Modeling and Analysis of Hybrid Systems
|
|
- Erick Edwards
- 5 years ago
- Views:
Transcription
1 Modeling and Analysis of Hybrid Systems 7. Linear hybrid automata II Prof. Dr. Erika Ábrahám Informatik 2 - LuFG Theory of Hybrid Systems RWTH Aachen University Szeged, Hungary, 27 September - 6 October 217 Ábrahám - Hybrid Systems 1 / 26
2 Contents 1 Reachability analysis algorithms and tools 2 Reachability analysis for linear hybrid automata I 3 Reachability analysis for linear hybrid automata II Ábrahám - Hybrid Systems 2 / 26
3 Contents 1 Reachability analysis algorithms and tools 2 Reachability analysis for linear hybrid automata I 3 Reachability analysis for linear hybrid automata II Ábrahám - Hybrid Systems 3 / 26
4 Some tools for hybrid automata reachability analysis Tool Ariadne C2D2 Cora dreach Flow HSolver HyCreate HyPro HyReach HySon isat-ode KeYmaera NLTOOLBOX SoapBox SpaceEx Characteristics non-linear ODEs; Taylor models, boxes; interval constraint propagation, deduction non-linear ODEs; guaranteed simulation non-linear ODEs; geometric representations; several algorithms, linear abstraction non-linear ODEs; logical representation; interval constraint propagation, δ-reachability, bounded model checking non-linear ODEs; Taylor models; owpipe construction non-linear ODEs; logical representation; interval constraint propagation non-linear ODEs; boxes; owpipe construction linear ODEs; several representations; owpipe construction linear ODEs; support functions; owpipe construction non-linear ODEs; guaranteed simulation non-linear ODEs; logical representation; interval constraint propagation, bounded model checking dierential dynamic logic; logical representation; theorem proving, computer algebra non-linear ODEs; Bernstein expansion, hybridisation linear ODEs; symbolic orthogonal projections; owpipe construction linear ODEs; geometric and symbolic representations; owpipe construction We will learn how owpipe-construction-based methods work. Flow and HyPro were/are developed in our group. Besides them, most closely related is the SpaceEx tool. Ábrahám - Hybrid Systems 4 / 26
5 Forward reachability computation Input: Set Init of initial states. Output: Set R of reachable states. Algorithm: R new := Init; R := ; while (R new ){ }; return R R := R R new ; R new := Reach(R new )\R; Ábrahám - Hybrid Systems 5 / 26
6 Most well-known state set representations Geometric objects: hyperrectangles [Moore et al., 29] oriented rectangular hulls [Stursberg et al., 23] convex polyhedra [Ziegler, 1995] [Chen at el, 211] orthogonal polyhedra [Bournez et al., 1999] template polyhedra [Sankaranarayanan et al., 28] ellipsoids [Kurzhanski et al., 2] zonotopes [Girard, 25]) Other symbolic representations: support functions [Le Guernic et al., 29] Taylor models [Berz and Makino, 1998, 29] [Chen et al., 212] Ábrahám - Hybrid Systems 6 / 26
7 Reminder: Polytopes Halfspace: set of points satisfying c T x z Polyhedron: an intersection of nitely many halfspaces Polytope: a bounded polyhedron representation union intersection Minkowski sum V-representation by vertices easy hard easy H-representation by facets hard easy hard Ábrahám - Hybrid Systems 7 / 26
8 Contents 1 Reachability analysis algorithms and tools 2 Reachability analysis for linear hybrid automata I 3 Reachability analysis for linear hybrid automata II Ábrahám - Hybrid Systems 8 / 26
9 Linear hybrid automata I Linear hybrid automata of type I (LHA I) are hybrid automata with the following restrictions: All derivatives are dened by intervals. All invariants and jump guards are dened by polytopes. All jump resets are dened by linear transformations of the form x := Ax + b. Hybrid automata of this type have linear behaviour, i.e., when time passes by in a location, the values of the variables evolve according to a linear function. (To be more precise, each reachable state can be reached by such a linear evolution.) Ábrahám - Hybrid Systems 9 / 26
10 Reminder: Minkowski sum x P x Q = x P Q x x x 1 P Q = {p + q p P and q Q} Ábrahám - Hybrid Systems 1 / 26
11 Linear hybrid automata I: Time evolution Ábrahám - Hybrid Systems 11 / 26
12 Linear hybrid automata I: Time evolution x 2 initial state set X x 1 Ábrahám - Hybrid Systems 11 / 26
13 Linear hybrid automata I: Time evolution x 2 initial state set X x 1 ẋ 2 derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
14 Linear hybrid automata I: Time evolution x 2 initial state set X x 1 ẋ 2 cone(q) derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
15 Linear hybrid automata I: Time evolution x 2 initial state set X x 1 ẋ 2 cone(q) derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
16 Linear hybrid automata I: Time evolution x 2 x 2 initial state set X x 1 x 1 ẋ 2 cone(q) derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
17 Linear hybrid automata I: Time evolution x 2 x 2 X cone(q) initial state set X x 1 x 1 ẋ 2 cone(q) derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
18 Linear hybrid automata I: Time evolution x 2 x 2 X cone(q) initial state set X x 1 x 1 ẋ 2 cone(q) derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
19 Linear hybrid automata I: Time evolution x 2 x 2 X cone(q) initial state set X x 1 x 1 ẋ 2 cone(q) derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
20 Linear hybrid automata I: Time evolution x 2 x 2 X cone(q) initial state set X x 1 x 1 ẋ 2 cone(q) derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
21 Linear hybrid automata I: Time evolution x 2 (X cone(q)) Inv(l) x 2 initial state set X x 1 x 1 ẋ 2 cone(q) derivatives Q ẋ 1 Ábrahám - Hybrid Systems 11 / 26
22 Linear hybrid automata I: Discrete steps (jumps) x 2 Example jump: ( l, 4 x 2 5, }{{} x 2 : [2, 4], }{{} l ) Edge G R [4,5] R R [2,4] P x 1 l Ábrahám - Hybrid Systems 12 / 26
23 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2, 4], }{{} l ) Edge G R [4,5] R R [2,4] P x 1 l Ábrahám - Hybrid Systems 12 / 26
24 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2, 4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 P G P x 1 x 1 l l Ábrahám - Hybrid Systems 12 / 26
25 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2, 4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 (P G) x1 P x 1 x 1 l l Ábrahám - Hybrid Systems 12 / 26
26 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2, 4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 ((P G) x1 ) R P x 1 x 1 l l Ábrahám - Hybrid Systems 12 / 26
27 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2, 4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 4 (((P G) x1 ) R) R P 2 x 1 x 1 l l Ábrahám - Hybrid Systems 12 / 26
28 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2, 4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 4 (((P G) x1 ) R) R P 2 x 1 x 1 l l Ábrahám - Hybrid Systems 12 / 26
29 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2, 4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 4 (((P G) x1 ) R) R P 2 x 1 x 1 l l Additionally, we intersect the result with the target location's invariant. Computed via projection, Minkowski sum and intersection. Ábrahám - Hybrid Systems 12 / 26
30 Contents 1 Reachability analysis algorithms and tools 2 Reachability analysis for linear hybrid automata I 3 Reachability analysis for linear hybrid automata II Ábrahám - Hybrid Systems 13 / 26
31 Linear hybrid automata II Linear hybrid automata of type II (LHA II) are hybrid automata with the following restrictions: All derivatives are dened by linear ordinary dierential equations (ODEs) of the form ẋ = Ax + Bu, where x = (x 1,..., x n ) T are the (continuous) variables, A is a matrix of dimension n n, u = (u 1,..., u m ) T are disturbance/input/control variables with rectangular domain U, and B is a matrix of dimension n m. All invariants and jump guards are dened by polytopes. All jump resets are dened by linear transformations of the form x := Ax + b. When time passes by in a location, the values of the continuous variables evolve according to linear ODEs. N.B.: Now the values follow non-linear functions! Ábrahám - Hybrid Systems 14 / 26
32 Approximating a owpipe Consider a dynamical system with state equation ẋ = f(x(t)). Ábrahám - Hybrid Systems 15 / 26
33 Approximating a owpipe Consider a dynamical system with state equation ẋ = f(x(t)). We assume f to be Lipschitz continuous. Ábrahám - Hybrid Systems 15 / 26
34 Approximating a owpipe Consider a dynamical system with state equation ẋ = f(x(t)). We assume f to be Lipschitz continuous. Wikipedia: Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is no greater than a denite real number [... ].. Ábrahám - Hybrid Systems 15 / 26
35 Approximating a owpipe Consider a dynamical system with state equation ẋ = f(x(t)). We assume f to be Lipschitz continuous. Wikipedia: Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is no greater than a denite real number [... ].. Lipschitz continuity implies the existence and uniqueness of the solution to an initial value problem, i.e., for every initial state x there is a unique solution x(t, x ) to the state equation. Ábrahám - Hybrid Systems 15 / 26
36 Approximating a owpipe The set of reachable states at time t from a set of initial states X is dened as R t (X ) = {x t x X. x t = x(t, x )}. Ábrahám - Hybrid Systems 16 / 26
37 Approximating a owpipe The set of reachable states at time t from a set of initial states X is dened as R t (X ) = {x t x X. x t = x(t, x )}. The set of reachable states, the owpipe, from X in the time interval [, T ] is dened as R [,T ] (X ) = t [,T ] R t (X ). Ábrahám - Hybrid Systems 16 / 26
38 Approximating a owpipe The set of reachable states at time t from a set of initial states X is dened as R t (X ) = {x t x X. x t = x(t, x )}. The set of reachable states, the owpipe, from X in the time interval [, T ] is dened as R [,T ] (X ) = t [,T ] R t (X ). We describe a solution which approximates the owpipe by a sequence of convex polytopes. Ábrahám - Hybrid Systems 16 / 26
39 Problem statement for polyhedral approximation of owpipes Given a set X of initial states which is a polytope, and a nal time T, compute a polyhedral approximation ˆR [,T ] (X ) to the owpipe R [,T ] (X ) such that R [,T ] (X ) ˆR [,T ] (X ). Ábrahám - Hybrid Systems 17 / 26
40 Flowpipe segmentation Since a single convex polyhedron would strongly overapproximate the owpipe, we compute a sequence of convex polyhedra, each approximating a owpipe segment. Ábrahám - Hybrid Systems 18 / 26
41 Segmented owpipe approximation Let the time interval [, T ] be divided into < N N time segments [, t 1 ], [t 1, t 2 ],..., [t N 1, T ] with t i = i δ for δ = T N. Ábrahám - Hybrid Systems 19 / 26
42 Segmented owpipe approximation Let the time interval [, T ] be divided into < N N time segments [, t 1 ], [t 1, t 2 ],..., [t N 1, T ] with t i = i δ for δ = T N. We generate an approximation ˆR [t1,t 2 ](X ) for each owpipe segment: R [t1,t 2 ](X ) ˆR [t1,t 2 ](X ). Ábrahám - Hybrid Systems 19 / 26
43 Segmented owpipe approximation Let the time interval [, T ] be divided into < N N time segments [, t 1 ], [t 1, t 2 ],..., [t N 1, T ] with t i = i δ for δ = T N. We generate an approximation ˆR [t1,t 2 ](X ) for each owpipe segment: R [t1,t 2 ](X ) ˆR [t1,t 2 ](X ). The complete owpipe approximation is the union of the approximation of all N pipe segments: R [,T ] (X ) ˆR [,T ] (X ) = ˆR [tk 1,t k ](X ) k=1,...,n Ábrahám - Hybrid Systems 19 / 26
44 Approaches Next we discuss one possible approach for owpipe approximation, but there are dierent other techniques, too. Ábrahám - Hybrid Systems 2 / 26
45 Linear hybrid automata II: Time evolution Ábrahám - Hybrid Systems 21 / 26
46 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Ábrahám - Hybrid Systems 21 / 26
47 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu R [,δ] R [δ,2δ] R [2δ,3δ] Ábrahám - Hybrid Systems 21 / 26
48 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i Ω R [,δ] R [δ,2δ] Ω 1 Ω 2 R [2δ,3δ] Ábrahám - Hybrid Systems 21 / 26
49 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Ábrahám - Hybrid Systems 21 / 26
50 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: δ 2δ t X Ábrahám - Hybrid Systems 21 / 26
51 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! e Aδ X δ 2δ t X Ábrahám - Hybrid Systems 21 / 26
52 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! e Aδ X δ 2δ t X convex hull of X e Aδ X Ábrahám - Hybrid Systems 21 / 26
53 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! e Aδ X δ 2δ t X bloating with B 1 to include non-linear behaviour Ábrahám - Hybrid Systems 21 / 26
54 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! e Aδ X δ 2δ t X (e Aδ X ) B 1 Ábrahám - Hybrid Systems 21 / 26
55 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! e Aδ X δ 2δ t X convex hull of X ((e Aδ X ) B 1 ) covers the behaviour R [,δ] under ẋ = Ax Ábrahám - Hybrid Systems 21 / 26
56 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! e Aδ X δ 2δ t X disturbance! Ábrahám - Hybrid Systems 21 / 26
57 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! e Aδ X δ 2δ t X bloating with B 2 Ábrahám - Hybrid Systems 21 / 26
58 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! e Aδ X δ 2δ t X (e Aδ X ) B 1 B 2 Ábrahám - Hybrid Systems 21 / 26
59 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! Ω e Aδ X δ 2δ t X Ω = conv(x ((e Aδ X ) B 1 B 2 )) covers the behaviour R [,δ] under ẋ = Ax + Bu Ábrahám - Hybrid Systems 21 / 26
60 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! Ω δ 2δ t Ábrahám - Hybrid Systems 21 / 26
61 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! The remaining ones: Ω δ 2δ t Ábrahám - Hybrid Systems 21 / 26
62 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! The remaining ones: Ω δ 2δ t e Aδ Ω covers the behaviour R [δ,2δ] under ẋ = Ax Ábrahám - Hybrid Systems 21 / 26
63 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! The remaining ones: Ω δ 2δ t e Aδ Ω disturbance! Ábrahám - Hybrid Systems 21 / 26
64 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! The remaining ones: Ω δ 2δ t e Aδ Ω bloating with B 2 Ábrahám - Hybrid Systems 21 / 26
65 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! The remaining ones: Ω δ 2δ t Ω 1 Ω 1 = (e Aδ Ω ) B 2 covers the behaviour R [δ,2δ] under ẋ = Ax + Bu Ábrahám - Hybrid Systems 21 / 26
66 Linear hybrid automata II: Time evolution Assume ẋ = Ax + Bu Compute polytopes Ω, Ω 1,... such that R [iδ,(i+1)δ] Ω i The rst owpipe segment: Reminder matrix exponential: e X = k= Xk k! The remaining ones: Ω δ 2δ t Ω 1 Ω 1 = (e Aδ Ω ) B 2 covers the behaviour R [δ,2δ] under ẋ = Ax + Bu Ábrahám - Hybrid Systems 21 / 26
67 Linear hybrid automata II: Discrete steps (jumps) The same procedure as for LHA I, extended with possibilities for aggregation and/or clustering. Ω 3 Ω 2 Ω 1 Ω Ábrahám - Hybrid Systems 22 / 26
68 Linear hybrid automata II: Discrete steps (jumps) The same procedure as for LHA I, extended with possibilities for aggregation and/or clustering. Ω 3 Ω 2 Ω 1 Ω Ábrahám - Hybrid Systems 22 / 26
69 Linear hybrid automata II: Discrete steps (jumps) The same procedure as for LHA I, extended with possibilities for aggregation and/or clustering. Ω 3 Ω 2 Ω 1 Ω Ábrahám - Hybrid Systems 22 / 26
70 Linear hybrid automata II: Discrete steps (jumps) The same procedure as for LHA I, extended with possibilities for aggregation and/or clustering. Ω 3 Ω 1 Ω 2 Ábrahám - Hybrid Systems 22 / 26
71 Linear hybrid automata II: Discrete steps (jumps) The same procedure as for LHA I, extended with possibilities for aggregation and/or clustering. Ω 3 Π 3 Ω 1 Ω 2 Π 1 Π 2 Ábrahám - Hybrid Systems 22 / 26
72 Linear hybrid automata II: Discrete steps (jumps) The same procedure as for LHA I, extended with possibilities for aggregation and/or clustering. Optionally aggregation (convex hull V 1 ) (alternative: clustering) Ω 3 Π 3 Ω 1 Ω 2 Π 1 Π 2 Ábrahám - Hybrid Systems 22 / 26
73 Linear hybrid automata II: The global picture Ábrahám - Hybrid Systems 23 / 26
74 Linear hybrid automata II: The global picture Ábrahám - Hybrid Systems 23 / 26
75 Linear hybrid automata II: The global picture Ábrahám - Hybrid Systems 23 / 26
76 Linear hybrid automata II: The global picture Ábrahám - Hybrid Systems 23 / 26
77 Linear hybrid automata II: The global picture Ábrahám - Hybrid Systems 23 / 26
78 Linear hybrid automata II: The global picture Ábrahám - Hybrid Systems 23 / 26
79 Linear hybrid automata II: The global picture Ábrahám - Hybrid Systems 23 / 26
80 Example Van der Pol equation: ẋ 1 = x 2 ẋ 2 =.2(x 2 1 1)x 2 x 1. Intial set: X = {(x 1, x 2 ).8 x 1 1 x 2 = }. Time: T = 1. Segments: 2 Ábrahám - Hybrid Systems 24 / 26
81 Other geometries for approximation Van der Pol equation with a third variable being a clock. Approximation with convex polyhedra and with oriented rectangular hull: Ábrahám - Hybrid Systems 25 / 26
82 Partitioning the initial set Var der Pol system with initial set X = {(x 1, x 2 ) 5 x 1 45 x 2 = }. Ábrahám - Hybrid Systems 26 / 26
Modeling and Analysis of Hybrid Systems
Modeling and Analysis of Hybrid Systems Linear hybrid automata II: Approximation of reachable state sets Prof. Dr. Erika Ábrahám Informatik 2 - Theory of Hybrid Systems RWTH Aachen University SS 2015 Ábrahám
More informationModeling and Analysis of Hybrid Systems
Modeling and Analysis of Hybrid Systems 5. Linear hybrid automata I Prof. Dr. Erika Ábrahám Informatik 2 - LuFG Theory of Hybrid Systems RWTH Aachen University Szeged, Hungary, 27 September - 06 October
More informationModeling and Analysis of Hybrid Systems Linear hybrid automata I Prof. Dr. Erika Ábrahám Informatik 2 - LuFG Theory of Hybrid Systems RWTH Aachen University Szeged, Hungary, 27 September - 06 October 2017
More informationUSING EIGENVALUE DECOMPOSITION
The present work was submitted to the LuFG Theory of Hybrid Systems BACHELOR OF SCIENCE THESIS USING EIGENVALUE DECOMPOSITION IN HYBRID SYSTEMS REACHABILITY ANALYSIS Jan Philipp Hafer Examiners: Prof.
More informationModeling and Analysis of Hybrid Systems
Modeling and Analysis of Hybrid Systems Algorithmic analysis for linear hybrid systems Prof. Dr. Erika Ábrahám Informatik 2 - Theory of Hybrid Systems RWTH Aachen University SS 2015 Ábrahám - Hybrid Systems
More informationAn Introduction to Hybrid Automata, Numerical Simulation and Reachability Analysis
An Introduction to Hybrid Automata, Numerical Simulation and Reachability Analysis Goran Frehse SyDe Summer School, September 10, 2015 Univ. Grenoble Alpes Verimag, 2 avenue de Vignate, Centre Equation,
More informationReachability Analysis of Hybrid Systems using Support Functions
Reachability Analysis of Hybrid Systems using Support Functions Colas Le Guernic 1 and Antoine Girard 2 1 Verimag, Université de Grenoble 2 Laboratoire Jean Kuntzmann, Université de Grenoble {Colas.Le-Guernic,Antoine.Girard}@imag.fr
More informationModel Checking of Hybrid Systems
Model Checking of Hybrid Systems Goran Frehse AVACS Autumn School, October 1, 2015 Univ. Grenoble Alpes Verimag, 2 avenue de Vignate, Centre Equation, 38610 Gières, France, Overview Hybrid Automata Set-Based
More informationUser s Manual of Flow* Version 2.0.0
User s Manual of Flow* Version 2.0.0 Xin Chen University of Colorado, Boulder 1 Introduction Flow* is a tool for safety verification of hybrid systems. Given a real-valued interval, a natural number m,
More informationNumerical Analysis and Reachability
Numerical Analysis and Reachability Hybrid Systems: Reachability Goran Frehse February 22, 2017 Ph.D. Course: Hybrid Systems Ph.D. in Information Technology Dipartimento di Elettronica, Informazione e
More informationWork in Progress: Reachability Analysis for Time-triggered Hybrid Systems, The Platoon Benchmark
Work in Progress: Reachability Analysis for Time-triggered Hybrid Systems, The Platoon Benchmark François Bidet LIX, École polytechnique, CNRS Université Paris-Saclay 91128 Palaiseau, France francois.bidet@polytechnique.edu
More informationDecidability Results for Probabilistic Hybrid Automata
Decidability Results for Probabilistic Hybrid Automata Prof. Dr. Erika Ábrahám Informatik 2 - Theory of Hybrid Systems RWTH Aachen SS09 - Probabilistic hybrid automata 1 / 17 Literatur Jeremy Sproston:
More informationEECS 144/244: System Modeling, Analysis, and Optimization
EECS 144/244: System Modeling, Analysis, and Optimization Continuous Systems Lecture: Hybrid Systems Alexandre Donzé University of California, Berkeley April 5, 2013 Alexandre Donzé: EECS 144/244 Hybrid
More informationDiscrete abstractions of hybrid systems for verification
Discrete abstractions of hybrid systems for verification George J. Pappas Departments of ESE and CIS University of Pennsylvania pappasg@ee.upenn.edu http://www.seas.upenn.edu/~pappasg DISC Summer School
More informationReachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, / 56
Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes Matthias Althoff Carnegie Mellon Univ. May 7, 2010 Reachability Analysis of Nonlinear and Hybrid Systems using Zonotopes May 7, 2010
More informationas support functions [18] and polynomials [34].
Decomposed Reachability Analysis for Nonlinear Systems Xin Chen University of Colorado, Boulder, CO xinchen@colorado.edu Sriram Sankaranarayanan University of Colorado, Boulder, CO srirams@colorado.edu
More informationEfficient Geometric Operations on Convex Polyhedra, with an Application to Reachability Analysis of Hybrid Systems
Efficient Geometric Operations on Convex Polyhedra, with an Application to Reachability Analysis of Hybrid Systems Willem Hagemann Abstract. We present a novel representation class for closed convex polyhedra,
More informationAnalysis of a Boost Converter Circuit Using Linear Hybrid Automata
Analysis of a Boost Converter Circuit Using Linear Hybrid Automata Ulrich Kühne LSV ENS de Cachan, 94235 Cachan Cedex, France, kuehne@lsv.ens-cachan.fr 1 Introduction Boost converter circuits are an important
More informationReachability Analysis: State of the Art for Various System Classes
Reachability Analysis: State of the Art for Various System Classes Matthias Althoff Carnegie Mellon University October 19, 2011 Matthias Althoff (CMU) Reachability Analysis October 19, 2011 1 / 16 Introduction
More informationReachability Analysis of Hybrid Systems Using Support Functions
Reachability Analysis of Hybrid Systems Using Support Functions Colas Le Guernic 1 and Antoine Girard 2 1 Verimag, Université de Grenoble 2 Laboratoire Jean Kuntzmann, Université de Grenoble {Colas.Le-Guernic,Antoine.Girard}@imag.fr
More informationThe algorithmic analysis of hybrid system
The algorithmic analysis of hybrid system Authors: R.Alur, C. Courcoubetis etc. Course teacher: Prof. Ugo Buy Xin Li, Huiyong Xiao Nov. 13, 2002 Summary What s a hybrid system? Definition of Hybrid Automaton
More informationHybrid systems and computer science a short tutorial
Hybrid systems and computer science a short tutorial Eugene Asarin Université Paris 7 - LIAFA SFM 04 - RT, Bertinoro p. 1/4 Introductory equations Hybrid Systems = Discrete+Continuous SFM 04 - RT, Bertinoro
More informationModel Predictive Real-Time Monitoring of Linear Systems
Model Predictive Real-Time Monitoring of Linear Systems Xin Chen and Sriram Sankaranarayanan University of Colorado, Boulder, CO Email: {xinchen,srirams}@colorado.edu Easy to Reuse * Consistent * Well
More informationScalable Static Hybridization Methods for Analysis of Nonlinear Systems
Scalable Static Hybridization Methods for Analysis of Nonlinear Systems Stanley Bak Air Force Research Laboratory Information Directorate, USA Taylor T. Johnson University of Texas at Arlington, USA Sergiy
More informationModeling & Control of Hybrid Systems. Chapter 7 Model Checking and Timed Automata
Modeling & Control of Hybrid Systems Chapter 7 Model Checking and Timed Automata Overview 1. Introduction 2. Transition systems 3. Bisimulation 4. Timed automata hs check.1 1. Introduction Model checking
More informationLarge-Scale Linear Systems from Order-Reduction (Benchmark Proposal)
EPiC Series in Computing Volume 43, 2017, Pages 60 67 ARCH16. 3rd International Workshop on Applied Verification for Continuous and Hybrid Systems Large-Scale Linear Systems from Order-Reduction (Benchmark
More informationSet- membership es-ma-on of hybrid dynamical systems.
Set- membership es-ma-on of hybrid dynamical systems. Towards model- based FDI for hybrid systems Prof. Nacim RAMDANI Université d Orléans, Bourges. France. nacim.ramdani@univ- orleans.fr!! ECC14 Pre-
More informationLarge-Scale Linear Systems from Order-Reduction (Benchmark Proposal)
Large-Scale Linear Systems from Order-Reduction (Benchmark Proposal) Hoang-Dung Tran, Luan Viet Nguyen, and Taylor T. Johnson v0.1, 2016-02-18 Abstract This benchmark suite is composed of nine examples
More informationSatisfiability Checking
Satisfiability Checking Interval Constraint Propagation Prof. Dr. Erika Ábrahám RWTH Aachen University Informatik 2 LuFG Theory of Hybrid Systems WS 14/15 Satisfiability Checking Prof. Dr. Erika Ábrahám
More informationVerification of Hybrid Systems
Verification of Hybrid Systems Goran Frehse Universite Grenoble 1, Verimag - with work from Thao Dang, Antoine Girard and Colas Le Guernic - MOVEP 08, June 25, 2008 1 Boeing & Tupolew Collision B757-200
More informationREGLERTEKNIK AUTOMATIC CONTROL LINKÖPING
Invariant Sets for a Class of Hybrid Systems Mats Jirstrand Department of Electrical Engineering Linkoping University, S-581 83 Linkoping, Sweden WWW: http://www.control.isy.liu.se Email: matsj@isy.liu.se
More informationc 2011 Kyoung-Dae Kim
c 2011 Kyoung-Dae Kim MIDDLEWARE AND CONTROL OF CYBER-PHYSICAL SYSTEMS: TEMPORAL GUARANTEES AND HYBRID SYSTEM ANALYSIS BY KYOUNG-DAE KIM DISSERTATION Submitted in partial fulfillment of the requirements
More informationVerification of Nonlinear Hybrid Systems with Ariadne
Verification of Nonlinear Hybrid Systems with Ariadne Luca Geretti and Tiziano Villa June 2, 2016 June 2, 2016 Verona, Italy 1 / 1 Outline June 2, 2016 Verona, Italy 2 / 1 Outline June 2, 2016 Verona,
More informationMath 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv
Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium
More informationVerification of Hybrid Systems with Ariadne
Verification of Hybrid Systems with Ariadne Davide Bresolin 1 Luca Geretti 2 Tiziano Villa 3 1 University of Bologna 2 University of Udine 3 University of Verona An open workshop on Formal Methods for
More informationAn Introduction to Hybrid Systems Modeling
CS620, IIT BOMBAY An Introduction to Hybrid Systems Modeling Ashutosh Trivedi Department of Computer Science and Engineering, IIT Bombay CS620: New Trends in IT: Modeling and Verification of Cyber-Physical
More informationThe partial-fractions method for counting solutions to integral linear systems
The partial-fractions method for counting solutions to integral linear systems Matthias Beck, MSRI www.msri.org/people/members/matthias/ arxiv: math.co/0309332 Vector partition functions A an (m d)-integral
More informationARCH-COMP18 Category Report: Continuous and Hybrid Systems with Linear Continuous Dynamics
EPiC Series in Computing Volume 54, 2018, Pages 23 52 ARCH18. 5th International Workshop on Applied Verification of Continuous and Hybrid Systems ARCH-COMP18 Category Report: Continuous and Hybrid Systems
More informationFrom Electrical Switched Networks to Hybrid Automata. Alessandro Cimatti 1, Sergio Mover 2, Mirko Sessa 1,3
1 2 3 From Electrical Switched Networks to Hybrid Automata Alessandro Cimatti 1, Sergio Mover 2, Mirko Sessa 1,3 Multidomain physical systems Multiple physical domains: electrical hydraulic mechanical
More informationAlgorithmic Algebraic Model Checking III: Approximate Methods
INFINITY 2005 Preliminary Version Algorithmic Algebraic Model hecking III: Approximate Methods Venkatesh Mysore 1,2 omputer Science epartment, ourant Institute New York University, New York, USA Bud Mishra
More informationHybrid Automata and ɛ-analysis on a Neural Oscillator
Hybrid Automata and ɛ-analysis on a Neural Oscillator A. Casagrande 1 T. Dreossi 2 C. Piazza 2 1 DMG, University of Trieste, Italy 2 DIMI, University of Udine, Italy Intuitively... Motivations: Reachability
More informationEllipsoidal Toolbox. TCC Workshop. Alex A. Kurzhanskiy and Pravin Varaiya (UC Berkeley)
Ellipsoidal Toolbox TCC Workshop Alex A. Kurzhanskiy and Pravin Varaiya (UC Berkeley) March 27, 2006 Outline Problem setting and basic definitions Overview of existing methods and tools Ellipsoidal approach
More informationAn Introduction to Hybrid Systems Modeling
CS620, IIT BOMBAY An Introduction to Hybrid Systems Modeling Ashutosh Trivedi Department of Computer Science and Engineering, IIT Bombay CS620: New Trends in IT: Modeling and Verification of Cyber-Physical
More informationAachen. Reachability Analysis of Non-Linear Hybrid Systems Using Taylor Models. Department of Computer Science. Technical Report.
Aachen Department of Computer Science Technical Report Reachability Analysis of Non-Linear Hybrid Systems Using Taylor Models Xin Chen ISSN 0935 3232 Aachener Informatik-Berichte AIB-2015-09 RWTH Aachen
More informationLinear Algebra Review: Linear Independence. IE418 Integer Programming. Linear Algebra Review: Subspaces. Linear Algebra Review: Affine Independence
Linear Algebra Review: Linear Independence IE418: Integer Programming Department of Industrial and Systems Engineering Lehigh University 21st March 2005 A finite collection of vectors x 1,..., x k R n
More informationAPPROXIMATING SWITCHED CONTINUOUS SYSTEMS BY RECTANGULAR AUTOMATA
European Control Conference 99, Karlsruhe (Germany), August 31 st - September 3 rd, 1999 APPROXIMATING SWITCHED CONTINUOUS SYSTEMS BY RECTANGULAR AUTOMATA O. Stursberg, S. Kowalewski Keywords: Approximation,
More informationOn the Chvatál-Complexity of Binary Knapsack Problems. Gergely Kovács 1 Béla Vizvári College for Modern Business Studies, Hungary
On the Chvatál-Complexity of Binary Knapsack Problems Gergely Kovács 1 Béla Vizvári 2 1 College for Modern Business Studies, Hungary 2 Eastern Mediterranean University, TRNC 2009. 1 Chvátal Cut and Complexity
More informationLecture 6: Reachability Analysis of Timed and Hybrid Automata
University of Illinois at Urbana-Champaign Lecture 6: Reachability Analysis of Timed and Hybrid Automata Sayan Mitra Special Classes of Hybrid Automata Timed Automata ß Rectangular Initialized HA Rectangular
More informationAutomata-theoretic analysis of hybrid systems
Automata-theoretic analysis of hybrid systems Madhavan Mukund SPIC Mathematical Institute 92, G N Chetty Road Chennai 600 017, India Email: madhavan@smi.ernet.in URL: http://www.smi.ernet.in/~madhavan
More informationARCH-COMP17 Category Report: Continuous and Hybrid Systems with Linear Continuous Dynamics
EPiC Series in Computing Volume 48, 2017, Pages 143 159 ARCH17. 4th International Workshop on Applied Verification of Continuous and Hybrid Systems ARCH-COMP17 Category Report: Continuous and Hybrid Systems
More informationHigh-Order Representation of Poincaré Maps
High-Order Representation of Poincaré Maps Johannes Grote, Martin Berz, and Kyoko Makino Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA. {grotejoh,berz,makino}@msu.edu
More informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 3: Linear Programming, Continued Prof. John Gunnar Carlsson September 15, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 15, 2010
More informationAPPROXIMATE SIMULATION RELATIONS FOR HYBRID SYSTEMS 1. Antoine Girard A. Agung Julius George J. Pappas
APPROXIMATE SIMULATION RELATIONS FOR HYBRID SYSTEMS 1 Antoine Girard A. Agung Julius George J. Pappas Department of Electrical and Systems Engineering University of Pennsylvania Philadelphia, PA 1914 {agirard,agung,pappasg}@seas.upenn.edu
More informationNUMERICAL ANALYSIS 2 - FINAL EXAM Summer Term 2006 Matrikelnummer:
Prof. Dr. O. Junge, T. März Scientific Computing - M3 Center for Mathematical Sciences Munich University of Technology Name: NUMERICAL ANALYSIS 2 - FINAL EXAM Summer Term 2006 Matrikelnummer: I agree to
More informationStep Simulation Based Verification of Nonlinear Deterministic Hybrid System
Step Simulation Based Verification of Nonlinear Deterministic Hybrid System Ratnesh Kumar, Professor, IEEE Fellow PhD Student: Hao Ren Electrical and Computer Engineering Iowa State University Verification
More informationVerifying Safety Properties of Hybrid Systems.
Verifying Safety Properties of Hybrid Systems. Sriram Sankaranarayanan University of Colorado, Boulder, CO. October 22, 2010. Talk Outline 1. Formal Verification 2. Hybrid Systems 3. Invariant Synthesis
More informationA Automatic Synthesis of Switching Controllers for Linear Hybrid Systems: Reachability Control
A Automatic Synthesis of Switching Controllers for Linear Hybrid Systems: Reachability Control Massimo Benerecetti, University of Naples Federico II, Italy Marco Faella, University of Naples Federico II,
More informationRigorous Simulation-Based Analysis of Linear Hybrid Systems
Rigorous Simulation-Based Analysis of Linear Hybrid Systems Stanley Bak 1 and Parasara Sridhar Duggirala 2 1 Air Force Research Laboratory 2 University of Connecticut Abstract. Design analysis of Cyber-Physical
More informationLimited-Information Control of Hybrid Systems via Reachable Set Propagation
Limited-Information Control of Hybrid Systems via Reachable Set Propagation Daniel Liberzon Coordinated Science Laboratory University of Illinois at Urbana-Champaign Urbana, IL 61821, U.S.A. liberzon@uiuc.edu
More informationParametric Verification and Test Coverage for Hybrid Automata Using the Inverse Method
Parametric Verification and Test Coverage for Hybrid Automata Using the Inverse Method Laurent Fribourg and Ulrich Kühne LSV ENS de Cachan, 94235 Cachan, France {kuehne,fribourg}@lsv.ens-cachan.fr Abstract.
More informationLinear Programming Inverse Projection Theory Chapter 3
1 Linear Programming Inverse Projection Theory Chapter 3 University of Chicago Booth School of Business Kipp Martin September 26, 2017 2 Where We Are Headed We want to solve problems with special structure!
More informationMinimizing Cubic and Homogeneous Polynomials over Integers in the Plane
Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane Alberto Del Pia Department of Industrial and Systems Engineering & Wisconsin Institutes for Discovery, University of Wisconsin-Madison
More informationCEGAR:Counterexample-Guided Abstraction Refinement
CEGAR: Counterexample-guided Abstraction Refinement Sayan Mitra ECE/CS 584: Embedded System Verification November 13, 2012 Outline Finite State Systems: Abstraction Refinement CEGAR Validation Refinment
More information7. F.Balarin and A.Sangiovanni-Vincentelli, A Verication Strategy for Timing-
7. F.Balarin and A.Sangiovanni-Vincentelli, A Verication Strategy for Timing- Constrained Systems, Proc. 4th Workshop Computer-Aided Verication, Lecture Notes in Computer Science 663, Springer-Verlag,
More informationSymbolic Orthogonal Projections: A New Polyhedral Representation for Reachability Analysis of Hybrid Systems
Symbolic Orthogonal Projections: A New Polyhedral Representation for Reachability Analysis of Hybrid Systems Willem Hagemann Dissertation zur Erlangung des Grades des Doktors der Ingenieurwissenschaften
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear
More informationGeometric Programming Relaxations for Linear System Reachability
Geometric Programg Relaxations for Linear System Reachability Hakan Yazarel and George J. Pappas Abstract One of the main obstacles in the safety analysis of continuous and hybrid systems has been the
More informationHybrid Systems - Lecture n. 3 Lyapunov stability
OUTLINE Focus: stability of equilibrium point Hybrid Systems - Lecture n. 3 Lyapunov stability Maria Prandini DEI - Politecnico di Milano E-mail: prandini@elet.polimi.it continuous systems decribed by
More informationOperational models of piecewise-smooth systems
Operational models of piecewise-smooth systems Andrew Sogokon Carnegie Mellon University Pittsburgh, PA, USA asogokon@cs.cmu.edu Khalil Ghorbal INRIA Rennes, France khalil.ghorbal@inria.fr Taylor T. Johnson
More informationCalculus and linear algebra for biomedical engineering Week 3: Matrices, linear systems of equations, and the Gauss algorithm
Calculus and linear algebra for biomedical engineering Week 3: Matrices, linear systems of equations, and the Gauss algorithm Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen
More informationEigenvalue problems and optimization
Notes for 2016-04-27 Seeking structure For the past three weeks, we have discussed rather general-purpose optimization methods for nonlinear equation solving and optimization. In practice, of course, we
More informationReachable set computation for solving fuel consumption terminal control problem using zonotopes
Reachable set computation for solving fuel consumption terminal control problem using zonotopes Andrey F. Shorikov 1 afshorikov@mail.ru Vitaly I. Kalev 2 butahlecoq@gmail.com 1 Ural Federal University
More informationA Rigorous Implementation of Taylor Model-Based Integrator in Chebyshev Basis Representation 1
A Rigorous Implementation of Taylor Model-Based Integrator in Chebyshev Basis Representation 1 Tomáš Dzetkulič Institute of Computer Science Czech Academy of Sciences December 15, 2011 1 This work was
More informationParameter iden+fica+on with hybrid systems in a bounded- error framework
Parameter iden+fica+on with hybrid systems in a bounded- error framework Moussa MAIGA, Nacim RAMDANI, & Louise TRAVE- MASSUYES Université d Orléans, Bourges, and LAAS CNRS Toulouse, France.!! SWIM 2015,
More informationA Decidable Class of Planar Linear Hybrid Systems
A Decidable Class of Planar Linear Hybrid Systems Pavithra Prabhakar, Vladimeros Vladimerou, Mahesh Viswanathan, and Geir E. Dullerud University of Illinois at Urbana-Champaign. Abstract. The paper shows
More informationIntrinsic diculties in using the. control theory. 1. Abstract. We point out that the natural denitions of stability and
Intrinsic diculties in using the doubly-innite time axis for input-output control theory. Tryphon T. Georgiou 2 and Malcolm C. Smith 3 Abstract. We point out that the natural denitions of stability and
More informationAutomatic Generation of Polynomial Invariants for System Verification
Automatic Generation of Polynomial Invariants for System Verification Enric Rodríguez-Carbonell Technical University of Catalonia Talk at EPFL Nov. 2006 p.1/60 Plan of the Talk Introduction Need for program
More informationMTH 306 Spring Term 2007
MTH 306 Spring Term 2007 Lesson 3 John Lee Oregon State University (Oregon State University) 1 / 27 Lesson 3 Goals: Be able to solve 2 2 and 3 3 linear systems by systematic elimination of unknowns without
More informationHyTech: A Model Checker for Hybrid Systems y. Thomas A. Henzinger Pei-Hsin Ho Howard Wong-Toi
HyTech: A Model Checker for Hybrid Systems y Thomas A. Henzinger Pei-Hsin Ho Howard Wong-Toi EECS Department Strategic CAD Labs Cadence Berkeley Labs Univ. of California, Berkeley Intel Corp., Hillsboro,
More informationSafe Over- and Under-Approximation of Reachable Sets for Delay Differential Equations
Safe Over- and Under-Approximation of Reachable Sets for Delay Differential Equations Bai Xue 1, Peter N. Mosaad 1, Martin Fränzle 1, Mingshuai Chen 2,3, Yangjia Li 2, and Naijun Zhan 2,3 1 Dpt. of Computing
More informationAutomatic reachability analysis for nonlinear hybrid models with C2E2
Automatic reachability analysis for nonlinear hybrid models with C2E2 Chuchu Fan 1, Bolun Qi 1, Sayan Mitra 1, Mahesh Viswanathan 1, and Parasara Sridhar Duggirala 2 1 University of Illinois, Urbana-Champaign
More informationEffective Synthesis of Switching Controllers for Linear Systems
PROCEEDINGS OF THE IEEE 1 Effective Synthesis of Switching Controllers for Linear Systems Eugene Asarin, Olivier Bournez, Thao Dang, Oded Maler and Amir Pnueli Abstract In this work we suggest a novel
More informationAlgorithmic Lie Symmetry Analysis and Group Classication for Ordinary Dierential Equations
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 1 / 25 Algorithmic Lie Symmetry Analysis and Group Classication for Ordinary Dierential Equations Dmitry A. Lyakhov 1 1 Computational Sciences
More informationVerification of analog and mixed-signal circuits using hybrid systems techniques
FMCAD, November 2004, Austin Verification of analog and mixed-signal circuits using hybrid systems techniques Thao Dang, Alexandre Donze, Oded Maler VERIMAG Grenoble, France Plan 1. Introduction 2. Verification
More informationHybrid Systems Course Lyapunov stability
Hybrid Systems Course Lyapunov stability OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata OUTLINE Focus: stability
More informationDiscrete Mathematics and Logic II. Regular Sets
Discrete Mathematics and Logic II. Regular Sets SFWR ENG 2FA3 Ryszard Janicki Winter 24 Acknowledgments: Material based on Automata and Computability by Dexter C. Kozen (Chapter 4). Ryszard Janicki Discrete
More informationNecessary and Sufficient Conditions for Reachability on a Simplex
Necessary and Sufficient Conditions for Reachability on a Simplex Bartek Roszak a, Mireille E. Broucke a a Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto,
More informationG : Statistical Mechanics Notes for Lecture 3 I. MICROCANONICAL ENSEMBLE: CONDITIONS FOR THERMAL EQUILIBRIUM Consider bringing two systems into
G25.2651: Statistical Mechanics Notes for Lecture 3 I. MICROCANONICAL ENSEMBLE: CONDITIONS FOR THERMAL EQUILIBRIUM Consider bringing two systems into thermal contact. By thermal contact, we mean that the
More informationChapter 1. Preliminaries
Introduction This dissertation is a reading of chapter 4 in part I of the book : Integer and Combinatorial Optimization by George L. Nemhauser & Laurence A. Wolsey. The chapter elaborates links between
More informationESC794: Special Topics: Model Predictive Control
ESC794: Special Topics: Model Predictive Control Discrete-Time Systems Hanz Richter, Professor Mechanical Engineering Department Cleveland State University Discrete-Time vs. Sampled-Data Systems A continuous-time
More informationComputable Analysis, Hybrid Automata, and Decision Procedures (Extended Thesis Abstract)
Computable Analysis, Hybrid Automata, and Decision Procedures (Extended Thesis Abstract) Sicun Gao 1 Introduction 1.1 Overview Cyber-Physical Systems (CPS) are systems that integrate digital computation
More informationTEMPORAL LOGIC [1], [2] is the natural framework for
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 1, FEBRUARY 2008 287 A Fully Automated Framework for Control of Linear Systems from Temporal Logic Specifications Marius Kloetzer, Student Member, IEEE,
More informationA Finite Element Method for an Ill-Posed Problem. Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D Halle, Abstract
A Finite Element Method for an Ill-Posed Problem W. Lucht Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D-699 Halle, Germany Abstract For an ill-posed problem which has its origin
More informationOptimization WS 13/14:, by Y. Goldstein/K. Reinert, 9. Dezember 2013, 16: Linear programming. Optimization Problems
Optimization WS 13/14:, by Y. Goldstein/K. Reinert, 9. Dezember 2013, 16:38 2001 Linear programming Optimization Problems General optimization problem max{z(x) f j (x) 0,x D} or min{z(x) f j (x) 0,x D}
More information3.1 Basic properties of real numbers - continuation Inmum and supremum of a set of real numbers
Chapter 3 Real numbers The notion of real number was introduced in section 1.3 where the axiomatic denition of the set of all real numbers was done and some basic properties of the set of all real numbers
More informationStatic-Dynamic Analysis of Security Metrics
Static-Dynamic Analysis of Security Metrics for Cyber-Physical Systems Sayan Mitra (PI), Geir Dullerud (co-pi), Swarat Chaudhuri (co-pi) University of Illinois at Urbana Champaign NSA SoS Quarterly meeting,
More information3. Linear Programming and Polyhedral Combinatorics
Massachusetts Institute of Technology 18.433: Combinatorial Optimization Michel X. Goemans February 28th, 2013 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the introductory
More informationSTABILITY ANALYSIS OF SWITCHED COUPLED DYNAMICAL SYSTEMS MAHESH SUNKULA (08023)
STABILITY ANALYSIS OF SWITCHED COUPLED DYNAMICAL SYSTEMS A REPORT submitted in partial fulllment of the requirements for the award of the dual degree of Bachelor of Science-Master of Science in MATHEMATICS
More informationVerifying Global Convergence for a Digital Phase-Locked Loop
Verifying Global Convergence for a Digital Phase-Locked Loop Jijie Wei & Yan Peng & Mark Greenstreet & Grace Yu University of British Columbia Vancouver, Canada October 22, 2013 Wei & Peng & Greenstreet
More informationAN INTRODUCTION TO CONVEXITY
AN INTRODUCTION TO CONVEXITY GEIR DAHL NOVEMBER 2010 University of Oslo, Centre of Mathematics for Applications, P.O.Box 1053, Blindern, 0316 Oslo, Norway (geird@math.uio.no) Contents 1 The basic concepts
More information