# Equation-free, coarse-grained feedback linearization

Abstract: We 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.

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### "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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4,138 citations

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### "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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790 citations

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### "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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