Exported functions and types

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.

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.

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.

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)