Received: 1 March 2016 Revised: 8 March 2017 Accepted: 9 March 2017
GLOBAL AND ROBUST OPTIMIZATION OF DYNAMIC SYSTEMS
Affine relaxations for the solutions of constrained parametric
ordinary differential equations
Stuart M. Harwood Paul I. Barton
Process Systems Engineering Laboratory,
Massachusetts Institute of Technology,
Cambridge, MA, 02139, USA
Paul I. Barton, Process Systems Engineering
Laboratory, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA.
Stuart M. Harwood, ExxonMobil Research
and Engineering, Annandale, NJ 08801,
Novartis Pharmaceuticals ; Novartis-MIT
Center for Continuous Manufacturing
This work presents a numerical method for evaluating affine relaxations of the solu-
tions of parametric ordinary differential equations. This method is derived from a
general theory for the construction of a polyhedral outer approximation of the reach-
able set (“polyhedral bounds”) of a constrained dynamic system subject to uncertain
time-varying inputs and initial conditions. This theory is an extension of differential
inequality-based comparison theorems. The new affine relaxation method is capa-
ble of incorporating information from simultaneously constructed interval bounds as
well as other constraints on the states; not only does this improve the quality of the
relaxations but it also yields numerical advantages that speed up the computation of
the relaxations. Examples demonstrate that tight affine relaxations can be computed
efficiently with this method.
nonlinear dynamics, parametric differential equations, relaxations
The focus of this work and one of its main contributions is a new method for the construction of affine relaxations of the solutions
of initial value problems (IVPs) in parametric ordinary differential equations (ODEs) (see Section 2.2 for an exact statement).
Convex and concave relaxations of the solutions of parametric ODEs are vital to the global solution of optimal control problems
via the direct method of control parameterization combined with branch and bound. Previous work dealing with this includes
The relaxation theory developed here is inspired the most by Villanueva et al.
What distinguishes the present
approach is that the basic theorem in this work deals with state-constrained control systems (time-varying inputs) and derives
relaxations for the solutions of parametric ODEs as a special case. Neither theory is more general than the other, and they provide
different approaches to similar problems. Further, a novel feature of the proposed method is that it combines the evaluation
of interval and affine relaxations; the result is that interval bounds can be improved by tight affine relaxations and vice versa.
This is in contrast with the methods in previous studies,
and it is demonstrated that this improves the overall tightness of the
resulting relaxations or speeds up their calculation.
This work states a general theory that gives sufficient conditions for a polyhedral-valued mapping to enclose all solutions
of a constrained dynamic system subject to uncertain inputs and initial conditions. Such a mapping is called “bounds” on the
reachable sets of the dynamic system. This theory is in the vein of a comparison theorem involving differential inequalities.
Such theorems have a long history, going back to “Müller’s theorem,”
which was subsequently generalized to control systems
These theorems give conditions under which one can construct componentwise upper and lower, or interval,
bounds on the solutions. Recent work in previous studies
has expanded these theorems. At the heart of the numerical methods
resulting from these theories is the construction of an auxiliary system of ODEs, whose solution yields the parameters describing
Optim Control Appl Meth. 2018;39:427–448. wileyonlinelibrary.com/journal/oca Copyright © 2017 John Wiley & Sons, Ltd. 427