Eigenstructure assignment in vibrating systems through active and passive approaches

The dynamic behaviour of a vibrating system depends on its eigenstructure, which consists of the eigenvalues and the eigenvectors. In fact, eigenvalues define natural frequencies, damping and settling time, while eigenvectors define the spatial distribution of vibrations, i.e. the mode shape, and also affect the sensitivity of eigenvalues with respect to the system parameters. Therefore, eigenstructure assignment, which is aimed at modifying the system in such a way that it features the desired set of eigenvalues and eigenvectors, is of fundamental importance in mechanical design. However, similarly to several other inverse problems, eigenstructure assignment is inherently challenging, due to its ill-posed nature. Despite the recent advancements of the state of the art in eigenstructure assignment, in fact, there are still important open issues. The available methods for eigenstructure assignment can be grouped into two classes: passive approaches, which consist in modifying the physical parameters of the system, and active approaches, which consist in employing actuators and sensors to exert suitable control forces as determined by a specified control law. Since both these approaches have advantages and drawbacks, it is important to choose the most appropriate strategy for the application of interest. In the present thesis, in fact, are collected passive, active, and even hybrid methods, in which active and passive techniques are concurrently employed. All the methods proposed in the thesis are aimed at solving open issues that emerged from the literature and which have applicative relevance, as well as theoretical. In contrast to several state-of-the-art methods, in fact, the proposed ones implement strategies that enable to ensure that the computed solutions are meaningful and feasible. Moreover, given that in modern mechanical design large-scale systems are increasingly common, computational issues have become a major concern and thus have been adequately addressed in the thesis. The proposed methods have been developed to be general and broadly applicable. In order to demonstrate the versatility of the methods, in the thesis it is provided an extensive numerical assessment, hence diverse test-cases have been used for validation purposes. In order to evaluate without bias the performances of the proposed methods, it has been chosen to employ well-established benchmarks from the literature. Moreover, selected experimental applications are presented in the thesis, in order to determine the capabilities of the developed methods when critically challenged. Given the focus on these issues, it is expected that the methods here proposed can constitute effective tools to improve the dynamic behaviour of vibrating systems and it is hoped that the present work could contribute to spread the use of eigenstructure assignment in the solution of engineering design problems

Multi-domain optimization of the eigenstructure of controlled underactuated vibrating systems

The paper proposes a multi-domain approach to the optimization of the dynamic response of an underactuated vibrating linear system through eigenstructure assignment, by exploiting the concurrent design of the mechanical properties, the regulator and state observers. The approach relies on handling simultaneously mechanical design and controller synthesis in order to enlarge the set of the achievable performances. The underlying novel idea is that structural properties of controlled mechanical systems should be designed considering the presence of the controller through a concurrent approach: this can considerably improve the optimization possibilities. The method is, first, developed theoretically. Starting from the definition of the set of feasible system responses, defined through the feasible mode shapes, an original formulation of the optimality criterion is proposed to properly shape the allowable subspace through the optimal modification of the design variables. A proper choice of the modifications of the elastic and inertial parameters, indeed, changes the space of the allowable eigenvectors that can be achieved through active control and allows obtaining the desired performances. The problem is then solved through a rank-minimization with constraints on the design variables: a convex optimization problem is formulated through the \u201csemidefinite embedding lemma\u201d and the \u201ctrace heuristics\u201d. Finally, experimental validation is provided through the assignment of a mode shape and of the related eigenfrequency to a cantilever beam controlled by a piezoelectric actuator, in order to obtain a region of the beam with negligible oscillations and the other one with large oscillations. The results prove the effectiveness of the proposed approach that outperforms active control and mechanical design when used alone

Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification

This paper proposes a novel method for the robust partial assignment of natural frequencies and antiresonances, together with the partial assignment of the related eigenvectors, in lightly damped linear vibrating systems. Dynamic structural modification is exploited to assign the eigenvalues, either of the system or of the adjoint system, together with their sensitivity with respect to some parameters of interest. To handle with constraints on the feasible modifications, the inverse eigenvalue problem is cast as a minimization problem and a solution method is proposed through homotopy optimization. Variables lifting for bilinear and trilinear terms, together with bilinear and double-McCormick's constraints, are exploited to provide a convexification of the problem and to boost the attainment of the global optimum. The effectiveness of the proposed method is assessed through four numerical examples

A Novel Algorithm to Minimize the Excitation of Undesirable Oscillations-State Space

A new state-space type stabilizer to minimize low frequency oscillation is proposed. The proposed Power System Stabilizer (PSS) design not only damps the oscillations by moving the Eigen-values to desired left hand plane locations but additionally reduces the excitation of the mode itself through optimization of the left eigenvector of the poorly damped electro-mechanical mode. The stabilizer is implemented as a State-Space (SS) type controller which is optimized using a constrained optimization procedure. Two possible inputs, speed and electrical power are considered as candidate inputs to the stabilizers. The state-space design is also compared with one based on traditional lead-lag compensation. The SS type stabilizer with electrical power input is shown to perform better compared to any other type of PSS by minimizing low frequency oscillations

Observer based active fault tolerant control of descriptor systems

The active fault tolerant control (AFTC) uses the information provided by fault detection and fault diagnosis (FDD) or fault estimation (FE) systems offering an opportunity to improve the safety, reliability and survivability for complex modern systems. However, in the majority of the literature the roles of FDD/FE and reconfigurable control are described as separate design issues often using a standard state space (i.e. non-descriptor) system model approach. These separate FDD/FE and reconfigurable control designs may not achieve desired stability and robustness performance when combined within a closed-loop system.This work describes a new approach to the integration of FE and fault compensation as a form of AFTC within the context of a descriptor system rather than standard state space system. The proposed descriptor system approach has an integrated controller and observer design strategy offering better design flexibility compared with the equivalent approach using a standard state space system. An extended state observer (ESO) is developed to achieve state and fault estimation based on a joint linear matrix inequality (LMI) approach to pole-placement and H∞ optimization to minimize the effects of bounded exogenous disturbance and modelling uncertainty. A novel proportional derivative (PD)-ESO is introduced to achieve enhanced estimation performance, making use of the additional derivative gain. The proposed approaches are evaluated using a common numerical example adapted from the recent literature and the simulation results demonstrate clearly the feasibility and power of the integrated estimation and control AFTC strategy. The proposed AFTC design strategy is extended to an LPV descriptor system framework as a way of dealing with the robustness and stability of the system with bounded parameter variations arising from the non-linear system, where a numerical example demonstrates the feasibility of the use of the PD-ESO for FE and compensation integrated within the AFTC system.A non-linear offshore wind turbine benchmark system is studied as an application of the proposed design strategy. The proposed AFTC scheme uses the existing industry standard wind turbine generator angular speed reference control system as a “baseline” control within the AFTC scheme. The simulation results demonstrate the added value of the new AFTC system in terms of good fault tolerance properties, compared with the existing baseline system

Robust control of redundantly actuated dynamical systems

The eigenstructure assignment scheme for robust multivariable feedback control is extended to redundantly actuated dynamical systems. It is shown that an orthonormal set of close loop eigenvectors is always exactly assignable in the case of redundant actuation proving the inherent robustness in the control design methodology. A choice of close loop eigenvector set to minimize the feedback gain matrix is suggested. Partial Eigenstructure Assignment methodology is proposed for second order mechanical systems. A methodology for coordinated actuation of redundant actuator sets by a trained weighted minimum norm solution is presented. To apply the methodology to hyper-redundant actuator arrays, for application to smart actuator arrays, a novel adaptive discretization algorithm is proposed. The adaptive aggregation strategy, based on the physics of the system, introduces nodes, to optimize a performance index of the overall plant model. The dimensionality of the inputs thus reduces to a finite number, making it a candidate plant for control by the robust redundant control scheme. The adaptive aggregation together with robust redundant control methodology is demonstrated on a finite element model of a novel morphing wing. This schema unifies the traditionally disparate methods of modeling and controller design

Optimal control with structure constraints and its application to the design of passive mechanical systems

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering; and, (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2002.Page 214 blank.Includes bibliographical references.Structured control (static output feedback, reduced-order control, and decentralized feedback) is one of the most important open problems in control theory and practice. In this thesis, various techniques for synthesis of structured controllers are surveyed and investigated, including H2 optimization, H[infinity] optimization, L1 control, eigenvalue and eigenstructure treatment, and multiobjective control. Unstructured control-full- state feedback and full-order control-is also discussed. Riccati-based synthesis, linear matrix inequalities (LMI), homotopy methods, gradient- and subgradientbased optimization are used. Some new algorithms and extensions are proposed, such as a subgradient-based method to maximize the minimal damping with structured feedback, a multiplier method for structured optimal H2 control with pole regional placement, and the LMI-based H2/H[infinity]/pole suboptimal synthesis with static output feedback. Recent advances in related areas are comprehensively surveyed and future research directions are suggested. In this thesis we cast the parameter optimization of passive mechanical systems as a decentralized control problem in state space, so that we can apply various decentralized control techniques to the parameter design which might be very hard traditionally. More practical constraints for mechanical system design are considered; for example, the parameters are restricted to be nonnegative, symmetric, or within some physically-achievable ranges. Marginally statable systems and hysterically damped systems are also discussed. Numerical examples and experimental results are given to illustrate the successful application of decentralized control techniques to the design of passive mechanical systems, such as multi-degree-of-freedom tuned-mass dampers, passive vehicle suspensions, and others.by Lei Zuo.S.M

Structural Damage Identification from Limited Measurement Data

The research focused on the development of a new method to identify damaged structural elements from a large flexible space structure on-orbit, using limited measured modal data. Limited measured modal data is loosely defined as measured data containing only a few modal frequencies and less than 10% of the total structural degrees-of-freedom. This effort was decomposed into four specific tasks. The first is the identification of partial modal properties from measured data of the nominal space structure. Second, the finite element model must be adjusted to match the measured nominal partial data. The third task is an analysis of the extent to which structural damage can be localized to individual structural elements using the measured data. In conjunction with this task is the determination of where to best place the limited number of sensors on the structure. Lastly, the identification of structural damage must be performed using the limited measured modal data from a damaged space structure. Identification of the modal parameters was accomplished using the Eigensystem Realization Algorithm, a time domain based method, adopted for use with averaged measured frequency response functions. Model tuning was performed using the Automated Structural Optimization Software package, adapted for model tuning. The method minimizes a cost function based on the mismatch between the measured and analytical eigenstructure. The minimization is solved using the eigenvalue and eigen-vector sensitivities at each iteration step. The determination of prioritized sensor locations and damage localization is performed using the eigenvalue and eigenvector sensitivities