ImplicitDiscreteSystem

struct ImplicitDiscreteSystem <: ModelingToolkit.AbstractDiscreteSystem

An implicit system of difference equations.

Fields

• tag: A tag for the system. If two systems have the same tag, then they are structurally identical.

• eqs: The difference equations defining the discrete system.

• iv: Independent variable.

• unknowns: Dependent (state) variables. Must not contain the independent variable.

• ps: Parameter variables. Must not contain the independent variable.

• tspan: Time span.

• var_to_name: Array variables.

• observed: Observed states.

• name: The name of the system

• description: A description of the system.

• systems: The internal systems. These are required to have unique names.

• defaults: The default values to use when initial conditions and/or parameters are not supplied in ImplicitDiscreteProblem.

• guesses: The guesses to use as the initial conditions for the initialization system.

• initializesystem: The system for performing the initialization.

• initialization_eqs: Extra equations to be enforced during the initialization sequence.

• preface: Inject assignment statements before the evaluation of the RHS function.

• connector_type: Type of the system.

• parameter_dependencies: Topologically sorted parameter dependency equations, where all symbols are parameters and the LHS is a single parameter.

• metadata: Metadata for the system, to be used by downstream packages.

• gui_metadata: Metadata for MTK GUI.

• tearing_state: Cache for intermediate tearing state.

• substitutions: Substitutions generated by tearing.

• namespacing: If false, then sys.x no longer performs namespacing.

• complete: If true, denotes the model will not be modified any further.

• index_cache: Cached data for fast symbolic indexing.

• parent: The hierarchical parent system before simplification.

• isscheduled

Example

using ModelingToolkit
using ModelingToolkit: t_nounits as t
@parameters σ=28.0 ρ=10.0 β=8/3 δt=0.1
@variables x(t)=1.0 y(t)=0.0 z(t)=0.0
k = ShiftIndex(t)
eqs = [x ~ σ*(y-x(k-1)),
       y ~ x(k-1)*(ρ-z(k-1))-y,
       z ~ x(k-1)*y(k-1) - β*z]
@named ide = ImplicitDiscreteSystem(eqs,t,[x,y,z],[σ,ρ,β]; tspan = (0, 1000.0))

structural_simplify(sys; ...)
structural_simplify(
    sys,
    io;
    additional_passes,
    simplify,
    split,
    allow_symbolic,
    allow_parameter,
    conservative,
    fully_determined,
    kwargs...
)

Structurally simplify algebraic equations in a system and compute the topological sort of the observed equations in sys.

Optional Arguments:

• optional argument io may take a tuple (inputs, outputs). This will convert all inputs to parameters and allow them to be unconnected, i.e., simplification will allow models where n_unknowns = n_equations - n_inputs.

Optional Keyword Arguments:

• When simplify=true, the simplify function will be applied during the tearing process.
• allow_symbolic=false, allow_parameter=true, and conservative=false limit the coefficient types during tearing. In particular, conservative=true limits tearing to only solve for trivial linear systems where the coefficient has the absolute value of $1$.
• fully_determined=true controls whether or not an error will be thrown if the number of equations don't match the number of inputs, outputs, and equations.
• sort_eqs=true controls whether equations are sorted lexicographically before simplification or not.

ImplicitDiscreteProblem(
    sys::ImplicitDiscreteSystem;
    ...
) -> Any
ImplicitDiscreteProblem(
    sys::ImplicitDiscreteSystem,
    u0map;
    ...
) -> Any
ImplicitDiscreteProblem(
    sys::ImplicitDiscreteSystem,
    u0map,
    tspan;
    ...
) -> Any
ImplicitDiscreteProblem(
    sys::ImplicitDiscreteSystem,
    u0map,
    tspan,
    parammap;
    eval_module,
    eval_expression,
    kwargs...
) -> Any

Generates an ImplicitDiscreteProblem from an ImplicitDiscreteSystem.

struct Shift <: Symbolics.Operator

Represents a shift operator.

Fields

• t: Fixed Shift

• steps

Examples

julia> using Symbolics

julia> Δ = Shift(t)
(::Shift) (generic function with 2 methods)