Partial Eigenvalue Assignment for LTI Systems with $$\mathbb {D}$$ D -Stability and LMI

Pole Assignment for Active Vibration Control of Linear Vibrating Systems through Linear Matrix Inequalities

“…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. Both active feedback control [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and passive approaches (i.e., parameter modifications, see e.g., [19][20][21]) or are exploited to accomplish this important task. Indeed, the system poles, which are often denoted as the eigenvalues, define the system stability as well as the properties of the transient response such as the damping ratio, the rise time, and the settling time.…”

“…In contrast, only a few and recent works develop RPP in the field of vibrating systems due to some numerical challenges of these systems, although their goals differ from those of this work. 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.…”

“…The third test-case is another common benchmark in the literature on pole assignment. It consists of an aircraft wing under an air stream, as proposed in [15,28,29]. The system matrices are:…”

“…The control design task prescribed to assign the dominant complex conjugate pole pair µ 1,2 = −1.500 ± 3.000i, while the remaining closed-loop poles µ i must belong to the LMI region proposed in [15], which is a disk of radius equal to 1 and center at coordinates (−3,0).…”

“…The gains are reported in Table 6 while the open-loop and closed-loop poles are reported in Table 7 and shown in Figure 9. It should be pointed out that, unlike [15], the method proposed here is able to modify the open-loop pole in −0.918 ± 1.7606i, by moving it, for example, to −1.500 ± 3.000i. This modification is effective in speeding up the closed-loop system response due to the increase of the pole natural frequency compared to the open-loop poles.…”