pole assignment theorem

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Pole assignment problem

Let $R$ be a commutative ring and let $(A,B)$ be a pair of matrices of sizes $(n \times n)$ and $(n \times m)$, respectively, with coefficients in $R$. The pole assignment problem asks the following. Given $r_1,\ldots,r_n$, does there exist an $(m \times n)$-matrix $F$, called a feedback matrix, such that the characteristic polynomial of $A+BF$ is precisely $(X-r_1)\cdots(X - r_n)$? The pair $(A,B)$ is then called a pole assignable pair of matrices. The terminology derives from the "interpretation" of $(A,B)$ as (the essential data of) a discrete-time time-invariant linear control system: \begin{equation}\label{eq:a1} x(t+1) = Ax(t) + Bu(t) \end{equation} where $x(t) \in R^n$, $u(t) \in R^m$, or also, when $R = \mathbf{R}$ or $\mathbf{C}$, a continuous-time time-invariant linear control system: \begin{equation}\label{eq:a2} \dot x(t) = Ax(t) + Bu(t) \end{equation} where $x(t) \in R^n$, $u(t) \in R^m$.

In both cases, state feedback (see Automatic control theory ), $u \mapsto u + Fx$, changes the pair $(A,B)$ to $(A+BF,B)$.

The transfer function of a system \eqref{eq:a1} or \eqref{eq:a2} with output $y(t) = C x(t)$ is equal to \begin{equation}\label{eq:a3} T(s) = C(sI-A)^{-1}B \end{equation} and therefore the terminology "pole assignment" is used.

The pair $(A,B)$ is a coefficient assignable pair of matrices if for all $a_1,\ldots,a_n \in R$ there is an $(m\times n)$-matrix $F$ such that $A+BF$ has characteristic polynomial $X^n + a_1X^{n-1} + \cdots + a_n$.

The pair $(A,B)$ is completely reachable , reachable , completely controllable , or controllable if the columns of the $(n\times nm)$-reachability matrix \begin{equation}\label{eq:a4} (B,AB,\ldots,A^{n-1}B) \end{equation} span all of $R^n$. All four mentioned choices of terminology are used in the literature. The reachability matrix \eqref{eq:a4} is also called the controllability matrix. This terminology also derives from the "interpretations" \eqref{eq:a1} and \eqref{eq:a2} of a pair $(A,B)$, see again Automatic control theory .

A cyclic vector for an $(n\times n)$-matrix $M$ is a vector $v\in R^n$ such that $(v,MV,\ldots,M^{n-1}v)$ is a basis for $R^n$, i.e., such that $(M,v)$ is completely reachable. Now consider the following properties for a pair of matrices $(A,B)$:

a) there exist a matrix $F$ and a vector $w \in R^m$ such that $Bw$ is cyclic for $A+BF$;

b) $(A,B)$ is coefficient assignable;

c) $(A,B)$ is pole assignable;

d) $(A,B)$ is completely reachable.

Over a field these conditions are equivalent and, in general, a)$\Rightarrow$b)$\Rightarrow$c)$\Rightarrow$d). In control theory, the implication d)$\Rightarrow$a) for a field $R$ is called the Heyman lemma, and the implication d)$\Rightarrow$c) for a field $R$ is termed the pole shifting theorem.

A ring $R$ is said to have the FC-property (respectively, the CA-property or the PA-property) if for that ring d) implies a) (respectively, d) implies b), or d) implies c)). Such a ring is also called, respectively, an FC-ring, a CA-ring or a PA-ring. As noted, each field is an FC-ring (and hence a CA-ring and a PA-ring). Each Dedekind domain (cf. also Dedekind ring ) is a PA-ring. The ring of polynomials in one indeterminate over an algebraically closed field is a CA-ring, but the ring of polynomials in two or more indeterminates over any field is not a PA-ring (and hence not a CA-ring) [a4] .

For a variety of related results, see [a1] , [a2] , [a3] , [a5] .

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A corollary of Pole assignment theorem

The well-known Pole Assignment Theorem is

Pole Assignment Theorem. Given $A\in \mathbb{R}^{n\times n},B\in \mathbb{R}^{n\times m}$ , for any target pole set $P=\{\lambda_1,...,\lambda_n\}$ which is closed under complex conjugate, there exists a matrix $F\in \mathbb{R}^{m\times n}$ such that $\lambda(A+BF)=P$ if and only if $(A,B)$ is controllable.

If $\mu$ is an eigenvalue of $A$ and $\mu$ is still an eigenvalue of $A+BF$ for any $F\in \mathbb{R}^{m\times n}$ , $\mu$ is called an uncontrollable pole of $A$ , my question is

Question. The pole assignment problem has a solution i.e. given a target pole set $P$ , there exists a matrix $F\in \mathbb{R}^{m\times n}$ such that $$\lambda(A+BF)=P$$ if and only if all uncontrollable poles of $A$ are in the target pole set $P$ .

by real Schur decomposition, we can choose an orthogonal matrix $Q$ such that $$Q^TAQ=\begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix}$$ where $\lambda(A_{11})$ consists of all the uncontrollable poles of $A$ . Let $$Q^TB=\begin{bmatrix} B_1 \\B_2 \end{bmatrix}$$ but I do not have idea that how to continue... any help will be appreciated.

• linear-algebra
• matrix-calculus
• control-theory

This is closely related to the Kalman decomposition , from which it can be noted that $\lambda(A_{22})$ instead of $\lambda(A_{11})$ should contain all uncontrollable poles of $A$ . You want to show that the eigenvalues of $A + B\,F$ and thus also $Q^\top(A + B\,F)\,Q$ always contain the uncontrollable poles of $A$ . Using this what can you say about $B_1$ and $B_2$ ? Hint: use $F\,Q = \begin{bmatrix}F_1 & F_2\end{bmatrix}$ and look at the eigenvalues of $Q^\top(A + B\,F)\,Q$ .

• $\begingroup$ Thank you very much! I got it! The key is Kalman decomposition... $\endgroup$ –  Xin Fu Commented Dec 11, 2019 at 3:32

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Not the answer you're looking for browse other questions tagged linear-algebra matrices matrix-calculus control-theory ..

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Pole assignment for systems over rings

1982, Systems & Control Letters

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Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey

• TO THE MEMORY OF G.A. LEONOV
• Published: 23 December 2019
• Volume 52 , pages 349–367, ( 2019 )

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• M. M. Shumafov 1  

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This paper reviews the problems of stabilization and pole assignment in the control of linear time-invariant systems using the static state and output feedback. The main results are presented, along with a literature review. Stationary and nonstationary stabilizations with the static feedback are considered. The algorithms of low- and high-frequency stabilization of linear systems for solving Brockett’s stabilization problem are provided. The effective necessary and sufficient conditions for stabilization of two- and three-dimensional controllable linear systems are given in terms of the system parameters. The pole assignment problem and the related issues for linear systems are considered.

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Robust Stabilization of Linear Control Systems Using a Frequency Domain Approach

Feedback Stabilization of Nonlinear Systems

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Shumafov, M.M. Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey. Vestnik St.Petersb. Univ.Math. 52 , 349–367 (2019). https://doi.org/10.1134/S1063454119040095

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Received : 09 May 2019

Revised : 13 June 2019

Accepted : 13 June 2019

Published : 23 December 2019

Issue Date : October 2019

DOI : https://doi.org/10.1134/S1063454119040095

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T1 - Pole assignment : a new proof and algorithms

AU - Eising, R.

N2 - In this paper a new proof of the pole assignment theorem is given. This proof is a very straightforward one. It is not based on canonical forms and also the reduction to the single input case (Heymann's Lemma) is not used. Furthermore, an algorithm is given which is based on structural properties and also an algorithm is presented which allows to take into account numerical aspects. This latter algorithm uses only unitary matrices as transforming matrices preceding the pole assignment.

AB - In this paper a new proof of the pole assignment theorem is given. This proof is a very straightforward one. It is not based on canonical forms and also the reduction to the single input case (Heymann's Lemma) is not used. Furthermore, an algorithm is given which is based on structural properties and also an algorithm is presented which allows to take into account numerical aspects. This latter algorithm uses only unitary matrices as transforming matrices preceding the pole assignment.

M3 - Report

T3 - Memorandum COSOR

BT - Pole assignment : a new proof and algorithms

PB - Technische Hogeschool Eindhoven

CY - Eindhoven