mpt_mpqp

MPT_MPQP Explicitly solves the given quadratic program (QP)

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

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 DESCRIPTION
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 Solves the following QP as a multiparametric program:
 min_U  0.5 U' H U + (x(0)' F + Cf) U + x(0)' Y x(0) + Cx x + Cc
 subj. to  GU <= W + Ex     (constraints)
        bndA*x<= bndb       (bound exploration space)
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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.G=G;
    Matrices.E=E;             
    Matrices.W=W;
    Matrices.H=H;
    Matrices.F=F;
    Matrices.Y=Y;
    Matrices.Cf=Cf;
    Matrices.Cx=Cx;
    Matrices.Cc=Cc;
    Matrices.bndA=bndA;
    Matrices.bndb=bndb;

 Options.verbose     - Optional: level of verbosity
 Options.lpsolver    - Optional: which LP solver to use (help mpt_solveLP)
 Options.qpsolver    - Optional: which QP solver to use (help mpt_solveQP)
 Options.step_size   - Optional: length of step over a facet; Making this
                          value larger often mitigates numerical problems but
                          may produce small gaps in the partition.
                          Default is 1e-4;

 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.

 Options.zero_tol    - everything below this value is considered zero
                       (default is 1e-13)

 Note: If Options is missing or some of the fiels 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) the optimal input is U=Fi{i}*x+Gi{i} 
 activeConstraints  - Cell Array which stores the active constraints 
                      of the optimizer in each region.
 Phard              - The set of feasible states (i.e. union of
                      all regions) as Phard.H*x<=Phard.K
 details            - structure with fields Ai, Bi, Ci in which the value
                      function associated to each region is stored
                      V = x' Ai{i} x + Bi{i} x + Ci{i}

 see also MPT_OPTCONTROL, MPT_CONSTRUCTMATRICES, MPT_MPLP