Nyquist criterion

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A necessary and sufficient condition for the stability of a linear closed-loop system formulated in terms of properties of the open-loop system.

Consider the linear single input-linear single output system with the following transfer function:

$$W(p)=\frac{M(p)}{N(p)},$$

where it is assumed that the degree of the polynomial $M(z)$ does not exceed that of the polynomial $N(z)$ (i.e. $W(p)$ is a proper rational function). The original Nyquist criterion gives necessary and sufficient conditions for the stability of the closed-loop system with unity feedback $u=y$. This is done in terms of the complex-valued function $z=W(i\omega)$ of the real variable $\omega\in[0,\infty)$ (the amplitude-phase characteristic of the open-loop system) which describes a curve in the complex $z$-plane, known as the Nyquist diagram. Suppose that the characteristic polynomial $N(z)$ of the open-loop system has $k$, $0\leq k\leq n$, roots with positive real part and $n-k$ roots with negative real part. The Nyquist criterion is as follows: The closed-loop system is stable if and only if the Nyquist diagram encircles the point $z=-1$ in the counter-clockwise sense $k/2$ times. (An equivalent formulation is: The vector drawn from $-1$ to the point $W(i\omega)$ describes an angle $\pi k$ in the positive sense as $\omega$ goes from $0$ to $+\infty$.)

This criterion was first proposed by H. Nyquist [1] for feedback amplifiers; it is one of the frequency criteria for the stability of linear systems (similar, e.g., to the Mikhailov criterion, see [2], [3]). It is important to note that if the equations of some of the elements of the systems are unknown, the Nyquist diagram can be constructed experimentally, by feeding a harmonic signal of variable frequency to the input of the open feedback [4].

Generalizations of this criterion have since been developed for multivariable, infinite-dimensional and sampled-data systems, e.g. [5], , , .

References

 [1] H. Nyquist, "Regeneration theory" Bell System Techn. J. , 11 : 1 (1932) pp. 126–147 [2] B.V. Bulgakov, "Oscillations" , Moscow (1954) (In Russian) [3] M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian) [4] Ya.N. Roitenberg, "Automatic control" , Moscow (1978) (In Russian) [5] L.S. Gnoenskii, G.A. Kamenskii, L.E. El'sgol'ts, "Mathematical foundations of the theory of control systems" , Moscow (1969) (In Russian)