Toolbox (max,+) Linear Systems

(max,+) Linear System constructor

MaxPlus.MPSysLin — Type

MPSysLin(A::MPAbstractVecOrMat, B::MPAbstractVecOrMat, C::MPAbstractVecOrMat [, D::MPAbstractVecOrMat, [x0::MPAbstractVecOrMat]])

Structure for state space representation of Max-Plus linear systems. Creation of max-plus dynamical linear systems in implicit state form:

    X(n) = D ⨂ X(n) ⨁ A ⨂ X(n-1) ⨁ B ⨂ U(n),
    Y(n) = C ⨂ X(n)

MPSysLin is the equivalent to ScicosLab function mpsyslin. It accepts dense or sparse Max-Plus arrays. D and x0 can be omited: they will be filled with Max-Plus zeros ε.

Arguments

A shall be squared.
B shall has its row dimension in accordance with dimensions of A.
C shall has its column dimension in accordance with dimensions of A.
D shall has its dimension in accordance with dimensions of A.
x0 shall has its dimensions in accordance with dimensions of A.

Sizes are checked and an error is thrown during the compilation.

Elements of the system can access directly ie. S.A, S.B, S.C, S.D, S.x0.

Examples

julia> S = MPSysLin(MP([1.0 2 3;  4 5 6;  7 8 9]), # Max-Plus matrix A
                    MP([    0.0;      0;      0]), # Max-Plus column vector B
                    MP([    0.0       0       0]), # Max-Plus row vector C
                    mpeye(3, 3),                   # Max-Plus matrix D
                    full(zeros(MP, 3, 1)))         # Max-Plus column vector x0

Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 3×3 Max-Plus dense matrix:
  0   .   .
  .   0   .
  .   .   0

A = 3×3 Max-Plus dense matrix:
  1   2   3
  4   5   6
  7   8   9

B = 3-element Max-Plus vector:
  0
  0
  0

C = 1×3 Max-Plus dense matrix:
  0   0   0

x0 = 3-element Max-Plus vector:
  .
  .
  .

julia> typeof(S)
MPSysLin

julia> S.A
3×3 Max-Plus dense matrix:
  1   2   3
  4   5   6
  7   8   9

(max,+) Linear System operations

MaxPlus.mpexplicit — Method

mpexplicit(S::MPSysLin)

Convert to an explicit linear system.

Examples

julia> S = MPSysLin(MP([1.0 2; 3 4]),
                    MP([0.0; 0]),
                    MP([0.0 0]),
                    mpeye(Float64, 2,2))

Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 2×2 Matrix{MP{Float64}}:
  0   .
  .   0

A = 2×2 Matrix{MP{Float64}}:
  1   2
  3   4

B = 2-element Vector{MP{Float64}}:
  0
  0

C = 1×2 Matrix{MP{Float64}}:
  0   0

x0 = 2-element Vector{MP{Float64}}:
  .
  .

julia> mpexplicit(S)

Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 2×2 Matrix{MP{Float64}}:
  0   .
  .   0

A = 2×2 Matrix{MP{Float64}}:
  1   2
  3   4

B = 2-element Vector{MP{Float64}}:
  0
  0

C = 1×2 Matrix{MP{Float64}}:
  0   0

x0 = 2-element Vector{MP{Float64}}:
  .
  .

MaxPlus.mpsimul — Method

mpsimul(S::MPSysLin, u::MPAbstractVecOrMat, history::Bool)

Compute states X of an autonomous linear max-plus system:

    x(n+1) = A ⨂ x(n)`  for n = 0 .. k

Where:

A is a system matrix
x0 is a initial state vector,
k is the number of iterations
when history is set to true save all computed states, else return the last one.

Arguments

u: inputs to inject in the system
history: if true then return a vector holding all results, else return the last result.

Examples

julia> S = MPSysLin(MP([1.0 2; 3 4]),
                    MP([0.0; 0]),
                    MP([0.0 0]),
                    mpeye(Float64, 2,2))

Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 2×2 Matrix{MP{Float64}}:
  0   .
  .   0

A = 2×2 Matrix{MP{Float64}}:
  1   2
  3   4

B = 2-element Vector{MP{Float64}}:
  0
  0

C = 1×2 Matrix{MP{Float64}}:
  0   0

x0 = 2-element Vector{MP{Float64}}:
  .
  .

julia> mpsimul(S, MP(1:10))
1×10 Array{MP{Float64},2}:
 1  5  9  13  17  21  25  29  33  37

julia> mpsimul(S1, MP(1:10), history=false)
1×1 Array{MP{Float64},2}:
 37

Base.:+ — Method

Base.:(+)(y::MPSysLin, x::MPSysLin)

Parallel composition of two Max-Plus linear systems.

Examples

julia> S1 = MPSysLin(MP([1.0 2 3;  4 5 6;  7 8 9]),
                     MP([    1.0;      2;      3]),
                     MP([    4.0       5       6]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S2 = MPSysLin(MP([10.0 11 12;  13 14 15;  16 17 18]),
                     MP([    0.0;      0;      0]),
                     MP([    0.0       0       0]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S1 + S2
Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 6×6 Max-Plus dense matrix:
  0   .   .   .   .   .
  .   0   .   .   .   .
  .   .   0   .   .   .
  .   .   .   0   .   .
  .   .   .   .   0   .
  .   .   .   .   .   0

A = 6×6 Max-Plus dense matrix:
  1   2   3    .    .    .
  4   5   6    .    .    .
  7   8   9    .    .    .
  .   .   .   10   11   12
  .   .   .   13   14   15
  .   .   .   16   17   18

B = 6-element Max-Plus vector:
  1
  2
  3
  0
  0
  0

C = 1×6 Max-Plus dense matrix:
  4   5   6   0   0   0

x0 = 6-element Max-Plus vector:
  .
  .
  .
  .
  .
  .

Base.:* — Method

Base.:(*)(y::MPSysLin, x::MPSysLin)

Series composition of two Max-Plus linear systems.

Examples

julia> S1 = MPSysLin(MP([1.0 2 3;  4 5 6;  7 8 9]),
                     MP([    1.0;      2;      3]),
                     MP([    4.0       5       6]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S2 = MPSysLin(MP([10.0 11 12;  13 14 15;  16 17 18]),
                     MP([    0.0;      0;      0]),
                     MP([    0.0       0       0]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S1 * S2
TODO

Base.:| — Method

Base.:(|)(y::MPSysLin, x::MPSysLin)

Diagonal composition of two Max-Plus linear systems.

Examples

julia> S1 = MPSysLin(MP([1.0 2 3;  4 5 6;  7 8 9]),
                     MP([    1.0;      2;      3]),
                     MP([    4.0       5       6]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S2 = MPSysLin(MP([10.0 11 12;  13 14 15;  16 17 18]),
                     MP([    0.0;      0;      0]),
                     MP([    0.0       0       0]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S1 | S2

Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 6×6 Matrix{MP{Float64}}:
  0   .   .   .   .   .
  .   0   .   .   .   .
  .   .   0   .   .   .
  .   .   .   0   .   .
  .   .   .   .   0   .
  .   .   .   .   .   0

A = 6×6 Matrix{MP{Float64}}:
  1   2   3    .    .    .
  4   5   6    .    .    .
  7   8   9    .    .    .
  .   .   .   10   11   12
  .   .   .   13   14   15
  .   .   .   16   17   18

B = 6×2 Matrix{MP{Float64}}:
  1   .
  2   .
  3   .
  .   0
  .   0
  .   0

C = 2×6 Matrix{MP{Float64}}:
  4   5   6   .   .   .
  .   .   .   0   0   0

x0 = 6-element Vector{MP{Float64}}:
  .
  .
  .
  .
  .
  .

Base.vcat — Method

Base.:vcat(x::MPSysLin, y::MPSysLin)

Composition of two Max-Plus linear systems: inputs in common, concatenation of outputs.

Examples

julia> S1 = MPSysLin(MP([1.0 2 3;  4 5 6;  7 8 9]),
                     MP([    1.0;      2;      3]),
                     MP([    4.0       5       6]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S2 = MPSysLin(MP([10.0 11 12;  13 14 15;  16 17 18]),
                     MP([    0.0;      0;      0]),
                     MP([    0.0       0       0]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> [S1 ; S2]
Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 6×6 Matrix{MP{Float64}}:
  0   .   .   .   .   .
  .   0   .   .   .   .
  .   .   0   .   .   .
  .   .   .   0   .   .
  .   .   .   .   0   .
  .   .   .   .   .   0

A = 6×6 Matrix{MP{Float64}}:
  1   2   3    .    .    .
  4   5   6    .    .    .
  7   8   9    .    .    .
  .   .   .   10   11   12
  .   .   .   13   14   15
  .   .   .   16   17   18

B = 6-element Vector{MP{Float64}}:
  1
  2
  3
  0
  0
  0

C = 2×6 Matrix{MP{Float64}}:
  4   5   6   .   .   .
  .   .   .   0   0   0

x0 = 6-element Vector{MP{Float64}}:
  .
  .
  .
  .
  .
  .

Base.hcat — Method

Base.:hcat(x::MPSysLin, y::MPSysLin)

Composition of two Max-Plus linear systems: concatenation of inputs, addition of outputs.

Examples

julia> S1 = MPSysLin(MP([1.0 2 3;  4 5 6;  7 8 9]),
                     MP([    1.0;      2;      3]),
                     MP([    4.0       5       6]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S2 = MPSysLin(MP([10.0 11 12;  13 14 15;  16 17 18]),
                     MP([    0.0;      0;      0]),
                     MP([    0.0       0       0]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> [S1 S2]

Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 6×6 Matrix{MP{Float64}}:
  0   .   .   .   .   .
  .   0   .   .   .   .
  .   .   0   .   .   .
  .   .   .   0   .   .
  .   .   .   .   0   .
  .   .   .   .   .   0

A = 6×6 Matrix{MP{Float64}}:
  1   2   3    .    .    .
  4   5   6    .    .    .
  7   8   9    .    .    .
  .   .   .   10   11   12
  .   .   .   13   14   15
  .   .   .   16   17   18

B = 6×2 Matrix{MP{Float64}}:
  1   .
  2   .
  3   .
  .   0
  .   0
  .   0

C = 1×6 Matrix{MP{Float64}}:
  4   5   6   0   0   0

x0 = 6-element Vector{MP{Float64}}:
  .
  .
  .
  .
  .
  .

Base.:/ — Method

Base.:(/)(y::MPSysLin, x::MPSysLin)

Feedback composition: computes mpstar(S1*S2)*S1 in state-space form.

Examples

julia> S1 = MPSysLin(MP([1.0 2 3;  4 5 6;  7 8 9]),
                     MP([    1.0;      2;      3]),
                     MP([    4.0       5       6]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S2 = MPSysLin(MP([10.0 11 12;  13 14 15;  16 17 18]),
                     MP([    0.0;      0;      0]),
                     MP([    0.0       0       0]),
                     mpeye(3, 3),
                     full(zeros(MP, 3, 1)));

julia> S1 / S2

Implicit dynamic linear Max-Plus system:
Implicit dynamic linear Max-Plus system:
  x(n) = D*x(n) + A*x(n-1) + B*u(n)
  y(n) = C*x(n)
  x(0) = x0

with:
D = 6×6 Matrix{MP{Float64}}:
  0   .   .   1   1   1
  .   0   .   2   2   2
  .   .   0   3   3   3
  4   5   6   0   .   .
  4   5   6   .   0   .
  4   5   6   .   .   0

A = 6×6 Matrix{MP{Float64}}:
  1   2   3    .    .    .
  4   5   6    .    .    .
  7   8   9    .    .    .
  .   .   .   10   11   12
  .   .   .   13   14   15
  .   .   .   16   17   18

B = 6-element Vector{MP{Float64}}:
  1
  2
  3
  .
  .
  .

C = 1×6 Matrix{MP{Float64}}:
  4   5   6   .   .   .

x0 = 6-element Vector{MP{Float64}}:
  .
  .
  .
  .
  .
  .

Base.:== — Method

Base.:(==)(x::MPSysLin, y::MPSysLin)

Test whether two Max-Plus linear system are equal. ```

MaxPlus.mpshow — Method

mpshow(io::IO, S::MPSysLin)

Base function for displaying a Max-Plus linear system.

Examples

julia>

Base.show — Method

show(io::IO, S::MPSysLin)

Display a Max-Plus linear system.

Examples

julia>

Base.show — Method

show(io::IO, ::MIME"text/plain", S::MPSysLin)

Display a Max-Plus linear system.

Examples

julia>

MaxPlus.LaTeX — Method

LaTeX(S::MPSysLin)

Return the LaTeX code as string from the given Max-Plus linear system.

Examples

julia>

MaxPlus.LaTeX — Method

LaTeX(io::IO, S::MPSysLin)

Display the LaTeX code from the given Max-Plus linear system.

Examples

julia>