# Exported functions and types

## Docstrings

### Integrators

SeeToDee.Rk4Type
f_discrete = Rk4(f, Ts; supersample = 1)

Discretize a continuous-time dynamics function f using RK4 with sample time Tₛ. f is assumed to have the signature f : (x,u,p,t)->ẋ and the returned function f_discrete : (x,u,p,t)->x(t+Tₛ).

supersample determins the number of internal steps, 1 is often sufficient, but this can be increased to make the interation more accurate. u is assumed constant during all steps.

If called with StaticArrays, this integrator is allocation free.

source
SeeToDee.SimpleCollocType
SimpleColloc(dyn, Ts, nx, na, nu; n = 5, abstol = 1.0e-8, solver=SimpleNewtonRaphson(), residual=false)
SimpleColloc(dyn, Ts, x_inds, a_inds, nu; n = 5, abstol = 1.0e-8, solver=SimpleNewtonRaphson(), residual=false)

A simple direct-collocation integrator that can be stepped manually, similar to the function returned by SeeToDee.Rk4.

This integrator supports differential-algebraic equations (DAE), the dynamics is expected to be on either of the forms

• nx,na provided: (xz,u,p,t)->[ẋ; res] where xz is a vector [x; z] contaning the differential state x and the algebraic variables z in this order. res is the algebraic residuals, and u is the control input. The algebraic residuals are thus assumed to be the last na elements of of the arrays returned by the dynamics (the convention used by ModelingToolkit).
• x_inds, a_inds provided: (xz,u,p,t)->xzd where xzd[x_inds] = ẋ and xzd[a_inds] = res.

The returned function has the signature f_discrete : (x,u,p,t)->x(t+Tₛ).

This integrator also supports a fully implicit form of the dynamics

$$$0 = F(ẋ, x, u, p, t)$$$

When using this interface, the dynamics is called using an additional input ẋ as the first argument, and the return value is expected to be the residual of the entire state descriptor. To use the implicit form, pass residual = true.

A Gauss-Radau collocation method is used to discretize the dynamics. The resulting nonlinear problem is solved using (by default) a Newton-Raphson method. This method handles stiff dynamics.

!!! Info "Differentiation" For fast automatic differentiation through this solver, use solver=NonlinearSolve.NewtonRaphson() instead of the default solver=SimpleNonlinearSolve.SimpleNewtonRaphson()

Arguments:

• dyn: Dynamics function (continuous time)
• Ts: Sample time
• nx: Number of differential state variables
• na: Number of algebraic variables
• x_inds, a_inds: If indices are provided instead of nx and na, the mass matrix is assumed to be diagonal, with ones located at x_inds and zeros at a_inds. For maximum efficiency, provide these indices as unit ranges or static arrays.
• nu: Number of inputs
• n: Number of collocation points. n=2 corresponds to trapezoidal integration.
• abstol: Tolerance for the root finding algorithm
• residual: If true the dynamics function is assumed to return the residual of the entire state descriptor and have the signature (ẋ, x, u, p, t) -> res. This is sometimes called "fully implicit form".
• solver: Any compatible SciML Nonlinear solver to use for the root finding problem

Extended help

• Super-sampling is not supported by this integrator, but you can trivially wrap it in a function that does super-sampling by stepping supersample times in a loop with the same input and sample time Ts / supersample.
• To use trapezoidal integration, set n=2 and nodetype=SeeToDee.FastGaussQuadrature.gausslobatto.
source

### Utilities

SeeToDee.linearizeFunction
A,B = linearize(f, x0, u0, p, t)

Linearize dynamics function f(x, u, p, t) w.r.t., state x, input u. Returns Jacobians A,B in

$$$ẋ = A\, Δx + B\, Δu$$$

Works for both continuous and discrete-time dynamics.

source
SeeToDee.initializeFunction
initialize(integ, x0, p, t = 0.0; solver=integ.solver, abstol=integ.abstol)

Given the differential state variables in x0, initialize the algebraic variables by solving the nonlinear problem f(x,u,p,t) = 0 using the provided solver.

Arguments:

• integ: An intergrator like SeeToDee.SimpleColloc
• x0: Initial state descriptor (differential and algebraic variables, where the algebraic variables comes last)
source