Partial Pole Placement in LMI Region

A new approach for pole placement of single-input system is proposed in this paper. Noncritical closed loop poles can be placed arbitrarily in a specified convex region when dominant poles are fixed in anticipant locations. The convex region is expressed in the form of linear matrix inequality (LMI), with which the partial pole placement problem can be solved via convex optimization tools. The validity and applicability of this approach are illustrated by two examples. 1. Introduction In classic control theory and application, pole placement (PP) of linear system is a well-known method to reach some desired transient performances [1] in terms of settling time, overshooting, and damping ratio. Indeed, the shape of the transient response strongly depends on the locations of the closed loop poles in complex plane. A strict PP is always achievable by a state feedback control law if the system is controllable. PP can be performed in a transfer function or state-space context, through a classical eigenvalue assignment based on characteristic polynomial of the closed loop system. However, when the system suffers uncertainties, strict PP in desirable locations is no longer suitable. For this reason, nonstrict placement in a subregion of the complex plane, such as a sector or a disc, is developed. A slight migration of the closed loop poles around desirable location may not induce a strong modification of the transient response, so the robust performance of the system can be assured. Chilali et al. proposed a linear matrix inequality (LMI) region [2, 3], which is convenient to depict typical convex subregion symmetric about real axis. With it, the robust controller is easy to design via solving some LMI [4] problems. The LMI region was expanded to quadratic matric inequality (QMI) region by Peaucelle et al. [5], and controller design for system suffering to different uncertainties was studied [6, 7]. Henrion et al. researched PP in QMI region with respect to polynomial system [8–10]. Maamri et al. [11–13] proposed a novel strategy with respect to PP in nonconnected QMI regions, while Yang placed poles in union of disjointed circular regions [14]. Partial pole placement by full state feedback is a new strategy for single-input linear system proposed by Datta et al. [15, 16]. critical closed loop poles of an th order single-input linear system are placed at prespecified locations in the complex plane, while the remaining noncritical poles can be placed arbitrarily inside a QMI region defined by a real symmetric matrix. These noncritical poles are optimized with