# Ellipsoidal Toolbox

### Alex A. Kurzhanskiy and Pravin Varaiya

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2006-46

May 6, 2006

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-46.pdf

Ellipsoidal Toolbox (ET) implements in MATLAB the ellipsoidal calculus and its application to the reachability analysis of continuous- and discrete-time, possibly time-varying linear systems, and linear systems with disturbances, for which ET calculates both open-loop and close-loop reach sets. The ellipsoidal calculus provides the following benefits:

- The complexity of the ellipsoidal representation is quadratic in the dimension of the state space, and linear in the number of time steps.
- It is possible to exactly represent the reach set of linear system through both external and internal ellipsoids.
- It is possible to single out individual external and internal approximating ellipsoids that are optimal to some given criterion (e.g. trace, volume, diameter), or combination of such criteria.
- It gives simple analytical expressions for the control that steers the state to a desired target.

BibTeX citation:

@techreport{Kurzhanskiy:EECS-2006-46, Author = {Kurzhanskiy, Alex A. and Varaiya, Pravin}, Title = {Ellipsoidal Toolbox}, Institution = {EECS Department, University of California, Berkeley}, Year = {2006}, Month = {May}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-46.html}, Number = {UCB/EECS-2006-46}, Abstract = {Ellipsoidal Toolbox (ET) implements in MATLAB the ellipsoidal calculus and its application to the reachability analysis of continuous- and discrete-time, possibly time-varying linear systems, and linear systems with disturbances, for which ET calculates both open-loop and close-loop reach sets. The ellipsoidal calculus provides the following benefits: <UL> <LI>The complexity of the ellipsoidal representation is quadratic in the dimension of the state space, and linear in the number of time steps. <LI>It is possible to exactly represent the reach set of linear system through both external and internal ellipsoids. <LI>It is possible to single out individual external and internal approximating ellipsoids that are optimal to some given criterion (e.g. trace, volume, diameter), or combination of such criteria. <LI>It gives simple analytical expressions for the control that steers the state to a desired target. </UL>} }

EndNote citation:

%0 Report %A Kurzhanskiy, Alex A. %A Varaiya, Pravin %T Ellipsoidal Toolbox %I EECS Department, University of California, Berkeley %D 2006 %8 May 6 %@ UCB/EECS-2006-46 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-46.html %F Kurzhanskiy:EECS-2006-46