mpt_getPWQLyapFct

MPT_GETPWQLYAPFCT Calculates PWQ Lyapunov function

 [dQ,dL,dC,feasible,drho]=mpt_getPWQLyapFct(ctrlStruct,Options)

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 DESCRIPTION
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 This function attempts to compute a piecewise quadratic Lyapunov function
 PWQ(x) which guarantees exponential stability. The following is satisfied
 PWQ(x) = x'dQ{r}x + x'dL{r}+ dC{r}
 PWQ(x(k+1)) - PWQ(x(k))<= rho * x^2      
 (rho must be negative to guarantee exponential stability)

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 INPUT
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 ctrl                        - Explicit controller (EXPCTRL object)
 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
 Options.useTmap      - If set to true (default), transition map will
                        be computed to rule out certain transitions
 Options.sphratio     - Gives factor which governs maximum number of separating
                        hyperplanes computed in transition maps. Number of
                        separating  hyperplnaes computed at each step is given
                        by length(Pn)^2 / Options.ratio
                        Default value is 20.
                        Set this option to 0 if you don't want to impose any
                        limit on number of separating hyperplanes.

 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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 dQ,dL,dC   - PWQ Lyapunov function:
                PWQ(x)=x'dQ{r}x+x'*dL{r}+C{r}
                iff x is in region r, i.e. Pn{r}.H*x<=Pn{r}.K
    
 feasible   - 1: stable or 0: no statement about stability possible
 drho       - the maximum Lyapunov decay rate over the partition
              (is this is greater than zero, stability cannot be guaranteed)
              The Lyapunov value decrease Delta V <= drho * ||x||^2

 see also MPT_GETPWALYAPFCT, MPT_GETPWPLYAPFCT, MPT_GETCOMMONLYAPFCT, MPT_GETSTABFEEDBACK