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# mpt_mplp_ver2

## PURPOSE

MPT_MPLP Explicitly solves the given linear program (LP)

## SYNOPSIS

function [Pn,Fi,Gi,activeConstraints,Phard,details]=mpt_mplp_ver2(Matrices,Options)

## DESCRIPTION

```MPT_MPLP Explicitly solves the given linear program (LP)

[Pn,Fi,Gi,activeConstraints,Phard,details]=mpt_mplp(Matrices,Options)

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DESCRIPTION
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Multiparametric linear programming

Solves the problem
V(x) = min H U
U
s.t.   G U <= W + E x
bndA*x <= bndb

As a solution we get 'nR' regions
Pn(i)={x : H x <= K}

with the optimal control law
U = Fi{i} x + Gi{i}

and the corresponding cost function expression
V(x) = Bi{i} x + Ci{i}

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INPUT
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Matrices - a struct with all the parameters which are needed.
See description above for explanation.
Matrices.H=H;
Matrices.G=G;
Matrices.W=W;
Matrices.E=E;
Matrices.bndA=bndA;   Limits on exploration space, i.e. bndA*x<=bndb
Matrices.bndb=bndb;

Options.verbose      - level of verbosity
Options.lpsolver     - which LP solver to use (help mpt_solveLP)
Options.max_iter     - maximum number of iterations of the algorithm
Options.step_size    - length of step over a facet
Options.f_perturb    - Perturbation of the optimization direction
Options.nu           - How many elements to extract from the optimizer (to
deal with slacks)
Options.debug_level
Due to numerical problems tiny regions are sometimes difficult to
calculate, i.e. are not identified at all. This may create "gaps"
in the computed control law. For the exploration, these will be
jumped over and the exploration in the state space will continue.
"debug_level" can have three values:

0: No debug done
1: A tolerance is given to find gap in the region partition,
small empty regions inside the region partition will be discarded.
Note that this is generally not a problem, since the feedback law
is continuous and can therefore be interpolated easily.
Correction to the calculation of the outer hull.
2: Zero tolerance to find gap in the region partition, empty regions
if they exist, will be detected, i.e. the user will be notified.
Correction to the calculation of the outer hull.

Note: If Options is missing or some of the fields are not defined, the default
values from mptOptions will be used

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OUTPUT
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Pn,Fi,Gi           - for region Pn(i).H*x <= Pn(i).K computed input is
U=Fi{i}*x+Gi{i}
activeConstraints  - Cell Array which stores the active constraints
of the optimizer in each region.
Phard              - Defines the feasible state space partition (i.e. union of
all regions) as Phard.H*x<=Phard.K
details            - a structure with the following fields:
nRegions  number of regions
Pn        polyhedral partition
Fi        control law
Gi        control law
BC        connection list
Bi        value function
Ci        value function
nHard     number of hard constraints
Phard     polytope given by hard constraints
nb        number of constraints for each region
LISTa     list of active constraints

## CROSS-REFERENCE INFORMATION

This function calls:
• length LENGTH Returns number of regions over which the explicit control law is defined
• bounding_box BOUNDING_BOX Compute a bounding box for a given polytope
• 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
• facetcircle FACETCIRCLE Returns largest circle inside facet 'ind' of polytope P
• isfulldim ISFULLDIM Checks if a polytope is full dimensional
• isinside ISINSIDE Checks if a given point lies in the interior of a given polytope
• 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
• projection PROJECTION Projection of a polytope or a polytope array
• size SIZE Returns size of the given polytope object
• mpt_error MPT_ERROR Function called if MPT toolbox is not initialized
• mpt_solveLPi MPT_SOLVELPi Interface to various LP solvers (version without errorchecks)
• mpt_solveQP MPT_SOLVEQP Interface to various QP solvers
This function is called by:
• mpt_mplp MPT_MPLP Explicitly solves the given linear program (LP)

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