Constructors for periodic matrices

PeriodicMatrix(M, T; nperiod = k) -> A::PeriodicMatrix

Discrete-time periodic matrix representation.

The discrete-time periodic matrix object A is built from a p-vector M of real matrices, the associated time period T and the number of subperiods specified via the keyword argument nperiod = k.

M contains the cyclic component matrices M[i], i = 1,..., p, where M[i] represents the value M(Δ(i-1)) of a time periodic matrix M(t) of period T/k, with Δ := T/(k*p), the associated sample time. It is assumed that M[i] := M[mod(i-1,p)+1] for arbitrary i. All component matrices are allowed to have arbitrary (time-varying) dimensions. The component matrices M, the period T, the number of subperiods k, the discrete period p and the sample time Δ can be accessed via A.M, A.period, A.nperiod, A.dperiod and A.Ts, respectively.

SwitchingPeriodicMatrix(M, ns, T; nperiod = k) -> A::SwitchingPeriodicMatrix

Discrete-time switching periodic matrix representation.

The discrete-time switching periodic matrix object A is built from a p-vector M of real matrices, a p-vector ns of increasing positive integers representing the discrete switching moments, the associated time period T and the number of subperiods specified via the keyword argument nperiod = k.

M contains the component matrices M[i], i = 1,..., p, which defines a sequence of N := ns[p] of matrices S[1], ..., S[N], such that S[j] = M[i] for j ∈ [ns[i-1]+1, ..., ns[i]] with ns[0] := 0. S[j] is the j-th value A(Δ(j-1)) of a time periodic matrix A(t) of subperiod T′ := T/k, with Δ := T′/N, the associated sample time. All component matrices are allowed to have arbitrary (time-varying) dimensions. The component matrices M, the integer vector ns, the period T, the number of subperiods k, the discrete period N and the sample time Δ can be accessed via A.M, A.ns, A.period, A.nperiod, A.dperiod and A.Ts, respectively.

The j-th time value A(Δ(j-1)) can be determined as A[j]. It is assumed that A[j] := A[mod(j-1,N)+1] for arbitrary j.

size(A::PeriodicMatrix)
size(A::SwitchingPeriodicMatrix)
size(A::PeriodicMatrix[, dim])
size(A::SwitchingPeriodicMatrix[, dim])

Return a tuple of two vectors containing the dimensions of the components of the discrete-time periodic matrix A. Optionally you can specify a dimension dim to just get the vector of lengths of that dimension.

length(A::PeriodicMatrix)
length(A::SwitchingPeriodicMatrix)

Return the number of component matrices (also called the discrete period) of the discrete-time periodic matrix A.

eltype(A::PeriodicMatrix)
eltype(A::SwitchingPeriodicMatrix)

Determine the type of the elements of component matrices of the discrete-time periodic matrix A.

getindex(A::PeriodicMatrix, i)
getindex(A::SwitchingPeriodicMatrix, i)

Return the i-th component matrix of the discrete-time periodic matrix A. Equivalent to the syntax A[i].

getindex(A::PeriodicMatrix, ind1, ind2)
getindex(A::SwitchingPeriodicMatrix, ind1, ind2)

Return the discrete-time periodic matrix built from the selected ranges [ind1,ind2] of elements of the component matrices. ind1 and ind2 may be integers, integer ranges or colons.

getindex(A::PeriodicMatrix, ind1, ind2)
getindex(A::SwitchingPeriodicMatrix, ind1, ind2)

Return the discrete-time periodic matrix built from the selected ranges [ind1,ind2] of elements of the component matrices. ind1 and ind2 may be integers, integer ranges or colons.

lastindex(A::PeriodicMatrix)
lastindex(A::SwitchingPeriodicMatrix)

Return the last index of the component matrices of the discrete-time periodic matrix A. The syntax A[end] is equivalent to A[lastindex(A)].

lastindex(A::PeriodicMatrix,dim)
lastindex(A::SwitchingPeriodicMatrix,dim)

Return the vector of last indices along dimension dim of the component matrices of the discrete-time periodic matrix A.

PeriodicArray(M, T; nperiod = k) -> A::PeriodicArray

Discrete-time periodic array representation.

The discrete-time periodic array object A is built from a m×n×p real array M, the associated time period T and the number of subperiods specified via the keyword argument nperiod = k. M contains the cyclic component matrices M[:,:,i], i = 1,..., p, where M[:,:,i] represents the value M(Δ(i-1)) of a time periodic matrix M(t) of period T/k, with Δ := T/(kp), the associated sample time. It is assumed that M[:,:,k] := M[:,:,mod(k-1,p)+1] for arbitrary k. The component matrices M, the period T, the number of subperiods k, the discrete period p and the sample time Δ can be accessed via A.M, A.period, A.nperiod, A.dperiod and A.Ts, respectively.

SwitchingPeriodicArray(M, ns, T; nperiod = k) -> A::SwitchingPeriodicArray

Discrete-time switching periodic array representation.

The discrete-time switching periodic array object A is built from a m×n×p real array M, a p-vector ns of increasing positive integers representing the discrete switching moments, the associated time period T and the number of subperiods specified via the keyword argument nperiod = k.

M contains the cyclic component matrices M[:,:,i], i = 1,..., p, which defines a sequence of N := ns[p] of matrices S[1], ..., S[N], such that S[j] =M[:,:,i]forj ∈ [ns[i-1]+1, ..., ns[i]]withns[0] := 0.S[j]is thej-th valueA(Δ(j-1))of a time periodic matrixA(t)of subperiodT′ := T/k, withΔ := T′/N, the associated sample time. The component matricesM, the integer vectorns, the periodT, the number of subperiodsk, the discrete periodNand the sample timeΔcan be accessed viaA.M,A.ns,A.period,A.nperiod,A.dperiodandA.Ts`, respectively.

The j-th time value A(Δ(j-1)) can be determined as A[j]. It is assumed that A[j] := A[mod(j-1,N)+1] for arbitrary j.

size(A::PeriodicArray)
size(A::SwitchingPeriodicArray)
size(A::PeriodicArray[, dim])
size(A::SwitchingPeriodicArray[, dim])

Return a tuple of two integers containing the common row and column dimensions of the components of the discrete-time periodic matrix A. Optionally you can specify a dimension dim to just get length of that dimension.

length(A::PeriodicArray)
length(A::SwitchingPeriodicArray)

Return the number of component matrices (also called the discrete period) of the discrete-time periodic matrix A.

eltype(A::PeriodicArray)
eltype(A::SwitchingPeriodicArray)

Determine the type of the elements of component matrices of the discrete-time periodic matrix A.

getindex(A::PeriodicArray, i)
getindex(A::SwitchingPeriodicArray, i)

Return the i-th component matrix of the discrete-time periodic matrix A. Equivalent to the syntax A[i].

getindex(A::PeriodicArray, ind1, ind2)
getindex(A::SwitchingPeriodicArray, ind1, ind2)

Return the discrete-time periodic matrix built from the selected ranges [ind1,ind2] of elements of the component matrices. ind1 and ind2 may be integers, integer ranges or colons.

lastindex(A::PeriodicArray)
lastindex(A::SwitchingPeriodicArray)

Return the last index of the component matrices of the discrete-time periodic matrix A. The syntax A[end] is equivalent to A[lastindex(A)].

lastindex(A::PeriodicArray,dim)
lastindex(A::SwitchingPeriodicArray,dim)

Return the last index along dimension dim of the component matrices of the discrete-time periodic matrix A.

PeriodicFunctionMatrix(f, T; nperiod = k) -> A::PeriodicFunctionMatrix

Continuous-time periodic function matrix representation.

The continuous-time periodic function matrix object A is built from the real matrix function f(t) of real time variable t, the associated time period T and the associated number of subperiods specified via the keyword argument nperiod = k. It is assumed that f(t) = f(t+T/k) for any real time value t. The function f(t), the period T, the row and column dimensions of f(t), the number of subperiods k can be accessed via A.f, A.period, the tuple A.dims and A.nperiod, respectively.

PeriodicSymbolicMatrix(F, T; nperiod = k) -> A::PeriodicSymbolicMatrix

Continuous-time periodic symbolic matrix representation.

The continuous-time periodic symbolic matrix object A is built from F, a symbolic real matrix or vector of symbolic variable t, the associated time period T and the associated number of subperiods specified via the keyword argument nperiod = k. It is assumed that F(t) = F(t+T/k) for any real time value t. The symbolic matrix F, the period T and the number of subperiods k can be accessed via A.F, A.period and A.nperiod, respectively.

PeriodicTimeSeriesMatrix(At, T; nperiod = k) -> A::PeriodicTimeSeriesMatrix

Continuous-time periodic time series matrix representation.

The continuous-time periodic time series matrix object A of period T is built from a p-vector At of real matrices and the associated subperiod T′ = T/k, where k ≥ 1 is the number of subperiods (default: k = 1). At contains the cyclic component matrices At[i], i = 1,..., p, where At[i] represents the value A(Δ*(i-1)) of a time periodic matrix A(t) of period T′, with Δ := T′/p, the associated sampling time. It is assumed that At[i] := At[mod(i-1,p)+1] for arbitrary i. All component matrices must have the same dimensions. The component matrices At, the period T and the number of subperiods k can be accessed via A.values, A.period, and A.nperiod, respectively.

 HarmonicArray(Ahr, T; nperiod = k) -> A::HarmonicArray

Continuous-time harmonic array representation.

The harmonic array object A of period T is built using the harmonic representation of a periodic matrix Ahr(t) of subperiod T′ = T/k in the form

                 p
 Ahr(t) = A_0 +  ∑ ( Ac_i*cos(i*t*2*π/T′)+As_i*sin(i*2*π*t/T′) ) ,
                i=1

where k ≥ 1 is the number of subperiods (default: k = 1). The m×n×(p+1) complex array Ahr contains the harmonic components as follows: Ahr[:,:,1] contains the constant term A_0 (the mean value) and the real and imaginary parts of Ahr[:,:,i+1] for i = 1, ..., p contain the coefficient matrices Ac_i and As_i, respectively. The complex matrix Ahr containing the harmonic components, the period T and the number of subperiods k can be accessed via A.values, A.period and A.nperiod, respectively.

 HarmonicArray(A0, Ac, As, T) -> A::HarmonicArray

Construct a harmonic array representation from the harmonic components.

The harmonic array object A is built for the harmonic representation Ahr(t) of a periodic matrix of period T in the form

                 p
 Ahr(t) = A_0 +  ∑ ( Ac_i*cos(i*t*2*π/T)+As_i*sin(i*2*π*t/T) ) ,
                i=1

where the constant term A_0 is contained in the real matrix A0, and Ac and As are vectors of real matrices such that the i-th (cosinus) coefficient matrix Ac_i is contained in Ac[i] and the i-th (sinus) coefficient matrix As_i is contained in As[i]. p is the maximum of length of the vectors of matrices Ac and As. If the length of Ac or As is less than p, then zero trailing matrices are assumed in the respective matrix. All component matrices must have the same dimensions. The complex matrix containing the harmonic components and the period T can be accessed via A.values and A.period, respectively.

 HarmonicArray(A0, Ac, As, T) -> A::HarmonicArray

Construct a harmonic array representation from the harmonic components.

The harmonic array object A is built for the harmonic representation Ahr(t) of a periodic matrix of period T in the form

                 p
 Ahr(t) = A_0 +  ∑ ( Ac_i*cos(i*t*2*π/T)+As_i*sin(i*2*π*t/T) ) ,
                i=1

where the constant term A_0 is contained in the real matrix A0, and Ac and As are vectors of real matrices such that the i-th (cosinus) coefficient matrix Ac_i is contained in Ac[i] and the i-th (sinus) coefficient matrix As_i is contained in As[i]. p is the maximum of length of the vectors of matrices Ac and As. If the length of Ac or As is less than p, then zero trailing matrices are assumed in the respective matrix. All component matrices must have the same dimensions. The complex matrix containing the harmonic components and the period T can be accessed via A.values and A.period, respectively.

 FourierFunctionMatrix(Afun, T) -> A::FourierFunctionMatrix

Continuous-time Fourier function matrix representation.

The Fourier function matrix object A of period T is built using the Fourier series representation of a periodic matrix Afun(t) of subperiod T′ = T/k, where each entry of Afun(t) has the form

         p
  a_0 +  ∑ ( ac_i*cos(i*t*2*π/T′)+as_i*sin(i*2*π*t/T′) ) ,
        i=1

where k ≥ 1 is the number of subperiods (default: k = 1). The matrix Afun containing the Fourier representation, the period T and the number of subperiods k can be accessed via A.M, A.period and A.nperiod, respectively.

PeriodicSwitchingMatrix(At, ts, T; nperiod = k) -> A::PeriodicSwitchingMatrix

Continuous-time periodic switching matrix representation.

The continuous-time periodic switching matrix object A of period T is built from a p-vector At of real matrices, a p-vector ts of increasingly ordered switching time values with ts[1] = 0, and the associated subperiod T′ = T/k, where k ≥ 1 is the number of subperiods (default: k = 1). At contains the cyclic component matrices At[i], i = 1,..., p, where At[i] is the constant value of a time periodic matrix A(t) of period T′ for t ∈ [ts[i],ts[i+1]), if i < p, or t ∈ [ts[i],T′), if i = p. It is assumed that At[i] := At[mod(i-1,p)+1] and ts[i] := ts[mod(i-1,p)+1] for arbitrary i. All component matrices must have the same dimensions. The component matrices At, the switching times ts, the period T and the number of subperiods k can be accessed via A.values, A.ts, A.period, and A.nperiod, respectively.

size(A::PM)
size(A::PM[, dim])

Return a tuple of two integers containing the dimensions of the continuous-time periodic matrix A of type PM, where PM is one of the types PeriodicFunctionMatrix, HarmonicArray, PeriodicTimeSeriesMatrix, PeriodicSwitchingMatrix, PeriodicSymbolicMatrix or PeriodicFunctionMatrix. Optionally you can specify a dimension dim to just get the length of that dimension.

eltype(A::PM)

Determine the type of the elements of the continuous-time periodic matrix A of type PM, where PM is one of the types PeriodicFunctionMatrix, HarmonicArray, PeriodicTimeSeriesMatrix, PeriodicSwitchingMatrix, PeriodicSymbolicMatrix or PeriodicFunctionMatrix.

getindex(A::PM, ind1, ind2)

Return the continuous-time periodic matrix built from the selected ranges [ind1,ind2] of the elements of the continuous-time periodic matrix Aof type PM, where PM is one of the types PeriodicFunctionMatrix, HarmonicArray, PeriodicTimeSeriesMatrix, PeriodicSwitchingMatrix, PeriodicSymbolicMatrix or PeriodicFunctionMatrix. ind1 and ind2 may be integers, integer ranges or colons.

lastindex(A::PM,dim)

Return the last index along dimension dim of the continuous-time periodic matrix A of type PM, where PM is one of the types PeriodicFunctionMatrix, HarmonicArray, PeriodicTimeSeriesMatrix, PeriodicSwitchingMatrix, PeriodicSymbolicMatrix or PeriodicFunctionMatrix.