Set-Oriented Dimension Reduction For Reachability Computations Bai Xue and Martin Fr¨ anzle Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de November 3, 2016 Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 1 / 32
Outline Background 1 Preliminaries 2 Reachable Sets Computation 3 Examples 4 Discussions and Comparisons 5 Conclusion and Future Work 6 Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 2 / 32
Background Cyber-Physical Systems: a new generation of systems with integrated computational and physical capabilities that can interact with humans through many new modalities Applications : Space vehicles 1 Hybrid gas-electric vehicles 2 Fully autonomous urban driving 3 . . . 4 Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 3 / 32
Background Cyber-physical systems are routinely subject to deviations coming from sensor noise 1 actuation noise 2 delays over unreliable network channels 3 . . . 4 ⇓ Requirement: Robustness ⇓ system acts safely in the context of disturbances Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 4 / 32
Background Reachability Analysis: the process of computing the set of reachable states for a system with uncertain initial states, parameters and inputs ⇓ Validate this robustness property of cyber-physical systems Problems inherent in Reachability Analysis: Wrapping Effects [J. Lunze and F. Lamnabhi-Lagarrigue (2009)] Definition 1.1 The wrapping effect is the propagation and accumulation of over-approximation error through the iterative computation in the construction of reachable sets. ⇓ Poorly Pessimistic Approximation Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 5 / 32
Background Observations and Difficulties for Overcoming Wrapping Effects Observations: The extent of wrapping effects correlates strongly with the volume of 1 the initial set Classical Techniques: Partition the initial state space Difficulties: Such partitions ⇒ 2 extensive demand on computation time and memory, especially for large 1 initial sets and/or large time horizons the number of partitions: exponential increase with the dimensions of 2 systems Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 6 / 32
Background Facilitating the reduction of the wrapping effect in applying existing reachability analysis techniques to perform reachability computations: A set-oriented dimension reduction based method Its Features: a collection of subsets extracted from the initial set in a low dimensional space is 1 explored for computations the number of subsets increases polynomially with the dimensions of systems 2 Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 7 / 32
Outline Background 1 Preliminaries 2 Reachable Sets Computation 3 Examples 4 Discussions and Comparisons 5 Conclusion and Future Work 6 Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 8 / 32
2.1 Problem of Interest Consider the system modelled by an ordinary differential equation: x = f ( x ) , x (0) = x 0 = ( x 1 , 0 , . . . , x n , 0 ) ′ ∈ X 0 , ˙ (1) where f ( · ) = ( f 1 , . . . , f n ) ′ : R n �− → R n is Lipschitz continuous; 1 the trajectory over the time interval [0 , T ] is defined to be φ ( t ; x 0 , 0) = x ( t ), in 2 which x ( t ) is the solution to the system (1) with the initial condition x 0 ∈ X 0 at time instant t 0 and φ ( · ) = ( φ 1 ( · ) , . . . , φ n ( · )) ′ . Definition 2.1 (Problem of Interest) Compute a set Ψ( t ; X 0 , t 0 ) such that this set encloses all states reached by system (1) initialised in the set X 0 of the interval form for the duration of t ∈ [0 , T ]. Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 9 / 32
2.2 Sensitivity Analysis Theory The sensitivity to initial conditions at time t is defined by s x 0 ( t ) = ∂ φ ( t ; x 0 , t 0 ) , ∂ x 0 where s x 0 is a square matrix of order n . Sensitivity Equation [A. Donz´ e and O. Maler (2007)]: s x 0 ( t ) = D f s x 0 , s x 0 (0) = I ∈ R n × n , ˙ (2) where D f is the Jacobian matrix of f along the trajectory φ ( t ; x 0 , t 0 ) and I is the identity matrix. Remark: The ij th element of s x 0 basically represents the influence of variations in the i th coordinate x i , 0 of x 0 on the j th coordinate x j ( t ) of φ ( t ; x 0 , t 0 ). Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 10 / 32
2.3 Interval Linear Systems An interval linear system is of the form: y = A y , y ( t 0 ) = y 0 ∈ Y 0 , ˙ where y ∈ R n , Y 0 ⊂ R n and A is a n × n matrix, the elements of which are independent intervals of real numbers. Theorem 2.2 (M. Althoff et al. (2007)) The over-approximation of the interval matrix exponential with order p, denoted ⌈ e A t ⌉ , where A ∈ I n × n , the elements of which p are independent intervals of real numbers, and t ≥ 0 , is obtained by p (( �A� ∞ ) t ) p +1 1 1 �A� ∞ t ( A t ) i + E ( t ) with E ( t ) = < − 1 , 1 > ⌈ e A t � ⌉ = I + W ( t ) + , ǫ = < 1 , p i ! ( p + 1)! 1 − ǫ p + 2 i =3 2 A 2 t 2 can be computed exactly by the following procedure using interval arithmetic: ∀ i � = j : w ij = where W ( t ) = A t + 1 a ii ( t + 1 2 ( a ii + a jj ) t 2 ) + 1 k : k � = i , k � = j a ik a kj t 2 , ∀ i : w ii = [ κ ( a ii , t ) , max ( { a ii t + 1 2 a 2 ii t 2 , a ii t + 1 2 a 2 ii t 2 } )] + 1 k : k � = i a ik a ki t 2 , � � 2 2 � min ( { a ii t + 1 2 a 2 ii t 2 , a ii t + 1 2 a 2 ii t 2 } ) , − 1 t / ∈ a ii κ ( a ii , t ) = and �A� ∞ = � max ( | A | , | A | ) � ∞ , in which the absolute − 1 2 , − 1 t ∈ a ii value and the maximum in the above expression are determined elementwise. Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 11 / 32
Outline Background 1 Preliminaries 2 Reachable Sets Computation 3 Examples 4 Discussions and Comparisons 5 Conclusion and Future Work 6 Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 12 / 32
3 Reachable Sets Computation The sensitivity matrix s x 0 = ∂φ 1 ( t ; x 0 , 0) ∂φ 1 ( t ; x 0 , 0) ∂φ 1 ( t ; x 0 , 0) . . . ∂ x 1 , 0 ∂ x 2 , 0 ∂ x n , 0 ∂φ 2 ( t ; x 0 , 0) ∂φ 2 ( t ; x 0 , 0) ∂φ 2 ( t ; x 0 , 0) . . . ∂ x 1 , 0 ∂ x 2 , 0 ∂ x n , 0 . . . ∂φ n ( t ; x 0 , 0) ∂φ n ( t ; x 0 , 0) ∂φ n ( t ; x 0 , 0) . . . ∂ x 1 , 0 ∂ x 2 , 0 ∂ x n , 0 Underlying Theory Behind Our Method: Theorem 3.1 Given a system (1) with an initial set X 0 = [ x 1 , 0 , x 1 , 0 ] × . . . × [ x n , 0 , x n , 0 ] ⊂ R n and a time instant t ∈ [ t 0 , t 1 ] , if ∂φ i ( t ; x 0 , t 0 ) , j = l i , . . . , m i , i ∈ { 1 , . . . , n } , has an uniquely ∂ x j , 0 determined sign for all x 0 = ( x 1 , 0 , . . . , x n , 0 ) ′ ∈ X 0 , where { l i , . . . , m i } ⊆ { 1 , . . . , n } and l i , m i are positive integers with 1 ≤ l i ≤ m i ≤ n, then there exist two sets Y i , 0 and Z i , 0 of dimension n − k i such that the mapping φ i ( t ; X 0 , t 0 ) can be bounded by performing computations on these two sets respectively, where k i is the number of the elements in the set { l i , . . . , m i } . Bai Xue and Martin Fr¨ anzle (Universit¨ at Oldenburg, Germany { bai.xue, martin.fraenzle } @uni-oldenburg.de) Set-Oriented Dimension Reduction For Reachability Computations November 3, 2016 13 / 32
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