Schur stability of polynomials and matrices

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