View source for Schur stability of polynomials and matrices


{{TEX|done}}

Consider the linear discrete-time
[[Dynamical system|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|characteristic polynomial]] for the
dynamical system. The polynomial $w(z)$ (or, equivalently, the
[[Matrix|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
[[#References|[a2]]].

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|Eigen value]]) of absolute value less than one
[[#References|[a1]]],
[[#References|[a4]]]. 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
[[#References|[a1]]].

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|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
[[#References|[a2]]]: 
$$S_1=\begin{pmatrix}a_0 & \dots &a_{n-2}&a_{n-1}\\
0&\ddots&\vdots&a_{n-2}\\
\vdots&\ddots&\ddots&\vdots\\
0 & \dots & 0 & a_0 \end{pmatrix}$$

$$S_2=\begin{pmatrix}a_n & \dots &a_{2}&a_{1}\\
0&\ddots&\vdots&a_{2}\\
\vdots&\ddots&\ddots&\vdots\\
0 & \dots & 0 & a_n \end{pmatrix}$$
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
[[#References|[a2]]]: 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
[[#References|[a2]]] 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|pole assignment problem]] for linear control
systems
[[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD>
 <TD valign="top"> R. Bhatia, "Matrix analysis" , Springer (1997)</TD>
</TR><TR><TD valign="top">[a2]</TD>
 <TD valign="top"> T. Kaczorek, "Theory of control and systems" , PWN (1993) (In Polish)</TD>
</TR><TR><TD valign="top">[a3]</TD>
 <TD valign="top"> A. Varga, "A Schur method for pole assignment" ''IEEE Trans. Autom. Control'' , '''AC-26''' : 2 (1981) pp. 517–519</TD>
</TR><TR><TD valign="top">[a4]</TD>
 <TD valign="top"> "Comprehensive dictionary of electrical engineering" , CRC (1999) (Dictionary)</TD>
</TR></table>