Formulas for inverting square matrices divided into rectangular blocks and their application in modal control
New formulas for inverting square matrices divided into rectangular blocks by rows or columns are obtained. Along with Frobenius formulas, where diagonal blocks are square, the new formulas constructed using matrix zero divisors make it possible to simplify inverting a block matrix of large dimensions by inverting two matrices of smaller dimensions. The formulas are applicable for inverting matrices written both in numerical (real or complex) and analytical (symbolic) form. For a certain wide class of linear time-invariant dynamic systems, compact analytical algorithms for calculating feedback matrices at solving control problems and evaluating components of the state vector are obtained using new formulas of block inversion. These algorithms are to simplify the generalized formulas of Bass -Gura and Ackermann both in direct and in dual versions. Examples of inverting some matrices divided into rectangular blocks are given for both a numerical matrix with complex-valued elements and a symbolic matrix. The problem of modal control of an aircraft spatial motion is solved using the proposed simplification of the generalized Ackermann formula and zeroing the required components of the controller matrix due to convenient parameterization, which does not affect the poles location. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.
Keywords
block matrix,
rectangular block,
block inversion,
matrix zero divisor,
modal control,
generalized Bass-Gura formula,
generalized Ackermann formulaAuthors
| Zubov Nikolay E. | Bauman Moscow State Technical University | nik.zubov@gmail.com |
| Lapin Alexey V. | Bauman Moscow State Technical University | AlexeyPoeme@yandex.ru |
| Ryabchenko Vladimir N. | Bauman Moscow State Technical University | ryabchenko.vn@mail.ru |
Всего: 3
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