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# projection

## PURPOSE

PROJECTION Projection of a polytope or a polytope array

## SYNOPSIS

function [P]= projection(PA,dim,Options)

## DESCRIPTION

PROJECTION Projection of a polytope or a polytope array

[P] = projection(P,dim,Options)

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DESCRIPTION
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[P] = PROJECTION(P,DIM,OPTIONS) projects polytope P on dimensions defined
in vector 'dim'

Three different algorithms can be used:
- Vertex enumeration/Convex hull based method
- Fourier-Motzkin Elimination
- Iterative Hull
- Block Elimination
- Equality Set Projection

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INPUT
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P                     - Polytope
dim                   - Dimensions on which to project
Options.projection=0  - Vertex enumeration/Convex-hull based method
Options.projection=1  - Fourier-Motzkin Elimination
Options.projection=2  - Iterative Hull
Options.projection=3  - Block Elimination
Options.projection=4  - Equality Set Projection (ESP)
Options.projection=5  - Fourier-Motzkin Elimination (mex implementation)
Options.projection=6  - Fourier-Motzkin Elimination (mex implementation) -
fast but eventualy unreliable
Options.projection=7  - Use approach based on mpLPs
Options.iterhull_maxiter  - Maximum number of iterations for the Iterative
Hull algorithm

Note: If Options.projection is not set, best method is selected automatically

Note: Options.projection can also be a vector of prefered methods

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OUTPUT
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P   - Projected Polytope

## CROSS-REFERENCE INFORMATION

This function calls:
• length LENGTH Returns number of regions over which the explicit control law is defined
• chebyball CHEBYBALL Computes center and radius of the largest ball inscribed in a polytope
• dimension DIMENSION Returns dimension of the given polytope
• double DOUBLE Function used to access internal properties of the given polytope
• end END Returns the last element in a given polytope array
• extreme EXTREME Calculates extreme points of a given polytope
• hull HULL Convex hull of n polytopes
• isbounded ISBOUNDED Checks if a polytope is bounded
• isfulldim ISFULLDIM Checks if a polytope is full dimensional
• isinside ISINSIDE Checks if a given point lies in the interior of a given polytope
• isminrep ISMINREP Checks if a given polytope is in minimal representation
• length LENGTH Returns number of elements in a polytope array
• nconstr NCONSTR Returns number of constraints that form an H-representation of a polytope
• polytope POLYTOPE Default constructor for the POLYTOPE object
• reduce REDUCE Reduces the polytope by removing redundant inequalities
• size SIZE Returns size of the given polytope object
• unique UNIQUE Removes redundant entries from a polytope array
• mpt_error MPT_ERROR Function called if MPT toolbox is not initialized
• mpt_solveLPi MPT_SOLVELPi Interface to various LP solvers (version without errorchecks)
• hull HULL Converts vertices/polytope array into an H-representation polytope
This function is called by:
• plus PLUS Minkowski sum of two polytopes
• mpt_maxCtrlSet MPT_MAXCTRLSET Computes the maximal robust control invariant set C_inf
• mpt_reachSets MPT_REACHSETS Computes sets of reachable states for a given system / controller
• mpt_reachXU MPT_REACHXU One-step reachability computation for an affine system
• mpt_feasibleStates MPT_FEASIBLESTATES returns equidistantly spaced data points in feasible set
• mpt_mplp_ver2 MPT_MPLP Explicitly solves the given linear program (LP)
• mpt_mplp_ver3 MPT_MPLP Explicitly solves the given linear program (LP)
• mpt_mplp_ver4 MPT_MPLP Explicitly solves the given linear program (LP)
• mpt_mplp_ver5 MPT_MPLP Explicitly solves the given linear program (LP)
• mpt_mplp_ver6 MPT_MPLP Explicitly solves the given linear program (LP)
• mpt_mplp_ver7 MPT_MPLP Explicitly solves the given linear program (LP)
• mpt_iterativePWA MPT_ITERATIVEPWA Computes a time-optimal or low-complexity explicit controller for PWA systems
• mpt_mixedMinTime MPT_MIXEDMINTIME Computes minimum time controller for systems with discrete and continuous inputs
• mpt_simplexContr MPT_SIMPLEXCONTR Computes a piecewise affine feedback law defined over simplices

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