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Difference between revisions of "Schur stability of polynomials and matrices"

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{{MSC|15|47}}
 
{{TEX|done}}
 
{{TEX|done}}
  
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system is said to be asymptotically stable if its characteristic
 
system is said to be asymptotically stable if its characteristic
 
polynomial $w(z)$ is stable
 
polynomial $w(z)$ is stable
[[#References|[a2]]].
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{{Cite|Ka}}.
  
 
Asymptotic stability of the polynomial or dynamical system is strongly
 
Asymptotic stability of the polynomial or dynamical system is strongly
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square matrix with real entries and with eigenvalues (cf. also
 
square matrix with real entries and with eigenvalues (cf. also
 
[[Eigen value|Eigen value]]) of absolute value less than one
 
[[Eigen value|Eigen value]]) of absolute value less than one
[[#References|[a1]]],
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{{Cite|Bh}},
[[#References|[a4]]]. Schur's theorem states that every matrix is
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{{Cite|}}. Schur's theorem states that every matrix is
 
unitarily similar to a triangular matrix. It has been noted that the
 
unitarily similar to a triangular matrix. It has been noted that the
 
triangular matrix is not unique
 
triangular matrix is not unique
[[#References|[a1]]].
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{{Cite|Bh}}.
  
 
A consequence of this theorem is the following. Let a matrix $A$ have
 
A consequence of this theorem is the following. Let a matrix $A$ have
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possible to associate to the characteristic polynomial $w(z)$ the
 
possible to associate to the characteristic polynomial $w(z)$ the
 
symmetric matrix $\def\tr{\mathrm{tr}} P = S_1^\tr S_1-S_2^\tr S_2$, where
 
symmetric matrix $\def\tr{\mathrm{tr}} P = S_1^\tr S_1-S_2^\tr S_2$, where
[[#References|[a2]]]:  
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{{Cite|Ka}}:  
 
$$S_1=\begin{pmatrix}a_0 & \dots &a_{n-2}&a_{n-1}\\
 
$$S_1=\begin{pmatrix}a_0 & \dots &a_{n-2}&a_{n-1}\\
 
0&\ddots&\vdots&a_{n-2}\\
 
0&\ddots&\vdots&a_{n-2}\\
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$$p_{ij} = \sum_{t=0}^{i-1}(a_{i-t-1}a_{j-t-1} -  a_{n+t-i+1}a_{n+t-j+1},\; j\ge i.$$
 
$$p_{ij} = \sum_{t=0}^{i-1}(a_{i-t-1}a_{j-t-1} -  a_{n+t-i+1}a_{n+t-j+1},\; j\ge i.$$
 
The following main stability theorem holds
 
The following main stability theorem holds
[[#References|[a2]]]: The polynomial $w(z)$ is asymptotically stable if
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{{Cite|Ka}}: The polynomial $w(z)$ is asymptotically stable if
 
and only if the matrix $P$ is positive definite, i.e. $P_k > 0$ for $k=1,\dots,n$,
 
and only if the matrix $P$ is positive definite, i.e. $P_k > 0$ for $k=1,\dots,n$,
 
where  
 
where  
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p_{k1} & \dots & p_{kk}\end{pmatrix},\dots, P_n = \det P.$$
 
p_{k1} & \dots & p_{kk}\end{pmatrix},\dots, P_n = \det P.$$
 
Using this theorem, one can prove
 
Using this theorem, one can prove
[[#References|[a2]]] that if $P_k \ne 0$ for $k=1,\dots,n$, then the characteristic
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{{Cite|Ka}} that if $P_k \ne 0$ for $k=1,\dots,n$, then the characteristic
 
polynomial $w(z)$ has $m$ roots inside and $n-m$ roots outside the unit
 
polynomial $w(z)$ has $m$ roots inside and $n-m$ roots outside the unit
 
circle, where $m = n-v(1,P_1,\dots,P_n)$ and $v$ denotes the number of sign changes in the
 
circle, where $m = n-v(1,P_1,\dots,P_n)$ and $v$ denotes the number of sign changes in the
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[[Pole assignment problem|pole assignment problem]] for linear control
 
[[Pole assignment problem|pole assignment problem]] for linear control
 
systems
 
systems
[[#References|[a3]]].
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{{Cite|Va}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD>
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{|
<TD valign="top"> R. Bhatia, "Matrix analysis" , Springer (1997)</TD>
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|-
</TR><TR><TD valign="top">[a2]</TD>
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|valign="top"|{{Ref|Bh}}||valign="top"| R. Bhatia, "Matrix analysis", Springer (1997) {{MR|1477662}}  {{ZBL|1088.90049}} {{ZBL|0863.15001}}
<TD valign="top"> T. Kaczorek, "Theory of control and systems" , PWN (1993) (In Polish)</TD>
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|-
</TR><TR><TD valign="top">[a3]</TD>
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|valign="top"|{{Ref|Ka}}||valign="top"| T. Kaczorek, "Theory of control and systems", PWN (1993) (In Polish)  
<TD valign="top"> A. Varga, "A Schur method for pole assignment" ''IEEE Trans. Autom. Control'' , '''AC-26''' : 2 (1981) pp. 517–519</TD>
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|-
</TR><TR><TD valign="top">[a4]</TD>
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|valign="top"|{{Ref|Va}}||valign="top"| A. Varga, "A Schur method for pole assignment" ''IEEE Trans. Autom. Control'', '''AC-26''' : 2 (1981) pp. 517–519 {{MR|0613566}}  {{ZBL|0475.93040}}
<TD valign="top"> "Comprehensive dictionary of electrical engineering" , CRC (1999) (Dictionary)</TD>
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|-
</TR></table>
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|}

Revision as of 19:45, 5 March 2012

2020 Mathematics Subject Classification: Primary: 15-XX Secondary: 47-XX [MSN][ZBL]

Consider the linear discrete-time dynamical system described by the difference equation $$x_{t+1} = Ax_t,\; t=0,1,2,\dots,$$ where $x_t\in \R^n$ and $A=(a_{ij})$, $i,j=1,\dots,n$, is an $(n\times n)$-matrix with real coefficients. Let $w(z)=a_0z^n+\cdots+a_{n-1}z+a_n = \det(zE_n - A)$ be the characteristic polynomial for the dynamical system. The polynomial $w(z)$ (or, equivalently, the matrix $A$) is said to be stable if all its roots are inside the unit circle on the complex plane. Similarly, the dynamical system is said to be asymptotically stable if its characteristic polynomial $w(z)$ is stable [Ka].

Asymptotic stability of the polynomial or dynamical system is strongly connected with Schur matrices and Schur's theorem. A Schur matrix is a square matrix with real entries and with eigenvalues (cf. also Eigen value) of absolute value less than one [Bh], . Schur's theorem states that every matrix is unitarily similar to a triangular matrix. It has been noted that the triangular matrix is not unique [Bh].

A consequence of this theorem is the following. Let a matrix $A$ have eigenvalues $s_1,\dots,s_n$. Then $$\sum_{k=1}^n |s_k|^2 \le \sum_{i,j=1}^n |a_{ij}|,$$ with equality if and only if $A$ is normal (cf. also Normal matrix). This leads to the estimate $$|s_k| \le n\;\max_{i,j} | a_{ij}|,$$ which can be directly used in asymptotic stability investigations for the dynamical system.

However, it should be stressed that it is possible to use also a different method in asymptotic stability considerations. Namely, it is possible to associate to the characteristic polynomial $w(z)$ the symmetric matrix $\def\tr{\mathrm{tr}} P = S_1^\tr S_1-S_2^\tr S_2$, where [Ka]: $$S_1=\begin{pmatrix}a_0 & \dots &a_{n-2}&a_{n-1}\\ 0&\ddots&\vdots&a_{n-2}\\ \vdots&\ddots&\ddots&\vdots\\ 0 & \dots & 0 & a_0 \end{pmatrix}$$

$$S_2=\begin{pmatrix}a_n & \dots &a_{2}&a_{1}\\ 0&\ddots&\vdots&a_{2}\\ \vdots&\ddots&\ddots&\vdots\\ 0 & \dots & 0 & a_n \end{pmatrix}$$ and the symbol $\tr$ denotes transposition. Therefore, the matrix $P=(p_{ij})$, $i=1,\dots,n$, where $$p_{ij} = \sum_{t=0}^{i-1}(a_{i-t-1}a_{j-t-1} - a_{n+t-i+1}a_{n+t-j+1},\; j\ge i.$$ The following main stability theorem holds [Ka]: The polynomial $w(z)$ is asymptotically stable if and only if the matrix $P$ is positive definite, i.e. $P_k > 0$ for $k=1,\dots,n$, where $$P_1 = p_{11},\; P_2 = \det\begin{pmatrix}p_{11} & p_{12}\\p_{21}&p_{22}\end{pmatrix},\dots$$

$$\dots, P_k = \det\begin{pmatrix}p_{11} & \dots & p_{1k}\\ \vdots& \dots & \vdots\\ p_{k1} & \dots & p_{kk}\end{pmatrix},\dots, P_n = \det P.$$ Using this theorem, one can prove [Ka] that if $P_k \ne 0$ for $k=1,\dots,n$, then the characteristic polynomial $w(z)$ has $m$ roots inside and $n-m$ roots outside the unit circle, where $m = n-v(1,P_1,\dots,P_n)$ and $v$ denotes the number of sign changes in the sequence $1,P_1,\dots,P_n$.

Moreover, it should be pointed out that Schur's matrix and Schur's theorem can be also used in the solution of the pole assignment problem for linear control systems [Va].

References

[Bh] R. Bhatia, "Matrix analysis", Springer (1997) MR1477662 Zbl 1088.90049 Zbl 0863.15001
[Ka] T. Kaczorek, "Theory of control and systems", PWN (1993) (In Polish)
[Va] A. Varga, "A Schur method for pole assignment" IEEE Trans. Autom. Control, AC-26 : 2 (1981) pp. 517–519 MR0613566 Zbl 0475.93040
How to Cite This Entry:
Schur stability of polynomials and matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_stability_of_polynomials_and_matrices&oldid=21529
This article was adapted from an original article by J. Klamka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article