86 C. Smith et al. output energy are finite; hence shaker configurations and control strategies should ensure that the available energy is evenly distributed across all the shakers [2]. A general framework for approaching the optimal source placement problem is as follows: i) select a control strategy, ii) determine a quantitative measure of the quality of the source configuration, and iii) optimize the source locations by maximizing the selected metric. In the context of structural dynamics, optimal control and source placement applications has largely focused in the area of active structural damping for mitigating vibrations [3]. Conceptually, MIMO control and active damping are similar in that the goal is to achieve a desired structural state with minimal energy. Much of this theory is based on state-space formulations with a critical focus on placing source that maximize controllability [4]. Research on source placement for MIMO vibration testing is less prolific in the literature. In [5], a greedy-based approach was taken to select actuators that minimized the output energy of the actuator in order to achieve a desired random vibration specification. In [6], another greedy search method was proposed, which aimed to minimize the random vibration frequency response match to some target response. This method performed well compared to a brute force exhaustive search. Regardless of the method, source placement optimization involves solving an NP-hard mixed integer optimization problem. While branch and bound methods can be employed, they are significantly constrained by problem size due to their computational complexity. As a more efficient alternative, greedy optimization approaches can be used, offering faster solutions at the expense of optimality. Greedy search algorithms involve iteratively adding sensors one at a time, each time choosing the sensor that minimizes some user-specified objective function. In this work, we develop a MIMO control and source placement optimization framework. The core of the approach involves solving a regularized inverse problem, which ensures stability and robustness in the presence of measurement errors, while providing a user-specified regularization parameter that controls the trade-off between the fidelity to the target dynamic response and the smoothness and magnitude of the control inputs. We propose two source placement optimization strategies that use a greedy search algorithm and compare these designs by applying them to the regularized inverse problem control strategy. The first method seeks to directly minimize the objective function corresponding to the l2-norm regularized inverse problem, with the solution depending on the user-selected regularization parameter. Recognizing that the greedy search is not guaranteed to provide a near-optimal or optimal solution, we compare this method to an equivalent mixed-integer quadratic programming (MIQP) problem and to randomized brute force search. Inspired by the optimal sensor placement literature, the second proposed source optimal experiment design (OED) framework aims to minimize the E-criterion. The E-criterion benefits from guarantees on the quality of the greedy-generated solution [7], and has the added advantage of being independent of the choice of regularization parameter and of the target response data. In this work, we show that minimizing the E-criterion also indirectly minimizes the input energy of the least squares control solution. We compare the two source placement OED frameworks in terms of their performance in solving thel2-norm regularized inverse problem. The control and source placement framework is applied to a MIMO problem that seeks to reproduce the dynamic response of a sharp cone flight vehicle subjected to a turbulence-induced, fluctuating pressure field using a limited number of shakers. This work is broken down into the following sections. The Background and Methodology section describes the MIMO control formulation followed by the proposed source placement optimization algorithms. In the MIQP Greedy Search Comparison section, we compare the performance of the l2-norm regularized greedy search method to the MIQP problem in terms of solution accuracy and computational runtime. In the Greedy OED Performance on Full-Scale Model section, we investigate the performance of the greedy method on a full-scale wind tunnel model. In this section, we also compare the E-optimal design andl2-norm regularized designs, and investigate their robustness to changes in the regularization parameter used to solve the control inverse problem. Background and Methodology Assume we are given a target response ˆuo(ω) where ω is angular frequency and a maximum number of actuators N. The goal is to find sources s(ω) ∈Cd,withd ≤N, and their locations such that the output response u(ω) of the MIMO vibration test is close to ˆuo(ω). Assuming the structural response is defined by a Linear Time Invariant system, the map from the sources to the acceleration is taken as u(ω)=H(ω)s(ω), (1) whereH(ω): Cd →Cn is the transfer operator (or system frequency response function matrix), u(ω)∈Cn is the acceleration response, ands(ω) ∈Cd is the source vector.
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