C. V. Hollot and R. Tempo, "On the Nyquist envelope of an
interval plant family", <I> IEEE Transactions on Automatic Control</I>,
1994, Vol. 39, No. 2, pp. 391-396.
<P>

The recent boom in applications of intervals to control theory started
with a 1978 result of Kharitonov who proved that for all polynomials from
an interval
family $p(s)=a_0+a_1s+a_2s^2+\ldots+a_ns^n$, $a_i\in
[a^-_i,a^+_i]$ to be 
{\it stable} (i.e., to have only roots with negative real parts) it is
necessary and sufficient that the
following four polynomials are stable:<BR> 
$$p_1(s)=a_0^+ +a_1^+s+a_2^-s^2+a_3^-s^3+a_4^+s^4+a^+_5s^5+\dots;$$<BR>
$$p_2(s)=a_0^- +a_1^-s+a_2^+s^2+a_3^+s^3+a_4^-s^4+a^-_5s^5+\dots;$$<BR>
$$p_3(s)=a_0^- +a_1^+s+a_2^+s^2+a_3^-s^3+a_4^-s^4+a^+_5s^5+\dots;$$<BR>
$$p_4(s)=a_0^+ +a_1^-s+a_2^-s^2+a_3^+s^3+a_4^+s^4+a^-_5s^5+\dots$$<BR>
These polynomials are called {\it Kharitonov polynomials} of the
interval polynomial family. 
<P>

In the paper under review, Kharitonov polynomials are used for the
following control problem:
A general linear system whose state can be described by only
one parameter $x$ can be described by 
differential equations of the following type: 
$${d^nx\over dt^n}+b_{n-1}{d^{n-1}x\over dt^{n-1}}+...+b_0=
a_m{d^mu\over dt^m}+a_{m-1}{d^{m-1}u\over dt^{m-1}}+...+a_0,$$ 
where $x(t)$ is the state
of the system at the moment $t$, and $u(t)$ is the value of the control
applied at the moment $t$. The solution of the system can be
explicitly given if we apply Fourier transform to both sides. Then, we
get $x(\omega)=P(\omega)u(\omega)$, where
$P(\omega)=N(\omega)/D(\omega)$, 
$N(\omega)=a_0+a_1(i\omega)+\ldots+a_m(i\omega)^m$, and 
$D(\omega)=b_0+b_1(i\omega)+\ldots+b_{n-1}(i\omega)^{n-1}+b_n$. 
This ratio $P(\omega)$ is called a {\it Nyquist characteristic}. 
<P>

Several important properties of control systems can be determined from
the {\it Nyquist plot}, i.e., from the set of all the values of
$P(\omega)$ for all $\omega$. 
For example, if the Nyquist plot does not
contain 0, this means that the system reacts to all controls; if the
plot is bounded, this means that the reaction is bounded for bounded
control, etc. 
<P>

Of course, we do not need to know all the
points from this plot: it is sufficient to know the {\it boundary} of
this plot, that is called the {\it Nyquist envelope}.
<P>

For real life systems, we only know {\it intervals} ${\bf a}_i$, ${\bf
b}_j$ of possible values of $a_i$ and $b_j$. For such situations, to
make similar conclusions, we need to know to know the union of
Nyquist plots that correspond to all possible values $a_i\in {\bf
a}_i$ and $b_j\in {\bf b}_j$. So, the Nyquist envelope for an
interval system is defined as a boundary of this union. 
<P>

In the paper under review, it is shown that many (but not all) important
characteristics of the Nyquist envelope can be determined if we know
the Nyquist plots of 16 systems $N_i(s)/D_j(s)$, where $N_i$ and
$D_j$, $1\le i,j\le 4$, are Kharitonov polynomials corresponding to
interval families $N(s)$ and $D(s)$.