Discretization of continuous-time periodic systems

 psc2d([PMT,] psysc, Ts; solver, reltol, abstol, dt) -> psys::PeriodicStateSpace{PMT}

Compute for the continuous-time periodic system psysc = (A(t),B(t),C(t),D(t)) of period T and for a sampling time Ts, the corresponding discretized periodic system psys = (Ad,Bd,Cd,Dd) using a zero-order hold based discretization method. The resulting discretized system psys has the matrices of type PeriodicArray by default, or of type PMT, where PMT is one of the types PeriodicMatrix, PeriodicArray, SwitchingPeriodicMatrix or SwitchingPeriodicArray.

The discretization is performed by determining the monodromy matrix as a product of K = T/Ts state transition matrices of the extended state-space matrix [A(t) B(t); 0 0] by integrating numerically the corresponding homogeneous linear ODE. The ODE solver to be employed can be specified using the keyword argument solver, together with the required relative accuracy reltol (default: reltol = 1.e-3), absolute accuracy abstol (default: abstol = 1.e-7) and/or the fixed step length dt (default: dt = Ts/10). Depending on the desired relative accuracy reltol, lower order solvers are employed for reltol >= 1.e-4, which are generally very efficient, but less accurate. If reltol < 1.e-4, higher order solvers are employed able to cope with high accuracy demands.

The following solvers from the OrdinaryDiffEq.jl package can be selected:

solver = "non-stiff" - use a solver for non-stiff problems (Tsit5() or Vern9());

solver = "stiff" - use a solver for stiff problems (Rodas4() or KenCarp58());

solver = "linear" - use a special solver for linear ODEs (MagnusGL6()) with fixed time step dt;

solver = "symplectic" - use a symplectic Hamiltonian structure preserving solver (IRKGL16());

solver = "" - use the default solver, which automatically detects stiff problems (AutoTsit5(Rosenbrock23()) or AutoVern9(Rodas5())).

For large values of K, parallel computation of factors can be alternatively performed by starting Julia with several execution threads. The number of execution threads is controlled either by using the -t/--threads command line argument or by using the JULIA_NUM_THREADS environment variable.