Frequency Response and Bode Plots

Everything in Parts 2 and 3 was built on the pole locations of the closed loop. Chapter 8 read the transient off \(\zeta\) and \(\omega_n\); Chapter 11 tested for right-half-plane roots with the Routh array; Chapter 13 watched those roots migrate as the gain was raised. All of it assumed that we could write the closed-loop characteristic polynomial down and factor it. In the laboratory that assumption often fails. A hydraulic actuator, a large motor with an unmodelled load, a long pneumatic line — these have transfer functions nobody has derived, and no amount of algebra will produce one.

But every one of them can be driven with a sinusoid. Apply \(r(t)=A\sin\omega t\), wait for the transient to die, and measure two numbers from the output: the ratio of its amplitude to the input's, and the phase by which it lags. Repeat at a hundred frequencies and the system has been characterised completely, without ever writing an equation. The frequency response is the one description of a plant that can always be measured.

That practical advantage would be enough on its own. What makes the frequency response indispensable is that it also carries the stability information. Chapter 16 will show that the closed loop's stability can be decided entirely from the open-loop frequency response — from data measured on the plant with the loop open, where nothing can run away. This chapter builds the two plots; the next two chapters learn to read them.

The claim to be proved is that a stable linear system answers a sinusoid with a sinusoid of the same frequency, altered only in amplitude and phase. Nothing about linearity makes this obvious — a linear system could in principle answer with any waveform. The proof is a partial-fraction argument, and it shows exactly where the two altering numbers come from.

Let the input be \(r(t)=A\sin\omega t\), whose transform is \(R(s)=A\omega/(s^2+\omega^2)\). Then

If \(G(s)\) is stable every \(p_i\) lies in the left half-plane, so the first group contributes terms \(r_i e^{p_i t}\) that vanish as \(t\to\infty\). Only the two residues at \(s=\pm j\omega\) survive. Evaluate the first by covering up its own factor:

Because \(G(s)\) has real coefficients, \(G(-j\omega)\) is the complex conjugate of \(G(j\omega)\), and the second residue is \(\bar a\), the conjugate of the first. Write \(G(j\omega)=|G(j\omega)|e^{j\phi}\) with \(\phi=\angle G(j\omega)\) and assemble the surviving time function:

Two conditions deserve emphasis. First, \(G(s)\) must be stable: if any \(p_i\) sits in the right half-plane its term grows instead of dying, no steady state exists, and \(|G(j\omega)|\) describes nothing observable. Second, the result is a steady-state statement. Immediately after the sinusoid is switched on the output is a mixture of the forced sinusoid and the natural modes; only after several time constants does the clean sinusoid emerge.

We now have two functions to plot. Plotting them naively is unhelpful: a servo may matter from \(0.1\) to \(1000\) rad/s, and on a linear axis the first three decades are crushed into the leftmost pixel. Worse, \(|G(j\omega)|\) is a product of the magnitudes of its factors, and products do not decompose into anything a sketch can exploit.

Both problems are cured by taking logarithms. Define the logarithmic magnitude in decibels and plot it against \(\log_{10}\omega\):

The second point is the working one. If \(G(j\omega)=G_1 G_2 / (G_3 G_4)\), then

Phase was always additive; the decibel makes magnitude additive too. So we need only learn the shape of a handful of elementary factors, then add their curves. Because each of those shapes is asymptotically a straight line, the addition can be done with a pencil.

Two units of frequency span appear constantly. A decade is a tenfold change in \(\omega\); an octave is a doubling. A slope of \(-20\) dB/decade is the same thing as \(-6.02\) dB/octave, which is usually quoted as \(-6\) dB/octave. Slopes in Bode work are always integer multiples of \(20\) dB/decade, because each pole or zero contributes exactly one such multiple.

Any rational \(G(s)\) with real coefficients factors into terms of just four kinds. Put the transfer function into time-constant form first — every factor written so that its constant term is unity — because that is the form in which the low-frequency gain \(K\) appears as an isolated constant:

The constant \(K\). Its magnitude is \(20\log_{10}K\) decibels at every frequency — a horizontal line — and its phase is \(0^\circ\) for \(K>0\) or \(-180^\circ\) for \(K<0\). Changing the loop gain therefore slides the entire magnitude curve up or down bodily and leaves the phase curve untouched. That single fact is what makes gain design in the frequency domain so direct.

The pole or zero at the origin, \((j\omega)^{\pm N}\). For a single integrator, \(20\log|1/j\omega| = -20\log\omega\): on a \(\log\omega\) axis this is an exact straight line of slope \(-20\) dB/decade, passing through \(0\) dB at \(\omega=1\). Its phase is a constant \(-90^\circ\). With \(N\) integrators the slope is \(-20N\) dB/decade and the phase \(-90N^\circ\). A differentiator \((j\omega)\) mirrors this: \(+20\) dB/decade, \(+90^\circ\).

The first-order factor \(1/(1+j\omega T)\). This is the workhorse, and its treatment defines the whole asymptotic method. Its exact magnitude is \(-10\log_{10}(1+\omega^2T^2)\) dB. Examine the two extremes:

The low-frequency asymptote is the \(0\) dB axis; the high-frequency asymptote is a line of slope \(-20\) dB/decade. They meet where \(\omega T=1\), that is at the corner frequency \(\omega_c=1/T\). The asymptotic sketch is simply those two lines. The price of that simplicity is a known, bounded error: at the corner the true value is \(-10\log 2=-3.01\) dB while the asymptote reads \(0\), and one octave either side the error has already fallen to \(0.97\) dB. Nowhere does it exceed \(3\) dB.

The phase runs \(\angle = -\tan^{-1}(\omega T)\), from \(0^\circ\) at low frequency through exactly \(-45^\circ\) at the corner to \(-90^\circ\) at high frequency. The usual straight-line approximation holds the phase at \(0^\circ\) up to \(\omega=0.1/T\), runs a \(-45^\circ\)-per-decade line to \(\omega=10/T\), and holds \(-90^\circ\) thereafter; its worst error is about \(5.7^\circ\).

The quadratic factor. A complex pole pair cannot be split into two real first-order factors, so it is handled whole. With the standard second-order denominator of Chapter 8,

For \(u\ll1\) this tends to \(0\) dB; for \(u\gg1\) it behaves as \(1/u^2\), that is \(-40\log u\) dB — a slope of \(-40\) dB/decade. The two asymptotes again meet at \(\omega=\omega_n\), so a complex pair contributes a single corner with a doubled slope change. The phase falls from \(0^\circ\) through exactly \(-90^\circ\) at \(\omega_n\) to \(-180^\circ\).

What the asymptotes cannot show is the damping. At the corner the exact magnitude is \(1/(2\zeta)\), which is \(+14\) dB for \(\zeta=0.1\) and \(-6\) dB for \(\zeta=1\). A lightly damped pair therefore produces a tall resonant peak that the straight-line sketch misses entirely, and the sketch must always be corrected by hand near \(\omega_n\) whenever \(\zeta\) is small. This is the one place where blind asymptotic drawing is genuinely dangerous.

With the factors known, the composite sketch follows a fixed routine. The key realisation is that the low-frequency end is settled entirely by \(K\) and the system type \(N\) of Chapter 9, and every corner thereafter only changes the slope — never the value discontinuously.

The phase plot is assembled the same way by adding the individual phase contributions, but it is safer computed than sketched: evaluate \(\angle G = -90N - \sum\tan^{-1}(\omega T_i) + \sum\tan^{-1}(\omega T_a)\) at half a dozen frequencies around the region of interest and join the points. The frequencies worth spending arithmetic on are the corners and the gain crossover.

Take \(G(s)=\dfrac{50}{s(s+10)}\) as a specimen. Time-constant form first: \(G(s)=\dfrac{5}{s(1+0.1s)}\), so \(K=5\), \(N=1\), one corner at \(\omega=10\). The low-frequency asymptote is \(-20\) dB/dec through \(0\) dB at \(\omega=5\); beyond \(\omega=10\) it turns to \(-40\) dB/dec.

Why should the slope at crossover matter? Consider what instability means for the open loop. If at some frequency the loop transfer function has magnitude exactly \(1\) and phase exactly \(-180^\circ\), then \(G(j\omega)H(j\omega)=-1\), the closed-loop denominator \(1+GH\) vanishes on the imaginary axis, and the system sustains an oscillation without any input. The point \(-1\) in the complex plane is the brink. Chapter 16 will make this precise; for now we measure distance from it along two separate directions.

Two frequencies do the measuring. The gain crossover frequency \(\omega_{gc}\) is where the magnitude curve passes through \(0\) dB, so \(|GH|=1\); the phase crossover frequency \(\omega_{pc}\) is where the phase curve passes through \(-180^\circ\). At each, one of the two conditions for \(GH=-1\) is met and we ask how badly the other is missed.

Sign conventions follow from the definitions and repay one moment's thought. If \(|GH|\) at \(\omega_{pc}\) is less than unity, the gain margin in decibels is positive and the system is stable in the gain sense. If the magnitude curve is still above \(0\) dB when the phase reaches \(-180^\circ\), the gain margin is negative — the loop has already passed the critical point and, for a minimum-phase system, is unstable. Similarly a phase curve that has fallen past \(-180^\circ\) by the time gain crossover arrives yields a negative phase margin.

A system whose phase never reaches \(-180^\circ\) has infinite gain margin: no finite increase in gain can destabilise it. Any first- or second-order system with no dead time is of this kind, which is exactly why the second-order results of Chapter 8 held for all \(K>0\).

A designer who has spent Part 2 thinking in overshoot and settling time needs a bridge back. It is supplied by the closed-loop frequency response \(M(\omega)=|C(j\omega)/R(j\omega)|\), whose shape is described by three numbers. Apply them to the standard second-order closed loop \(\omega_n^2/(s^2+2\zeta\omega_n s+\omega_n^2)\) of Chapter 8 and each becomes an explicit function of \(\zeta\).

The resonant peak \(M_r\) is the largest value of \(M(\omega)\), and the resonant frequency \(\omega_r\) is where it occurs. Both come from minimising the denominator \((1-u^2)^2+(2\zeta u)^2\) with respect to \(u=\omega/\omega_n\): differentiating gives \(-4u(1-u^2)+8\zeta^2u=0\), so \(u^2=1-2\zeta^2\).

The bandwidth \(\omega_B\) is the frequency at which \(M\) has fallen to \(1/\sqrt2\), i.e. \(-3\) dB, of its zero-frequency value. Setting \((1-u^2)^2+(2\zeta u)^2=2\) and solving the resulting quadratic in \(u^2\) gives

Bandwidth is proportional to \(\omega_n\), and Chapter 8 gave rise time and settling time as inversely proportional to \(\omega_n\). So bandwidth is speed: a wider band means a faster response, and the two are reciprocally related. The cost is equally direct — a wide band admits more measurement noise and more high-frequency disturbance. Every bandwidth choice is a bargain struck between speed and noise rejection.

Finally, the open-loop phase margin correlates tightly with the closed-loop damping. Working out \(\text{PM}\) for the standard second-order loop \(\omega_n^2/[s(s+2\zeta\omega_n)]\) gives an exact but unmemorable expression; over the useful range it is very nearly linear in \(\zeta\).

A transfer function is called minimum phase when all of its poles and all of its zeros lie in the left half-plane. The name comes from a comparison: among all transfer functions sharing a given magnitude curve, the minimum-phase one has the smallest phase lag at every frequency. Bode proved that for such a function the magnitude and phase are not independent — the phase at any frequency is determined by the slope of the log-magnitude curve, and near a region of constant slope the phase is roughly \(n\times 90^\circ\) for a slope of \(n\times 20\) dB/decade.

That rigid link is what allows a Bode magnitude sketch alone to be used for design, and it is why the "\(-20\) dB/decade at crossover" habit works: a slope of \(-20\) dB/decade implies about \(-90^\circ\) of phase, hence roughly \(90^\circ\) of phase margin, while \(-40\) dB/decade implies about \(-180^\circ\) and hence almost none.

A non-minimum-phase factor breaks the link. The simplest example is the right-half-plane zero \((1-j\omega T)\). Its magnitude \(\sqrt{1+\omega^2T^2}\) is identical to that of the left-half-plane zero \((1+j\omega T)\), but its phase is \(-\tan^{-1}(\omega T)\) instead of \(+\tan^{-1}(\omega T)\) — a lag where the magnitude promised a lead. The all-pass factor \((1-j\omega T)/(1+j\omega T)\) makes the point starkly: unit magnitude at every frequency, and a phase that slides from \(0^\circ\) to \(-180^\circ\). Pure phase lag, invisible on the magnitude plot, and utterly destructive of phase margin. Transport lag, \(e^{-j\omega T_d}\), does the same and worse, since its phase \(-\omega T_d\) radians grows without bound.

Put every transfer function into time-constant form before drawing anything; almost every mistake in Bode work is made before the pencil touches the paper. Assume unity feedback unless told otherwise. Difficulty rises down the list.