Abstract
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.
Highlights
A ground-breaking advancement in the field of pole placement in active vibration control is the receptance method proposed by Ram and Mottershead in [1] that uses only the measured receptances of the system, in lieu of a first-order model
The goal is to compute the control gains that exactly assign the set of dominant poles to the prescribed locations of the complex plane, while the remaining non-dominant poles are clustered into Linear Matrix Inequalities (LMI) regions to feature some dynamic properties
In the first stage the receptance method for symmetric or asymmetric systems is applied to compute the gains that exactly assign a subset of the system poles
Summary
The assignment of the dynamic response of vibrating mechanical systems, such as structures, mechanisms, or multibody systems, is often performed by properly assigning the poles of the controlled systems. A ground-breaking advancement in the field of pole placement in active vibration control is the receptance method proposed by Ram and Mottershead in [1] that uses only the measured receptances of the system, in lieu of a first-order model. An example of RPP to vibrating systems is developed in [15], where some poles of the open-loop systems are kept unchanged by the controller, while the remaining ones are clustered within prescribed regions of the convex plane. In [16] a kind of RPP is developed to place the latent roots of a time-delayed vibrating system in the left half-plane All these examples of RPP exploit the powerful theory of Linear Matrix Inequalities (LMI), since such a mathematical theory represents complicated requirements on the location of poles in the complex plane
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