28-08-2012, 01:14 PM
A VLSI Analog Computer / Math Co-processor for a Digital Computer
Overview of this Thesis.pdf (Size: 3.04 MB / Downloads: 130)
ABSTRACT
This dissertation investigates the utility of a programmable VLSI analog com-
puter for the solution of di®erential equations. To investigate the capability of analog
computing in a modern context, a large VLSI circuit (100 mm2) was designed and
fabricated in a standard mixed-signal CMOS technology. It contains 416 analog
functional blocks, switches for their interconnection, and circuitry for the system's
program and control. This chip is controlled and programmed by a PC via a data
acquisition card. This arrangement has been used to solve di®erential equations with
acceptable accuracy, as much as 400x faster than a modern workstation.
The utility of a VLSI analog computer has been demonstrated by solving sto-
chastic di®erential equations, partial di®erential equations, and ordinary di®erential
equations (ODEs). Additionally, techniques for using the digital computer to re¯ne
the solution from the analog computer are presented. Solutions from the analog com-
puter have been used to accelerate a digital computer's solution of the periodic steady
state of an ODE by more than an order of magnitude.
Introduction
Motivation
Analog computers of the 1960s were physically large, tedious to program and required
signi¯cant user expertise. However, once programmed, they rapidly solved a variety
of mathematical problems, notably di®erential equations, without time-domain dis-
cretization artifacts, albeit with only moderate accuracy [1]. Analog computers were
superseded by digital ones long ago; the technical community has hardly considered
what advantages modern VLSI techniques could impart to analog computers.
Digital computers can solve di®erential equations with very high accuracy.
However, they may su®er from a variety of convergence problems. Further, some
simulations may take a long time. This can preclude repeatedly solving the equations
as part of design optimization. In real-time control applications, long simulation time
may require that a simpler and possibly inferior model be simulated.
History of Analog Computation
Historically analog computers primarily solved the di®erential equations necessary for
system simulation. Analog circuits can solve ordinary di®erential equations (ODEs)
written in the form x_ = f(x; u; t), where x is a vector of state variables of length
n, u is a vector of inputs of length m, f is a vector of possibly nonlinear functions
of length n, and t is time. To solve this, an AC needs n integrators, m inputs and
su±cient circuitry to implement f, or an adequate approximation to it. In addition,
techniques exist (e.g. method of lines [3]) for converting partial di®erential equations
(PDEs) to ODEs of the above form.
Interest in analog computers decreased in the 1960s and 1970s as digital com-
puters became more advanced. One of the few remaining applications of analog
computers today is a back-up system in the Soyuz spacecraft.
Overview of this Thesis
Chapter 2 describes how analog and digital computers each solve di®erential equa-
tions, and their respective strengths and weaknesses. The design of a large VLSI
analog computer is outlined in Chapter 3 and the hybrid computation environment
in which it is used is described in Chapter 4. The performance of the individual
circuit blocks is summarized in Chapter 5. Chapter 6 gives representative examples
of di®erential equations solved by the analog computer and Chapter 7 describes how
the analog computer's solution can be used and re¯ned by a digital computer in a
way that speeds up the digital computer's solution. Chapter 8 compares this ana-
log computer to a digital computer in terms of power consumption and computation
speed. Chapter 9 gives some suggestions for future work in this ¯eld.