# CDS 240, Spring 2016: HW 4

R. Murray, J. Doyle | Issued: 17 May 2016 (Tue) |

CDS 240, Spring 2016 | Due: 25 May 2016 (Wed) |

Much of dynamical systems theory is devoted to understanding the boundary between stability and instability in ODEs, as many interesting phenomena such as turbulence are traditionally studied as bifurcations from stable to unstable. A variety of results in control theory show that plant instability can combine with delays, actuator saturation, unstable zeros, noise, etc to make control more difficult. What this block of material argued using high shear flow turbulence as a case study, is that instability (e.g. eigenvalue locations) is just one mechanism to create "interesting" dynamics, and that "amplification" (large singular values or operator norms) with or without instability is more fundamental. While the relative importance of singular vs eigen values has been a theme in robust control for decades, it is relatively newer in science and other parts of engineering, and largely unexplored.

The homework for this block is to find examples other than high shear turbulence where interesting dynamics or difficult control is created by increased amplification and not just instability, and model and analyze a case study that illustrates this issue. A more ambitious goal would be to find an example where the "high gain, low rank" amplification creates highly predictable though fragile system behavior that is robust to a variety of uncertainties, as it does in turbulence.

Examples can be drawn from any domain of science or engineering where dynamics are involved.