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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.10489 |
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| _version_ | 1866916128143441920 |
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| author | Munteanu, Ionut |
| author_facet | Munteanu, Ionut |
| contents | It is well known that if $A\in\mathbb{C}^{N\times N}$ and $B\in\mathbb{C}^{N\times M}$ form a controllable pair (in the sense that the Kalman matrix $[B\ |\ AB\ | \ \dots\ |\ A^{N-1}B]$ has full rank) then, there exists $K\in\mathbb{C}^{M\times N}$ such that the matrix $A+BK$ has only eigenvalues with negative real parts. The matrix $K$ is not unique, and is usually defined by a solution of a Lyapunov equation, which, in case of large $N$, is not easily manageable from the computational point of view. In this work, we show that, for general matrices $A$ and $B$, if they satisfy the controllability Kalman rank condition, then $$K=-\overline{B}^\top\sum_{k=1}^N\left[(\overline{A}^\top+γ_kI)^{-1}\right]\left\{\sum_{k=1}^N\left[(A+γ_kI)^{-1}B\overline{B}^\top(\overline{A}^\top+γ_kI)^{-1}\right]\right\}^{-1}$$ ensures that the matrix $A+BK$ has all the eigenvalues with the real part less than $-γ_1$. Here, $0<γ_1<γ_2<\dots<γ_N$ are $N$ positive numbers, large enough such that $A+γ_kI$ is invertible, for each $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10489 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A simple pole-shifting gain matrix $K$ which avoids solving Lyapunov equations Munteanu, Ionut Optimization and Control It is well known that if $A\in\mathbb{C}^{N\times N}$ and $B\in\mathbb{C}^{N\times M}$ form a controllable pair (in the sense that the Kalman matrix $[B\ |\ AB\ | \ \dots\ |\ A^{N-1}B]$ has full rank) then, there exists $K\in\mathbb{C}^{M\times N}$ such that the matrix $A+BK$ has only eigenvalues with negative real parts. The matrix $K$ is not unique, and is usually defined by a solution of a Lyapunov equation, which, in case of large $N$, is not easily manageable from the computational point of view. In this work, we show that, for general matrices $A$ and $B$, if they satisfy the controllability Kalman rank condition, then $$K=-\overline{B}^\top\sum_{k=1}^N\left[(\overline{A}^\top+γ_kI)^{-1}\right]\left\{\sum_{k=1}^N\left[(A+γ_kI)^{-1}B\overline{B}^\top(\overline{A}^\top+γ_kI)^{-1}\right]\right\}^{-1}$$ ensures that the matrix $A+BK$ has all the eigenvalues with the real part less than $-γ_1$. Here, $0<γ_1<γ_2<\dots<γ_N$ are $N$ positive numbers, large enough such that $A+γ_kI$ is invertible, for each $k$. |
| title | A simple pole-shifting gain matrix $K$ which avoids solving Lyapunov equations |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2402.10489 |