Nyquist stability criterion

The Nyquist plot for $G(s)=\frac{1}{s^2+s+1}$.

When designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. 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, is a graphical technique for determining the stability of a system. Because it only looks at the Nyquist plot 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, such as systems with delays. In contrast to Bode plots, 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, such as control systems for airplanes.

While Nyquist is one of the most general stability tests, it is still restricted to linear, time-invariant systems. Non-linear systems must use more complex stability criteria, such as 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 G(s); when placed in a closed loop with feedback H(s), the closed loop transfer function (CLTF) then becomes $\frac{G}{1+GH}$. The case where H=1 is usually taken, when investigating stability, and then the characteristic equation, used to predict stability, becomes G + 1 = 0. 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 plots or, as here, polar plot of the OLTF using the Nyquist criterion, as follows.

Any Laplace domain transfer function $\mathcal{T}(s)$ can be expressed as the ratio of two polynomials

$\mathcal{T}(s) = \frac{N(s)}{D(s)}.$

We define:

• Zero: the zeros of $\mathcal{T}(s)$ are the roots of N(s) = 0, and

• Pole: the poles of $\mathcal{T}(s)$ are the roots of D(s) = 0.

Stability of $\mathcal{T}(s)$ is determined by its poles or simply the roots of the characteristic equation: D(s) = 0. For stability, the real part of every pole must be negative. If $\mathcal{T}(s)$ is formed by closing a negative unity feedback loop around the open-loop transfer function $G(s) = \frac{A(s)}{B(s)}$, then the roots of the characteristic equation are also the zeros of 1 + G(s), or simply the roots of A(s) + B(s).

Cauchy's argument principle

From complex analysis, specifically the argument principle, we know that a contour Γ_s drawn in the complex s plane, encompassing but not passing through any number of zeros and poles of a function F(s), can be mapped to another plane (the F(s) plane) by the function F(s). The resulting contour Γ_F(s) will encircle the origin of the F(s) plane N times, where N = Z − P. Z and P are respectively the number of zeros and poles of F(s) inside the contour Γ_s. Note that we count encirclements in the F(s) plane in the same sense as the contour Γ_s and that encirclements in the opposite direction are negative encirclements.

Instead of Cauchy's argument principle, the original paper by Harry Nyquist 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 jω axis, from $0 - j\infty$ to $0 + j\infty$.
• a semicircular arc, with radius $r \to \infty$, that starts at $0 + j\infty$ and travels clock-wise to $0 - j\infty$.

The Nyquist Contour mapped through the function 1 + G(s) yields a plot of 1 + G(s) in the complex plane. By the Argument Principle, the number of clock-wise encirclements of the origin must be the number of zeros of 1 + G(s) in the right-half complex plane minus the poles of 1 + G(s) in the right-half complex plane. If instead, the contour is mapped through the open-loop transfer function G(s), the result is the Nyquist Plot of G(s). 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 1 + G(s). Recalling that the zeros of 1 + G(s) are the poles of the closed-loop system, and noting that the poles of 1 + G(s) are same as the poles of G(s), we now state The Nyquist Criterion:

Given a Nyquist contour Γ_s, let P be the number of poles of G(s) encircled by Γ_s, and Z be the number of zeros of 1 + G(s) encircled by Γ_s. Alternatively, and more importantly, Z is the number of poles of the closed loop system in the right half plane. The resultant contour in the G(s)-plane, Γ_G(s) shall encircle (clock-wise) the point ( − 1 + j0) N times such that N = Z − P.

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 − 1 + j0 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 G(s) does not have any pole on the imaginary axis (i.e. poles of the form 0 + jω). This results from the requirement of the argument principle 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 0 + jω. One way to do it is to construct a semicircular arc with radius $r \to 0$ around 0 + jω, that starts at 0 + j(ω − r) and travels anticlockwise to 0 + j(ω + r). Such a modification implies that the phasor G(s) travels along an arc of infinite radius by − lπ, where l is the multiplicity of the pole on the imaginary axis.

Summary

• If the open-loop transfer function G(s) has a zero pole of multiplicity l, then the Nyquist plot has a discontinuity at ω = 0. During further analysis it should be assumed that the phasor travels l 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 G(s) should be considered stable.
• If the open-loop transfer function G(s) is stable, then the closed-loop system is unstable for any encirclement of the point -1.
• If the open-loop transfer function G(s) is unstable, then there must be one counter clock-wise encirclement of -1 for each pole of G(s) 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 − 1 + j0, then deciding upon even the marginal stability of the system becomes difficult and the only conclusion that can be drawn from the graph is that there exist zeros on the jω axis.

See also

• What the Nyquist Criterion Means to Your Sampled Data System Design [1] - Analog Devices
• Routh–Hurwitz stability criterion
• Control engineering
• Phase margin
• Barkhausen stability criterion

References

• Faulkner, E.A. (1969): Introduction to the Theory of Linear Systems; Chapman & Hall; ISBN 0-412-09400-2
• Pippard, A.B. (1985): Response & Stability; Cambridge University Press; ISBN 0-521-31994-3
• Gessing, R. (2004): Control fundamentals; Silesian University of Technology; ISBN 83-7335-176-0
• Franklin, G. (2002): Feedback Control of Dynamic Systems; Prentice Hall, ISBN 0-130-32393-4