CTDirect.jl
The CTDirect.jl package is part of the control-toolbox ecosystem.
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An optimal control problem with fixed initial and final times, denoted (OCP), can be described as minimising the cost functional
\[g(x(t_0), x(t_f)) + \int_{t_0}^{t_f} f^{0}(t, x(t), u(t))~\mathrm{d}t\]
where the state $x$ and the control $u$ are functions subject, for $t \in [t_0, t_f]$, to the differential constraint
\[ \dot{x}(t) = f(t, x(t), u(t))\]
and other constraints such as
\[\begin{array}{llcll} ~\xi_l &\le& \xi(t, u(t)) &\le& \xi_u, \\ \eta_l &\le& \eta(t, x(t)) &\le& \eta_u, \\ \psi_l &\le& \psi(t, x(t), u(t)) &\le& \psi_u, \\ \phi_l &\le& \phi(t_0, x(t_0), t_f, x(t_f)) &\le& \phi_u. \end{array}\]
The so-called direct approach transforms the infinite dimensional optimal control problem (OCP) into a finite dimensional optimization problem (NLP). This is done by a discretization in time by Runge-Kutta methods applied to the state and control variables, as well as the dynamics equation. These methods are usually less precise than indirect methods based on Pontryagin’s Maximum Principle, but more robust with respect to the initialization. Also, they are more straightforward to apply, hence their wide use in industrial applications. We refer the reader to for instance[1] and [2] for more details on direct transcription methods and NLP algorithms.
Example of the time discretization by the trapezoidal rule:
\[\begin{array}{lcl} t \in [t_0,t_f] & \to & \{t_0, \ldots, t_N=t_f\}\\[0.2em] x(\cdot),\, u(\cdot) & \to & X=\{x_0, \ldots, x_N, u_0, \ldots, u_N\} \\[1em] \hline \\ \text{step} & \to & h = (t_f-t_0)/N\\[0.2em] \text{criterion} & \to & \min\ g(x_0, x_N) \\[0.2em] \text{dynamics} & \to & x_{i+i} = x_i + (h/2)\, (f(t_i, x_i, u_i) + f(t_{i+1}, x_{i+1}, u_{i+1})) \\[0.2em] \text{control constraints} &\to& \xi_l \le \xi(t_i, u_i) \le \xi_u \\[0.2em] \text{path constraints} &\to& \eta_l \le \eta(t_i, x_i) \le \eta_u \\[0.2em] \text{mixed constraints} &\to& \psi_l \le \psi(t_i, x_i, u_i) \le \psi_u \\[0.2em] \text{limit conditions} &\to& \phi_l \le \phi(x_0, x_N) \le \phi_u \end{array}\]
We therefore obtain a nonlinear programming problem on the discretized state and control variables of the general form:
\[(NLP)\quad \left\{ \begin{array}{lr} \min \ F(X) \\ LB \le C(X) \le UB \end{array} \right.\]
Solving the (NLP) problem is done using packages from JuliaSmoothOptimizers, with Ipopt as the default solver.
On the input side of this package, we use an OptimalControlModel structure from CTBase to define the (OCP).
The direct transcription to build the (NLP) can use discretization schemes such as trapeze (default), midpoint, or Gauss-Legendre collocations.
This package is equivalent to the bocop software.