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

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:
$$p_1(s)=a_0^+ +a_1^+s+a_2^-s^2+a_3^-s^3+a_4^+s^4+a^+_5s^5+\dots;$$
$$p_2(s)=a_0^- +a_1^-s+a_2^+s^2+a_3^+s^3+a_4^-s^4+a^-_5s^5+\dots;$$
$$p_3(s)=a_0^- +a_1^+s+a_2^+s^2+a_3^-s^3+a_4^-s^4+a^+_5s^5+\dots;$$
$$p_4(s)=a_0^+ +a_1^-s+a_2^-s^2+a_3^+s^3+a_4^+s^4+a^-_5s^5+\dots$$
These polynomials are called {\it Kharitonov polynomials} of the interval polynomial family.

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}.

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.

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}.

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.

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 $D(s)$.