Consistent Approximations for Optimal Control Problems
Based on Runge-Kutta Integration
A. Schwartz and E. Polak
This paper explores the use of Runge-Kutta integration methods in the
construction of families of finite dimensional, consistent
approximations to non-smooth, control and state constrained optimal
control problems. Consistency is defined in terms of epiconvergence
of the approximating problems and hypoconvergence of their optimality
functions. A significant consequence of this concept of consistency is
that stationary points and global solutions of the approximating
discrete time optimal control problems can only converge to stationary
points and global solutions of the original optimal control problem.
The construction of consistent approximations requires the introduction
of appropriate finite dimensional subspaces of the space of controls
and the extension of the standard Runge-Kutta methods to piecewise
continuous functions.
It is shown that in solving discrete time optimal control problems that
result from Runge-Kutta integration, a non-Euclidean inner product and
norm must be used on the control space to avoid potentially serious
ill-conditioning effects.
Optimal Nonlinear PI Compensators
S.M. Shahruz and A.L. Schwartz
In this paper, linear time-invariant single-input single-output
(SISO) systems that are stabilizable by (linear) proportional and
integral (PI) compensators are considered. For such systems a
five-parameter nonlinear compensator, to be called the nonlinear PI,
is proposed. The parameters of the proposed nonlinear compensator are
tuned by solving an optimization problem. The optimization problem
always has a solution. Additionally, a general nonlinear PI-type
compensator is proposed and is approximated by an easy-to-compute
compensator, for instance, a six-parameter nonlinear compensator. The
parameters of the approximate compensator are tuned to satisfy an
optimality condition. To ensure the stability of the closed-loop
system, a term is added to the cost function of the optimization
problem. The added term incorporates a measure of the stability of
the linearized closed-loop system in a neighborhood of the system
equilibrium point. The superiority of the proposed nonlinear PI
compensators over linear PI compensators is discussed and is
demonstrated for two feedback systems. Finally, the potential of
extending the design methodology proposed in the paper to the design
of nonlinear proportional, integral, and derivative (PID) compensators
for nonlinear unstable systems is shown in an example.
Approximate Solution for Boundary-Value Problems with
slowly-Varying Coefficients
S.M. Shahruz and A.L. Schwartz
In this paper, an approximate closed-form solution for linear
boundary-value problems with slowly varying coefficient matrices is
obtained. The derivation of the approximate solution is based on the
freezing technique, which is commonly used in analyzing the stability
of slowly varying initial-value problems as well as solving them. The
error between the approximate and the exact solutions is given, and an
upper bound on the norm of the error is obtained. This upper bound is
proportional to the rate of change of the coefficient matrix of the
boundary-value problem. The proposed approximate solution is obtained
for a two-point boundary-value problem and is compared to its solution
obtained numerically. Good agreement is observed between the
approximate and the numerical solutions, when the rate of change of
the coefficient matrix is small.
Comments on Fuzzy Logic for Control
of Roll and Moment for a Flexible Wing Aircraft
Adam L. Schwartz
A recent article explores the use of fuzzy logic in certain control
problems as an alternative to conventional control methodologies. In
that article, an attempt is made to demonstrate the usefulness of a
fuzzy control design by example. There are two notable features about
this article that deserve a closer look. First, it is a simple matter
to design a linear controller that appears to outperform the fuzzy
logic controller presented in that article. Second, some of the
claims made in the article about the capabilities of fuzzy logic are
vague and unsubstantiated. It is this second problem that is a
particularly troubling feature of many recent publication on fuzzy
control.
Comparison of Compensators for Double Integrator Plants
Adam L. Schwartz
Modern control theory has become very popular because of its
mathematical compactness and computational power. Unlike classical
design methods, however, modern control paradigms often do not
directly relate to the basic performance objectives of the
servomechanism design problem. The goal of this thesis is to provide
insight into the relationships that exist between classical and modern
control methodologies, particularly the LQG design methodology. In
this way, classical concepts can be incorporated into LQG designs. A
case study of the double integrator plant model, along with the common
design issues associated with it, provides the basis for the results
in this thesis.