Demonstration of “Stabilizing an Unstable System”

In a recent talk at the CMU Robotics Institute, I decided to give a quick impromptu demonstration of what it means to “stabilize an unstable system,” thereby making it easier to apply system identification.  This was in relation to my presentation of my previous PhD student Manu Madhav’s paper:

 

M. S. Madhav, S. A. Stamper, E. S. Fortune, and N. J. Cowan. “Closed-loop stabilization of the jamming avoidance response reveals its locally unstable and globally nonlinear dynamics”. J Exp Biol, 216:4272-4284, 2013.

Time for Control

In engineering and mathematics, t is the quintessential independent variable: an immutable quantity in terms of which all other variables depend. Most control systems do require a clock, but clocks have been engineered with such low drift rates that for all practical purposes, imperfections in chronometry have been largely ignored.

Biological systems do not have it so easy. Biological clocks were not “engineered” on top of a physical phenomenon like the oscillation of a quartz crystal. Rather, biological wetware must keep time over many scales using physiological, neural, and biochemical mechanisms. Biological clocks are typically described as nonlinear dynamical systems exhibiting limit cycle behavior where the phase of the system advances monotonically with the passage of time. For example, circadian rhythms and other longer-term processes highlight the importance of external cues in the timekeeping process. Circadian and circannual rhythms, for example, are regulated by changes in daylength, temperature, and other environmental cues.

So, while uncertainty in time is justifiably neglected in the design and analysis of most engineering control systems, perfect timekeeping is a poor assumption for the modeling and analysis of biological control systems. Indeed, timekeeping during simple human motor control tasks involves errors of around 10% of the movement cycle duration. Despite this extremely high level of temporal imprecision, the overwhelming majority of computational models of the human motor control makes the implicit assumptions that time is known. Who knows what happens to any of these analyses when our assumption about perfect timekeeping is relaxed?

These three papers begin to address this question:

S. M. LaValle and M. B. Egerstedt, “On time: Clocks, chronometers, and open-loop control,” in Proc. IEEE Int. Conf. on Decision Control, 2007, pp. 1916–1922.

S. G. Carver, E. S. Fortune, and N. J. Cowan, “State-estimation and cooperative control with uncertain time,” in Proc. Amer. Control Conf., 2013, in press.

A. Lamperski and N. J. Cowan, “Time-changed linear quadratic regulators,” in Proc. Euro. Control Conf., 2013, in press.

Written by Noah J. Cowan with Eric S. Fortune, Andrew Lamperski and Sean G. Carver.