ODESystem

struct ODESystem <: ModelingToolkit.AbstractODESystem

A system of ordinary differential equations.

Fields

• eqs

  The ODEs defining the system.

• iv

  Independent variable.

• states

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

• ps

  Parameter variables. Must not contain the independent variable.

• var_to_name

  Array variables.

• ctrls

  Control parameters (some subset of ps).

• observed

  Observed states.

• tgrad

  Time-derivative matrix. Note: this field will not be defined until calculate_tgrad is called on the system.

• jac

  Jacobian matrix. Note: this field will not be defined until calculate_jacobian is called on the system.

• ctrl_jac

  Control Jacobian matrix. Note: this field will not be defined until calculate_control_jacobian is called on the system.

• Wfact

  Wfact matrix. Note: this field will not be defined until generate_factorized_W is called on the system.

• Wfact_t

  Wfact_t matrix. Note: this field will not be defined until generate_factorized_W is called on the system.

• name

  Name: the name of the system

• systems

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

• defaults

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

• structure

  structure: structural information of the system

• connector_type

  connector_type: type of the system

• connections

  connections: connections in a system

• preface

  preface: inject assignment statements before the evaluation of the RHS function.

• continuous_events

  continuous_events: A Vector{SymbolicContinuousCallback} that model events. The integrator will use root finding to guarantee that it steps at each zero crossing.

Example

using ModelingToolkit

@parameters σ ρ β
@variables t x(t) y(t) z(t)
D = Differential(t)

eqs = [D(x) ~ σ*(y-x),
       D(y) ~ x*(ρ-z)-y,
       D(z) ~ x*y - β*z]

@named de = ODESystem(eqs,t,[x,y,z],[σ,ρ,β])

ode_order_lowering(sys::ODESystem) -> Any

Takes a Nth order ODESystem and returns a new ODESystem written in first order form by defining new variables which represent the N-1 derivatives.

dae_index_lowering(sys::ODESystem; kwargs...) -> ODESystem

Perform the Pantelides algorithm to transform a higher index DAE to an index 1 DAE. kwargs are forwarded to pantelides!. End users are encouraged to call structural_simplify instead, which calls this function internally.

liouville_transform(sys::ModelingToolkit.AbstractODESystem)

Generates the Liouville transformed set of ODEs, which is the original ODE system with a new variable trJ appended, corresponding to the -tr(Jacobian). This variable is used for properties like uncertainty propagation from a given initial distribution density.

For example, if $u'=p*u$ and p follows a probability distribution $f(p)$, then the probability density of a future value with a given choice of $p$ is computed by setting the inital trJ = f(p), and the final value of trJ is the probability of $u(t)$.

Example:

using ModelingToolkit, OrdinaryDiffEq, Test

@parameters t α β γ δ
@variables x(t) y(t)
D = Differential(t)

eqs = [D(x) ~ α*x - β*x*y,
       D(y) ~ -δ*y + γ*x*y]

sys = ODESystem(eqs)
sys2 = liouville_transform(sys)
@variables trJ

u0 = [x => 1.0,
      y => 1.0,
      trJ => 1.0]

prob = ODEProblem(sys2,u0,tspan,p)
sol = solve(prob,Tsit5())

Where sol[3,:] is the evolution of trJ over time.

Sources:

Probabilistic Robustness Analysis of F-16 Controller Performance: An Optimal Transport Approach

Abhishek Halder, Kooktae Lee, and Raktim Bhattacharya https://abhishekhalder.bitbucket.io/F16ACC2013Final.pdf

function DiffEqBase.ODEFunction{iip}(sys::AbstractODESystem, dvs = states(sys),
                                     ps = parameters(sys);
                                     version = nothing, tgrad=false,
                                     jac = false,
                                     sparse = false,
                                     kwargs...) where {iip}

Create an ODEFunction from the ODESystem. The arguments dvs and ps are used to set the order of the dependent variable and parameter vectors, respectively.

function DiffEqBase.ODEProblem{iip}(sys::AbstractODESystem,u0map,tspan,
                                    parammap=DiffEqBase.NullParameters();
                                    version = nothing, tgrad=false,
                                    jac = false,
                                    checkbounds = false, sparse = false,
                                    simplify=false,
                                    linenumbers = true, parallel=SerialForm(),
                                    kwargs...) where iip

Generates an ODEProblem from an ODESystem and allows for automatically symbolically calculating numerical enhancements.

function DiffEqBase.SteadyStateProblem(sys::AbstractODESystem,u0map,
                                    parammap=DiffEqBase.NullParameters();
                                    version = nothing, tgrad=false,
                                    jac = false,
                                    checkbounds = false, sparse = false,
                                    linenumbers = true, parallel=SerialForm(),
                                    kwargs...) where iip

Generates an SteadyStateProblem from an ODESystem and allows for automatically symbolically calculating numerical enhancements.

ODAEProblem{iip}(sys, u0map, tspan, parammap = DiffEqBase.NullParameters(); kw...)

This constructor acts similar to the one for ODEProblem with the following changes: ODESystems can sometimes be further reduced if structural_simplify has already been applied to them. This is done this constructor. In these cases, the constructor uses the knowledge of the strongly connected components calculated during the process of simplification as the basis for building pre-simplified nonlinear systems in the implicit solving. In summary: these problems are structurally modified, but could be more efficient and more stable. Note, the returned object is still of type ODEProblem.