1,134 research outputs found
Synthesis of Minimal Error Control Software
Software implementations of controllers for physical systems are at the core
of many embedded systems. The design of controllers uses the theory of
dynamical systems to construct a mathematical control law that ensures that the
controlled system has certain properties, such as asymptotic convergence to an
equilibrium point, while optimizing some performance criteria. However, owing
to quantization errors arising from the use of fixed-point arithmetic, the
implementation of this control law can only guarantee practical stability:
under the actions of the implementation, the trajectories of the controlled
system converge to a bounded set around the equilibrium point, and the size of
the bounded set is proportional to the error in the implementation. The problem
of verifying whether a controller implementation achieves practical stability
for a given bounded set has been studied before. In this paper, we change the
emphasis from verification to automatic synthesis. Using synthesis, the need
for formal verification can be considerably reduced thereby reducing the design
time as well as design cost of embedded control software.
We give a methodology and a tool to synthesize embedded control software that
is Pareto optimal w.r.t. both performance criteria and practical stability
regions. Our technique is a combination of static analysis to estimate
quantization errors for specific controller implementations and stochastic
local search over the space of possible controllers using particle swarm
optimization. The effectiveness of our technique is illustrated using examples
of various standard control systems: in most examples, we achieve controllers
with close LQR-LQG performance but with implementation errors, hence regions of
practical stability, several times as small.Comment: 18 pages, 2 figure
A Comparative Study of a Class of Mean Field Theories of the Glass Transition
In a recently developed microscopic mean field theory, we have shown that the
dynamics of a system, when described only in terms of its pair structure, can
predict the correct dynamical transition temperature. Further, the theory
predicted the difference in dynamics of two systems (the Lennard-Jones and the
WCA) despite them having quite similar structures. This is in contrast to the
Schweizer-Saltzman (SS) formalism which predicted the dynamics of these two
systems to be similar. The two theories although similar in spirit have certain
differences. Here we present a comparative study of these two formalisms to
find the origin of the difference in their predictive power. We show that not
only the dynamics in the potential energy surface, as described by our earlier
study, but also that in the free energy surface, like in the SS theory, can
predict the correct dynamical transition temperature. Even an approximate one
component version of our theory, similar to the system used in the SS theory,
can predict the transition temperature reasonably well. According to our
analysis, the absence of the Vineyard approximation in the SS formalism led it
to predict similar dynamics for the two systems. Interestingly, we show here
that despite the above mentioned shortcomings the SS theory can actually
predict the correct transition temperatures. Thus microscopic mean field
theories of this class which express dynamics in terms of the pair structure of
the liquid while being unable to predict the actual dynamics of the system are
successful in predicting the correct dynamical transition temperature.Comment: 12 pages, 5 figure
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