Nyquist stability criterion

Nyquist stability criterion

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When designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable
BIBO stability
In 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 Nyquist
Harry Nyquist
Harry 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 plot
Nyquist plot
A 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 functions
Rational function
In 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 plots
Bode plot
A 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 outputs
MIMO
In 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 linear
Linear system
A 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-invariant
Time-invariant system
A 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 Lyapunov
Lyapunov stability
Various 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 plot
Bode plot
A 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 analysis
Complex analysis
Complex 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 principle
Argument principle
In 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 Nyquist
Harry Nyquist
Harry 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 Plot
Nyquist plot
A 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 principle
Argument principle
In 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
    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
    Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors...

  • Phase margin
    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
    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...