Methods

statedim(s::AbstractSystem)

Returns the dimension of the state space of system s.

stateset(s::AbstractSystem)

Returns the set of allowed states of system s.

inputdim(s::AbstractSystem)

Returns the dimension of the input space of system s.

inputset(s::AbstractSystem)

Returns the set of allowed inputs of system s.

nextinput(input::ConstantInput, n::Int=1)

Returns the first n elements of this input.

Input

• input – a constant input
• n – (optional, default: 1) the number of desired elements

Output

A repeated iterator that generates n equal samples of this input.

nextinput(input::VaryingInput, n::Int=1)

Returns the first n elements of this input.

Input

• input – varying input
• n – (optional, default: 1) number of desired elements

Output

An iterator of type Base.Iterators.Take that represents at most the first n elements of this input.

noisedim(s::AbstractSystem)

Returns the dimension of the noise space of system s.

noiseset(s::AbstractSystem)

Returns the set of allowed noises of system s.

outputdim(m::AbstractMap)

Returns the dimension of the output space of the map m.

outputmap(s::AbstractSystem)

Returns the output map of a system with output.

islinear(s::AbstractSystem)

Determines whether the dynamics of system s is specified by linear equations.

Notes

We adopt the notion from [Section 2.7, 1]. For example, the system with inputs $x' = f(t, x, u) = A x + B u$ is linear, since the function $f(t, ⋅, ⋅)$ is linear in $(x, u)$ for each $t ∈ \mathbb{R}$. On the other hand, $x' = f(t, x, u) = A x + B u + c$ is affine but not linear, since it is not linear in $(x, u)$.

The result of this function only depends on the system type, not the value, and can also be applied to typeof(s). So, if a system type allows an instance that is not linear, it returns false by default. For example, polynomial systems can be nonlinear; hence islinear returns false.

[1] Sontag, Eduardo D. Mathematical control theory: deterministic finite dimensional systems. Vol. 6. Springer Science & Business Media, 2013.

islinear(m::AbstractMap)

Determines whether the map m is linear or not.

Notes

A map is linear if it preserves the operations of scalar multiplication and vector addition.

isaffine(s::AbstractSystem)

Determines whether the dynamics of system s is specified by affine equations.

Notes

An affine system is the composition of a linear system and a translation. See islinear(::AbstractSystem) for the notion of linear system adopted in this library.

The result of this function only depends on the system type, not the value, and can also be applied to typeof(s). So, if a system type allows an instance that is not affine, it returns false by default. For example, polynomial systems can be nonlinear; hence isaffine is false.

isaffine(m::AbstractMap)

Determines whether the map m is affine or not.

Notes

An affine map is the composition of a linear map and a translation. See also islinear(::AbstractMap).

ispolynomial(s::AbstractSystem)

Determines whether the dynamics of system s is specified by polynomial equations.

The result of this function only depends on the system type, not the value, and can also be applied to typeof(s). Hence, e.g. a LinearContinuousSystem is not considered to be of polynomial type.

isblackbox(s::AbstractSystem)

Determines whether no specific structure is assumed for the dynamics of system s.

The result of this function only depends on the system type, not the value, and can also be applied to typeof(s).

isnoisy(s::AbstractSystem)

Determines if the dynamics of system s contains a noise term w.

The result of this function only depends on the system type, not the value, and can also be applied to typeof(s).

iscontrolled(s::AbstractSystem)

Determines if the dynamics of system s contains a control input u.

The result of this function only depends on the system type, not the value, and can also be applied to typeof(s).

isconstrained(s::AbstractSystem)

Determines if the system s has constraints on the state, input and noise, respectively (those that are available).

The result of this function only depends on the system type, not the value, and can also be applied to typeof(s).

state_matrix(s::AbstractSystem)

Returns the state matrix of an affine system.

Notes

The state matrix is the matrix proportional to the state, e.g. the matrix A in the linear continuous system $x' = Ax$.

input_matrix(s::AbstractSystem)

Returns the input matrix of a system with linear input.

Notes

The input matrix is the matrix proportional to the input, e.g. the matrix B in the linear continuous system with input, $x' = Ax + Bu$.

noise_matrix(s::AbstractSystem)

Returns the noise matrix of a system with linear noise.

Notes

The noise matrix is the matrix proportional to the noise, e.g. the matrix D in the linear system with noise, $x' = Ax + Dw$.

affine_term(s::AbstractSystem)

Returns the affine term in an affine system.

Notes

The affine term is e.g. the vector $c$ in the affine system $x' = Ax + c$.

mapping(s::AbstractSystem)

Returns the mapping of a black-box system.

apply(m::AbstractMap, args...)

Apply the rule specified by the map to the given arguments.

successor(system::DiscreteIdentitySystem, x::AbstractVector)

Return the successor state of a DiscreteIdentitySystem.

Input

• system – DiscreteIdentitySystem
• x – state (it should be any vector type)

Output

The same state x.

successor(system::ConstrainedDiscreteIdentitySystem, x::AbstractVector;
          [check_constraints]=true)

Return the successor state of a ConstrainedDiscreteIdentitySystem.

Input

• system – ConstrainedDiscreteIdentitySystem
• x – state (it should be any vector type)
• check_constraints – (optional, default: true) check if the state belongs to the state set

Output

The same state x.

successor(system::AbstractDiscreteSystem, x::AbstractVector;
          [check_constraints]=true)

Return the successor state of an AbstractDiscreteSystem.

Input

• system – AbstractDiscreteSystem
• x – state (it should be any vector type)
• check_constraints – (optional, default: true) check if the state belongs to the state set

Output

The result of applying the system to state x.

successor(system::AbstractDiscreteSystem, x::AbstractVector, u::AbstractVector;
          [check_constraints]=true)

Return the successor state of an AbstractDiscreteSystem.

Input

• system – AbstractDiscreteSystem
• x – state (it should be any vector type)
• u – input (it should be any vector type) or noise, if system is not controlled
• check_constraints – (optional, default: true) check if the state belongs to the state set

Output

The result of applying the system to state x and input u.

Notes

If the system is not controlled but noisy, the input u is interpreted as noise.

successor(system::AbstractDiscreteSystem,
          x::AbstractVector, u::AbstractVector, w::AbstractVector; [check_constraints]=true)

Return the successor state of an AbstractDiscreteSystem.

Input

• system – AbstractDiscreteSystem
• x – state (it should be any vector type)
• u – input (it should be any vector type)
• w – noise (it should be any vector type)
• check_constraints – (optional, default: true) check if the state belongs to the state set

Output

The result of applying the system to state x, input u and noise w.

vector_field(system::ContinuousIdentitySystem, x::AbstractVector)

Return the vector field state of a ContinuousIdentitySystem.

Input

• system – ContinuousIdentitySystem
• x – state (it should be any vector type)

Output

A zeros vector of dimension statedim.

vector_field(system::ConstrainedContinuousIdentitySystem, x::AbstractVector;
          [check_constraints]=true)

Return the vector field state of a ConstrainedContinuousIdentitySystem.

Input

• system – ConstrainedContinuousIdentitySystem
• x – state (it should be any vector type)
• check_constraints – (optional, default: true) check if the state belongs to the state set

Output

A zeros vector of dimension statedim.

vector_field(system::AbstractContinuousSystem, x::AbstractVector;
          [check_constraints]=true)

Return the vector field state of an AbstractContinuousSystem.

Input

• system – AbstractContinuousSystem
• x – state (it should be any vector type)
• check_constraints – (optional, default: true) check if the state belongs to the state set

Output

The vector field of the system at state x.

vector_field(system::AbstractContinuousSystem, x::AbstractVector, u::AbstractVector;
          [check_constraints]=true)

Return the vector field state of an AbstractContinuousSystem.

Input

• system – AbstractContinuousSystem
• x – state (it should be any vector type)
• u – input (it should be any vector type) or noise, if system is not controlled
• check_constraints – (optional, default: true) check if the state belongs to the state set

Output

The vector field of the system at state x and applying input u.

Notes

If the system is not controlled but noisy, the input u is interpreted as noise.

vector_field(system::AbstractContinuousSystem,
          x::AbstractVector, u::AbstractVector, w::AbstractVector; [check_constraints]=true)

Return the vector field state of an AbstractContinuousSystem.

Input

• system – AbstractContinuousSystem
• x – state (it should be any vector type)
• u – input (it should be any vector type)
• w – noise (it should be any vector type)
• check_constraints – (optional, default: true) check if the state belongs to the state set

Output

The vector field of the system at state x and applying input u and noise w.

discretize(system::AbstractContinuousSystem, ΔT::Real,
           algorithm::AbstractDiscretizationAlgorithm=ExactDiscretization(),
           constructor=_default_complementary_constructor(system))

Discretization of a isaffine AbstractContinuousSystem to a AbstractDiscreteSystem with sampling time ΔT using the discretization method algorithm.

Input

• system – an affine continuous system
• ΔT – sampling time
• algorithm – (optional, default: ExactDiscretization()) discretization algorithm
• constructor – (optional, default: _default_complementary_constructor(system)) construction method

Output

Returns a discretization of the input system system with discretization method algorithm and sampling time ΔT.

_discretize(::AbstractDiscretizationAlgorithm, ΔT::Real
            A::AbstractMatrix, B::AbstractMatrix, c::AbstractVector, D::AbstractMatrix)

Implementation of the discretization algorithm defined by the first input argument with sampling time ΔT.

Input

• `` – discretization algorithm, used for dispatch
• ΔT – sampling time
• A – state matrix
• B – input matrix
• c – vector
• D – noise matrix

Output

Returns a vector containing the discretized input arguments A, B, c and D.

Notes

See discretize for more details.

_discretize(algorithm::AbstractDiscretizationAlgorithm, ΔT::Real,
            A::AbstractMatrix)

Discretize the state matrix A with sampling time ΔT and discretization method algorithm.

Input

• algorithm – discretization algorithm
• ΔT – sampling time
• A – state matrix

Output

Returns a vector containing the discretized input argument A.

Notes

See discretize for more details.

_discretize(algorithm::AbstractDiscretizationAlgorithm, ΔT::Real,
            A::AbstractMatrix, B::AbstractMatrix)

Discretize the state matrix A and input or noise matrix B with sampling time ΔT and discretization method algorithm.

Input

• algorithm – discretization algorithm
• ΔT – sampling time
• A – state matrix
• B – input or noise matrix

Output

Returns a vector containing the discretized input arguments A and B.

Notes

This method signature with two arguments of type AbstractMatrix works both for a noisy system with fields (:A,:D) and a controlled system with fields (:A,:B).

See discretize for more details.

_discretize(algorithm::AbstractDiscretizationAlgorithm, ΔT::Real,
            A::AbstractMatrix,c::AbstractVector)

Discretize the state matrix A and vector c with sampling time ΔT and discretization method algorithm.

Input

• algorithm – discretization algorithm
• ΔT – sampling time
• A – state matrix
• c – vector

Output

Returns a vector containing the discretized input arguments A and c.

Notes

See discretize for more details.

_discretize(algorithm::AbstractDiscretizationAlgorithm, ΔT::Real,
            A::AbstractMatrix, B::AbstractMatrix, c::AbstractVector)

Discretize the state matrix A, input matrix B and vector c with sampling time ΔT and discretization method algorithm.

Input

• algorithm – discretization algorithm
• ΔT – sampling time
• A – state matrix
• B – input matrix
• c – vector

Output

Returns a vector containing the discretized input arguments A, B and c.

Notes

See discretize for more details.

_discretize(algorithm::AbstractDiscretizationAlgorithm, ΔT::Real,
            A::AbstractMatrix, B::AbstractMatrix, D::AbstractMatrix)

Discretize the state matrix A, input matrix B and noise matrix C with sampling time ΔT and discretization method algorithm.

Input

• algorithm – discretization algorithm
• ΔT – sampling time
• A – state matrix
• B – input matrix
• D – noise matrix

Output

Returns a vector containing the discretized input arguments A, B and D.

Notes

See discretize for more details.

typename(system::AbstractSystem)

Returns the base type of system without parameter information.

Input

• system – AbstractSystem

Output

The base type of system.

_complementary_type(system_type::Type{<:AbstractSystem})

Return the complementary type of a system type system_type.

Input

• system_type – type of AbstractSystem

Output

The complementary type of system_type.

Notes

There are two main subclasses of abstract types: continuous types and discrete types. A complementary type of system_type has the same fields as system_type but belongs to the other subclass, e.g. for a LinearContinuousSystem which is a subtype of AbstractContinuousSystem and has the field :A, the subtype of AbstractDiscreteSystem with the field :A, i.e. LinearDiscreteSystem, is returned.

To get the complementary type of system type, use _complementary_type(typename(system)).