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Aerospace engineering graduate student Takashi Tanaka has won the 2011 Best Student Paper Award from the recent Conference on Decision and Control (CDC) of the Institute of Electrical and Electronics Engineers (IEEE).
Advised by CSL researcher Cedric Langbort, an assistant professor of aerospace engineering, Tanaka presented the paper, “Symmetric Formulation of the Kalman-Yakubovich-Popov Lemma and Exact Losslessness Condition,” as a contribution to control theory at the level of fundamental mathematics and abstraction.
According to Tanaka, control theory refers to a variety of mathematical frameworks used in engineering when “controlling” the behaviors of dynamical systems is desired. The dynamical systems in question can be varied in nature, including airplanes, robots, electrical circuits, and even biochemical systems or financial systems, as long as mathematical models are available.
An important goal of control theory is to achieve stability in dynamical systems in spite of unavoidable modeling errors. The motivation is clear since engineers want to stabilize airplanes even though mathematical models of the crafts are imperfect. This desire led to the development of the “robust control theory” over the past three decades.
The Kalman-Yakubovich-Popov (KYP) Lemma studied by Tanaka plays a central role in robust control theory and provides many connections to other sub-fields as well.
Tanaka’s research originated in the field of decentralized control, in which scientists use cooperative agents to achieve robust stability. Decentralized control is extremely important for controlling modern large-scale systems such as internet or traffic systems. Unfortunately, a definitive theory of decentralization is still elusive.
In order to make progress on this difficult question, Tanaka and Langbort focused on this problem at the level of the KYP lemma, and tried to determine when the lemma is "lossless", i.e., what types of robust stability problems are exactly solvable using known computational techniques.
After working with the equations, Tanaka noticed that two variables corresponding to the system frequency and the system uncertainty entered the KYP lemma almost in a symmetric manner. Moreover, he then realized that the losslessness of the KYP lemma was well characterized by this symmetry.
Discovering the symmetric structure in the KYP lemma doesn't solve the decentralized control problem. However, this structure is useful because it unifies some of the recently reported results on robust control theory, and provides a simplified view, which had up to now been absent from the literature.