Time evolution of reachability sets for dynamical systems...Time evolution of reachability sets for dynamical systems Sergiy Zhuk joint work with T.Tcharakian and S. Tirupathi IBM

Time evolution of reachability sets for dynamicalsystems

Sergiy Zhukjoint work with T.Tcharakian and S. Tirupathi

IBM Research - Ireland

Edinburgh University

December 11, 2014

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Outline

Reachability setsLinear systemsNon-linear systemsRelations to Markov diffusions

Approximation of reachability setsLinear systemsClass of non-linear systems

ExamplesTraffic density estimation: 1D conservation lawsFlood modelling: 1D Saint Venant equations (with S.Tirupathi)2D incompressible Euler equations (with T.Tcharakian)

1 / 35 Dynamics of reachability sets (Sergiy Zhuk) IBM Research

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Reachability set for x ′ = −x , 0 ≤ x(0) ≤ 1.


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Liouville equations

Dynamical system with uncertain but bounded parameters:

dX

dt=M(X ), X (t0) = X0 ,

ϕ(X0) ≤ 1 .(1)

The reachability set can be represented as:

R(t) := {x ∈ Rn : V (t, x) ≤ 1}where V solves (in a viscosity sense) the followingHamilton-Jacobi-Bellman equation:

∂tV +M ·∇V = 0,V (t0, x) = ϕ(x) , x ∈ Rn .


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Hamilton-Jacobi-Bellman (HJB) equations

Dynamical system with uncertain but bounded parameters:

dX

dt=M(X (t)) + f (t), X (t0) = X0 ,

Y (t) = H(X (t)) + η(t) ,

ϕ(X0) +

∫ Tt0

‖f (t)‖2 + ‖η(t)‖2dt ≤ 1 .

(2)

The reachability set can be represented as:

R(t) := {x ∈ Rn : V (t, x) ≤ 1}where V solves (in a viscosity sense) the followingHamilton-Jacobi-Bellman equation:

∂tV +M ·∇V = ‖Y (t)− H(x)‖2,V (t0, x) = ϕ(x) , x ∈ Rn .


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Kolmogorov equations: probabilistic approach

Diffusion process:

dX =M(X )dt + dW , X (t0) = X0, (3)W is a standard Wiener process, M is “the model”.

Density:

Assume that the following linear parabolic PDE has a uniquesolution:

∂tV +n∑

i=1

∂

∂xi(Mi (x)V )−

1

2∆V = 0, V (t0, x) = ρ(x) , (4)

where ρ is the density of X0. Then V (t, ·) is the density of X (t).


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Kolmogorov equations: probabilistic approach

Diffusion process

dX =M(X )dt + dW , X (t0) = X0,Y (t) = H(X (t)) + η(t) .

(5)

“Conditional” moments:

Let us define a conditional expectation:

V (s, x) := E (

∫ Ts‖Y (τ)− H(X (τ))‖2dτ + ϕ(X (T )),X (s) = x) .

Then under some conditions (M is Lipschitz):∂Vs +M ·∇V +

1

2∆V + ‖Y (s)− H(x)‖2 = 0 ,

V (T , x) = ϕ(x) , x ∈ Rn .(6)


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Kolmogorov equations vs HJB equations

HJB equation

∂tV +M·∇V = ‖Y (t)−H(x)‖2,V (t0, x) = ϕ(x) , x ∈ Rn . (7)

Backward Kolmogorov equation

∂sV +M ·∇V +1

2∆V + ‖Y (s)− H(x)‖2 = 0 ,

V (T , x) = ϕ(x) , x ∈ Rn .(8)


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Reachability set for a linear model

Assume that M(X ) = MX and H(X ) = HX . The reachability setis given by an ellipsoid

R(t) = {X : V(t,X ) = 〈P(t)(X−X̂ (T )), (X−X̂ (t))〉Rn ≤ 1−β2(t)}

where P solves the following Riccati equationdP

dt= −MP − PMT − P2 + HTH, P(t0) = I .

The dynamics of the minimax center X̂ of R(t) is described bydX̂ (t)

dt= MX̂ (t) + P(t)HT (Y (t)− HX̂ (t)), X̂ (0) = 0 .

The dynamics of the ”observation error” is

dβ2

dt= ‖Ŷ (t)− HX̂ (t)‖2, β2(0) = 0


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Reachability set for a linear model

Assume that M(X ) = MX and H(X ) = HX . The reachability setis given by an ellipsoid

R(t) = {X : V(t,X ) = 〈P(t)(X−X̂ (T )), (X−X̂ (t))〉Rn ≤ 1−β2(t)}where P solves the following Riccati equation

dP

dt= −MP − PMT − P2 + HTH, P(t0) = I .

The dynamics of the minimax center X̂ of R(t) is described bydX̂ (t)

dt= MX̂ (t) + P(t)HT (Y (t)− HX̂ (t)), X̂ (0) = 0 .

The dynamics of the ”observation error” is

dβ2

dt= ‖Ŷ (t)− HX̂ (t)‖2, β2(0) = 0


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Reachability set for bilinear models

Assume that M(X ) = A(X )X , A is linear in X and H(X ) = HX .

Then the reachability set is contained in the ellipsoid:

R(t) = {X : V(t,X ) = 〈P(t)(X − X̂ (T )), (X − X̂ (t))〉Rn ≤ 1}where P solves the following Riccati equation:

dP

dt= −A(X̂ )P − PAT (X̂ )− P2 + HTH, P(t0) = I .

The dynamics of the minimax center X̂ of R(t) is described by:dX̂ (t)

dt= A(X̂ )X̂ (t) + P(t)HT (Y (t)− HX̂ (t)), X̂ (0) = 0 .


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Reachability set for bilinear models

Assume that M(X ) = A(X )X , A is linear in X and H(X ) = HX .Then the reachability set is contained in the ellipsoid:

R(t) = {X : V(t,X ) = 〈P(t)(X − X̂ (T )), (X − X̂ (t))〉Rn ≤ 1}where P solves the following Riccati equation:

dP

dt= −A(X̂ )P − PAT (X̂ )− P2 + HTH, P(t0) = I .

The dynamics of the minimax center X̂ of R(t) is described by:dX̂ (t)

dt= A(X̂ )X̂ (t) + P(t)HT (Y (t)− HX̂ (t)), X̂ (0) = 0 .


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Macroscopic traffic flow models


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Lighthill-Whitham-Richards (LWR) model

The standard equilibrium traffic flow model consists of a scalarconservation law:

∂tu(x , t) + ∂x f (u(x , t)) = 0 (9)

with periodic boundary conditions on the interval (0, 1) and initialdata

u0(x) = u(x , 0) (10)

where u : R×R+ → R is the traffic density, x ∈ R and t ∈ R+ arethe independent variables, space and time respectively, andf : R→ R is the flux function. A typical flux function is that ofGreenshields, given by

f (u) = uVm

(1− u

um

)(11)

where the constants Vm and um are the maximum speed and themaximum density respectively.


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Spectral viscosity for LWR model

We approximate u by uN , which is the N + 1-term truncated series:

uN(x , t) =

N/2∑n=−N/2

an(t)einx , (12)

define the residual:

RN(x , t) =∂uN∂t

+∂f (uN)

∂x(13)

and require it to be orthogonal to span{e inx}|n|≤N/2. This gives usequations for the coefficients an(t):

dandt

=

N/2∑k=−N/2|n−k|≤N/2

2ikan−k(t)ak(t)− inan(t),

n = −N/2 . . .N/2.

(14)


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Spectral viscosity for LWR model

However, solutions of LWR model can developshock-discontinuities, even for smooth initial data. This will giverise to strong oscillations which will spread to the entire spatialdomain. We overcome this by adding “spectral viscosity” on thehigher modes:

dandt

=

N/2∑k=−N/2|n−k|≤N/2

2ikan−k(t)ak(t)− inan(t) −εn2an(t) ,

n = −N/2 . . .N/2.

(15)

The viscosity term is only activated for |n/2| ≥ m, where m issome threshold wave number.


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Data assimilation for LWR model

Figure : Tchrakian,Zhuk, IEEE Trans. on Intelligent TransportationSystems, 2014


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Ensemble Kalman filter for LWR model

0 1 2 3 4 5 6−0.2

0

0.2

0.4

0.6

0.8

1

time = 0.50

x

u

EnKF estimate

Truth

Perturbed observations

Figure : EnKF: 41 sensors


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0 1 2 3 4 5 6−0.2

0

0.2

0.4

0.6

0.8

1

time = 2.00

x

u

EnKF estimate

Truth




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0 1 2 3 4 5 6−0.2

0

0.2

0.4

0.6

0.8

1

time = 5.00

x

u

EnKF estimate

Truth




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0 1 2 3 4 5 6−0.2

0

0.2

0.4

0.6

0.8

1

time = 1.00

x

u

EnKF estimate

Truth




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0 1 2 3 4 5 6−0.2

0

0.2

0.4

0.6

0.8

1

time = 3.00

x

u

EnKF estimate

Truth




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0 1 2 3 4 5 6−0.2

0

0.2

0.4

0.6

0.8

1

time = 8.00

x

u

EnKF estimate

Truth




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Saint Venant (StV) equations

The standard equilibrium flood model consists of a system of scalarconservation laws:

∂th + ∂x(hu) = 0 ,

∂t(hu) + ∂x(hu2 +

gh2

2) = 0 .

(16)

with boundary conditions u(0, t) = ul(t) and h(0, t) = hl(t) on(0, 1), where h is the fluid depth, u is the averaged velocity and gis the gravitational constant.

Let U = (h, hu)T . Then (16) may be rewritten as a system ofconservation laws:

∂tU + ∂x f (U) = 0

provided f is chosen appropriately.


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Saint Venant (StV) equations

The standard equilibrium flood model consists of a system of scalarconservation laws:

∂th + ∂x(hu) = 0 ,

∂t(hu) + ∂x(hu2 +

gh2

2) = 0 .

(16)

with boundary conditions u(0, t) = ul(t) and h(0, t) = hl(t) on(0, 1), where h is the fluid depth, u is the averaged velocity and gis the gravitational constant.Let U = (h, hu)T . Then (16) may be rewritten as a system ofconservation laws:

∂tU + ∂x f (U) = 0

provided f is chosen appropriately.


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Discontinuous Galerkin method for StV equations

We split the domain Ω into cells Ij = [xj− 12, xj+ 1

2] and integrate by

parts to obtain the weak form:∫Ij

∂tU(x , t)v(x)dx −∫Ij

f (U(x , t))∂xv(x)dx

+ F (U(x−j+ 1

2

),U(x+j+ 1

2

))v(x−j+ 1

2

)− F (U(x−j− 1

2

),U(x+j− 1

2

))v(x+j− 1

2

) = 0

with F (a, b) representing a Lax-Friedrichs numerical flux, i.e.

F (a, b) =f (b) + f (a)− C (b − a)

2, C > 0 .

Here v belongs to a subspace generated by Lagrange polynomials(up to 2nd order). To maintain stability and track shocks correctlywe use Total Variation Diminishing flux limiter.


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Euler equations

We consider 2D incompressible Euler equation in vorticity-streamfunction form:

∂tω + u∂xω + v∂yω = 0 , u = −∂yψ , v = ∂xψ ,−∆ψ = ω , ψ(x , y) = 0 , (x , y) ∈ ∂Ω . (17)

Here ω denotes vorticity function and the initial vorticity functionis obtained from the Matlab peaks function.


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Euler equations: discretization

• to compute ω(x , y , t + h) given ω(x , y , t) and u(x , y , t),v(x , y , t) we applied a 4th-order explicit RK methodevaluating ∂xω, ∂yω on a 128× 128 uniform grid Γ using fastFourier transform;

• to get u(x , y , t), v(x , y , t) given ω(x , y , t) we approximatedω(x , y , t) by its projection ω̃ =

∑N 12k,s=1〈ω, ϕks〉L2(Ω)ϕks , where

ϕks := sin(kx2 ) sin(

sy2 ) denotes the eigenfunction of the

Laplacian −∆ on Ω; this allowed us to find the exact solutionof the Poisson equation −∆ψ = ω̃, namelyψ =

∑N 12k,s=1〈ω, ϕks〉L2(Ω)λ−1ks ϕks ; we then computed

u = −∂yψ, v = ∂xψ differentiating the representation for thestream function ψ.


Documents

Time evolution of reachability sets for dynamical systems...Time evolution of reachability sets for dynamical systems Sergiy Zhuk joint work with T.Tcharakian and S. Tirupathi IBM