Vector Barrier Certificates and Comparison Systems Andrew Sogokon 1 - PowerPoint PPT Presentation

Vector Barrier Certificates and Comparison Systems Andrew Sogokon 1 Khalil Ghorbal 2 Yong Kiam Tan 1 André Platzer 1 1 - Carnegie Mellon University, Pittsburgh, USA 2 - Inria, Rennes, France 16 July 2018 , FM 2018, Oxford, UK 1/26

Preliminaries: Systems of ODEs An autonomous n -dimensional system of ODEs has the general form: x ′ 1 = f 1 ( x 1 , . . . , x n ) , . . . x ′ n = f n ( x 1 , . . . , x n ) , i denotes the time derivative dx i where x ′ dt and f i are continuous functions. 2/26

Preliminaries: Systems of ODEs An autonomous n -dimensional system of ODEs has the general form: x ′ 1 = f 1 ( x 1 , . . . , x n ) , . . . x ′ n = f n ( x 1 , . . . , x n ) , i denotes the time derivative dx i where x ′ dt and f i are continuous functions. We write x ′ = f ( x ) . The vector field f : R n → R n gives the direction of motion at each point in space. 2/26

Preliminaries: Systems of ODEs An autonomous n -dimensional system of ODEs has the general form: x ′ 1 = f 1 ( x 1 , . . . , x n ) , . . . x ′ n = f n ( x 1 , . . . , x n ) , i denotes the time derivative dx i where x ′ dt and f i are continuous functions. We write x ′ = f ( x ) . The vector field f : R n → R n gives the direction of motion at each point in space. A solution x ( x 0 , t ) : R n × R → R n exactly describes the motion of a particle x 0 under the influence of the vector field. 2/26

Example: Van der Pol oscillator The Van der Pol system oscillator evolves according to the following ODEs: x ′ 1 = x 2 , 2 = (1 − x 2 x ′ 1 ) x 2 − x 1 3/26

Example: Van der Pol oscillator The Van der Pol system oscillator evolves according to the following ODEs: x ′ 1 = x 2 , 2 = (1 − x 2 x ′ 1 ) x 2 − x 1 x 0 3/26

Example: Van der Pol oscillator The Van der Pol system oscillator evolves according to the following ODEs: x ′ 1 = x 2 , 2 = (1 − x 2 x ′ 1 ) x 2 − x 1 x 0 x ( x 0 , t ) 3/26

Barrier certificates Lyapunov-like safety verification method, due to Prajna & Jadbabaie (2004). MAIN IDEA: Find a differentiable function B : R n → R such that B ( x ) > 0 holds for every x ∈ Unsafe , For all x 0 ∈ Init , B ( x ( x 0 , t )) ≤ 0 holds for all future t . 4/26

Barrier certificates Lyapunov-like safety verification method, due to Prajna & Jadbabaie (2004). MAIN IDEA: Find a differentiable function B : R n → R such that B ( x ) > 0 holds for every x ∈ Unsafe , For all x 0 ∈ Init , B ( x ( x 0 , t )) ≤ 0 holds for all future t . Init x 2 Unsafe x 1 4/26

Barrier certificates Lyapunov-like safety verification method, due to Prajna & Jadbabaie (2004). MAIN IDEA: Find a differentiable function B : R n → R such that B ( x ) > 0 holds for every x ∈ Unsafe , For all x 0 ∈ Init , B ( x ( x 0 , t )) ≤ 0 holds for all future t . B > 0 B ≤ 0 Init x 2 Unsafe B > 0 x 1 4/26

Barrier certificates (more formally) Lemma (Safety with semantic barrier certificates) Given : a system of n first-order ODEs x ′ = f ( x ) , 5/26

Barrier certificates (more formally) Lemma (Safety with semantic barrier certificates) Given : a system of n first-order ODEs x ′ = f ( x ) , possibly an evolution constraint Q ⊆ R n , 5/26

Barrier certificates (more formally) Lemma (Safety with semantic barrier certificates) Given : a system of n first-order ODEs x ′ = f ( x ) , possibly an evolution constraint Q ⊆ R n , a set of initial states Init ⊆ R n , 5/26

Barrier certificates (more formally) Lemma (Safety with semantic barrier certificates) Given : a system of n first-order ODEs x ′ = f ( x ) , possibly an evolution constraint Q ⊆ R n , a set of initial states Init ⊆ R n , and a set of unsafe states Unsafe ⊆ R n , 5/26

Barrier certificates (more formally) Lemma (Safety with semantic barrier certificates) Given : a system of n first-order ODEs x ′ = f ( x ) , possibly an evolution constraint Q ⊆ R n , a set of initial states Init ⊆ R n , and a set of unsafe states Unsafe ⊆ R n , if a differentiable (barrier) function B : R n → R satisfies the following conditions, then the system is safe : ∀ x ∈ Unsafe . B ( x ) > 0 , 1 � � ∀ x 0 ∈ Init . ∀ t ≥ 0 . ( ∀ τ ∈ [0 , t ] . x ( x 0 , τ ) ∈ Q ) ⇒ B ( x ( x 0 , t )) ≤ 0 . 2 5/26

Kinds of barrier certificates Recall the (semantic) conditions: ∀ x ∈ Unsafe . B ( x ) > 0 , 1 � � ∀ x 0 ∈ Init . ∀ t ≥ 0 . ( ∀ τ ∈ [0 , t ] . x ( x 0 , τ ) ∈ Q ) ⇒ B ( x ( x 0 , t )) ≤ 0 . 2 6/26

Kinds of barrier certificates Recall the (semantic) conditions: ∀ x ∈ Unsafe . B ( x ) > 0 , 1 � � ∀ x 0 ∈ Init . ∀ t ≥ 0 . ( ∀ τ ∈ [0 , t ] . x ( x 0 , τ ) ∈ Q ) ⇒ B ( x ( x 0 , t )) ≤ 0 . 2 Several direct sufficient conditions have been proposed to ensure the last requirement. Observe that the solutions x ( x 0 , t ) are not explicit. Convex Exponential-type ‘General’ (Prajna & Jadbabaie, 2004) (Kong et al., 2013) (Dai et al., 2017) Q → B ′ ≤ 0 . Q → B ′ ≤ λB. Q → B ′ ≤ ω ( B ) , ∀ t ≥ 0 . b ( t ) ≤ 0 , b is the solution to b ′ = ω ( b ) . 6/26

Kinds of barrier certificates Recall the (semantic) conditions: ∀ x ∈ Unsafe . B ( x ) > 0 , 1 � � ∀ x 0 ∈ Init . ∀ t ≥ 0 . ( ∀ τ ∈ [0 , t ] . x ( x 0 , τ ) ∈ Q ) ⇒ B ( x ( x 0 , t )) ≤ 0 . 2 Several direct sufficient conditions have been proposed to ensure the last requirement. Observe that the solutions x ( x 0 , t ) are not explicit. Convex Exponential-type ‘General’ (Prajna & Jadbabaie, 2004) (Kong et al., 2013) (Dai et al., 2017) Q → B ′ ≤ 0 . Q → B ′ ≤ λB. Q → B ′ ≤ ω ( B ) , ∀ t ≥ 0 . b ( t ) ≤ 0 , b is the solution to b ′ = ω ( b ) . All these conditions are instantiations of the comparison principle . 6/26

Comparison principle Used by R. Conti (1956), F. Brauer, C. Corduneanu (1960s), many others. Not a new idea in applied mathematics; used in stability theory. 7/26

Comparison principle Used by R. Conti (1956), F. Brauer, C. Corduneanu (1960s), many others. Not a new idea in applied mathematics; used in stability theory. MAIN IDEA: Given x ′ = f ( x ) , if a positive definite differentiable function V : R n → R satisfies the differential inequality V ′ ≤ ω ( V ) , where ω : R → R is an appropriate scalar function, one may infer the stability of x ′ = f ( x ) from the stability of the one-dimensional system v ′ = ω ( v ) . 7/26

Comparison principle Used by R. Conti (1956), F. Brauer, C. Corduneanu (1960s), many others. Not a new idea in applied mathematics; used in stability theory. MAIN IDEA: Given x ′ = f ( x ) , if a positive definite differentiable function V : R n → R satisfies the differential inequality V ′ ≤ ω ( V ) , where ω : R → R is an appropriate scalar function, one may infer the stability of x ′ = f ( x ) from the stability of the one-dimensional system v ′ = ω ( v ) . One obtains an abstraction of the system by another one-dimensional system. 7/26

Comparison theorem (scalar majorization) The comparison principle hinges on an appropriate comparison theorem . Theorem (Scalar comparison theorem) Let V ( t ) and v ( t ) be real valued functions differentiable on [0 , T ] . If V ′ ≤ ω ( V ) v ′ = ω ( v ) and holds on [0 , T ] for some locally Lipschitz continuous function ω and if V (0) = v (0) , then for all t ∈ [0 , T ] one has V ( t ) ≤ v ( t ) . Informally, Solutions to the ODE v ′ = ω ( v ) act as upper bounds (i.e. majorize ) solutions to V ′ ≤ ω ( V ) . 8/26

Comparison principle 1. Introduce a fresh variable v (really a function of time v ( t ) ), 9/26

Comparison principle 1. Introduce a fresh variable v (really a function of time v ( t ) ), 2. Replace the scalar differential inequality V ′ ≤ ω ( V ) by an equality. 9/26

Comparison principle 1. Introduce a fresh variable v (really a function of time v ( t ) ), 2. Replace the scalar differential inequality V ′ ≤ ω ( V ) by an equality. V ′ ≤ ω ( V ) v ′ = ω ( v ) − − − − − − − − − − − → 9/26

Comparison principle 1. Introduce a fresh variable v (really a function of time v ( t ) ), 2. Replace the scalar differential inequality V ′ ≤ ω ( V ) by an equality. V ′ ≤ ω ( V ) v ′ = ω ( v ) − − − − − − − − − − − → Obtain one-dimensional abstraction; 1-d systems are easy to study. ω ( v ) v 9/26

Kinds of barrier certificates Recall the (semantic) conditions: ∀ x ∈ Unsafe . B ( x ) > 0 , 1 � � ∀ x 0 ∈ Init . ∀ t ≥ 0 . ( ∀ τ ∈ [0 , t ] . x ( x 0 , τ ) ∈ Q ) ⇒ B ( x ( x 0 , t )) ≤ 0 . 2 Several direct sufficient conditions have been proposed to ensure the last requirement. Observe that the solutions x ( x 0 , t ) are not explicit. Convex Exponential-type ‘General’ (Prajna & Jadbabaie, 2004) (Kong et al., 2013) (Dai et al., 2017) Q → B ′ ≤ 0 . Q → B ′ ≤ λB. Q → B ′ ≤ ω ( B ) , ∀ t ≥ 0 . b ( t ) ≤ 0 , b is the solution to b ′ = ω ( b ) . All these conditions are instantiations of the comparison principle . 10/26

Convex barrier certificates (Prajna & Jadbabaie, 2004) Differential inequality B ′ ≤ 0 ω ( B ) 0 B Comparison system b ′ = 0 0 b 11/26

Exponential-type barrier certificates (Kong et al., 2013) Differential inequality B ′ ≤ λB ω ( B ) λB B Comparison system b ′ = λb 0 b 12/26