Abstract:
An obvious challenge implicit in computational methods for modeling, simulation, state estimation and/or optimization of nonlinear dynamical systems’ behavior is that the associated initial state and model parameters are only estimates of physical reality. In many settings, the need for simulation, estimation or optimization of the system state and other functions depending on the state requires one to make adjustments to a high-dimensioned vector of unknowns embedded in the system model. Meeting such challenges frequently leads to the need for a first, second or higher order Taylor series approximation of the system state or outputs as a function of variations of the initial states and other parameters in the system mathematical model. These Taylor series approximations hold in the some vicinity of current best estimates or nominal values. For a large class of systems described by nonlinear differential equations, with some system specific effort, ancillary differential equations can be derived whose solutions yield the partial derivatives. Depending on the number of unknowns and the state space dimension, these ancillary differential equations for the partial derivatives can have high dimensionality. For systems described by high dimensioned nonlinear differential equations, the auxiliary differential equations, even when available, are typically not parallelizable to take advantage of modern computer architectures. One frequently encounters challenges associated with the curse of dimensionality, and having only “one life to give” in the task of deriving these differential equations whose solutions theoretically yield the desired partial derivatives. Most important there is significant coding and debugging efforts to implement the formulations as efficient computer codes. The computational cost associated with solving these differential equations, especially if the degree of the Taylor expansions is 2 or higher is significant. When something is evidently wrong in the numerical solution, one may have difficulty discerning whether the derivation, the computer code, or both contain some unknown error, leading to a significant, iterative effort to track down the errors. A number of symbol manipulation and operator overloading tools have evolved that partially address the implicit challenges. But the expense, especially in terms of human effort and computational cost, remain significant. Days, weeks and occasionally, months that many of us, ahem, more mature analysts, have spent in this heartbreak hotel has caused a few early retirements J. Finding and resolving such problems, when time and effort are integrated across many analysts’ effort spent in this arena has, without a doubt, consumed multiple human lifetimes over the past 5 years.
This lecture will present some easy to understand results that will enable all in attendance to hereafter to obtain high precision partial derivatives for many high-dimensioned dynamical systems, with minimal derivation and coding. You may then consume less of your “precious fluid of your life” (i.e., your time). Using the results I present today, when you must visit “heartbreak hotel” associated with accurate partial derivatives, you may be able to check out early.
This clip from Meryl Streep’s Postcard from the Edge movie will help you understand. This movie is about a young actress who completed a painful stay in her own personal “heartbreak hotel” (an institution where she painfully recovered from a severe drug and alcohol addiction), and was back her first movie scene after leaving this rehab facility. Her song is a metaphorical statement of joy after her painful stay in “heartbreak hotel” during rehab. You may find this song a useful metaphorical way to celebrate when you converge on resolving some difficulty in your research.
Bio:
John L. Junkins, University Distinguished Professor of Aerospace Engineering and holder of the Royce E. Wisenbaker Chair in Innovation in Texas A&M University’s College of Engineering, is the Founding Director of the Hagler Institute for Advanced Study. Junkins led an effort to bring together the faculty and administration to launch the Institute in 2011. The Texas A&M Institute for Advanced Study was officially renamed the Hagler Institute for Advanced Study in 2017 after long-time A&M benefactor Jon Hagler endowed the Institute with a $20 million gift. Starting in 2011, Junkins and Norman Augustine created the External Advisory Board that has been instrumental in the rapid early evolution of the Hagler Institute for Advanced Study.
While an undergraduate at Auburn University, Junkins began his career at the age of 19 as a co-op student during the Apollo program at the National Aeronautics and Space Administration in Huntsville, Alabama. He then time-shared graduate study at UCLA with full-time employment at McDonnell-Douglas, where he supported numerous launches of satellites aboard Delta rockets. Following previous academic appointments at the University of Virginia and Virginia Polytechnic Institute, he joined the Texas A&M University faculty in 1985 as the first endowed chair holder in the College of Engineering. His expertise spans basic and applied research. He performs theoretical studies, computations, experiments, design, and supports space flight implementations.
Junkins recently graduated his 61st PhD student and has spawned two generations of professors and three generations of over 200 PhD graduates who play significant roles in aerospace engineering. His technical descendants are an important national resource. He remains an active professor, principal investigator and prolific scholar. Currently he directs six doctoral candidates and a post-doctoral researcher. The breadth of his interests and scholarship is evident in the following distribution of his publications, by field (from webofknowledge.com):