Semi-decentralized approximation of a LQR-based controller for a one-dimensional cantilever array

Abstract

We apply the method of semi-decentralized approximation, introduced in Lenczner and Yakoubi [2009] and Yakoubi [2010], to the linear quadratic regulation of a one-dimensional array of cantilevers with regularly spaced actuators and sensors. It is based on two mathematical concepts, namely on functions of operators, and on the Cauchy integral formula. We evaluate its performances and the errors of approximation. We also propose its implementation in terms of an analog processor, namely a periodic network of resistors. The presented application is based on a twoscale model representing an array of cantilevers. We shortly explain its genesis before to state it in details, and to show validation results.

Semi-decentralized approximation of optimal control of distributed systems based on a functional calculus

“…so by construction e(λ) = 0 and e(Λ) = 0. Multiplying the last equality by Φ Z to the left and by Φ −1 Z to the right, using Lemma 6.2 (3 ) and (4 ), and posing P = Φ Z p(Λ) Φ −1 Z we find that P satisfies the Riccati equation (13). Next, the nonnegativity and symmetry of p with Lemma 6.2 (1 ) and (5 ) yield the nonnegativity and self-adjointness of P .…”

“…In Section 7.3, there is an example of observation operator C that is not a function of Λ, while in the paper , it is the case for B the control operator. For boundary control or observation problems, it is impossible to find such isomorphisms. Nevertheless, in Section 7.4, we show how to proceed to address some boundary control problems. Multiscale models with controls at the microscale, as in and , are also possible applications.…”

“…In an example, we show how the method may also be applied to a particular boundary control problem. We view possible applications in the field of systems including a network of actuators and sensors, see for instance [13] dedicated to arrays of Atomic Force Microscopes.…”

Estimation of deflections by interferometry in a cantilever array and its optimization based on a two-scale model

“…The two-scale model governing elastic deflections in a one-dimensional array of AFMs, see Figure 4, introduced in [5], is restated in a way [7] appropriate to its implemen tation by the finite element method. The base and the can tilevers are modeled by the Euler-Bernoulli beam equation in the first component XI of the macroscopic variable and in the second component Y 2 of the microscopic variable respectively.…”

A cost effective AFM setup, combining interferometry and FPGA

“…Finally, the latency between the entrance of the first pixel of an image and the end of deflection computation must be as small as possi ble. All these requirements are stated in the perspective of implementing real-time active control for each cantilever (see [6], [7]). …”