OPTIMAL CONTROL IN HYBRID SYSTEMS: AN EFFICIENT DECISION MAKING TOOL
S. Lenhart1,2 and V. Protopopescu1
1 Oak Ridge National Laboratory, Oak Ridge, TN 37831
2University of Tennessee, Knoxville, TN 37901-1996
Abstract. Hybrid systems evolve simultaneously in continuous and discrete state spaces. An illustration of practical
interest is realized by continuous systems for which decisions have to be made at discrete times. We set this problem as
an optimal control problem, whereby the decisions become controls that are implemented in order to optimize a certain
desired outcome. Due to the intertwining between continuous and discrete dynamics, the derivation of the adjoint and
optimality systems is different from the purely continuous or discrete cases. We obtain the necessary optimality
conditions and, for a few typical illustrations, we obtain also explicit expressions for the optimal controls.
1. Introduction
Engineering design, materials processing, resource
allocation, investment strategies, etc. are complex
activities that pertain to large, distributed systems. In any
of these activities, the decision maker is repeatedly called
upon to choose among various possible and sometimes
competing prospective solutions to a practical question
with a consequential outcome. While the specifics of the
problem depend on application, context, and additional
constraints, the ultimate - albeit imprecise - goal in all
these activities is to "optimize performance", i.e., to have
maximal success, profit, or return with minimal time,
effort, or investment. Therefore, a crucial point in
decision-making is properly understanding and
quantifying the various trade-offs, including all their
future relevant consequences.
To illustrate the approach, we assume that one deals with
a system consisting of only two levels. At the lower
level, the underlying physical and engineering processes
are ruled by complex dynamics. This dynamics can be
described by continuous and/or discrete, analytic and/or
computer models. At the higher level, the process may
be viewed as comprised of a finite number of discrete
nodes, which represent its major articulations and crucial
decision points. Based upon these decisions, one may
alter the course of the system’s dynamics and steer it
towards the desired outcome. As mentioned, the ultimate
goal is to improve decision making (i.e., to reduce the use
of resources and to maximize the expected result) by
fully taking advantage of the information provided by the
dynamics at the lower level. Setting this requirement into
a mathematically well posed and hopefully tractable
problem, depends crucially on the selection of a suitable
objective functional (OF), which properly quantifies
“effort” and “success”. The fundamental difficulty in
choosing an appropriate objective functional stems from
the fact that “assigning value” to an object, action, or
feature is not a purely logical, but an axiologic process,
whereby subjective determinations are weighed against
each other [1 - 7].
The framework above leads naturally to a multi-criteria
optimization problem, where one seeks for the optimal
solution under a set of existing constraints. (We note
that, in general, this problem includes a probabilistic
component, dealing with risk and various sources of
uncertainty. This optimization problem can be solved
by various optimization algorithms, or by optimal control
(OC) methods. The complete solution of the OC problem
for continuous systems described by ordinary differential
equations (ODEs) was developed in the late ‘50’s by L.
S. Pontryagin and his co-workers [8, 9], in the form of
Pontryagin’s Maximum Principle (PMP). Later, the
PMP was generalized to cover also discrete time systems
[10]. Generalization to distributed systems, described by
partial differential equations is more difficult, but a
general framework was developed during the ‘60’s by J.-
L. Lions and co-workers in which OC problems for such
systems can be properly set and analyzed [11, 12]. An
additional challenge is presented by hybrid systems [13 –
16], in which continuous and discrete dynamics occur
simultaneously and oftentimes intertwined.
In this paper, we present a generalization of OC to hybrid
systems, with particular emphasis on decision making
processes. The lower level (underlying) dynamics may
be discrete or continuous, while the decision making
process usually takes place at discrete times and is
superimposed upon the original underlying dynamics.
.....