mpt_getPWPLyapFct

MPT_GETPWPLYAPFCT Calculates Piecewise Polynomial (sum of squares) Lyapunov function

  [dQ,feasible,drho]=mpt_getPWPLyapFct(ctrl,ndeg,Options)

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 DESCRIPTION
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 This function attempts to compute a piecewise higher order Sum of Squares Lyapunov function
 using YALMIP which guarantees exponential stability. The following is satisfied
 alpha * x^2 <= V(x(k))<= beta * x^2  
 V(x(k+1)) - V(x(k))<= - rho * x^2      
 (alpha,beta,rho>=0)

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 INPUT
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 ctrl                        - Explicit controller (EXPCTRL object)
 ndeg                        - Degree of Lyapunov Function(should be even)

 Options.debug_level  - If this is set to 1, the solution provided by the LMI
                        solver will be double-checked manually via SDP. We
                        strongly advise to set this to 1, since we've experienced
                        numerous numerical issues with certain LMI solvers.
                        If this value is set to 2, an additional check via
                        state trajectories will be performed. 
 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.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         - cell array of coefficients of Lyapunov polytope
              In order to see the output polynomial in a "nice" way, proceed as follows:
              x1=sdpvar(1,1), x2=sdpvar(1,1), ... , xn=sdpvar(1,1)
              xmon=monolist([x1 x2 ... xn], ndeg);
              sdisplay(dQ{i}'*xmon)

    
 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
 runtime    - structure that stores "setup_time", "solver_time" and "postproc_time"

 see also MPT_GETPWQLYAPFCT, MPT_GETPWALYAPFCT, MPT_GETCOMMONLYAPFCT, MPT_GETSTABFEEDBACK