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Proceedings ArticleDOI

Equation-free, coarse-grained feedback linearization

08 Jun 2005-Vol. 4, pp 2554-2558

AbstractWe explore a systematic computational approach to the feedback regulator synthesis problem based on the "equation-free" timestepper methodology [Theodoropoulos, K, et al., 2000], [Makeev, A, et al., 2002], [Kevrekidis, A. G., et al., 2003], [Siettos, C, et al., 2003], where both the closed-loop dynamics linearization and pole-placement objectives are simultaneously attained in a single design step [Kazantzis, N, 2001]. This is of particular interest in the case of systems/processes modeled via microscopic/stochastic simulations (e.g. kinetic Monte Carlo) for which coarse-grained, macroscopic models at the level we wish to control the behavior are not available in closed form.

Topics: Feedback linearization (52%), Linearization (51%)

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Citations
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Journal ArticleDOI
TL;DR: An overview of recently developed methods for control and optimization of complex process systems described by multiscale models using examples of thin film growth processes to motivate the development of these methods and illustrate their application.
Abstract: In this work, we present an overview of recently developed methods for control and optimization of complex process systems described by multiscale models. We primarily discuss methods developed in the context of our previous research work and use examples of thin film growth processes to motivate the development of these methods and illustrate their application.

98 citations


References
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Book
01 Jan 1985
Abstract: : The principal goal of this three years research effort was to enhance the research base which would support efforts to systematically control, or take advantage of, dominant nonlinear or distributed parameter effects in the evolution of complex dynamical systems. Such an enhancement is intended to support the development of flight controllers for increasing the high angle of attack or high agility capabilities of existing and future generations of aircraft and missiles. The principal investigating team has succeeded in the development of a systematic methodology for designing feedback control laws solving the problems of asymptotic tracking and disturbance rejection for nonlinear systems with unknown, or uncertain, real parameters. Another successful research project was the development of a systematic feedback design theory for solving the problems of asymptotic tracking and disturbance rejection for linear distributed parameter systems. The technical details which needed to be overcome are discussed more fully in this final report.

8,498 citations


"Equation-free, coarse-grained feedb..." refers background or methods in this paper

  • ...The first one is the exact input/output (I/O) feedback linearization approach, where the introduction of nonlinear state feedback induces linear I/O behavior of the system of interest, by forcing the system's output to follow a pre-specified linear and stable trajectory [9]....

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  • ...In the case of linear systems/processes, the synthesis of pole-placing state feedback control laws where the closed-loop eigenvalues (poles) are viewed as tunable parameters, has been very popular due to its intuitive appeal [9]....

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  • ...The second approach is the geometric exact feedback linearization approach, realized by the following twostep design procedure [9]: as a first step, a simultaneous implementation of a nonlinear coordinate transformation and a state feedback control law is proposed, in order to transform the original nonlinear system to a linear and controllable one....

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  • ...In the pertinent body of literature, two main model-based poleplacing controller synthesis methods can be identified, that are both rooted in the area of geometric control theory [9]....

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  • ...However, the geometric exact feedback linearization approach is based on a set of very restrictive conditions, that can hardly be met by any physical system [9]....

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Book
01 Jan 1987
TL;DR: Iterative Methods for Optimization does more than cover traditional gradient-based optimization: it is the first book to treat sampling methods, including the Hooke& Jeeves, implicit filtering, MDS, and Nelder& Mead schemes in a unified way.
Abstract: This book presents a carefully selected group of methods for unconstrained and bound constrained optimization problems and analyzes them in depth both theoretically and algorithmically. It focuses on clarity in algorithmic description and analysis rather than generality, and while it provides pointers to the literature for the most general theoretical results and robust software, the author thinks it is more important that readers have a complete understanding of special cases that convey essential ideas. A companion to Kelley's book, Iterative Methods for Linear and Nonlinear Equations (SIAM, 1995), this book contains many exercises and examples and can be used as a text, a tutorial for self-study, or a reference. Iterative Methods for Optimization does more than cover traditional gradient-based optimization: it is the first book to treat sampling methods, including the Hooke& Jeeves, implicit filtering, MDS, and Nelder& Mead schemes in a unified way.

1,845 citations


"Equation-free, coarse-grained feedb..." refers methods in this paper

  • ...Quasi_Newton optimization methods as the Broyden, Fletcher, Goldfarb, Shamo (BFGS) method and/or a line search method (direct method) [8] can be used to minimize the above expression....

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Journal ArticleDOI
TL;DR: A framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/stochastic) level of description to perform modeling tasks at a coarse (macroscopic, systems) level, and can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form is presented.
Abstract: We present and discuss a framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/stochastic) level of description to perform modeling tasks at a coarse (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains. Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in (1), when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales,"coarse" bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a system identification based, "closure-on-demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We will briefly survey the application of these "numerical enabling technology" ideas through examples including the computation of coarsely self-similar solutions, and discuss various features, limitations and potential extensions of the approach.

790 citations


Journal ArticleDOI
TL;DR: Over the last few years with several collaborators, a mathematically inspired, computational enabling technology is developed and validated that allows the modeler to perform macroscopic tasks acting on the microscopic models directly, and can lead to experimental protocols for the equation-free exploration of complex system dynamics.
Abstract: The best available descriptions of systems often come at a fine level (atomistic, stochastic, microscopic, agent based), whereas the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization, and control) are at a much coarser, macroscopic level. Traditional modeling approaches start by deriving macroscopic evolution equations from microscopic models, and then bringing an arsenal of computational tools to bear on these macroscopic descriptions. Over the last few years with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the “equation-free” approach, since it circumvents the step of obtaining accurate macroscopic descriptions. The backbone of this approach is the design of computational “experiments”. In traditional numerical analysis, the main code “pings“ a subroutine containing the model, and uses the returned information (time derivatives, etc.) to perform computer-assisted analysis. In our approach the same main code “pings“ a subroutine that runs an ensemble of appropriately initialized computational experiments from which the same quantities are estimated. Traditional continuum numerical algorithms can, thus, be viewed as protocols for experimental design (where “experiment“ means a computational experiment set up, and performed with a model at a different level of description). Ultimately, what makes it all possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized computational experimentation -through matrix-free numerical analysis, and systems theory tools like estimationbridge microscopic simulation with macroscopic modeling. If enough control authority exists to initialize laboratory experiments “at will” this computational enabling technology can lead to experimental protocols for the equation-free exploration of complex system dynamics.

364 citations


"Equation-free, coarse-grained feedb..." refers background in this paper

  • ...Over the past few years we have demonstrated that “coarse timesteppers” [2,3,4,5,6,7], establish a link between traditional continuum numerical analysis and microscopic/ stochastic simulation....

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