Decidability of the Reachability for a Family of Linear Vector - PowerPoint PPT Presentation

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Decidability of the Reachability for a Family of Linear Vector Fields Ting Gan 1 , Mingshuai Chen 2 , Yangjia Li 2 , Bican Xia 1 , and Naijun Zhan 2 1 LMAM & School of Mathematical Sciences, Peking University 2 State Key Lab. of Computer Science, Institute of Software, Chinese Academy of Sciences Aalborg, June 2016 Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 1 / 28

x x x x x x x x x x x x x x T with the initial set X x x x x x x . Is it possible for the temperature x getting over than F (unsafe) ? UNBOUNDED. Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 2 / 28

Is it possible for the temperature x getting over than F (unsafe) ? UNBOUNDED. Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. x 1 = 1 2 (45 − x 1 ) + 1 ˙ 2 ( x 2 − x 1 ) , x 2 = 1 2 ( x 1 − x 2 ) + 1 4 (35 − x 2 ) + 1 4 ( x 3 − x 2 ) + 20 , ˙ x 3 = 1 4 ( x 2 − x 3 ) + 3 4 (35 − x 3 ) , ˙ with the initial set X = { ( x 1 , x 2 , x 3 ) T | 1 − ( x 1 − 45) 2 − ( x 2 − 35) 2 − ( x 3 − 35) 2 > 0 } . Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 2 / 28

UNBOUNDED. Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. x 1 = 1 2 (45 − x 1 ) + 1 ˙ 2 ( x 2 − x 1 ) , x 2 = 1 2 ( x 1 − x 2 ) + 1 4 (35 − x 2 ) + 1 4 ( x 3 − x 2 ) + 20 , ˙ x 3 = 1 4 ( x 2 − x 3 ) + 3 4 (35 − x 3 ) , ˙ with the initial set X = { ( x 1 , x 2 , x 3 ) T | 1 − ( x 1 − 45) 2 − ( x 2 − 35) 2 − ( x 3 − 35) 2 > 0 } . Is it possible for the temperature x 2 getting over than 70 ◦ F (unsafe) ? Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 2 / 28

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Example : Home Heating x 3 ( t ) = Temperature in the attic, x 2 ( t ) = Temperature in the living area, x 1 ( t ) = Temperature in the basement, t = Time in hours. x 1 = 1 2 (45 − x 1 ) + 1 ˙ 2 ( x 2 − x 1 ) , x 2 = 1 2 ( x 1 − x 2 ) + 1 4 (35 − x 2 ) + 1 4 ( x 3 − x 2 ) + 20 , ˙ x 3 = 1 4 ( x 2 − x 3 ) + 3 4 (35 − x 3 ) , ˙ with the initial set X = { ( x 1 , x 2 , x 3 ) T | 1 − ( x 1 − 45) 2 − ( x 2 − 35) 2 − ( x 3 − 35) 2 > 0 } . Is it possible for the temperature x 2 getting over than 70 ◦ F (unsafe) ? UNBOUNDED. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 2 / 28

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Outline Background and Contributions 1 For Linear Systems with Purely Imaginary Eigenvalues 2 Abstraction of the General Cases 3 4 Concluding Remarks Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 3 / 28

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Outline Background and Contributions 1 Background and Preliminaries Reachability of the Linear Dynamical Systems (LDSs) with Inputs For Linear Systems with Purely Imaginary Eigenvalues 2 Preliminaries Decidability of the Reachability An Illustrating Example Abstraction of the General Cases 3 Preliminaries Abstraction of the Reachable Sets Examples Concluding Remarks 4 Conclusions Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 4 / 28

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Background and Preliminaries Hybrid Systems Hybrid systems exhibit combinations of discrete jumps and continuous evolution, many of which are Safety-critical. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 5 / 28

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Background and Preliminaries Safety Verification Using Reachable Set  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 Decidability of the Reachability for LDSs Aalborg, June 2016 6 / 28

Reachability problem (Unbounded) : x y t t x t with initial set : n p p J and unsafe set : n p J p J y X x X y Y y y x X Y x Y y x Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Reachability of LDSs LDSs with Inputs Linear dymamical systems (LDSs) with inputs : ˙ ξ = A ξ + u , (1) where ξ ( t ) ∈ R n , A ∈ R n × n , and u : R → R n . Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 7 / 28

with initial set : n p p J and unsafe set : n p J p J y X x x y Y x y Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Reachability of LDSs LDSs with Inputs Linear dymamical systems (LDSs) with inputs : ˙ ξ = A ξ + u , (1) where ξ ( t ) ∈ R n , A ∈ R n × n , and u : R → R n . Reachability problem (Unbounded) : F ( X , Y ) := ∃ x ∃ y ∃ t : x ∈ X ∧ y ∈ Y ∧ t ≥ 0 ∧ Φ( x , t ) = y . Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 7 / 28

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Reachability of LDSs LDSs with Inputs Linear dymamical systems (LDSs) with inputs : ˙ ξ = A ξ + u , (1) where ξ ( t ) ∈ R n , A ∈ R n × n , and u : R → R n . Reachability problem (Unbounded) : F ( X , Y ) := ∃ x ∃ y ∃ t : x ∈ X ∧ y ∈ Y ∧ t ≥ 0 ∧ Φ( x , t ) = y . with initial set : X = { x ∈ R n | p 1 ( x ) ≥ 0 , · · · , p J 1 ( x ) ≥ 0 } , and unsafe set : Y = { y ∈ R n | p J 1 +1 ( y ) ≥ 0 , · · · , p J ( y ) ≥ 0 } . Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 7 / 28

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Reachability of LDSs Decidability Results of the Reachability of LDSs In [LPY 2001], Lafferriere et al . proved the decidability of the reachability problems of the following three families of LDSs : 1 A is nilpotent , i.e. A n = 0 , and each component of u is a polynomial ; 2 A is diagonalizable with rational eigenvalues, and each component of u is of the form ∑ m i =1 c i e λ i t , where λ i s are rationals and c i s are subject to semi-algebraic constraints ; 3 A is diagonalizable with purely imaginary eigenvalues, and each component of u of the form ∑ m i =1 c i sin ( λ i t ) + d i cos ( λ i t ) , where λ i s are rationals and c i s and d i s are subject to semi-algebraic constraints. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 8 / 28

Background and Contributions LDSs with Purely Imaginary Eigenvalues Abstraction Conclusions . . . . . . . . . . . . . . . . . . . . . Reachability of LDSs Main Contributions Generalization of case 2 and case 3 : A has real eigenvalues, and each component of u is of the form ∑ m i =1 c i e λ it , where λ i s 2 are reals and c i s are subject to semi-algebraic constraints ; [Gan et al . 15] A has purely imaginary eigenvalues, and each component of u of the form 3 ∑ m i =1 c i sin ( λ i t ) + d i cos ( λ i t ) , where λ i s are reals and c i s and d i s are subject to semi-algebraic constraints. Abstraction of general dynamical systems where A may have complex eigenvalues, by reducing the problem to the reachability in the case 2. Mingshuai Chen Institute of Software, CAS Decidability of the Reachability for LDSs Aalborg, June 2016 9 / 28