When designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be

stableIn electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...

. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system overcorrects in the opposite direction, and even further back in the first, until the car goes off the road. In contrast, for a stable system the vehicle would continue to track the control input. The

**Nyquist stability criterion**, named after

Harry NyquistHarry Nyquist was an important contributor to information theory.-Personal life:...

, is a graphical technique for determining the stability of a system. Because it only looks at the

Nyquist plotA Nyquist plot is a parametric plot of a transfer function used in automatic control and signal processing. The most common use of Nyquist plots is for assessing the stability of a system with feedback. In Cartesian coordinates, the real part of the transfer function is plotted on the X axis. The...

of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane singularities must be known). As a result, it can be applied to systems defined by non-

rational functionsIn mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

, such as systems with delays. In contrast to

Bode plotsA Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response...

, it can handle transfer functions with right half-plane singularities. In addition, there is a natural generalization to more complex systems with

multiple inputs and multiple outputsIn radio, multiple-input and multiple-output, or MIMO , is the use of multiple antennas at both the transmitter and receiver to improve communication performance. It is one of several forms of smart antenna technology...

, such as control systems for airplanes.

While Nyquist is one of the most general stability tests, it is still restricted to

linearA linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case....

,

time-invariantA time-invariant system is one whose output does not depend explicitly on time.This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output....

systems. Non-linear systems must use more complex stability criteria, such as

LyapunovVarious types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...

. While Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable. Techniques like Bode plots, while less general, are sometimes a more useful design tool.

## Background

We consider a system whose open loop transfer function (OLTF) is

; when placed in a closed loop with feedback

, the closed loop transfer function (CLTF) then becomes

. The case where H=1 is usually taken, when investigating stability, and then the

*characteristic equation*, used to predict stability, becomes

. Stability can be determined by examining the roots of this equation e.g. using the Routh array, but this method is somewhat tedious. Conclusions can also be reached by examining the OLTF, using its

Bode plotA Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response...

s or, as here, polar plot of the OLTF using the Nyquist criterion, as follows.

Any Laplace domain transfer function

can be expressed as the ratio of two polynomials

We define:

**Zero**: the *zeros* of are the roots of , and

**Pole**: the *poles* of are the roots of .

Stability of

is determined by its poles or simply the roots of the

*characteristic equation*:

. For stability, the real part of every pole must be negative. If

is formed by closing a negative unity feedback loop around the open-loop transfer function

, then the roots of the characteristic equation are also the zeros of

, or simply the roots of

.

## Cauchy's argument principle

From

complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, specifically the

argument principleIn complex analysis, the argument principle determines the difference between the number of zeros and poles of a meromorphic function by computing a contour integral of the function's logarithmic derivative....

, we know that a contour

drawn in the complex

plane, encompassing but not passing through any number of zeros and poles of a function

, can be mapped to another plane (the

plane) by the function

. The resulting contour

will encircle the origin of the

plane

times, where

.

and

are respectively the number of zeros and poles of

inside the contour

. Note that we count encirclements in the

plane in the same sense as the contour

and that encirclements in the opposite direction are

*negative* encirclements.

Instead of Cauchy's argument principle, the original paper by

Harry NyquistHarry Nyquist was an important contributor to information theory.-Personal life:...

in 1932 uses a less elegant approach. The approach explained here is similar to the approach used by Leroy MacColl (Fundamental theory of servomechanisms 1945) or by Hendrik Bode (Network analysis and feedback amplifier design 1945), both of whom also worked for Bell Laboratories. This approach appears in most modern textbooks on control theory.

## The Nyquist criterion

We first construct

**The Nyquist Contour**, a contour that encompasses the right-half of the complex plane:

- a path traveling up the axis, from to .
- a semicircular arc, with radius , that starts at and travels clock-wise to .

The Nyquist Contour mapped through the function

yields a plot of

in the complex plane. By the Argument Principle, the number of clock-wise encirclements of the origin must be the number of zeros of

in the right-half complex plane minus the poles of

in the right-half complex plane. If instead,

the contour is mapped through the open-loop transfer function

, the result is the

Nyquist PlotA Nyquist plot is a parametric plot of a transfer function used in automatic control and signal processing. The most common use of Nyquist plots is for assessing the stability of a system with feedback. In Cartesian coordinates, the real part of the transfer function is plotted on the X axis. The...

of

. By counting the resulting contour's encirclements of -1, we find the difference between the number of poles and zeros in the right-half complex plane of

. Recalling that the zeros of

are the poles of the closed-loop system, and noting that the poles of

are same as the poles of

, we now state

**The Nyquist Criterion**:

Given a Nyquist contour

, let

be the number of poles of

encircled by

, and

be the number of zeros of

encircled by

. Alternatively, and more importantly,

is the number of poles of the closed loop system in the right half plane. The resultant contour in the

-plane,

shall encircle (clock-wise) the point

times such that

.

If the system is originally open-loop unstable, feedback is necessary to stabilize the system. Right-half-plane (RHP) poles represent that instability. For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero. Hence, the number of counter-clockwise encirclements about

must be equal to the number of open-loop poles in the RHP. Any clockwise encirclements of the critical point by the open-loop frequency response (when judged from low frequency to high frequency) would indicate that the feedback control system would be destabilizing if the loop were closed. (Using RHP zeros to "cancel out" RHP poles does not remove the instability, but rather ensures that the system will remain unstable even in the presence of feedback, since the closed-loop roots travel between open-loop poles and zeros in the presence of feedback. In fact, the RHP zero can make the unstable pole unobservable and therefor not stabilizable through feedback.)

## The Nyquist criterion for systems with poles on the imaginary axis

The above consideration was conducted with an assumption that the open-loop transfer function

does not have any pole on the imaginary axis (i.e. poles of the form

). This results from the requirement of the

argument principleIn complex analysis, the argument principle determines the difference between the number of zeros and poles of a meromorphic function by computing a contour integral of the function's logarithmic derivative....

that the contour cannot pass through any pole of the mapping function. The most common case are systems with integrators (poles at zero).

To be able to analyze systems with poles on the imaginary axis, the Nyquist Contour can be modified to avoid passing through the point

. One way to do it is to construct a semicircular arc with radius

around

, that starts at

and travels anticlockwise to

. Such a modification implies that the phasor

travels along an arc of infinite radius by

, where

is the multiplicity of the pole on the imaginary axis.

## Summary

- If the open-loop transfer function has a zero pole of multiplicity , then the Nyquist plot has a discontinuity at . During further analysis it should be assumed that the phasor travels times clock-wise along a semicircle of infinite radius. After applying this rule, the zero poles should be neglected, i.e. if there are no other unstable poles, then the open-loop transfer function should be considered stable.
- If the open-loop transfer function is stable, then the closed-loop system is unstable for
*any* encirclement of the point -1.
- If the open-loop transfer function is
*unstable*, then there must be one *counter* clock-wise encirclement of -1 for each pole of in the right-half of the complex plane.
- The number of surplus encirclements (greater than N+P) is exactly the number of unstable poles of the closed-loop system
- However, if the graph happens to pass through the point , then deciding upon even the marginal stability
In the theory of dynamical systems, and control theory, a continuous linear time-invariant system is marginally stable if and only if the real part of every eigenvalue in the system's transfer-function is non-positive, and all eigenvalues with zero real value are simple roots...

of the system becomes difficult and the only conclusion that can be drawn from the graph is that there exist zeros on the axis.

## See also

- What the Nyquist Criterion Means to Your Sampled Data System Design http://www.analog.com/static/imported-files/tutorials/MT-002.pdf - Analog Devices
- Routh–Hurwitz stability criterion
- Control engineering
Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors...

- Phase margin
In electronic amplifiers, phase margin is the difference between the phase, measured in degrees, of an amplifier's output signal and 180°, as a function of frequency. The PM is taken as positive at frequencies below where the open-loop phase first crosses 180°, i.e. the signal becomes inverted,...

- Barkhausen stability criterion
The Barkhausen stability criterion is a mathematical condition to determine when a linear electronic circuit will oscillate. It was put forth in 1921 by German physicist Heinrich Georg Barkhausen...