1. http://chess.eecs.berkeley.edu/ November 21, 2005 Center for Hybrid and Embedded Software Systems Homepage http://www.eecs.berkeley.edu/~akurzhan/ellipsoids Description Ellipsoidal 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. E llipsoidal 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 Computation Affine transformation special case of affine transformation is projection Geometric sum = Intersection of ellipsoid and hyperplane Geometric difference = Intersections – ellipsoid – halfspace – polytope 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 bounded control unknown bounded disturbance internal approximation external approximation good curves Tight approximations of reach sets good curves – trajectories along which external 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 bounded control fixed input Discrete-time reach set Forward reach sets can be computed also for singular A[k] Backward reach set computation allows only nonsingular A[k] cut at t = 5 Applications backward reach set forward reach set Reaching given point at given time initial state target state Switching systems system switches dynamics and inputs at apriori known times Affine hybrid systems continuous dynamics is affine; guards are hyperplanes and polyhedra system 1 system 2 system 3 state 1 state 2 guard phase plot use good curves of the forward and backward reach sets: switch between them at computed time