A package to solve multiple-phase optimal control problems

An p-phase optimal control problem can be stated in the following general form. Determine the state, ${\bf x^{(p)}}(t) \in \mathbb{R}^{n_{z}^{(p)}}$ control ${\bf u^{(p)}}(t) \in \mathbb{R}^{n_{u}^{(p)}}$, initial time $t_{0}^{(p)}\in\mathbb{R}$, final time $t_{f}^{(p)}\in\mathbb{R}$, integrals, ${\bf q}^{(p)}\in\mathbb{R}^{n_{q}^{(p)}}$, local parameters ${\bf k_{p}} \in \mathbb{R}^{n_{k}^{(p)}}$ in each phase $p\in[1,...,P]$, and the static parameters, ${\bf k_{g}}\in\mathbb{R}^{n_{kg}}$, that minimize the cost functional

$$J=\sum_{p=1}^{n}J_{p}$$

where,

$$J_{p}=\intop_{t_{i}^{p}}^{t_{f}^{p}}L^{p}({\bf x^{(p)}}(t),{\bf u^{(p)}}(t),t^{(p)},{\bf k},{\bf ap})\ dt^{p}+\phi(x^{(p)}(t_{f}^{p}),u^{(p)}(t_{f}^{p}),t_{f}^{(p)},{\bf k},{\bf ap})$$

Note: ${\bf ap}$ contains all the auxillary parameters used to define various functions, and ${\bf k} =[{\bf k_{p}}; {\bf k_{g}}]$ is a stacked vector of phase and global parameters which can be optimized.

The cost funtional $J$ must be minimized subject to the following constraints in each phase $p$:

Path constraints:

Path constraints are the constraints which the states and controls must obey at each instant $t^{(p)}$ .

$${\bf pf_{l}}^{(p)}\leq{\bf pathfun}^{(p)}({\bf x^{p}}(t),{\bf u^{p}}(t),t^{(p)},{\bf k},{\bf ap})\in\mathbb{R}^{n_{pf}^{k}}\leq{\bf pf_{u}}^{(p)}\ \forall\ t^{(p)}\in[t_{i}^{(p)},t_{f}^{(p)}]$$

Integral constraints:

Define:

$${\bf IF}^{(p)}=\intop_{t_{i}^{p}}^{t_{f}^{p}}{\bf integralfun}^{(p)}({\bf x^{(p)}}(t),{\bf u^{(p)}}(t),t^{(p)},{\bf k},{\bf ap})\ dt^{(p)}$$ $${\bf if_{l}}^{(p)}\leq{\bf pathfun}({\bf x^{p}}(t),{\bf u^{p}}(t),t^{(p)},{\bf k},{\bf ap})\in\mathbb{R}^{n_{pf}^{k}}\leq{\bf if_{u}}^{(p)}$$

Event constraints:

Events in time are what cause a change of phase. Event constraints represent linkages between initial times, states, inputs and final times, states, inputs between phases.

$${\bf psi_{u}}^{(p)}\leq{\bf psi}(t_{i}^{(1)},t_{f}^{(1)},{\bf x_{f}}^{(1)},{\bf u_{f}}^{(1)},t_{i}^{(2)},t_{f}^{(2)},{\bf x_{f}}^{(2)},{\bf u_{f}}^{(2)},\ldots t_{i}^{(P)},t_{f}^{(P)},{\bf x_{f}}^{(P)},{\bf u_{f}}^{(P)})\in\mathbb{R}^{n_{psi}^{(p)}}\leq{\bf psi_{u}}^{(p)}$$

Features

1. Multiple phases
2. Scaling
3. Mesh recomputation
4. Flexibility in formulating problems

Installation

add https://github.com/A-C1/DirectOptimalControl.jl