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Continuous-time systems. unknown bounded disturbance. bounded control. Tight approximations of reach sets. special case of affine transformation is projection. external approximation. good curves. Intersection of ellipsoid and hyperplane. internal approximation.

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Ellipsoidal toolbox

Continuous-time systems






Tight approximations of reach sets

special case of affine transformation is projection

external approximation

good curves

Intersection of ellipsoid and hyperplane

internal approximation

good curves – trajectories along which

external approximation touches the internal

Geometric sum

Backward reach set

  • if v(t) is fixed, then

  • open-loop reach set = closed-loop reach set

  • if v(t) is unknown disturbance, then

  • closed-loop reach set is computed

  • Forward and backward reach set computation


center trajectory

Geometric difference

Discrete-time systems





Discrete-time reach set


  • Forward reach sets can be computed

  • also for singular A[k]

  • Backward reach set computation

  • allows only nonsingular A[k]



Reaching given point

at given time

Switching systems

Affine hybrid systems

phase plot


reach set

state 1

– polytope



– ellipsoid

– halfspace

Additional functions



system 1

system 2

state 2

  • Distance – ellipsoid-to ellipsoid,

  • ellipsoid-to hyperplane, ellipsoid-to-polytope

  • Feasibility check – checks if intersection of given

  • objects is nonempty


reach set

system 3


use good curves of the forward

and backward reach sets: switch

between them at computed time

system switches dynamics

and inputs at apriori

known times

continuous dynamics

is affine; guards are

hyperplanes and polyhedra

  • Alex Kurzhanskiy

  • Advisor:

  • professor Pravin Varaiya

Ellipsoidal Toolbox

  • Homepage

  • http://www.eecs.berkeley.edu/~akurzhan/ellipsoids

  • Description

  • Ellipsoidal Toolboxis a standalone set of easy-to-use configurable MATLAB routines to perform operations with ellipsoids and hyperplanes of arbitrary dimensions. It computes the external and internal ellipsoidal approximations of geometric (Minkowski) sums and differences of ellipsoids, intersections of ellipsoids and intersections of ellipsoids with halfspaces and polytopes; distances between ellipsoids, between ellipsoids and hyperplanes, between ellipsoids and polytopes; and projections onto given subspaces.

  • Ellipsoidal methods are used to compute forward and backward reach sets of continuous- and discrete-time piecewise affine systems. Forward and backward reach sets can be also computed for continuous-time piece-wise linear systems with disturbances. It can be verified if computed reach sets intersect with given ellipsoids, hyperplanes, or polytopes

  • Software used by ET

  • YALMIP – high-level MATLAB toolbox for rapid

  • development of optimization code:

  • http://control.ee.ethz.ch/~joloef/yalmip.php

  • SeDuMi – MATLAB toolbox for solving optimization

  • problems over symmetric cones:

  • http://sedumi.mcmaster.ca

  • Both packages are included in the Ellipsoidal Toolbox distribution and need not be downloaded separately.

  • Ellipsoidal Toolbox supports polytope object of the Multi-Parametric Toolbox (MPT):

  • http://control.ee.ethz.ch/~mpt

Reach Set Computation

Ellipsoidal Calculus

Affine transformation


at t = 5

Center for Hybrid and Embedded Software Systems