mpt_getPWALyapFct

MPT_GETPWALYAPFCT Calculates PWA Lyapunov function

 [lyapunovL,lyapunovC,feasible,drho]=mpt_getPWALyapFct(ctrl,Options)

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   DESCRIPTION:
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 This function attempts to compute a piecewise affine Lyapunov matrix
 which guarantees stability.

 PWQ = L*x+ C
 Delta PWA <= rho * ||x||_1
 (rho must be negative to guarantee exponential stability)

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    INPUT:
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 ctrl                - Explicit controller (MPTCTRL object)
 Options.abs_tol     - Absolute tolerance
 Options.lpsolver    - Which LP solver to use (see help mpt_solveLP for details)
 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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 lyapunovL,lyapunovC       - PWA Lyapunov function:
                             PWA(x)=lyapunovL{r}*x+lyapunovC{r}
                             iff x is in region r, i.e. Pn{r}.H*x<=Pn{r}.K
    
 feasible   - 1: asymptotically stable 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||_1
              Note: If drho=0 then feasible=0, since the system is not exponentially stable 
                    (it is merely stable)