November 21, 2005 Center for Hybrid and Embedded Software Systems

http://chess.eecs.berkeley.edu/

Homepagehttp://www.eecs.berkeley.edu/~akurzhan/ellipsoids

DescriptionEllipsoidal Toolbox is 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

Ellipsoidal Toolbox

Alex Kurzhanskiy

Advisor:

professor Pravin Varaiya

Ellipsoidal Calculus Reach Set ComputationAffine transformation

special case of affine transformation is projection

Geometric sum

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2

-1

0

1

2

3

-8 -6 -4 -2 0 2 4 6 8-10

-8

-6

-4

-2

0

2

4

6

8

10

=

Intersection of ellipsoid and hyperplane

Geometric difference

-3 -2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

=

Intersections

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x1

x2

-6 -4 -2 0 2 4 6 8-4

-3

-2

-1

0

1

2

3

x1

x2

– ellipsoid – halfspace – polytope-2 -1 0 1 2 3 4

-3

-2

-1

0

1

2

3

Additional functions

• Distance – ellipsoid-to ellipsoid, ellipsoid-to hyperplane, ellipsoid-to-polytope• Feasibility check – checks if intersection of given objects is nonempty

Continuous-time systems

boundedcontrol

unknownbounded

disturbance

-20 -15 -10 -5 0 5 10 15-20

-15

-10

-5

0

5

10

15

20Reach set at time T = 5

x1

x2

internal approximation

external approximation

good curves

Tight approximations of reach sets

good curves – trajectories along whichexternal approximation touches the internal

Backward reach set

center trajectory

• 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

Discrete-time systems

boundedcontrol

fixedinput

02

46 8

10

-15

-10

-5

0

5

10

-15

-10

-5

0

5

10

k

Discrete-time reach tube

x1

x2

Discrete-time reach set

• Forward reach sets can be computed also for singular A[k]• Backward reach set computation allows only nonsingular A[k]

cutat t = 5

Applications

backwardreach set

forwardreach set

Reaching given pointat given time

initialstate

targetstate

Switching systems

system switches dynamicsand inputs at apriori

known times

Affine hybrid systems

continuous dynamicsis affine; guards are

hyperplanes and polyhedra

system 1system 2

system 3

-20 0 20 40 60 80 100

0

10

20

30

40

50

60

70

80

90

x1

x2

state 1

state 2

guard

phase plot

use good curves of the forwardand backward reach sets: switchbetween them at computed time