Hybrid Systems Modeling, Analysis and Control Radu Grosu Vienna University of Technology Lecture 10
HA with Affine Dynamics Flows of continuous variables: x = A x + B u Invariants and guards: A x ≤ c Actions: x : = A x Symbolic representation: convex sets (e.g. polytopes) Reachability: A semi algorithm Methodolgy: Exact time elapse only at discrete time Tools: SpaceEx
Bouncing Ball: HA with Affine Dynamics x = x 0 , v = 0 initial condition location freeFall x ≥ 0 invariant x = v, v = − g discrete transition flow label bounce: guard x = 0 ∧ v < 0 action ′ v = − c v
Bouncing Ball: Associated Program x = x 0 , v = 0 initial condition location freeFall x ≥ 0 invariant x = v, v = − g discrete transition flow label bounce: guard x = 0 action float x = x 0 , v = 0; d = d 0 ; // initial condition while true { // main loop while ( x ≥ 0) { // free fall x = x + v ∗ d ; // ! x = v v = v − g ∗ d ; } // ! v = − g v = − c ∗ v ; x = 0; } // bounce
Linear Dynamics Autonomous part of dynamics: x = A x , x ∈ R n Analytic solution: x( t ) = e A t x 0 Time discretization: x( t ) t = δ k x 3 x( δ ( k + 1 )) = e A δ x( δ k ) x 2 Recursive DT-Solution: x 1 x 0 x( 0 ) = x 0 x( k + 1 ) = e A δ x( k ) δ 0 2 δ 3 δ t Multiplication with constant matrix = linear transformation x = M x As a program instruction
Linear DT-Dynamics from Initial Set Recursive DT-Solution: Reach [0 , 3 δ ] ( X 0 ) X( 0 ) = X 0 X( t ) X 3 X( k + 1 ) = e A δ X( k ) X 2 X 1 Purely continuous systems: X 0 i Discrete solution acceptable i x (t) is in ε ( δ )-neighbourhood of some X k in the DT-solution δ 0 2 δ 3 δ t Not acceptable for hybrid systems: i Discrete transitions may fire between sampling times i If transitions are missed, x (t) is not in ε ( δ )-neighbourhood
Bouncing Ball: Error in the DT-Solution X 90
Polyhedral Flow-Pipe Approximation Divide: Reach [0,N δ ] ( X 0 ) into N δ -segments X( t ) t δ 0 2 δ 3 δ
Polyhedral Flow-Pipe Approximation Divide: Reach [0,N δ ] ( X 0 ) into N δ -segments X( t ) Enclose: segments with convex polytopes Ω i Ω 2 Ω 1 Ω 0 t δ 0 2 δ 3 δ
Polyhedral Flow-Pipe Approximation App ( X 0 ) Reach [0,3 δ ] Divide: Reach [0,N δ ] ( X 0 ) into N δ -segments X( t ) Enclose: segments with convex polytopes Ω i Ω 2 App ( X 0 ) = ∪ i = 0 N − 1 Ω i Ω 1 Union: Reach [0,N δ ] Ω 0 App ( X 0 ) Approx: Reach [0,N δ ] ( X 0 ) ⊆ Reach [0,N δ ] t Works for general nonlinear systems! δ 0 2 δ 3 δ
Wrapping Hyper-Planes Around a Set c 2 c 1 Step 1 : S − Choose: directions c 1 ...c n c 3 − Construct: C T = [c 1 ...c n ] c 4
Wrapping Hyper-Planes Around a Set x 2 c i c i .x = c i T x Step 2 : = x || x ||cos θ = − Compute optimal d: in C T x ≤ d d i S where C T = [c 1 ...c 4 ] and x ∈ S − Stepwise: For each direction i d i = max x ∈ S c i .x || x || θ x 1
Wrapping Hyper-Planes Around a Set c 2 c 1 Step 2 : − Compute optimal d: in C T x ≤ d S where C T = [c 1 ...c 4 ] and x ∈ S c 3 − Stepwise: For each direction i c 4 d i = max x ∈ S c i .x
Wrapping a Reached Set Given directions c 1 ...c n , wrap Reach [ t k ,t k + 1 ] ( X 0 ) within a polytope by solving for each i the optimisation problem d i = max T x ( t, x 0 ) x 0 ,t c i x 0 ∈ X 0 t ∈ [ t k ,t k + 1 ] Optimization problem is solved by embedding simulation into objective function computation
Flow Pipe Segment Approximation Step 1: Step 2: Vertices( X 0 ) Solve optimization a. Simulate trajectories at t k problem for each d i from each vertex of X 0 b. Take convex hull and identify outward normal vectors c i c 1 c 1 at t k + 1 Vertices( X 0 ) Flow-pipe segment approximated by { } T x ≤ d i , ∀ i x | c i
Improvements for Linear Systems App ( X 0 ) Step 1: Reach [0,3 δ ] i Compute Ω 0 as discussed before, but X( t ) Ω 2 X( t ) = e A t X 0 Ω 1 Step 2: Ω 0 i Compute Ω k+1 by using the recurrence: t Ω k+1 = e A δ Ω k (proof in Girard's Thesis) δ 0 2 δ 3 δ
Computation of Ω 0 x 00 e A δ X 1 x 01 e A δ x 00 x 01 X 0 Reach [0, δ ] ( X 0 ) Conv( X 0 , X 1 ) Bloat(Conv( X 0 , X 1 ))
Bouncing Ball: DT with Convex Hull X 1 X 0 Ω 0 X 1 Ω 0 X 0
Adding Input Flows of continuous state variables: x = A x + B u , x ∈ R n , u ∈ U ⊆ R p where x is the state and u is the input Input is treated as nondeterministic: x = A x + BU, x ∈ R n , U ⊆ R p used later for overapproximating nonlinear dynamics
Analytic Solution δ x ( t ) = e A δ x 0 + ∫ e A( δ − τ ) B u ( τ ) d τ 0 Autonomous dynamics Influence of inputs Reach [0 , 3 δ ] ( X 0 ) δ 0 2 δ 3 δ t
Nondeterminism Overapproximation How far can the input push the system within δ ? V = box with radius e || A|| δ − 1 sup u ∈ U || B u || || A|| Ω 0 = Bloat(Conv( X 0 ,e A δ X 0 )) ⊕ V Ω k+1 = e A δ Ω k ⊕ V Minkowski sum: A + B = { a + b | a ∈ A,b ∈ B }
Implementing Reachability Find representation for continuous sets with: i Linear transformation: Ω k+1 = ΦΩ k i Minkowski sum: Ω k+1 = ΦΩ k ⊕ V i Intersection with guards: A x ≤ c
Polyhedra Finite conjunction of linear constraints: P = { x | A x ≤ b }
Operations on Polyhedra Linear transformation: i Transform matrix i O(n 3 ) Minkowski sum: i Need to compute vertices i O(exp(n)) Intersection: i Join lists of constraints i O(1)
Zonotopes Central-symmetric polyhedron: ⎧ ⎫ m ∑ Z = ( c , < v 1 ,..., v m > ) = c + α i v i | -1 ≤ α i ≤ 1 ⎨ ⎬ ⎩ ⎭ i = 1 generators center
Operations on Zonotopes Linear transformation: i Transform generators: Φ Z = ( Φ c, < Φ v 1 ,..., Φ v m > ) i O(n 2 ) Minkowski sum: i Join generator lists: Z + Z = ( c + ′ c , < v 1 ,..., v m , ′ ′ v 1 ,..., ′ v m' > ) i O(1) Intersection: i Problem: intersection of zonotopes is not a zonotope i Overapproximate
Ellipsoids Quadratic form: { } E = x | x T Q x + A x ≤ b
Operations on Ellipsoids Linear transformation: i Transform generators i O(n 2 ) Minkowski sum: i Problem: result is not an ellipsoid i Overapproximate Intersection: i Problem: intersection of ellipsoids is not an ellipsoid i Overapproximate
Support Functions ρ Ω ( l ) = max x ∈Ω l.x Functional form (lazy evaluation): Ω = ∩ l ∈ R d { x ∈ R d | l.x ≤ ρ Ω ( l )}
Support Functions Unit ball for 2-norm B 2 = {x ∈ R d | ||x|| 2 ≤ 1}: ρ Ω ( l ) = || l || 2 Ellipsoid Ω = {x ∈ R d | x T Qx ≤ 1}, Q is positive definite: ρ Ω ( l ) = l T Ql Hyperrectangle Ω = [-h 1 ,h 1 ] × × [-h d ,h d ], h i ∈ R + d : ∑ d ρ Ω ( l ) = | h j l j | j = 1 ∑ r | α i ∈ [-1,1]}, generators g i ∈ R d : Zonotope Ω = { α i g i j = 1 ∑ r ρ Ω ( l ) = | g j .l | j = 1 Polytope Ω = { x ∈ R d | C x ≤ d}: ρ Ω ( l ) = max C x ≤ d l.x
Operations on Support Functions Linear transformation: i ρ A Ω ( l ) = ρ Ω (A T l ) i ρ λΩ ( l ) = ρ Ω ( λ l ) = λρ Ω ( l ) Minkowski sum: i ρ Ω⊕ Ω ( l ) = ρ Ω ( l ) + ρ Ω ( l ) ′ ′ Convex hull: i ρ CH( Ω , Ω ) ( l ) = max{ ρ Ω ( l ), ρ Ω ( l )} ′ ′ Intersection: i Reduces to an optimization problem
Implementing Reachability Complexity of 1-step of time elapse: i Polyhedra: O(exp(n)) i Zonotopes: O(n 2 ) Problem: i With each iteration, Ω k gets more complex i Ω k+1 = e A δ Ω k ⊕ V Minkowski sum increases the number of: i Polyhedra: constraints i Zonotopes: generators
Wrapping Effect Fight complexity by overapproximation Overapproximated sequence: Ω k+1 = Approx(e A δ ˆ i ˆ Ω k ⊕ V ) Accumulation of approximations: i Wrapping effect i Exponential increase in approximation error!
Wrapping Effect Exact versus overapproximation i Dimension 5 for 600 time steps i Overapproximation with 100 generators
Wrapping Effect Ω k+1 = Approx(e A δ ˆ ˆ Ω k ⊕ V ) How does error accumulate? i Linear transformation: Scaling error up (exponential) i Minkowski sum: adding V adds more error
Wrapping Effect Ω k+1 = Approx(e A δ ˆ ˆ Ω k ⊕ V )
Wrapping Effect Ω k+1 = Approx(e A δ ˆ ˆ Ω k ⊕ V )
Wrapping Effect Ω k+1 = Approx(e A δ ˆ ˆ Ω k ⊕ V )
Wrapping Effect Ω k+1 = Approx(e A δ ˆ ˆ Ω k ⊕ V )
Fighting the Wrapping Effect Separate transformations and Minkowski sum: Ω k+1 = e A(k+1) δ Ω 0 ⊕ e Ak δ V ⊕ (e A(k-1) δ V ⊕ ⊕ V ) R k+1 V k S k S k+1 Use four sequences: R 0 = Ω 0 V 0 = V S 0 = {0} R k+1 = e A δ R k Ω k+1 = R k+1 ⊕ S k+1 S k+1 = S k ⊕ V k V k+1 = e A δ V k
Four-Sequence Algorithm R k+1 = e A δ R k Ω k+1 = R k+1 ⊕ S k+1 S k+1 = S k ⊕ V k V k+1 = e A δ V k Transformations only in R k and V k : i Complexity independent of k i No overapproximation necessary Minkowski sum only in S k and Ω k : i Growing no of generators, but no longer transformed i O(Nn 3 ) instead of O(N 2 n 3 )
Four-Sequence Algorithm R k+1 = e A δ R k ˆ S k+1 = ˆ V k+1 = e A δ V k S k ⊕ App( V k ) Ω k+1 = R k+1 ⊕ S k+1 Use overapproximation with: i App( X ) ⊕ App( Y ) = App( X ⊕ Y ) i Bounding box, octogonal, etc. No accumulation of error: i ˆ S k = App( S k ) i ˆ Ω k ⊆ App( Ω k )
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