Partial Pole Placement with Controller Optimization

An arbitrary subset (n - m) of the (n) closed loop eigenvalues of an n(th) order continuous time single input linear time invariant system is to be placed using full state feedback, at pre-specified locations in the complex plane. The remaining closed loop eigenvalues can be placed anywhere inside a pre-defined region in the complex plane. This region constraint on the unspecified poles is translated into a linear matrix inequality constraint on the feedback gains through a convex inner approximation of the polynomial stability region. The closed loop locations for these eigenvalues are optimized to obtain a minimum norm feedback gain vector. This reduces the controller effort leading to less expensive actuators required to be installed in the control system. The proposed algorithm is illustrated on a linearized model of a 4-machine, 2-area power system example

Pole assignment for active vibration control of linear vibrating systems through Linear Matrix Inequalities

This paper proposes a novel method for pole placement in linear vibrating systems through state feedback and rank-one control. Rather than assigning all the poles to the desired locations of the complex plane, the proposed method exactly assigns just the dominant poles, while the remaining ones are free to assume arbitrary positions within a pre-specified region in the complex plane. Therefore, the method can be referred to as "regional pole placement". A two-stage approach is proposed to accomplish both the tasks. In the first stage, the subset of dominant poles is assigned to exact locations by exploiting the receptance method, formulated for either symmetric or asymmetric systems. Then, in the second stage, a first-order model formulated with a reduced state, together with the theory of Linear Matrix Inequalities, are exploited to cluster the subset of the unassigned poles into some stable regions of the complex plane while keeping unchanged the poles assigned in the first stage. The additional degrees of freedom in the choice of the gains, i.e., the non-uniqueness of the solution, is exploited through a semidefinite programming problem to reduce the control gains. The method is validated by means of four meaningful and challenging test-cases, also borrowed from the literature. The results are also compared with those of classic partial pole placement, to show the benefits and the effectiveness of the proposed approach

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