Compensation in the Time and Frequency Domains

By this point in the course a single feedback loop has been described in two entirely different vocabularies. Chapters 8 and 9 spoke of damping ratio, natural frequency, peak overshoot, settling time and steady-state error — quantities you could measure with a stopwatch and a ruler on an oscilloscope trace. Chapters 14 to 18 spoke of gain crossover frequency, phase margin, gain margin, resonant peak and bandwidth — quantities you would measure with a signal generator and a phase meter. The system never changed. Only the description did.

The reason both survive is that each is natural for a different question. A servomechanism specification arrives as "settle to within 2% in half a second with no more than 10% overshoot": that is a time-domain sentence, and the root-locus methods of Chapters 12 and 13, which place closed-loop poles directly, answer it directly. A specification for an amplifier or a process loop arrives as "at least 45° of phase margin and 6 dB of gain margin, with the loop responding out to 20 rad/s": that is a frequency-domain sentence, and the Bode methods of Chapters 14 and 17 answer it directly. In industry both arrive, often on the same page, and sometimes the plant model itself is available only as a measured frequency response with no transfer function at all.

Chapter 21 built the hardware — the lead network, the lag network, and their combination — and derived the pole, zero and gain of each. What it did not settle is which domain you should work in and how a requirement stated in one domain constrains the other. That is this chapter's business. We will find that the translation is exact for the second-order prototype, approximate but reliable for well-designed higher-order loops, and occasionally misleading in ways worth knowing about in advance.

The bridge is built on the one system for which both descriptions can be computed in closed form: the second-order prototype of Chapter 8, in unity feedback,

Everything in the time domain follows from \(\zeta\) and \(\omega_n\): the overshoot \(M_p=e^{-\pi\zeta/\sqrt{1-\zeta^{2}}}\), the peak time \(t_p=\pi/\omega_d\), the 2% settling time \(t_s\approx 4/\zeta\omega_n\). Now compute the frequency-domain quantities for the same pair. The gain crossover frequency is where the open-loop magnitude is unity:

The phase of \(G(j\omega)\) is \(-90^{\circ}-\tan^{-1}\!\left(\omega/2\zeta\omega_n\right)\), so the phase margin, defined in Chapter 17 as \(180^{\circ}+\angle G(j\omega_{gc})\), is

Evaluated over the useful range the relation is strikingly close to a straight line, which is the origin of the rule of thumb every control engineer carries:

Two more bridges complete the dictionary. The resonant peak of the closed-loop magnitude, derived in Chapter 18, is \(M_r=1/(2\zeta\sqrt{1-\zeta^{2}})\) for \(\zeta<0.707\), so a peak specification is a damping specification. And the closed-loop bandwidth, the frequency at which \(|T(j\omega)|\) falls to \(-3\) dB, is

Divide the bandwidth column of the table by the crossover column and something useful appears: the ratio \(\mathrm{BW}/\omega_{gc}\) is 1.57 at \(\zeta=0.2\), 1.62 at \(\zeta=0.5\) and 1.55 at \(\zeta=0.707\). It barely moves. Bandwidth is therefore about 1.6 times the gain crossover frequency for any sensibly damped loop, and both are proportional to \(\omega_n\) — which is to say, all three are measures of the same thing, the speed of the loop.

With the bridges in place, converting a specification is mechanical. The route always runs through \(\zeta\) and \(\omega_n\), because those are the two numbers both domains agree on. Given a time-domain specification, extract \(\zeta\) from the overshoot and \(\omega_n\) from the timing requirement, then push them across. Given a frequency-domain specification, do the reverse.

The steady-state row deserves a comment. Error constants live comfortably in both domains and need no translating at all: \(K_v=\lim_{s\to0}sG(s)\) is simultaneously the gain that fixes the ramp error in the time domain and the quantity that sets the height of the low-frequency \(-20\) dB/decade asymptote on the Bode plot. Because of that, the error specification is almost always dealt with first, in either method: it fixes the gain, and everything else is then shaped around a gain that can no longer be reduced.

The lead network of Chapter 21, \(G_c(s)=\dfrac{1+Ts}{1+\alpha Ts}\) with \(\alpha<1\), has a zero at \(-1/T\) and a pole at \(-1/\alpha T\), the zero always nearer the origin. Its maximum phase lead and the frequency at which it occurs are

Now read those same three numbers in the time domain. The zero at \(-1/T\) and the pole at \(-1/\alpha T\) are points in the s-plane, and what matters about a pole-zero pair at a trial closed-loop location \(s_d\) is the net angle it contributes there, \(\angle(s_d+1/T)-\angle(s_d+1/\alpha T)\). Because the zero is closer to the origin than the pole, that net angle is positive: the lead network supplies angle. Supplying angle at \(s_d\) is exactly what the root-locus angle criterion needs when the uncompensated locus misses the desired pole location, and it is exactly the same physical property as supplying phase at crossover. There is only one network; there are two ways of asking what it does.

The last two rows connect this chapter back to Chapters 19 and 20. A PD controller is a lead compensator whose pole has been pushed to infinity, and a PI controller is a lag compensator whose pole has been pulled all the way to the origin. Practical PID implementations put the derivative pole at a finite frequency precisely to recover the realisability that the ideal form throws away — which makes the practical PID a lead–lag network in all but name.

Design on the root locus starts from a point, not a curve. Translate the transient specification into a desired dominant pole \(s_d=-\zeta\omega_n+j\omega_n\sqrt{1-\zeta^{2}}\), then ask whether the uncompensated locus passes through it. The angle criterion of Chapter 12 answers that question with a number.

The construction leaves one equation, \(\theta_z-\theta_p=\phi_c\), and two unknowns, the zero and the pole location. That surplus freedom is not a defect; it is where engineering judgement enters, and three conventions are common. Cancellation places the zero on top of a real plant pole, which removes that pole from the loop and leaves a simple second-order design — clean, and the choice made in Example 3 below, though it should never be used to cancel a pole in the right half-plane or one whose location is poorly known. Bisection places the zero and pole symmetrically about the bisector of the angle between the negative real axis and the line \(Os_d\), which for a given \(\phi_c\) makes \(\alpha\) as large as possible and so keeps the pole-to-zero spread — and the noise amplification — as small as possible. Fixed-\(\alpha\) starts from an available network and asks what it can do.

Once the geometry is settled, the gain follows from the magnitude criterion, and the error constants follow from the gain:

The procedure in full, then, is: translate the specification into \(s_d\); evaluate \(\angle G(s_d)\) and take the deficiency; choose zero and pole by one of the three conventions; find \(K\) from the magnitude criterion; check the resulting \(K_v\) against the accuracy requirement, and if it falls short, add a lag section whose dipole sits close enough to the origin that it contributes under about 5° at \(s_d\) and therefore does not disturb the geometry just built.

Design on the Bode plot starts from a gain, not a point. The accuracy requirement fixes \(K\) first, because the low-frequency asymptote is the error constant. Draw the plot with that gain, read the phase margin it leaves, and the shortfall tells you what to add.

For a lead design the shortfall is made up directly. Let \(\mathrm{PM}_{\text{req}}\) be the requirement and \(\mathrm{PM}_{\text{unc}}\) what the uncompensated loop delivers. The compensator's maximum phase is

The safety allowance \(\varepsilon\) is not arbitrary padding. Adding lead lifts the magnitude curve, which moves the crossover to a higher frequency, where the plant's own phase is more negative than it was at the old crossover. The lead therefore has to cover both the original shortfall and the extra lag it creates by moving the crossover. A steeply falling phase curve needs a larger \(\varepsilon\); a flat one needs almost none.

The new crossover must land exactly where the lead peaks, or the phase is wasted. Since \(|G_c(j\omega_m)|=1/\sqrt{\alpha}\), the compensated magnitude is zero dB where the uncompensated magnitude equals \(-20\log_{10}(1/\sqrt{\alpha})\) dB. Solve for that frequency, call it \(\omega_m\), and the network is fixed:

For a lag design the logic runs the other way. The lag network's phase is a liability, not an asset, so it is placed far below the new crossover where its phase has died away and only its attenuation remains. Find the frequency \(\omega_2\) at which the uncompensated phase is \(-180^{\circ}+\mathrm{PM}_{\text{req}}+\varepsilon\); that is where the crossover must be moved to. The gain there must be brought down to unity, so

Putting the zero a decade below the new crossover is the standard compromise: at \(\omega_2=10/T\) the lag network still contributes about \(-5^{\circ}\) or so, which the allowance \(\varepsilon\) has already absorbed. Move the zero closer and the phase penalty grows; move it further away and the time constants become impractically long.

Everything so far has assumed the compensator sits in the forward path, in series with the plant. That is cascade compensation, and it is the default because it is easy to build and easy to analyse. But a compensator can equally be placed in a minor feedback loop wrapped around part of the plant, and in some applications that arrangement is strictly better.

The classic minor loop is rate feedback — a tachometer on a motor shaft feeding its output back through a gain \(K_t\). Take the servo plant \(G(s)=\dfrac{\omega_n^{2}}{s(s+2\zeta\omega_n)}\). Closing the inner loop with \(H_i(s)=K_t s\) gives

The inner loop adds damping without moving the natural frequency, and — this is the point — without introducing a closed-loop zero. A PD controller in cascade produces the same characteristic polynomial but leaves a zero at \(-K_p/K_d\) in the closed-loop numerator, and that zero raises the overshoot above what \(\zeta\) alone predicts. Example 6 works both designs on the same plant and puts numbers on the difference.

Against that, the minor loop costs accuracy. The inner feedback reduces the outer-loop gain at low frequency, so \(K_v\) falls, whereas cascade PD leaves it untouched. It also requires a second sensor. The choice is therefore a real engineering trade, summarised below.

Every translation in this chapter rests on the assumption that the closed-loop response is dominated by one complex pair. That assumption is a working hypothesis, not a theorem, and a compensator design is exactly the operation most likely to break it, because the compensator adds poles and zeros of its own. Three failure modes recur.

The compensator zero appears in the closed-loop numerator. Any cascade compensator's zero is a zero of \(T(s)\). A zero at \(-z\) in the left half-plane raises overshoot, and the closer it is to the dominant pair the worse the effect. The clean way to see it is to write \(T(s)=T_0(s)\left(1+s/z\right)\) with \(T_0\) the zero-free prototype; then \(y(t)=y_0(t)+\dot y_0(t)/z\), so the response acquires a scaled copy of its own derivative. The derivative peaks early, so the overshoot grows and the peak time shortens. A lead compensator designed for \(\zeta=0.5\) routinely delivers 19–25% overshoot instead of the tabulated 16.3%.

The third pole is not far enough away. The dominance rule of Chapter 8 asks the remaining poles to lie at least five times further left than the dominant pair. A lead compensator's own pole at \(-1/\alpha T\) is usually comfortably out there, but if \(\alpha\) is close to one, or the plant already has a pole in the neighbourhood, the response contains a visible third mode and the predicted overshoot is wrong.

The lag dipole leaves a tail. A lag network puts a pole at \(-1/\beta T\) and a zero at \(-1/T\) close together near the origin. The closed loop inherits a real pole very close to that zero. Its residue is small — that is why the transient shape survives — but its time constant is long, and the result is a slow, low-amplitude creep towards the final value that can easily take five or ten times the nominal settling time. If the specification measures settling to 2%, and the tail amplitude is 3%, the design fails a test the dominant-pole calculation said it passed.

Assume unity feedback throughout, and use the 2% criterion for settling time. Where a design is asked for, state the compensator transfer function and verify the achieved specification in the domain you did not design in.