  Robust control theory studies the effect of noise, disturbances, and other
uncertainty on system performance. Despite growing recognition across science
and engineering that robustness and efficiency tradeoffs dominate the evolution
and design of complex systems, the use of robust control theory remains
limited, partly because the mathematics involved is relatively inaccessible to
nonexperts, and the important concepts have been inexplicable without a fairly
rich mathematics background. This paper aims to begin changing that by
presenting the most essential concepts in robust control using human stick
balancing, a simple case study popular in both the sensorimotor control
literature and extremely familiar to engineers. With minimal and familiar
models and mathematics, we can explore the impact of unstable poles and zeros,
delays, and noise, which can then be easily verified with simple experiments
using a standard extensible pointer. Despite its simplicity, this case study
has extremes of robustness and fragility that are initially counter-intuitive
but for which simple mathematics and experiments are clear and compelling. The
theory used here has been well-known for many decades, and the cart-pendulum
example is a standard in undergrad controls courses, yet a careful
reconsidering of both leads to striking new insights that we argue are of great
pedagogical value.
