mpt_getQuadLyapFct

MPT_GETQUADLYAPFCT Computes common Lyapunov function for PWA system

 [lyapP,decay,feasible]=mpt_getQuadLyapFct(ctrlStruct,Options)

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 DESCRIPTION
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 This function attempts to compute a quadratic Lyapunov function V(x)=x'lPx
 which guarantees exponential stability.
 PWQ(x) = x'LPx
 PWQ(x(k+1)) - PWQ(x(k)) <= rho * x^2 

 (i.e. rho must be negative to guarantee exponential stability)

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 INPUT
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 ctrl                   - Explicit controller (MPTCTRL objects)
 Options.decayEllipsoid      
       - If set to 1, decay rate of PWQ function is constrained to be negative 
         over ellipsoidal regions(reduces number of free variables in LMI)
         If set to 0, decay rate is constrained to be negative 
         over polytopic regions(more free variables in LMI)
 Options.posEllipsoid        
       - If set to 1, positivity of PWQ function is constrained to be valid  
         over ellipsoidal regions(reduces number of free variables in LMI)
         If set to 0, positivity is constrained to be valid  
         over polytopic regions(more free variables in LMI)
 Options.enforcePositivity   
       - If set to zero, positivity constraints for PWQ function are not 
         included in LMI (reduces constraints and computation time)
         Post computation verification is performed to check if postivity holds
         If not, positivity constraints are added and solution recomputed.
 Options.abs_tol       - Absolute tolerance
 Options.lpsolver      - Which LP solver to use (see help mpt_solveLP for details)
 Options.epsilon       - This is a tolerance factor which is introduced to turn
                           LMI inequalities into strict inequalities.
 Options.debug_level   - If this is set to 1, the solution provided by the LMI
                         solver will be double-checked manually. We strongly
                         advise to set this to 1, since we've experienced
                         numerous numerical issues with certain LMI solvers.
 Options.nicescale     - This will add additional constraints to obtain a nicely scaled
                         Lyapunov function. There is no benefit in this apart from 
                         getting nicer plots afterwards. Therefore this is switched 
                         off by default

 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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 lyapP      - Quadratic Lyapunov function: Q(x)=x'*lyapP{r}*x
 decay      - the maximum Lyapunov decay rate over the partition
              (is this is greater than zero, stability cannot be guaranteed)
              The Lyapunov value decrease Delta V <= decay * ||x||^2
 feasible   - 1: stable or 0: no statement about stability possible


 see also MPT_GETPWQLYAPFCT, MPT_GETSTABFEEDBACK, MPT_GETCOMMONLYAPFCT