General Multiple-Phase Problem Solvable Using GPOPS-II

The general multiple-phase optimal control problem that can be solved by \(\mathbb{GPOPS-II}\) is given as follows. First, let \(p\in[1,\ldots,P]\) be the phase number where \(P\) as the total number of phases. The optimal control problem is to determine the state, \(\mathbf{y}^{(p)}(t)(t)\in\mathbb{R}^{n_y^{(p)}}\), the control, \(\mathbf{u}^{(p)}(t)(t)\in\mathbb{R}^{n_u^{(p)}}\), integrals, \(\mathbf{q}^{(p)}\in\mathbb{R}^{n_q^{(p)}}\), phase start times, \(t_0^{(p)}\in\mathbb{R}\), phase terminus times, \(t_f^{(p)}\in\mathbb{R}\), in all phases \(p\in[1,\ldots,P]\), along with the static parameters \(\mathbf{s}\in\mathbb{R}^{n_s}\), that minimize the objective functional
$$
J = \phi \left( \mathbf{e}^{(1)},\ldots,\mathbf{e}^{(P)}, \mathbf{s} \right),
$$
subject to the dynamic constraints
$$
\dot{\mathbf{y}}^{(p)}(t) = \mathbf{a}^{(p)}(\mathbf{y}^{(p)}(t),\mathbf{u}^{(p)}(t),t,\mathbf{s}), \quad (p=1,\ldots,P),
$$
the event constraints
$$ \label{general-event}
\mathbf{b}_{\min} \leq \mathbf{b} \left(\mathbf{e}^{(1)}, \ldots,\mathbf{e}^{(P)}, \mathbf{s} \right)\leq \mathbf{b}_{\max},
$$
where
$$\small
\mathbf{e}^{(p)}=\left[\mathbf{y}^{(p)}\left(t_0^{(p)}\right),t_0^{(p)},\mathbf{y}^{(p)}\left(t_f^{(p)}\right),t_f^{(p)},\mathbf{q}^{(p)}\right],\;(p=1,\ldots,P),$$
the inequality path constraints
$$
\mathbf{c}_{\min}^{(p)} \leq \mathbf{c}^{(p)}(\mathbf{y}^{(p)}(t),\mathbf{u}^{(p)}(t),t^{(p)},\mathbf{s})\leq \mathbf{c}_{\max}^{(p)}, \; (p=1,\ldots,P),
$$
the static parameter constraints
$$
\mathbf{s}_{\min} \leq \mathbf{s} \leq \mathbf{s}_{\max},
$$
and the integral constraints
$$
\mathbf{q}_{\min}^{(p)} \leq \mathbf{q} ^{(p)} \leq \mathbf{q}_{\max}^{(p)}, \; (p=1,\ldots,P),
$$
and the integrals in each phase are defined as
$$
q_i^{(p)} = \int_{t_0^{(p)}}^{t_f^{(p)}} g_i^{(p)} (\mathbf{y}^{(p)}(t),\mathbf{u}^{(p)}(t),t^{(p)},\mathbf{s})dt, \left[\begin{array}{c} i=1,\ldots n_q^{(p)},\\ p=1,\ldots,P \end{array} \right].
$$
It is important to note that the event constraints of Eq.~(\ref{general-event}) can contain any functions that relate information at the start and/or terminus of any phase (including relationships that include both static parameters and integrals) and that the phases themselves need not be sequential.

Image is in the Public Domain via Wikimedia Commons

Copyright © 2013-2015 RP Optimization Research LLC. All Rights Reserved. Contact Us