Methods

Methods

This section describes systems methods implemented in IntervalMatrices.jl.

Common functions

IntervalArithmetic.inf — Function.
inf(A::IntervalMatrix{T}) where {T}

Return the infimum of an interval matrix A, which corresponds to taking the element-wise infimum of A.

Input

  • A – interval matrix

Output

A scalar matrix whose coefficients are the infima of each element in A.

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IntervalArithmetic.sup — Function.
sup(A::IntervalMatrix{T}) where {T}

Return the supremum of an interval matrix A, which corresponds to taking the element-wise supremum of A.

Input

  • A – interval matrix

Output

A scalar matrix whose coefficients are the suprema of each element in A.

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IntervalArithmetic.mid — Function.
mid(A::IntervalMatrix{T}) where {T}

Return the midpoint of an interval matrix A, which corresponds to taking the element-wise midpoint of A.

Input

  • A – interval matrix

Output

A scalar matrix whose coefficients are the midpoints of each element in A.

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Base.rand — Function.
rand(::Type{IntervalMatrix}, m::Int=2, n::Int=2;
     N=Float64, rng::AbstractRNG=GLOBAL_RNG)

Return a random interval matrix of the given size and numeric type.

Input

  • IntervalMatrix – type, used for dispatch
  • m – (optional, default: 2) number of rows
  • n – (optional, default: 2) number of columns
  • rng – (optional, default: GLOBAL_RNG) random-number generator

Output

An interval matrix of size $m × n$ whose coefficients are normally-distributed intervals of type N with mean 0 and standard deviation 1.

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IntervalMatrices.sample — Function.
sample(A::IntervalMatrix{T}; rng::AbstractRNG=GLOBAL_RNG) where {T}

Return a sample of the given random interval matrix.

Input

  • A – interval matrix
  • m – (optional, default: 2) number of rows
  • n – (optional, default: 2) number of columns
  • rng – (optional, default: GLOBAL_RNG) random-number generator

Output

An interval matrix of size $m × n$ whose coefficients are normally-distributed intervals of type N with mean 0 and standard deviation 1.

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Base.split — Function.
split(A::IntervalMatrix{T}) where {T}

Split an interval matrix $A$ into two scalar matrices $C$ and $S$ such that $A = C + [-S, S]$.

Input

  • A – interval matrix

Output

A pair (C, S) such that the entries of C are the central points and the entries of S are the (nonnegative) radii of the intervals in A.

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Base.:∈ — Function.
∈(M::AbstractMatrix, A::AbstractIntervalMatrix)

Check whether a concrete matrix is an instance of an interval matrix.

Input

  • M – concrete matrix
  • A – interval matrix

Output

true iff M is an instance of A

Algorithm

We check for each entry in M whether it belongs to the corresponding interval in A.

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Exponentiation

IntervalMatrices.expm_overapproximation — Function.
expm_overapproximation(M::IntervalMatrix{T, Interval{T}}, t, p) where {T}

Overapproximation of the exponential of an interval matrix.

Input

  • A – interval matrix
  • t – non-negative time value
  • p – order of the approximation

Algorithm

See Theorem 1 in Reachability Analysis of Linear Systems with Uncertain Parameters and Inputs by M. Althoff, O. Stursberg, M. Buss.

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IntervalMatrices.expm_underapproximation — Function.
expm_underapproximation(M::IntervalMatrix{T, Interval{T}}, t, p) where {T}

Overapproximation of the exponential of an interval matrix.

Input

  • A – interval matrix
  • t – non-negative time value
  • p – order of the approximation

Algorithm

See Theorem 2 in Reachability Analysis of Linear Systems with Uncertain Parameters and Inputs by M. Althoff, O. Stursberg, M. Buss.

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Finite expansions

IntervalMatrices.quadratic_expansion — Function.
quadratic_expansion(A::IntervalMatrix, t)

Exactly compute the quadratic formula $At + \frac{1}{2}A^2t^2$ using interval arithmetics.

Input

  • A – interval matrix
  • t – non-negative time value

Algorithm

See Lemma 1 in Reachability Analysis of Linear Systems with Uncertain Parameters and Inputs by M. Althoff, O. Stursberg, M. Buss.

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Hulls

IntervalMatrices.correction_hull — Function.
correction_hull(A::IntervalMatrix{T}, t, p) where {T}

Compute the correction term for the convex hull of a point and its linear map with an interval matrix in order to contain all trajectories of a linear system.

Input

  • A – interval matrix
  • t – non-negative time value
  • p – order of the approximation

Output

An interval matrix representing the correction term.

Algorithm

See Theorem 3 in [1].

[1] M. Althoff, O. Stursberg, M. Buss. Reachability Analysis of Linear Systems with Uncertain Parameters and Inputs. CDC 2007.

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Norms

LinearAlgebra.opnorm — Function.
opnorm(A::IntervalMatrix, p::Real=Inf)

The matrix norm of an interval matrix.

Input

  • A – interval matrix
  • p – (optional, default: Inf) the class of p-norm

Notes

The matrix $p$-norm of an interval matrix $A$ is defined as

\[ ‖A‖_p := ‖\max(|\text{inf}(A)|, |\text{sup}(A)|)‖_p\]

where $\max$ and $|·|$ are taken elementwise.

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