Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Verification of Delayed Differential Dynamics Based on Validated Simulation Mingshuai Chen 1 , Martin Fränzle 2 , Yangjia Li 1 , Peter N. Mosaad 2 , Naijun Zhan 1 1 State Key Lab. of Computer Science, Institute of Software, Chinese Academy of Sciences 2 Dpt. of Computing Science, C. v. Ossietzky Universität Oldenburg Limassol, November 2016 Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 1 / 21
x t x t x Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Motivation : Why Delays ? { ˙ x ( t ) = − x ( t ) x (0) = 1 1 0.5 x 0 −0.5 0 5 10 15 t Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 2 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Motivation : Why Delays ? { ˙ { ˙ x ( t ) = − x ( t ) x ( t ) = − x ( t − 1) x ([ − 1 , 0]) ≡ 1 x (0) = 1 1 1 0.5 0.5 x x 0 0 −0.5 −0.5 0 5 10 15 0 5 10 15 t t Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 2 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Motivation : Why Delays ? Delayed logistic equation [G. Hutchinson, 1948] : ˙ N ( t ) = N ( t )[1 − N ( t − r )] Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 3 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Motivation : Why Delays ? Delayed logistic equation [G. Hutchinson, 1948] : ˙ N ( t ) = N ( t )[1 − N ( t − r )] r=0.25 r=1.52 r=1.65 2 2 2 1.8 1.8 1.8 1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 1 1 1 N N N 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 50 100 0 50 100 0 50 100 t t t Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 3 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Outline Problem Formulation 1 Simulation-Based Verification 2 Validated Simulation of Delayed Differential Dynamics 3 Experimental Results 4 Concluding Remarks 5 Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 4 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Outline 1 Problem Formulation Delayed Dynamical Systems Safety Verification Problem Simulation-Based Verification 2 Basic Idea Verification Algorithm Validated Simulation of Delayed Differential Dynamics 3 Local Error Bounds Simulation Algorithm Solving Optimization Correctness and Completeness Experimental Results 4 Delayed Logistic Equation Delayed Microbial Growth 5 Concluding Remarks Conclusions Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 5 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Delayed Dynamical Systems Delayed Dynamical Systems Delayed Dynamical Systems { x ( t ) f ( x ( t ) , x ( t − r 1 ) , . . . , x ( t − r k )) , t ∈ [0 , ∞ ) ˙ = x ( t ) t ∈ [ − r max , 0] ≡ x 0 ∈ Θ , The unique solution ( trajectory ) : ξ x 0 ( t ) : [ − r max , ∞ ) �→ R n . Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 6 / 21
System is safe, if no trajectory enters the unsafe set. Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Safety Verification Problem Safety Verification Problem Given T ∈ R , X 0 ⊆ Θ , U ⊆ R n , whether (∪ ) t ≤ T ξ x 0 ( t ) ∀ x 0 ∈ X 0 : ∩ U = ∅ ? 1. The figure is taken from [M. Althoff, 2010]. Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 7 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Safety Verification Problem Safety Verification Problem Given T ∈ R , X 0 ⊆ Θ , U ⊆ R n , whether (∪ ) t ≤ T ξ x 0 ( t ) ∀ x 0 ∈ X 0 : ∩ U = ∅ ? System is safe, if no trajectory enters the unsafe set. 1. The figure is taken from [M. Althoff, 2010]. Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 7 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Outline 1 Problem Formulation Delayed Dynamical Systems Safety Verification Problem Simulation-Based Verification 2 Basic Idea Verification Algorithm Validated Simulation of Delayed Differential Dynamics 3 Local Error Bounds Simulation Algorithm Solving Optimization Correctness and Completeness Experimental Results 4 Delayed Logistic Equation Delayed Microbial Growth 5 Concluding Remarks Conclusions Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 8 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Basic Idea Basic Idea ) e - E x 0 ,� ( t ) d ǫ e ξ x 0 ( t ) l - ǫ ) x 0 � � Reach = t B � ( x 0 ) � Figure : An Over-approximation of the reachable set by imply consists in finding Figure : A finite ϵ -cover of the initial set of states. bloating the simulation. 2. Figures are taken from [A. DonzDonzé & O. Maler, 2007]. Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 9 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Verification Algorithm Verification Algorithm Algorithm 1: Simulation-based Verification for Delayed Dynamical Systems input : The dynamics f ( x , u ) , delay term r , initial set X 0 , unsafe set U , time bound T , precision ǫ . /* initialization */ 1 R ← ∅ ; δ ← dia ( X 0 ) / 2 ; τ ← τ 0 ; 2 X ← δ -Partition ( X 0 ) ; 3 while X � = ∅ do if δ < ǫ then 4 return (UNKNOWN , R ); 5 for B δ ( x 0 ) ∈ X do 6 � t , y , d � ← Simulation ( B δ ( x 0 ) , f ( x , u ) , r, τ, T ) ; 7 T ← � N − 1 n =0 conv ( B d n ( y n ) ∪ B d n +1 ( y n +1 )) ; 8 if T ∩ U = ∅ then 9 X ← X\B δ ( x 0 ) ; R ← R ∪ T ; 10 else if ∃ i. B d i ( y i ) ⊆ U then 11 return ( UNSAFE , T ) ; 12 else 13 X ← X ∪ δ X ← X\B δ ( x 0 ) ; 2 -Partition ( B δ ( x 0 )) ; 14 δ ← δ/ 2 ; 15 16 return ( SAFE , R ) ; Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 10 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Outline 1 Problem Formulation Delayed Dynamical Systems Safety Verification Problem Simulation-Based Verification 2 Basic Idea Verification Algorithm Validated Simulation of Delayed Differential Dynamics 3 Local Error Bounds Simulation Algorithm Solving Optimization Correctness and Completeness Experimental Results 4 Delayed Logistic Equation Delayed Microbial Growth 5 Concluding Remarks Conclusions Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 11 / 21
Validation Property : t t i y i t i t y i t for each t t i t i E t t i t i x Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Local Error Bounds Local Error Bounds { d 0 , if t = 0 , E ( t ) = E ( t i ) + ( t − t i ) e i +1 , if t ∈ [ t i , t i +1 ] . Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 12 / 21
Problem Formulation Verification Shell Validated Simulation Experimental Results Concluding Remarks . . . . . . . . . . . . Local Error Bounds Local Error Bounds { d 0 , if t = 0 , E ( t ) = E ( t i ) + ( t − t i ) e i +1 , if t ∈ [ t i , t i +1 ] . Validation Property : ( ( t − t i ) y i + ( t i +1 − t ) y i +1 ) ξ x 0 ( t ) ∈ B E ( t ) , for each t ∈ [ t i , t i +1 ] . t i +1 − t i Mingshuai Chen Institute of Software, CAS Verification of Delayed Differential Dynamics Limassol, FM 2016 12 / 21
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