mpt_mplp_ver3

MPT_MPLP Explicitly solves the given linear program (LP)

 [Pn,Fi,Gi,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 toleranke 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

 see also MPT_CONSTRUCTMATRICES, MPT_MPQP, MPT_OPTCONTROL, MPT_OPTCONTROLPWA