A simple pole-shifting gain matrix $K$ which avoids solving Lyapunov equations

TL;DR

This paper introduces a pole-shifting gain matrix formula that avoids solving Lyapunov equations, simplifying the process of placing eigenvalues with negative real parts for controllable systems.

Contribution

The authors propose a new explicit formula for the gain matrix K that guarantees eigenvalue placement without Lyapunov equation solutions, applicable to general controllable systems.

Findings

Provides a closed-form expression for K avoiding Lyapunov equations.

Ensures eigenvalues of A+BK have real parts less than -γ₁.

Applicable to large systems with controllability condition.

Abstract

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 $$K=-\overline{B}^\top\sum_{k=1}^N\left[(\overline{A}^\top+\gamma_kI)^{-1}\right]\left\{\sum_{k=1}^N\left[(A+\gamma_kI)^{-1}B\overline{B}^\top(\overline{A}^\top+\gamma_kI)^{-1}\right]\right\}^{-1}$$ ensures that the matrix $A+BK$ has all the eigenvalues with the real part less than $-\gamma_1$. Here, $0<\gamma_1<\gamma_2<\dots<\gamma_N$ are $N$ positive numbers, large enough such that $A+\gamma_kI$ is invertible, for each $k$.