PeriodicMatrices.jl

Handling of periodic time-varying matrices

Compatibility

Julia 1.10 and higher.

About

PeriodicMatrices.jl provides the basic tools to handle periodic time-varying matrices.

For a real periodic matrix A(t) with period T, the dependence of the time variable t can be either continuous or discrete. The periodicity condition A(t) = A(t+T) is assumed for all values of t, but it is not assumed that T is the minimum value for which this condition holds. For each periodic matrix representation a subperiod Tsub := T/n can also be defined, such that A(t) = A(t+Tsub) for all t, where n is the number of subperiods. Usually n = 1 and thus Tsub = T, but in some cases values n > 1 allow substantial memory saving for some classes of periodic representations.

A continuous-time periodic matrix can be specified in one of the following forms:

• periodic matrix function, with A(t) a matrix function of the real variable t ∈ [0, T);

• periodic symbolic matrix, with A(t) a symbolic matrix as defined in the Symbolics.jl package depending on the (symbolic) real variable t ∈ [0, T);

• harmonic matrix series, with A(t) defined as

                     p
       A(t) = A_0 +  ∑ ( Ac_i*cos(iωt)+As_i*sin(iωt) ) ,
                    i=1 
  

  where ω = 2π/T and A_0, Ac_i, As_i for i = 1,..., p are real matrices;

• periodic matrix time series with constant dimensions on a uniform time grid;

• periodic matrix time series with constant dimensions on a non-uniform time grid;

• Fourier matrix series approximation, with A(t) a Fourier series representation (similar to the harmonic matrix series representation) as defined in the ApproxFun.jl package.

A discrete-time periodic matrix can be specified in the following forms:

• periodic matrix time series with time-varying dimensions on a uniform time grid;

• periodic matrix time series with time-varying dimensions on a non-uniform time grid;

• periodic matrix time series with constant dimensions on a uniform time grid;

• periodic matrix time series with constant dimensions on a non-uniform time grid.

All possible conversions between the above representations are supported. The provided classes of periodic representations significantly extend the classes used in the Periodic Systems Toolbox for Matlab (see [1]).

Several operations on periodic matrices are implemented, such as, inversion, transposing, norms, derivative/shifting, trace. All operations with two periodic matrices such as addition/substraction, multiplication, horizontal/vertical concatenation, block-diagonal appending, allow different, but commensurate, periods/subperiods.

Several advanced computational functions are provided to compute the characteristic multipliers and characteristic exponents of periodic matrices, using methods based on the periodic Schur decomposition of matrix products (provided in the SLICOT library or in the PeriodicSchurDecompositions.jl package) or structure exploitung fast algorithms requiring no external supporting packages. These functions are instrumental to apply Floquet theory to study the properties of solutions of various classes of differential equations (e.g., Mathieu, Hill, Meissner) and the stability of linear periodic systems (see PeriodicSystems.jl package). The implementations of several functions rely on the high performance ODE solvers available in the OrdinaryDiffEq and IRKGaussLegendre packages.

Examples of using some functions are available here.

References

[1] A. Varga. A Periodic Systems Toolbox for Matlab. Proc. of IFAC World Congress, Prague, Czech Republic, 2005.

[2] S. Bittanti and P. Colaneri. Periodic Systems - Filtering and Control, Springer Verlag, 2009.

[3] J. A. Richards. Analysis of Periodically Time-Varying Systems, Springer Verlag, 1983.