s + c3
Safety Verification: Motivation
Adaptive cruise control
Program controller such that v → vf while having 𝑠 > 𝑙𝑖𝑚𝑖𝑡.
CAV 2016 4
𝒔
𝐯𝐟𝐯
ODE model.ሶs = vf − v;ሶv = −kav + u;
Controller designu = c1v + c2s + c3
Closed loop systemሶsv
=a11 a12a21 a22
sv
+b1b2
represented asሶ𝑥 = 𝐴𝑥 + 𝐵( )
Safety Verification: Motivation
Adaptive cruise control
Program controller such that v → vf while having 𝑠 > 𝑙𝑖𝑚𝑖𝑡.
CAV 2016 5
𝒔
𝐯𝐟𝐯
ODE model.ሶs = vf − v;ሶv = −kav + u;
Controller designu = c1v + c2s + c3
Closed loop systemሶsv
=a11 a12a21 a22
sv
+b1b2
Safety verification problem for linear systems ሶ𝒙 = 𝑨𝒙 + 𝑩From initial set Θ (dis)prove that no trajectory enters the unsafe set U
represented asሶ𝑥 = 𝐴𝑥 + 𝐵( )
Solution: Reachable Set
System: ሶ𝑥 = 𝐴𝑥 + 𝐵, initial set Θ (polyhedra), unsafe set 𝑈.
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𝚯
. 𝜉 𝑥0, 𝑡 = 𝑒𝐴𝑡𝑥0 +න0
𝑡
𝑒𝐴(𝑡−𝜏)𝐵𝑑𝜏
𝑥0
Solution: Reachable Set
System: ሶ𝑥 = 𝐴𝑥 + 𝐵, initial set Θ (polyhedra), unsafe set 𝑈.
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𝚯
. 𝜉 𝑥0, 𝑡 = 𝑒𝐴𝑡𝑥0 +න0
𝑡
𝑒𝐴(𝑡−𝜏)𝐵𝑑𝜏
𝑥0
Procedure to compute reachable set1. Represent the set Θ using data structure
Data structureSpaceEx – Support FunctionsCORA – ZonotopesFlow* – Taylor Models
Solution: Reachable Set
System: ሶ𝑥 = 𝐴𝑥 + 𝐵, initial set Θ (polyhedra), unsafe set 𝑈.
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𝚯
. 𝜉 𝑥0, 𝑡 = 𝑒𝐴𝑡𝑥0 +න0
𝑡
𝑒𝐴(𝑡−𝜏)𝐵𝑑𝜏
𝑥0
Procedure to compute reachable set1. Represent the set Θ using data structure2. Select a time interval ℎ.3. Compute 𝑃𝑜𝑠𝑡(Θ, ℎ) for [0, ℎ]
Data structureSpaceEx – Support FunctionsCORA – ZonotopesFlow* – Taylor Models
Solution: Reachable Set
System: ሶ𝑥 = 𝐴𝑥 + 𝐵, initial set Θ (polyhedra), unsafe set 𝑈.
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𝚯
. 𝜉 𝑥0, 𝑡 = 𝑒𝐴𝑡𝑥0 +න0
𝑡
𝑒𝐴(𝑡−𝜏)𝐵𝑑𝜏
𝑥0
Procedure to compute reachable set1. Represent the set Θ using data structure2. Select a time interval ℎ.3. Compute 𝑃𝑜𝑠𝑡(Θ, ℎ) for [0, ℎ]4. Iterate for future intervals.
Data structureSpaceEx – Support FunctionsCORA – ZonotopesFlow* – Taylor Models
Solution: Reachable Set
System: ሶ𝑥 = 𝐴𝑥 + 𝐵, initial set Θ (polyhedra), unsafe set 𝑈.
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𝚯
. 𝜉 𝑥0, 𝑡 = 𝑒𝐴𝑡𝑥0 +න0
𝑡
𝑒𝐴(𝑡−𝜏)𝐵𝑑𝜏
𝑥0
Procedure to compute reachable set1. Represent the set Θ using data structure2. Select a time interval ℎ.3. Compute 𝑃𝑜𝑠𝑡(Θ, ℎ) for [0, ℎ]4. Iterate for future intervals.
Data structureSpaceEx – Support FunctionsCORA – ZonotopesFlow* – Taylor Models
Drawbacks1. Representation cost grows with n2. Only overapproximation3. Cannot be directly applied for
time varying linear systems
This Paper: Contributions
New simulation based verification for linear systems.
1. For 𝒏-dimensional system, 𝒏 + 𝟏 simulations suffice.
2. Works for both time invariant and time variant systems.
3. Works for non-convex and unbounded initial set.
4. Can compute over- and under-approximation.
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The How: Superposition Principle
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𝑥1
𝑥2
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
The How: Superposition Principle
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𝑥1
𝑥2
𝜆𝑥1 + (1 − 𝜆)𝑥2
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
.
The How: Superposition Principle
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𝑥1
𝑥2
𝜆𝑥1 + (1 − 𝜆)𝑥2
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
𝜉 𝜆𝑥1 + 1 − 𝜆 𝑥2, 𝑡=
𝜆𝜉(𝑥1, 𝑡) + 1 − 𝜆 𝜉(𝑥2, 𝑡)
.
Implications Of Superposition
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𝑥0
𝑥1
𝑥2
v2
v1
.
.
.
Implications Of Superposition
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𝜉(𝑥0, 𝑡)
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
.
..v1′
v2′
𝑥0
𝑥1
𝑥2
v2
v1
.
.
.
Implications Of Superposition
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𝜉(𝑥0, 𝑡)
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
.
..v1′
v2′
𝑥0
𝑥1
𝑥2
v2
v1
.
.
.
𝑥0 + 𝛼1v1 + 𝛼2v2
.
Implications Of Superposition
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𝜉(𝑥0, 𝑡)
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
𝜉(𝑥0 + 𝛼1𝑣1 + 𝛼2𝑣2, 𝑡)
.
..v1′
v2′
𝑥0
𝑥1
𝑥2
v2
v1
.
.
.
𝑥0 + 𝛼1v1 + 𝛼2v2
.
Implications Of Superposition
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𝜉(𝑥0, 𝑡)
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
𝜉(𝑥0 + 𝛼1𝑣1 + 𝛼2𝑣2, 𝑡)
.
..v1′
v2′
.𝛼1v1
′ + 𝛼2v2′
𝑥0
𝑥1
𝑥2
v2
v1
.
.
.
𝑥0 + 𝛼1v1 + 𝛼2v2
.
Implications Of Superposition
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𝜉(𝑥0, 𝑡)
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
𝜉(𝑥0 + 𝛼1𝑣1 + 𝛼2𝑣2, 𝑡)
.
..v1′
v2′
.𝛼1v1
′ + 𝛼2v2′
𝑥0
𝑥1
𝑥2
v2
v1
.
.
.
𝑥0 + 𝛼1v1 + 𝛼2v2
.
Why is this important?Given 𝜉0, 𝜉1, and 𝜉2, one can
compute any simulation startingfrom linear span of 𝑥0, 𝑣1, and 𝑣2.
Implications Of Superposition
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𝜉(𝑥0, 𝑡)
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
𝜉(𝑥0 + 𝛼1𝑣1 + 𝛼2𝑣2, 𝑡)
.
..v1′
v2′
.
𝑥0
𝑥1
𝑥2
v2
v1
.
.
.
𝜉(𝑥0 + 𝛼1′𝑣1 + 𝛼2
′𝑣2, 𝑡)
𝛼1′v1
′ + 𝛼2′ v2
′
Why is this important?Given 𝜉0, 𝜉1, and 𝜉2, one can
compute any simulation startingfrom linear span of 𝑥0, 𝑣1, and 𝑣2.
Implications Of Superposition
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𝜉(𝑥0, 𝑡)
𝜉(𝑥1, 𝑡)
𝜉(𝑥2, 𝑡)
𝜉(𝑥0 + 𝛼1𝑣1 + 𝛼2𝑣2, 𝑡)
.
..v1′
v2′
.
𝑥0
𝑥1
𝑥2
v2
v1
.
.
.
𝜉(𝑥0 + 𝛼1′𝑣1 + 𝛼2
′𝑣2, 𝑡)
𝛼1′v1
′ + 𝛼2′ v2
′
Why is this important?Given 𝜉0, 𝜉1, and 𝜉2, one can
compute any simulation startingfrom linear span of 𝑥0, 𝑣1, and 𝑣2.
Key Idea: Use a set representation that exploits this property
Representation: Generalized Stars
Generalized star is represented as ⟨𝑐, 𝑉, 𝑃⟩
𝑐 – center, 𝑉 – set of vectors, 𝑃 – predicate.
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𝑐, 𝑉, 𝑃 = 𝑥 ∃ ത𝛼 = (𝛼1, … , 𝛼𝑛), c + Σ𝑖𝛼𝑖𝑣𝑖 = 𝑥, 𝑃 ത𝛼 = ⊤}
𝑣1
𝑣2
𝑐1
𝑃 𝛼1, 𝛼2≜
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
𝑐1 + 𝛼1𝑣1 + 𝛼2𝑣2.
Representation: Generalized Stars
Generalized star is represented as ⟨𝑐, 𝑉, 𝑃⟩
𝑐 – center, 𝑉 – set of vectors, 𝑃 – predicate.
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𝑐, 𝑉, 𝑃 = 𝑥 ∃ ത𝛼 = (𝛼1, … , 𝛼𝑛), c + Σ𝑖𝛼𝑖𝑣𝑖 = 𝑥, 𝑃 ത𝛼 = ⊤}
𝑃 𝛼1, 𝛼2≜
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1 ∧ 𝛼1 + 𝛼2 ≤ 1.5𝑣1
𝑣2
𝑐1
Representation: Generalized Stars
Generalized star is represented as ⟨𝑐, 𝑉, 𝑃⟩
𝑐 – center, 𝑉 – set of vectors, 𝑃 – predicate.
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𝑐, 𝑉, 𝑃 = 𝑥 ∃ ത𝛼 = (𝛼1, … , 𝛼𝑛), c + Σ𝑖𝛼𝑖𝑣𝑖 = 𝑥, 𝑃 ത𝛼 = ⊤}
𝑃 𝛼1, 𝛼2≜
𝛼1 ≤ 1 − 𝛼22𝑣1
𝑣2
𝑐1
Representation: Generalized Stars
Generalized star is represented as ⟨𝑐, 𝑉, 𝑃⟩
𝑐 – center, 𝑉 – set of vectors, 𝑃 – predicate.
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𝑐, 𝑉, 𝑃 = 𝑥 ∃ ത𝛼 = (𝛼1, … , 𝛼𝑛), c + Σ𝑖𝛼𝑖𝑣𝑖 = 𝑥, 𝑃 ത𝛼 = ⊤}
𝑃 𝛼1, 𝛼2≜
𝑣1
𝑣2
𝑐1 𝟏. 𝟓*𝒔𝒒𝒓𝒕 -𝒂𝒃𝒔 𝒂𝒃𝒔 𝒙 – 𝟏 *𝒂𝒃𝒔 𝟑 – 𝒂𝒃𝒔 𝒙
𝒂𝒃𝒔 𝒙 – 𝟏 * 𝟑 – 𝒂𝒃𝒔 𝒙* 𝟏 +
𝒂𝒃𝒔 𝒂𝒃𝒔 𝒙 – 𝟑
𝒂𝒃𝒔 𝒙 - 𝟑* 𝒔𝒒𝒓𝒕 𝟏 –
𝒙
𝟕
𝟐
+
𝟒.𝟓 + 𝟎. 𝟕𝟓 * 𝒂𝒃𝒔 𝒙 – 𝟎.𝟓 + 𝒂𝒃𝒔 𝒙 + 𝟎. 𝟓 – 𝟐. 𝟕𝟓 * 𝒂𝒃𝒔 𝒙-𝟎. 𝟕𝟓 + 𝒂𝒃𝒔 𝒙 + 𝟎. 𝟕𝟓 * 𝟏 +𝒂𝒃𝒔 𝟏 – 𝒂𝒃𝒔 𝒙
𝟏 – 𝒂𝒃𝒔 𝒙,
-𝟑 *𝒔𝒒𝒓𝒕 𝟏 -𝒙
𝟕
𝟐
*𝒔𝒒𝒓𝒕𝒂𝒃𝒔 𝒂𝒃𝒔 𝒙 – 𝟒
𝒂𝒃𝒔 𝒙 -𝟒, 𝒂𝒃𝒔
𝒙
𝟐– 𝟎.𝟎𝟗𝟏𝟑𝟕𝟐𝟐 * 𝒙𝟐-𝟑 + 𝒔𝒒𝒓𝒕 𝟏 – 𝒂𝒃𝒔 𝒂𝒃𝒔 𝒙 – 𝟐 – 𝟏 𝟐 ,
(𝟐. 𝟕𝟏𝟎𝟓𝟐+ 𝟏. 𝟓 – 𝟎. 𝟓 * 𝒂𝒃𝒔(𝒙) – 𝟏. 𝟑𝟓𝟓𝟐𝟔 * 𝒔𝒒𝒓𝒕(𝟒 – (𝒂𝒃𝒔(𝒙) – 𝟏)^𝟐)) * 𝒔𝒒𝒓𝒕(𝒂𝒃𝒔(𝒂𝒃𝒔(𝒙) – 𝟏)/(𝒂𝒃𝒔(𝒙) – 𝟏))
Reachable Set Computation Using Simulations For Generalized Stars
Given Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩ to compute reachable set
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𝑐 𝑣1
𝑣2Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Reachable Set Computation Using Simulations For Generalized Stars
Given Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩ to compute reachable set1. Simulate from 𝑐2. Simulate from 𝑐 + 𝑣𝑖 for each 𝑖
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𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Reachable Set Computation Using Simulations For Generalized Stars
Given Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩ to compute reachable set1. Simulate from 𝑐2. Simulate from 𝑐 + 𝑣𝑖 for each 𝑖
Reachable set at time 𝑡 is given by ⟨𝑐′, 𝑉′, 𝑃⟩ where1. 𝑐′ is the simulation corresponding to 𝑐2. 𝑣𝑖′ is the difference of simulations from 𝑐 + 𝑣𝑖 and from 𝑐
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𝑐 𝑣1
𝑣2
𝑐′𝑣1′
𝑣2′
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Reachable Set Computation Using Simulations For Generalized Stars
Given Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩ to compute reachable set1. Simulate from 𝑐2. Simulate from 𝑐 + 𝑣𝑖 for each 𝑖
Reachable set at time 𝑡 is given by ⟨𝑐′, 𝑉′, 𝑃⟩ where1. 𝑐′ is the simulation corresponding to 𝑐2. 𝑣𝑖′ is the difference of simulations from 𝑐 + 𝑣𝑖 and from 𝑐
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𝑐 𝑣1
𝑣2
𝑐′𝑣1′
𝑣2′
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Reach(Θ, t) ≜ ⟨𝑐′, 𝑉′, 𝑃⟩
Reachable Set Computation Using Simulations For Generalized Stars
Given Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩ to compute reachable set1. Simulate from 𝑐2. Simulate from 𝑐 + 𝑣𝑖 for each 𝑖
Reachable set at time 𝑡 is given by ⟨𝑐′, 𝑉′, 𝑃⟩ where1. 𝑐′ is the simulation corresponding to 𝑐2. 𝑣𝑖′ is the difference of simulations from 𝑐 + 𝑣𝑖 and from 𝑐
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𝑐 𝑣1
𝑣2
𝑐′𝑣1′
𝑣2′
Observation: 𝑹𝒆𝒂𝒄𝒉 preservesthe “shape” of the initial set.𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Reach(Θ, t) ≜ ⟨𝑐′, 𝑉′, 𝑃⟩
Reachable Set Computation Using Simulations For Generalized Stars
Given Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩ to compute reachable set1. Simulate from 𝑐2. Simulate from 𝑐 + 𝑣𝑖 for each 𝑖
Reachable set at time 𝑡 is given by ⟨𝑐′, 𝑉′, 𝑃⟩ where1. 𝑐′ is the simulation corresponding to 𝑐2. 𝑣𝑖′ is the difference of simulations from 𝑐 + 𝑣𝑖 and from 𝑐
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𝑐 𝑣1
𝑣2
𝑐′𝑣1′
𝑣2′
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1 ∧ 𝛼1 + 𝛼2 ≤ 1.5
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1 ∧ 𝛼1 + 𝛼2 ≤ 1.5
Observation: 𝑹𝒆𝒂𝒄𝒉 preservesthe “shape” of the initial set.
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Reach(Θ, t) ≜ ⟨𝑐′, 𝑉′, 𝑃⟩
Reachable Set Computation Using Simulations For Generalized Stars
Given Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩ to compute reachable set1. Simulate from 𝑐2. Simulate from 𝑐 + 𝑣𝑖 for each 𝑖
Reachable set at time 𝑡 is given by ⟨𝑐′, 𝑉′, 𝑃⟩ where1. 𝑐′ is the simulation corresponding to 𝑐2. 𝑣𝑖′ is the difference of simulations from 𝑐 + 𝑣𝑖 and from 𝑐
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𝑐 𝑣1
𝑣2
𝑐′𝑣1′
𝑣2′
𝛼1 ≤ 1 − 𝛼22
𝛼1 ≤ 1 − 𝛼22
Observation: 𝑹𝒆𝒂𝒄𝒉 preservesthe “shape” of the initial set.
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Reach(Θ, t) ≜ ⟨𝑐′, 𝑉′, 𝑃⟩
Reachable Set Computation Using Simulations For Generalized Stars
Given Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩ to compute reachable set1. Simulate from 𝑐2. Simulate from 𝑐 + 𝑣𝑖 for each 𝑖
Reachable set at time 𝑡 is given by ⟨𝑐′, 𝑉′, 𝑃⟩ where1. 𝑐′ is the simulation corresponding to 𝑐2. 𝑣𝑖′ is the difference of simulations from 𝑐 + 𝑣𝑖 and from 𝑐
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𝑐 𝑣1
𝑣2
𝑐′𝑣1′
𝑣2′
Problem: Exact simulations requires computing 𝒆𝑨𝒕 and is not necessarily finitely representable
𝛼1 ≤ 1 − 𝛼22
𝛼1 ≤ 1 − 𝛼22
Observation: 𝑹𝒆𝒂𝒄𝒉 preservesthe “shape” of the initial set.
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Reach(Θ, t) ≜ ⟨𝑐′, 𝑉′, 𝑃⟩
Validated Simulations
CAV 2016 35
𝜉 𝑥0, 𝑡
𝑥0
𝑣𝑎𝑙𝑆𝑖𝑚(𝑥0, 𝑡) returns sequence of regions such that 𝜉 𝑥0, 𝑡 ∈ 𝑅𝑙 when 𝑡 ∈ [𝑡𝑙 , 𝑡𝑙+1]
𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑅𝑙 → 0 as |t𝑙+1 − 𝑡𝑙| → 0
.
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 36
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
𝑅0
𝑅1
𝑅2
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 37
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
𝑐′𝑣1′
𝑣2′
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 38
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 39
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
𝑐′ 𝑣1′
𝑣2′
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 40
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 41
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 42
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩Over–approximation is the
union of all such stars
Under–approximation is the intersection of all such stars
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 43
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩Over–approximation is the
union of all such stars
Under–approximation is the intersection of all such stars
𝑂𝐴 = 𝑥 ∃𝑐, ∃𝑣1, ∃𝑣2 ∃ ത𝛼, 𝑥 = 𝑐 + 𝛼1𝑣1 + 𝛼2𝑣2}𝑈𝐴 = 𝑥 ∀𝑐, ∀𝑣1, ∀𝑣2 ∃ ത𝛼, 𝑥 = 𝑐 + 𝛼1𝑣1 + 𝛼2𝑣2}
Over- and Under-Approximations Using Validated Simulations
Problem – exact value of 𝑐, 𝑣1′ , and 𝑣2
′ is not known!
CAV 2016 44
𝑐 𝑣1
𝑣2
𝛼1 ≤ 1 ∧ 𝛼2 ≤ 1
Θ ≜ ⟨𝑐, 𝑉, 𝑃⟩Provided in paper:1. Computing overapproximation2. Checking safety violation without using QE for bounded initial sets
Experimental Results - I
Comparison with SpaceEx on Linear Systems
CAV 2016 45
Benchmark Vars. TH Sim.Time. Verif.Time. C2E2 SpaceEx Result
Insulin 8 10 0.157 s 0.049 s 0.20 s 8.07 s Safe
Inslin 8 10 0.166 s 0.034 s 0.2 s 7.89 s Unsafe
Platoon 10 25 0.337 s 0.019 s 0.356 s TO Safe
Platoon 10 25 0.323 s 0.019 s 0.342 s TO Unsafe
Tank–10 10 20 0.745 s 0.206 s 0.951 s 4.886 s Safe
Tank–10 10 20 0.721 s 0.19 s 0.911 s 4.992 s Unsafe
Tank–15 15 20 1.325 s 0.363 s 1.688 s 8.176 s Safe
Tank–18 18 20 1.705 s 0.569 s 2.274 s 10.466 s Safe
Helicopter 28 20 3.192 s 1.634 s 4.826 s 2m 1.66 s Safe
Experimental Results - II
Comparison with Flow* for Linear Time Varying Systems
CAV 2016 46
Benchmark Vars C2E2 Flow*
Tank–TV 2 0.132 s 1.56 s
Tank–TV 4 0.198 s 4.28 s
Tank–TV 6 0.287 s 9.41 s
Tank–TV 8 0.356 s 18.73 s
Tank–TV 10 0.484 s 33.67 s
LTV 5 0.24 s 7.51 s
LTV 7 0.31 s 12.09 s
LTV 9 0.4 s 18.18 s
Experimental Results - III
Verification of Non-convex and Unbounded Initial Sets
CAV 2016 47
Benchmark Dim. TH. Init. Set. Res. Time
ACC 3 2 Non–Convex Safe 2.185
ACC 3 2 Unbounded Safe 1.774
ACC 3 2 Non–Convex Unsafe 1.11
ACC 3 2 Unbounded Unsafe 1.01
Tank 5 1 Non–Convex Safe 2.717
Tank 5 1 Unbounded Safe 2.145
Tank 5 1 Non–Convex Unsafe 1.722
Tank 5 1 Unbounded Unsafe 1.519
Conclusions
New simulation based verification for linear systems
1. For 𝑛–dimensional system, 𝑛 + 1 simulations suffice.2. Works for both time invariant and time variant systems.3. Works for non-convex and unbounded systems4. Can compute over- and under-approximation.5. Reuse the simulations for different initial sets.
CAV 2016 48
Thank You