Model reduction

Many systems in nature, such as airflow over bird wings, through wind-farms, blood flow through our bodies, flapping of wings, and other such examples, are modeled or described by differential-algebraic equations (ordinary or partial, often stochastic). Numerical simulations of such systems is complicated, requiring fine discretization of the physical domain, often on the order of a million grid-points. However, in many cases, the dynamics of such systems seems too simple to the "eye" to require such level of detail. Model reduction methods attempt to reduce the complexity and result in much simpler sets of equations while retaining the accuracy of the original models.

Model reduction is very important for control design of complex systems. Most techniques for control design (typically involving matrix manipulations) have complexity O(n³), where n is the number of grid-points. For a million grid-points, just the storage requirements for such manipulations runs into terabytes.

The movie below shows the use of control for suppression of vortex shedding in the wake of a flat-plate, at a Reynolds number of 100. The flow is actuated by a body force near the trailing edge; we think of it being a simplistic model of blowing and suction using, say, synthetic jet actuators. The numerical domain consists of O(10⁵) grid-points, but a low-order model with around 30 states is sufficient to develop a controller that suppresses the vortex shedding. In the movie, the control is turned on at around t=6500.

Z. Ma, S. Ahuja and C. W. Rowley. Reduced order models for control of fluids using the Eigensystem Realization Algorithm. Theoretical and Computational Fluid Dynamics, 25(1): 233-247, June 2011. (pdf)

S. Ahuja and C. W. Rowley. Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. Journal of Fluid Mechanics, 645, 447-478, 2010; arXiv:0902.1207v1. (pdf)

S. Ahuja. Reduction methods for feedback stabilization of fluid flows. Ph.D. dissertation, Princeton University, May 2009. (pdf)