(Global) Asymptotic Stability (AS) A system is AS with respect to 0 if it is Lyapunov stable and there exists a value δ > 0 such that every execution σ starting from B δ (0) converges to 0. δ 0 A system is GAS with respect to 0 if it is Lyapunov stable and every execution σ converges to 0. 13
(Global) Asymptotic Stability (AS) A system is AS with respect to 0 if it is Lyapunov stable and there exists a value δ > 0 such that every execution σ starting from B δ (0) converges to 0. δ 0 A system is GAS with respect to 0 if it is Lyapunov stable and every execution σ converges to 0. 13
(Global) Asymptotic Stability (AS) A system is AS with respect to 0 if it is Lyapunov stable and there exists a value δ > 0 such that every execution σ starting from B δ (0) converges to 0. δ 0 A system is GAS with respect to 0 if it is Lyapunov stable and every execution σ converges to 0. Global asymptotic stability 13
(Global) Asymptotic Stability (AS) A system is AS with respect to 0 if it is Lyapunov stable and there exists a value δ > 0 such that every execution σ starting from B δ (0) converges to 0. δ 0 A system is GAS with respect to 0 if it is Lyapunov stable and every execution σ converges to 0. Global asymptotic stability 13
(Global) Asymptotic Stability (AS) A system is AS with respect to 0 if it is Lyapunov stable and there exists a value δ > 0 such that every execution σ starting from B δ (0) converges to 0. δ 0 A system is GAS with respect to 0 if it is Lyapunov stable and every execution σ converges to 0. Global asymptotic stability Asymptotic stability 13
Challenges in Stability Verification for Hybrid Systems 14
Stability analysis Linear dynamical systems y y x x
Stability analysis Linear dynamical systems y y x x Stable Stable
Stability analysis Linear dynamical systems y y Stability can be determined by eigen values analysis x x Stable Stable
Stability analysis Linear dynamical systems y y Stability can be determined by eigen values analysis x x Stable Stable Linear hybrid systems y x
Stability analysis Linear dynamical systems y y Stability can be determined by eigen values analysis x x Stable Stable Linear hybrid systems y x Stable
Stability analysis Linear dynamical systems y y Stability can be determined by eigen values analysis x x Stable Stable Linear hybrid systems y y x x Stable
Stability analysis Linear dynamical systems y y Stability can be determined by eigen values analysis x x Stable Stable Linear hybrid systems y y x x Stable Unstable
Stability analysis Linear dynamical systems y y Stability can be determined by eigen values analysis x x Stable Stable Linear hybrid systems y y Eigen value analysis does not suffice for switched linear system x x Stable Unstable
Lyapunov’s second method x = F ( x ) ˙ Lyapunov function: ✤ Continuously differentiable V : R n → R + ✤ Positive definite V ( x ) ≥ 0 ∀ x ✤ Decreases along any trajectory ∂ V ( x ) ∂ x F ( x ) ≤ 0 ∀ x V x y 16
Lyapunov’s second method x = F ( x ) ˙ Lyapunov function: ✤ Continuously differentiable V : R n → R + ✤ Positive definite V ( x ) ≥ 0 ∀ x ✤ Decreases along any trajectory ∂ V ( x ) ∂ x F ( x ) ≤ 0 ∀ x V x y 16
Lyapunov’s second method x = F ( x ) ˙ Lyapunov function: ✤ Continuously differentiable V : R n → R + ✤ Positive definite V ( x ) ≥ 0 ∀ x ✤ Decreases along any trajectory ∂ V ( x ) ∂ x F ( x ) ≤ 0 ∀ x V x y 16
Lyapunov’s second method Template based automated search x = F ( x ) ˙ ✤ Choose a template Lyapunov function: ✤ Polynomial with coefficients as parameters ✤ Continuously differentiable V : R n → R + ✤ Encode (a relaxation) of the constraints as a sum-of- square programming problem ✤ Positive definite ✤ Use existing tools for SOS V ( x ) ≥ 0 ∀ x ✤ Decreases along any trajectory ∂ V ( x ) ∂ x F ( x ) ≤ 0 ∀ x V x y 16
Lyapunov’s second method Template based automated search x = F ( x ) ˙ ✤ Choose a template Lyapunov function: ✤ Polynomial with coefficients as parameters ✤ Continuously differentiable V : R n → R + ✤ Encode (a relaxation) of the constraints as a sum-of- square programming problem ✤ Positive definite ✤ Use existing tools for SOS V ( x ) ≥ 0 ∀ x ✤ Decreases along any trajectory Shortcomings: ∂ V ( x ) ∂ x F ( x ) ≤ 0 ∀ x ✤ Success depends crucially on the choice of the template V ✤ The current methods provide no insight into the reason for the failure, when a template fails to prove stability ✤ No guidance regarding the choice of the next template x y 16
Lyapunov’s second method Template based automated search x = F ( x ) ˙ ✤ Choose a template Lyapunov function: ✤ Polynomial with coefficients as parameters ✤ Continuously differentiable V : R n → R + ✤ Encode (a relaxation) of the constraints as a sum-of- square programming problem ✤ Positive definite ✤ Use existing tools for SOS V ( x ) ≥ 0 ∀ x ✤ Decreases along any trajectory Shortcomings: ∂ V ( x ) ∂ x F ( x ) ≤ 0 ∀ x ✤ Success depends crucially on the choice of the template V ✤ The current methods provide no insight into the reason for the failure, when a template fails to prove stability ✤ No guidance regarding the choice of the next template A CEGAR framework x y 16
Counter-example guided abstraction refinement 17
Abstraction 1 2 3 4 5 6 9 7 8 Safety Analysis 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis ✤ Every trajectory corresponds to a path in the graph 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety 18
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 Safety Analysis ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety 19
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety 19
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ The abstract graph has a counter-example ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety 19
Abstraction 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ The abstract graph has a counter-example ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety ✤ Right abstractions are hard to find! 19
Refinement 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ The abstract graph has a counter-example ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety ✤ Right abstractions are hard to find! ✤ Refine by analyzing the abstract counter-example 20
Refinement 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ The abstract graph has a counter-example ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety ✤ Right abstractions are hard to find! ✤ Refine by analyzing the abstract counter-example 20
Refinement 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ The abstract graph has a counter-example ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety ✤ Right abstractions are hard to find! ✤ Refine by analyzing the abstract counter-example 20
Refinement 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ The abstract graph has a counter-example ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety ✤ Right abstractions are hard to find! ✤ Refine by analyzing the abstract counter-example 20
Refinement 1 2 3 1 2 3 4 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ The abstract graph has a counter-example ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety ✤ Right abstractions are hard to find! ✤ Refine by analyzing the abstract counter-example 20
Refinement 1 2 3 1 2 3 5 6 4 5 6 8 9 7 9 7 8 ✤ The above system is safe Safety Analysis ✤ The abstract graph has a counter-example ✤ Every trajectory corresponds to a path in the graph ✤ Absence of a path from green to red node implies safety ✤ Right abstractions are hard to find! ✤ Refine by analyzing the abstract counter-example 20
Counter-example guided abstraction refinement Property Concrete Abstract Yes System System Property ✤ CEGAR for discrete systems Abstract Model-Check satisfied [Kurshan et al. 93, Clarke et al. 00, No Ball et al. 02] Abstraction Abstract Relation Counter-example ✤ CEGAR for hybrid systems safety verification [Alur et al 03, Clarke et No Yes Property al 03, Prabhakar et al 13] Refine Validate Analysis violated Results 21
Counter-example guided abstraction refinement Property Concrete Abstract Yes System System Property ✤ CEGAR for discrete systems Abstract Model-Check satisfied [Kurshan et al. 93, Clarke et al. 00, No Ball et al. 02] Abstraction Abstract Relation Counter-example ✤ CEGAR for hybrid systems safety verification [Alur et al 03, Clarke et No Yes Property al 03, Prabhakar et al 13] Refine Validate Analysis violated Results Template based search CEGAR framework ✤ Systematically iterate over the abstract ✤ Success depends crucially on the choice of the template systems ✤ Returns a counter-example in the case ✤ No insight into the reason for the failure, when a template fails to prove stability that the abstraction fails ✤ The counter-example can be used to ✤ No guidance regarding the choice of the next template guide the choice of the next abstraction 21
AVERIST: An Algorithmic VERIfier for STability Global Asymptotic Stability Analyzer Local Asymptotic Linear/Non- Stability Analyzer Linear Hybrid Automaton Quantitative GLPK Hybridization Predicate Abstraction NetworkX Model-Checking Stability Zone Computation Z3 Validation Stable/ Unstable Region Stability Analysis PPL Refinement Tool webpage: http://software.imdea.org/projects/averist/ 22
Abstraction based analysis: Lyapunov and asymptotic stability 23
Quantitative Predicate Abstraction p 2 p 1 p 3 p 2 p 1 C B A D p 3 p 6 p 4 E F p 6 p 4 p 5 p 5 24
Quantitative Predicate Abstraction p 2 p 1 p 3 p 2 p 1 C B A D p 3 p 6 p 4 E F p 6 p 4 p 5 p 5 p 1 p 2 w ( e ) = | d 2 | | d 1 | d 2 d 1 24
Quantitative Predicate Abstraction p 2 p 1 p 3 w 1 p 2 p 1 w 2 w 6 C B A D p 3 p 6 p 4 E F w 3 w 5 p 6 p 4 p 5 p 5 w 4 p 1 p 2 w ( e ) = | d 2 | | d 1 | d 2 d 1 24
Quantitative Predicate Abstraction p 2 p 1 p 3 w 1 p 2 p 1 w 2 w 6 C B A D p 3 p 6 p 4 E F w 3 w 5 p 6 p 4 p 5 p 5 w 4 p 1 p 2 Weights capture information w ( e ) = | d 2 | about distance to the origin | d 1 | d 2 along the executions d 1 24
Weighted Graph Construction p 2 p 2 p 2 p 2 p 2 p 3 p 1 p 3 p 1 p 3 p 3 p 3 p 1 p 1 p 1 p 4 p 4 p 4 p 4 p 4 p 2 p 2 p 2 1 1 1/2 1 2 1 p 1 p 1 p 1 p 3 p 3 p 3 1 1 1 1 1/2 2 p 4 p 4 p 4 25
Higher Dimensions 26
Higher Dimensions The weighted graph construction has a bisimulation like property for 2D. 26
Higher Dimensions The weighted graph construction has a bisimulation like property for 2D. p 1 p 2 d 2 d 1 w ( e ) = | d 2 | | d 1 | 26
Higher Dimensions The weighted graph construction has a bisimulation like property for 2D. z p 1 p 2 ~ b d 2 d 1 ~ a x w ( e ) = | d 2 | | d 1 | y 26
Higher Dimensions The weighted graph construction has a bisimulation like property for 2D. z p 1 | ~ b | p 2 | ~ a | ~ b d 2 d 1 ~ a x w ( e ) = | d 2 | | d 1 | y 26
Higher Dimensions The weighted graph construction has a bisimulation like property for 2D. z p 1 | ~ b | p 2 | ~ a | ~ b d 2 d 1 ~ a x w ( e ) = | d 2 | | d 1 | y 26
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