About an unusual case in the pole placement problem
- 作者: Mukhin A.V.1
-
隶属关系:
- Lobachevsky State University
- 期: 编号 112 (2024)
- 页面: 64-73
- 栏目: Mathematical control theory
- URL: https://journals.rcsi.science/1819-2440/article/view/284210
- ID: 284210
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The pole placement problem using a static output feedback is considered. If the problem is solvable, then the spectrum of the closed-loop system matrix can be located at any given points of the complex half-plane, symmetric with respect to the real axis. This makes it possible not only to stabilize the system, but also to set the required characteristics, such as stability margin, transition time, and others. It is known that if the multiplication of the number of inputs and outputs is greater than the dimension of the system, then the pole placement problem for a system in the form of a transfer matrix is solvable. The article shows that this ratio is not a sufficient condition for a system defined in the state space. There is an exceptional case in which the pole placement problem is fundamentally unsolvable. This case is simple discovered be means of multiplying of matrixes output and input. If this product gives a zero matrix, then due to the matrix trace consistency of the closed system matrix, the problem is unsolvable both in the real and in the complex domain. Moreover, the product of the output and input matrices is invariant with respect to the basis. A necessary condition for solvability is formulated.
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