Advances in Reachability Analysis with Applications to Safety Verification of Vehicle Control Systems Matthias Althoff, Colas Le Guernic, and Bruce H. Krogh Carnegie Mellon University New York University April 27, 2011 Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 1 / 24
Introduction Safety Verification Using Reachable Sets unsafe set exemplary trajectory x 2 reachable set initial set x 1 System is safe, if no trajectory enters the unsafe set. Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 2 / 24
Introduction Safety Verification Using Reachable Sets unsafe set exemplary trajectory x 2 overapproximated initial set x 1 reachable set System is safe, if no trajectory enters the unsafe set. Overapproximated system is safe → real system is safe. Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 2 / 24
Introduction Main Innovations Consideration of Time-Varying Parameters for Linear Systems There is much work for linear time invariant (LTI) Systems; a wrapping-free algorithm exists [A. Girard, C. Le Guernic, O. Maler; HSCC 2006]. Here : The system matrix is uncertain and time-varying. Novel Linearization Approach for Nonlinear Systems Before : The linearization error is considered by an additional uncertain input. Here : The linearization error is considered by adding parameter uncertainties. Continuization of Hybrid Systems Before : Hybrid dynamics requires intersection of reachable sets with guard sets. Here : The intersection can be eliminated by temporarily enlarging the set of uncertain parameters. Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 3 / 24
Reachability Analysis of Linear Time Varying Systems Considered Class of Systems Linear systems with uncertain time varying parameters x ( t ) = A ( t ) x ( t ) + u ( t ) , ˙ where A : R + → A , u : R + → U are piecewise continuous, and A ⊂ R n × n , U ⊂ R n . For reachability analysis, we consider all possible functions A ( t ) and u ( t ). � [ − 1 . 05 , − 0 . 95] � [ − 4 . 05 , − 3 . 95] A = [3 . 95 , 4 . 05] [ − 1 . 05 , − 0 . 95] Example: � 1 � U = [ − 0 . 05 , 0 . 05] 1 Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 4 / 24
Reachability Analysis of Linear Time Varying Systems Overview of Reachable Set Computation Compute reachable set H ( r ) at time r when there is no input. 1 Input not yet considered. Obtain convex hull of initial set R (0) and H ( r ). 2 Curvature of trajectories not yet considered. Enlarge reachable set to account for (1) uncertain inputs, (2) curvature of 3 trajectories. Continue with further time intervals [ kr , ( k + 1) r ], k ∈ N . 4 H ( r ) convex hull of R (0), H ( r ) R ([0 , r ]) R (0) enlargement ➁ ➀ ➂ Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 5 / 24
Reachability Analysis of Linear Time Varying Systems Peano Baker Series Superposition principle: First, consider only the initial state solution x ( t ) = Φ( A ( τ ) , t ) x 0 , where Φ( A ( τ ) , t ) is referred to as the Peano Baker Series. Peano Baker Series � t � t � σ 1 Φ( A ( τ ) , t ) = I + A ( σ 1 ) d σ 1 + A ( σ 1 ) A ( σ 2 ) d σ 2 d σ 1 0 0 0 � t � σ 1 � σ 2 + A ( σ 1 ) A ( σ 2 ) A ( σ 3 ) d σ 3 d σ 2 d σ 1 + . . . 0 0 0 How to compute the set { Φ( A ( τ ) , t ) | A ( τ ) ∈ A} ? Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 6 / 24
Reachability Analysis of Linear Time Varying Systems Overapproximation of the Peano Baker Series � t 0 A ( σ i ) d σ i ≈ � k Time discretization: l i =1 A ( l i ∆)∆, t = k ∆ (Riemann 1 integration). Approximate Φ( A ( τ ) , t ) iteratively as k � ˜ Φ 1 ( A ( τ ) , k , ∆) = I + A ( l 1 ∆)∆ , l 1 =1 � i � k l 2 � � � Φ i ( A ( τ ) , k , ∆) = ˜ ˜ ∆ i , Φ i − 1 ( t , ∆) + . . . A ( l q ∆) l i =1 l 1 =1 q =1 i =1 � �� � � t � t � σ 1 Reminder: Φ( A ( τ ) , t ) = I + A ( σ 1 ) d σ 1 + A ( σ 1 ) A ( σ 2 ) d σ 2 d σ 1 + . . . 0 0 0 � �� � i =2 Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 7 / 24
Reachability Analysis of Linear Time Varying Systems Overapproximation of the Peano Baker Series � t 0 A ( σ i ) d σ i ≈ � k Time discretization: l i =1 A ( l i ∆)∆, t = k ∆ (Riemann 1 integration). Replace concrete matrices by sets of matrices. 2 Approximate Φ( A ( τ ) , t ) iteratively as k � ˜ Φ 1 ( A ( τ ) , k , ∆) = I + A ( l 1 ∆)∆ , l 1 =1 � �� � ∈ � k l 1=1 A ∆ � i � l 2 k � � � Φ i ( A ( τ ) , k , ∆) = ˜ ˜ ∆ i Φ i − 1 ( t , ∆) + . . . A ( l q ∆) , q =1 l i =1 l 1 =1 � �� � li =1 ... � l 2 l 1=1 A i ∆ i ∈ � k where ⊕ represents the Minkowski addition: A ⊕ B = { A + B | A ∈ A , B ∈ B} . Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 7 / 24
Reachability Analysis of Linear Time Varying Systems Overapproximation of the Peano Baker Series � t 0 A ( σ i ) d σ i ≈ � k Time discretization: l i =1 A ( l i ∆)∆, t = k ∆ (Riemann 1 integration). Replace concrete matrices by sets of matrices. 2 Apply distributivity of convex matrix sets: a A ⊕ b A = ( a + b ) A 3 Approximate Φ( A ( τ ) , t ) iteratively as k � ˜ Φ 1 ( A ( τ ) , k , ∆) ∈ I ⊕ A ∆ , l 1 =1 � �� � ⊆ CH ( A ) t l 2 k � � Φ i ( A ( τ ) , k , ∆) ∈ ˜ ˜ A i ∆ i Φ i − 1 ( t , ∆) ⊕ . . . , l i =1 l 1 =1 � �� � ⊆ 1 i ! CH ( A i ) t i =: M i ( t ) where CH () is the convex hull operator, which ensures that the distributivity law can be applied. Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 7 / 24
Reachability Analysis of Linear Time Varying Systems Overapproximation of the State Transition Matrix The expressions M i ( t ) are independent of ∆. For lim ∆ → 0 we have that Overapproximation of the state transition matrix ∞ M i ( t ) = t i � i ! CH ( A i ) . Φ( A ( τ ) , t ) ∈ M i ( t ) , i =0 Overapproximation of the state transition matrix: time invariant case � ∞ � � t i � � i ! A i Φ( A , t ) ∈ � A ∈ A . � i =0 Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 8 / 24
Reachability Analysis of Linear Time Varying Systems Overapproximation of the State Transition Matrix The expressions M i ( t ) are independent of ∆. For lim ∆ → 0 we have that Overapproximation of the state transition matrix ∞ M i ( t ) = t i � i ! CH ( A i ) . Φ( A ( τ ) , t ) ∈ M i ( t ) , i =0 Overapproximation of the state transition matrix by a finite sum η � Φ i ( A ( τ ) , t ) ∈ M i ( t ) ⊕ [ − W ( t ) , W ( t )] , i =0 W ( t ): remainder bound Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 8 / 24
Reachability Analysis of Linear Time Varying Systems Overview of Reachable Set Computation Compute reachable set H ( r ) at time r when there is no input. 1 done Obtain convex hull of initial set R (0) and H ( r ). 2 trivial Enlarge reachable set to account for (1) uncertain inputs ( next slide ), (2) 3 curvature of trajectories ( skipped ). Continue with further time intervals [ kr , ( k + 1) r ], k ∈ N . 4 H ( r ) convex hull of R (0), H ( r ) R ([0 , r ]) R (0) enlargement ➁ ➀ ➂ Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 9 / 24
Reachability Analysis of Linear Time Varying Systems Input Solution Removing the input The differential equation ˙ x ( t ) = A ( t ) x ( t ) + u ( t ) can be rewritten as � x ( t ) � � A ( t ) � � x ( t ) � d u ( t ) = dt 1 0 0 1 � �� � A u ( t ) . . . analogous proofs . . . Reachable set due to the input η � � t i +1 t � ( i + 1)! CH ( A i U ) P ( t ) = ⊕ η + 2[ − W ( t ) , W ( t )] {|U|} . i =0 Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 10 / 24
Reachability Analysis of Linear Time Varying Systems Typical Types of Sets for Reachability Analysis Analogous definitions for vector and matrix sets: Polytopes: Convex hull of vertices � r A α i v ( i ) � � � � � v ( i ) ∈ R n , α i ≥ 0 , � α i = 1 v ( i ) i =1 i Zonotopes: Minkowski sum of line segments l 2 l i = [ − 1 , 1] g ( i ) l 3 l 1 κ A p i g ( i ) � � � � g (0) + � g ( i ) ∈ R n , p i ∈ [ − 1 , 1] � i =1 Interval Vector a , a ∈ R n . [ a , a ] , ∀ i : a i ≤ a i , Althoff, Le Guernic, Krogh (CMU,NYU) Reachability Analysis of Vehicle Ctrl Systems April 27, 2011 11 / 24
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