### Abstract

The partial pole assignment (PPA) problem is the one of reassigning a few unwanted eigenvalues of a control system by feedback to suitably chosen ones, while keeping the remaining large number of eigenvalues unchanged. The problem naturally arises in modifying dynamical behaviour of the system. The PPA has been considered by several authors in the past for standard state-space systems and for quadratic matrix polynomials associated with second-order systems. In this paper, we consider the PPA for a cubic matrix polynomial arising from modelling of a vibrating system with aerodynamics effects and derive explicit formulas for feedback matrices in terms of the coefficient matrices of the polynomial. Our results generalize those of a quadratic matrix polynomial by Datta et al. (Linear Algebra Appl. 1997;257:29) and is based on some new orthogonality relations for eigenvectors of the cubic matrix polynomial, which also generalize the similar ones reported in Datta et al. (Linear Algebra Appl. 1997;257:29) for the symmetric definite quadratic pencil. Besides playing an important role in our solution for the PPA, these orthogonality relations are of independent interests, and believed to be an important contribution to linear algebra in its own right.

Original language | English |
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Pages (from-to) | 41-58 |

Number of pages | 18 |

Journal | Numerical Linear Algebra with Applications |

Volume | 11 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2004 |

### Keywords

- Aerodynamic effect
- Cubic pencils
- Orthogonality relations
- Partial pole assignment

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## Cite this

*Numerical Linear Algebra with Applications*,

*11*(1), 41-58. https://doi.org/10.1002/nla.332