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

Ionut Munteanu

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 $$K=-\overline{B}^\top\sum_{k=1}^N\left[(\overline{A}^\top+γ_kI)^{-1}\right]\left\{\sum_{k=1}^N\left[(A+γ_kI)^{-1}B\overline{B}^\top(\overline{A}^\top+γ_kI)^{-1}\right]\right\}^{-1}$$ 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 $k$.

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

Abstract

It is well known that if and form a controllable pair (in the sense that the Kalman matrix has full rank) then, there exists such that the matrix has only eigenvalues with negative real parts. The matrix is not unique, and is usually defined by a solution of a Lyapunov equation, which, in case of large , is not easily manageable from the computational point of view. In this work, we show that, for general matrices and , if they satisfy the controllability Kalman rank condition, then ensures that the matrix has all the eigenvalues with the real part less than . Here, are positive numbers, large enough such that is invertible, for each .
Paper Structure (3 sections, 2 theorems, 39 equations)

This paper contains 3 sections, 2 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Theory and the main result
  3. Conclusions

Key Result

Theorem 2

Assume that the system e1 is controllable. Then, the control $u=Kx,$ where once plugged into e1 ensures the exponential asymptotic stability in e1, with the decay rate $-\gamma_1$. Here, $0<\gamma_1<\gamma_2<\dots<\gamma_N$ are $N$ positive numbers, large enough such that $A+\gamma_kI$ is invertible for each $k$.

Theorems & Definitions (3)

  • Definition 1
  • Theorem 2
  • Lemma 3