The Nyquist Stability Criterion

Chapter 10 settled what stability means: every pole of the closed-loop transfer function must lie strictly in the left half of the s-plane. The Routh–Hurwitz array that followed settled how to test it, and did so without ever computing a root. Root locus then showed how those poles migrate as a gain is varied. Between them these tools answer almost every stability question about a system whose closed-loop characteristic polynomial can be written down.

That last clause is the catch. Three situations arise constantly in practice where the characteristic polynomial is either unavailable or not a polynomial at all.

The first is measured data. For a large plant — a turbogenerator, a paper machine, a flight vehicle — no one writes down \(G(s)\). An engineer excites the loop with sinusoids and records gain and phase against frequency. That table of numbers is the entire model. Routh's array needs coefficients; there are none.

The second is transport delay. A pipeline, a rolling mill, a networked control link all introduce a dead time \(T\), contributing a factor \(e^{-sT}\). The characteristic equation \(1 + G(s)e^{-sT} = 0\) is transcendental and has infinitely many roots. Building a Routh array for it is not merely tedious; it is impossible.

The third is the question Routh answers only crudely. A Routh test returns a verdict — stable or not — and, at best, the gain at which the verdict flips. It says nothing about how close to instability a stable system sits. That "how close" is the whole subject of Chapter 17, and it needs a geometric picture of stability rather than an algebraic one.

Harry Nyquist supplied that picture in 1932. His criterion decides the stability of the closed loop entirely from the open-loop frequency response \(G(j\omega)H(j\omega)\) — the very quantity a frequency sweep measures, and the very quantity whose polar plot Chapter 15 taught us to draw. The bridge between the two is a theorem of complex analysis about how contours map.

Everything rests on one algebraic observation. Write the open-loop transfer function as a ratio of polynomials,

Read the numerator and denominator of \(F\) separately. The denominator is \(D(s)\), whose roots are the open-loop poles — the poles of \(G(s)H(s)\), which we know, because we are handed \(G\) and \(H\). The numerator is \(D(s)+N(s)\), which is precisely the closed-loop characteristic polynomial: the closed-loop transfer function \(G/(1+GH)\) has denominator \(D+N\). So the roots of the numerator of \(F\) are the closed-loop poles — the objects we want to locate but cannot see.

A single function therefore carries both sets of information, one in its zeros and one in its poles.

One more observation converts \(F\) into something measurable. Since \(F = 1 + GH\), the value \(F(s)=0\) corresponds to \(G(s)H(s) = -1\). Geometrically, the image of a point under \(F\) is just its image under \(GH\) shifted one unit to the right. Encirclements of the origin by the \(F\) plot are identical in number and sense to encirclements of the point \(-1+j0\) by the \(GH\) plot. This is why we never actually draw \(F\): we draw \(GH\), which a frequency sweep gives us directly, and we watch the point \(-1\).

The counting theorem is Cauchy's principle of the argument. Its proof for our purposes is nothing more than bookkeeping with angles, and it is worth doing rather than quoting, because the sign conventions that trip students up all come from this one calculation.

Write \(F(s)\) in factored form with zeros \(z_i\) and poles \(p_j\):

Each factor \(s-z_i\) is the vector drawn from the zero \(z_i\) to the test point \(s\). Now let \(s\) travel once around a closed contour \(\Gamma\) in the clockwise direction and watch each of those vectors.

If \(z_i\) lies inside \(\Gamma\), the vector from \(z_i\) to \(s\) must swing all the way round as \(s\) completes its circuit: its angle changes by exactly \(-360^\circ\), the minus sign because the traversal is clockwise. If \(z_i\) lies outside \(\Gamma\), the vector swings out and back again, and its net angle change over the closed circuit is zero. The same statement holds for each pole vector \(s-p_j\), except that pole angles enter \(\angle F\) with a minus sign, so an enclosed pole contributes \(+360^\circ\).

Let \(Z\) be the number of zeros of \(F\) inside \(\Gamma\) and \(P\) the number of poles of \(F\) inside \(\Gamma\), each counted with multiplicity. Summing the contributions:

An angle change of \(-360^\circ\) in \(F\) means its image point has gone once clockwise around the origin of the \(F\)-plane. So a total change of \(-360^\circ (Z-P)\) means the image curve encircles the origin \(Z-P\) times clockwise.

The principle of the argument works for any closed contour. To make it answer our question, we choose the contour that encloses exactly the region we care about: the entire right half of the s-plane. That contour is called the Nyquist contour, written \(\Gamma\), and it has three parts.

Along the imaginary axis we run from \(s=-j\infty\) up to \(s=+j\infty\). Along this segment \(s=j\omega\), so \(F(s)\) evaluates to \(1+G(j\omega)H(j\omega)\) — the measured frequency response. This is the whole point of the construction: the useful part of the contour is exactly the part we can obtain experimentally.

We close the path with a semicircle of radius \(R\to\infty\) swung through the right half-plane, from \(+j\infty\) back round to \(-j\infty\). Because every physical \(G(s)H(s)\) is strictly proper — the denominator degree exceeds the numerator degree — \(G(s)H(s)\to 0\) everywhere on this arc, so \(F(s)\to 1\). The entire infinite semicircle collapses to the single point \(1\) in the \(F\)-plane, or equivalently to the origin in the \(GH\)-plane. It contributes nothing to the winding count and is drawn only for completeness.

Traversed in this order — up the imaginary axis, then round to the right — the contour keeps the right half-plane on its right-hand side, which is the clockwise sense the principle of the argument requires.

The third part is a repair. The principle of the argument forbids the contour from passing through any pole or zero of \(F\), and many loop transfer functions have poles exactly on the imaginary axis: every type-1 or type-2 system has one or two poles at the origin, and a lightly damped resonant element can put a pair at \(\pm j\omega_0\). At such a point \(F\) is infinite and the mapping is undefined. The cure is to indent the contour with a small semicircle of radius \(\varepsilon\to 0\) that bulges into the right half-plane, stepping around the offending pole and leaving it outside \(\Gamma\).

The detour has a consequence that must be stated plainly: because the small semicircle passes to the right of the origin, a pole at the origin is not enclosed and does not count in \(P\). Only open-loop poles with strictly positive real part do. The price for that convenience is paid in the \(GH\)-plane, where the tiny semicircle maps to an enormous arc.

To see it, put \(s = \varepsilon e^{j\theta}\) with \(\theta\) running from \(-90^\circ\) to \(+90^\circ\) as we walk up around the origin. For a type-\(N\) system, \(G(s)H(s)\approx K/s^N\) near the origin, so

The magnitude \(K/\varepsilon^N\) is infinite; the angle runs from \(+N\!\cdot\!90^\circ\) down to \(-N\!\cdot\!90^\circ\). The infinitesimal detour therefore maps to an arc of infinite radius sweeping \(N\times 180^\circ\) in the clockwise direction. A type-1 system gets a single infinite semicircle; a type-2 system gets a full infinite circle. These arcs are not decoration — they are where most of the encirclements of \(-1\) come from in type-2 systems, as Example 5 shows.

Assemble the pieces. The contour \(\Gamma\) encloses the right half-plane clockwise. The zeros of \(F=1+GH\) inside \(\Gamma\) are the closed-loop poles in the right half-plane — call that number \(Z\). The poles of \(F\) inside \(\Gamma\) are the open-loop poles in the right half-plane — call that number \(P\), and note that we know it by inspection of \(G(s)H(s)\). The principle of the argument gives \(N = Z-P\), where \(N\) counts clockwise encirclements of the origin by the image of \(\Gamma\) under \(F\). Replacing \(F\) by \(GH\) shifts the target from the origin to \(-1\). Rearranging for the unknown:

Two corollaries carry nearly all the practical weight.

For an open-loop stable plant, \(P=0\), and stability requires \(N=0\): the Nyquist plot must not encircle \(-1\) at all. This is the everyday case, and it is why control engineers speak of "staying away from the \(-1\) point" as though it were the definition of stability.

For an open-loop unstable plant with \(P\) right-half-plane poles, stability requires \(N=-P\): the plot must encircle \(-1\) exactly \(P\) times in the counter-clockwise direction. Feedback must actively wrap around the critical point to pull the unstable poles into the left half-plane. This is a genuinely counter-intuitive result, and it is one Routh's array can reproduce but never explain.

Counting \(N\) reliably takes a little discipline. The safe method is the ray test: draw any ray from the point \(-1\) out to infinity in a direction that avoids awkward tangencies, then walk along the Nyquist plot in the direction of increasing \(\omega\) and tally each crossing of that ray, \(+1\) for a crossing in the clockwise sense about \(-1\) and \(-1\) for a counter-clockwise one. The net tally is \(N\), and it is independent of which ray you chose. For simple plots the same answer is obvious by eye; for plots that loop back on themselves the ray test is what keeps you honest.

One boundary case deserves naming. If the Nyquist plot passes exactly through the point \(-1\), then \(F(s)=0\) somewhere on the imaginary axis: the closed loop has a pole on the \(j\omega\) axis and is marginally stable. The encirclement count is undefined there, which is appropriate — the system is on the knife edge between the two verdicts.

Chapter 15 built the polar plot of \(G(j\omega)H(j\omega)\) for \(0\le\omega\le\infty\). The Nyquist plot is that curve plus the two pieces needed to close it. The procedure is mechanical.

Step 1 — the positive-frequency branch. Sketch the polar plot for \(\omega\) from \(0^{+}\) to \(\infty\), fixing the two ends by the low- and high-frequency behaviour and pinning the middle with the real- and imaginary-axis crossings.

Step 2 — the negative-frequency branch. Because \(G\) and \(H\) have real coefficients, \(G(-j\omega)H(-j\omega)\) is the complex conjugate of \(G(j\omega)H(j\omega)\). The \(\omega\lt 0\) branch is therefore the mirror image of the \(\omega\gt 0\) branch reflected in the real axis. It is never computed separately.

Step 3 — close the curve. The infinite outer semicircle maps to the origin, joining the \(\omega\to+\infty\) and \(\omega\to-\infty\) ends there. If there are poles at the origin, join the \(\omega=0^{-}\) end to the \(\omega=0^{+}\) end with the infinite clockwise arc of \(N\times 180^\circ\) found in Section 16-4.

Step 4 — count. Mark \(-1\), apply the ray test, and evaluate \(Z=N+P\).

The one quantity worth computing exactly is the real-axis crossing, because it is what the critical point is compared against. Set the imaginary part of \(G(j\omega)H(j\omega)\) to zero and solve for \(\omega\); this frequency is the phase crossover frequency \(\omega_{pc}\), where the phase is exactly \(-180^\circ\). Substituting \(\omega_{pc}\) back gives the crossing point on the negative real axis. Everything about the gain range follows from that single number.

In practice the algebra is cleanest if the whole denominator is rationalised first. For \(G(s)H(s)=K/[s(s+1)(s+2)]\), multiply out the denominator with \(s=j\omega\):

The result is real when the imaginary part of the denominator vanishes, that is when \(\omega(2-\omega^{2})=0\). Discarding \(\omega=0\) leaves \(\omega_{pc}=\sqrt{2}\), and there \(G H = K/(-3\cdot 2) = -K/6\). The whole stability question for this system now reduces to whether \(K/6\) is less than or greater than 1.

Both panels show the same shape; only the scale differs. Increasing \(K\) inflates the plot uniformly about the origin, so the critical point — which does not move — is eventually swallowed. The crossing at \(-K/6\) slides left past \(-1\) as \(K\) passes 6, and at that instant the closed loop hands over two poles to the right half-plane. This is the same information the root locus delivers when its branches cross the imaginary axis, seen from a different direction.

Because the loop gain \(K\) multiplies \(G(j\omega)H(j\omega)\) at every frequency, changing \(K\) rescales the whole Nyquist plot radially about the origin without altering its shape or its phase at any frequency. The critical point stays fixed. Stability limits are therefore read off the plot with no redrawing at all.

Suppose the plot drawn for a reference gain \(K_0\) crosses the negative real axis at \(-a\). Scaling the gain to \(K\) moves that crossing to \(-a K/K_0\). The crossing reaches \(-1\) when

For an open-loop stable system whose plot crosses the negative real axis exactly once, the loop is stable for \(0 \lt K \lt K_{\max}\) and unstable above. That single-crossing case covers most textbook systems, and it produces exactly the same answer as the Routh array — as it must.

The interesting systems are the ones with more than one crossing. If the Nyquist plot crosses the negative real axis at two points, say \(-a_1\) and \(-a_2\) with \(a_1 \lt a_2\), then as \(K\) rises the critical point is first passed by one crossing and later by the other. The encirclement count goes \(0 \to 2 \to 0\), and the system is stable, then unstable, then stable again. Such a loop is called conditionally stable.

Three families of system are where the Nyquist criterion earns its place in the syllabus rather than merely duplicating Routh.

Unstable open loops. An inverted pendulum, a magnetically suspended rotor, and an aircraft with relaxed static stability all have \(P\ge 1\). Stability then demands encirclements. The simplest instance is \(G(s)H(s)=K/(s-1)\), whose Nyquist plot is a circle of diameter \(K\) sitting on the negative real axis between \(0\) and \(-K\), traversed counter-clockwise. With \(P=1\) we need \(N=-1\), so \(-1\) must lie inside that circle, which happens when \(K\gt 1\). Direct solution of \(s-1+K=0\) gives \(s=1-K\), stable for \(K\gt 1\) — the same answer. The lesson is structural: for an unstable plant there is a minimum usable gain, below which no amount of care helps.

Transport delay. A dead time contributes the factor \(e^{-j\omega T}\) to the frequency response. Its magnitude is exactly 1 at every frequency and its phase is \(-\omega T\) radians, growing without bound. On the Nyquist plot a delay therefore rotates every point clockwise by an angle proportional to frequency, winding the curve into a spiral that eventually crosses the negative real axis no matter how benign the delay-free system was. This is why delay is so corrosive to feedback: it costs phase without costing gain, and it costs more of it the faster you try to go.

Non-minimum-phase elements. A zero in the right half-plane, \((s-z)\) with \(z\gt 0\), has the same magnitude curve as \((s+z)\) but contributes phase lag instead of lead. The extra lag again pushes the plot toward and past \(-180^\circ\), so a right-half-plane zero imposes a ceiling on achievable loop bandwidth. Note that a right-half-plane zero of \(G(s)H(s)\) never appears in \(P\); only right-half-plane poles do. It affects stability through the shape of the plot, not through the count.

Begin every problem by writing down \(P\) from the open-loop poles, before drawing anything. Then find the negative-real-axis crossings, then count. Assume unity feedback where no \(H(s)\) is given; difficulty rises down the list.