Received: 16 August 2016 Revised: 20 December 2016 Accepted: 16 January 2017
GLOBAL AND ROBUST OPTIMIZATION OF DYNAMIC SYSTEMS
Towards rigorous robust optimal control via generalized
high-order moment expansion
Jiaqi C. Li
School of Information Science and Technology,
ShanghaiTech University, Shanghai 200031, China
Centre for Process Systems Engineering,
Department of Chemical Engineering, Imperial
College London, South Kensington Campus,
London SW7 2AZ, United Kingdom
Benoit Chachuat, Centre for Process Systems
Engineering, Department of Chemical Engineering,
Imperial College London, South Kensington
Campus, London SW7 2AZ, United Kingdom.
This study is concerned with the rigorous solution of worst-case robust optimal
control problems having bounded time-varying uncertainty and nonlinear dynam-
ics with affine uncertainty dependence. We propose an algorithm that combines
existing uncertainty set-propagation and moment-expansion approaches. Specifi-
cally, we consider a high-order moment expansion of the time-varying uncertainty,
and we bound the effect of the infinite-dimensional remainder term on the system
state, in a rigorous manner, using ellipsoidal calculus. We prove that the error intro-
duced by the expansion converges to zero as more moments are added. Moreover,
we describe a methodology to construct a conservative, yet more computationally
tractable, robust optimization problem, whose solution values are also shown to
converge to those of the original robust optimal control problem. We illustrate the
applicability and accuracy of this approach with the robust time-optimal control of
a motorized robot arm.
moment expansion, robust optimal control, set propagation, time-varying uncer-
Classical robust control theory has a long history,
with enormous practical relevance that has led to countless contributions
over the last few decades.
In essence, the need for robust control arises whenever an uncertain process or system is subject
to critical safety or operational constraints. Worst-case robust optimal control aims to minimize or maximize a function of the
states and controls measuring the performance of a dynamic system, while satisfying the terminal and path constraints for all
possible realizations of all uncertain quantities. For dynamic processes described by differential equations, one may distinguish
two types of uncertainty: (1) parametric uncertainty, entering either the initial condition or the right-hand side function in the
form of time-invariant parameters; and (2) time-varying uncertainty, describing either exogenous disturbances or endogenous
disturbances such as structural plant-model mismatch. The main focus of this paper is on worst-case robust optimal control
methods that are applicable to bounded time-varying uncertainty, although these methods can be readily extended to encompass
time-invariant uncertainty as a special case.
Existing approaches to worst-case robust optimal control with bounded time-invariant or time-varying uncertainty may be
divided into two broad classes, namely, discretization and set-propagation methods. The former includes direct simultane-
ous optimal control methods,
which proceed by discretizing all the control and state trajectories along with the time-varying
uncertainties, and often neglects the discretization errors. This way, the original worst-case robust optimal control problem is
approximated by a finite-dimensional min-max optimization problem.
The robust counterpart methodology by Ben-Tal et al
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work is properly cited.
© 2017 The Authors. Optimal Control Applications and Methods published by John Wiley & Sons, Ltd.
Optim Control Appl Meth. 2018;39:489–502. wileyonlinelibrary.com/journal/oca 489