Lead, Lag, and Lead–Lag Compensator Design

Chapter 12 built the root locus as the set of points satisfying the angle condition, and that set depends only on the open-loop poles and zeros — not on the gain. Turning the gain knob slides the closed-loop poles along the locus; it cannot move them off it. If the locus never passes through a region where the transient specification is met, no value of \(K\) will meet it.

The frequency-domain statement is the same fact in different clothing. Raising \(K\) lifts the whole magnitude curve without touching the phase curve, so the gain crossover slides to a higher frequency where the plant's phase is more negative, and the phase margin of Chapter 17 falls. Chapter 9 closed with the resulting deadlock: accuracy wants a large \(K\), damping wants a small one, and a plant of any complexity soon reaches a point where the two demands cannot both be met.

A compensator breaks the deadlock by adding dynamics of its own — in the classical case one zero and one pole, cascaded with the plant.

Write the lead network in its two equivalent forms, one showing the corner frequencies, the other the pole and zero locations:

The zero sits at \(-1/T\) and the pole at \(-1/(\alpha T)\); since \(\alpha<1\) the pole is a factor \(1/\alpha\) further from the origin. Between the two corner frequencies the magnitude rises at \(20\) dB/decade and the phase is positive. The phase contribution is the difference of two arctangents,

which is zero at both extremes and therefore has a maximum somewhere between. Differentiating and setting the derivative to zero locates it.

The maximum falls at the geometric mean of the two corner frequencies — the midpoint on a logarithmic axis, which is why the phase bump looks symmetric on a Bode plot. Substituting \(\omega_mT=1/\sqrt{\alpha}\) into the phase expression gives the peak value.

The last column is the reason a single lead network is rarely pushed past about \(60^\circ\). The factor \(1/\alpha\) is a high-frequency gain applied to everything the sensor delivers, and a sensor delivers noise as well as signal. Chapter 19 raised this objection against derivative action; the lead network is that same derivative action with the amplification deliberately limited, and \(\alpha\) is the limit. Taking \(\alpha\) below \(0.05\) trades an unreliable measurement for a marginal gain in phase. When more than \(60^\circ\) is genuinely needed, two lead sections in cascade — each supplying half — cost far less in noise than one section supplying all of it.

The design problem is usually posed as a pair of specifications: an error constant, which fixes the low-frequency gain, and a phase margin, which fixes the behaviour at crossover. The procedure below satisfies the first exactly and the second by construction.

Its one subtlety is the reason \(\phi_m\) is not simply the shortfall in phase margin. Adding the lead network raises the magnitude curve near \(\omega_m\), which pushes the gain crossover to a higher frequency, where the plant contributes more phase lag than it did before. Some of the phase we just bought is therefore immediately spent. An allowance of \(5^\circ\) for a gently sloping plant, up to \(12^\circ\) for a steep one, covers the loss.

When the specification is given as overshoot and settling time rather than as a phase margin, Chapter 8 converts it into a pair of desired dominant poles \(s_d=-\zeta\omega_n\pm j\omega_n\sqrt{1-\zeta^2}\), and the design question becomes geometric: how must the locus be bent so that it passes through \(s_d\)?

The answer comes straight from the angle condition. Evaluate the open-loop transfer function at the desired point; if the angle is not \(-180^\circ\), the compensator must make up the difference.

Knowing how much angle is needed does not by itself fix the pole and the zero: infinitely many pairs supply the same angle at \(s_d\). Three choices are standard. The first is cancellation — put the compensator zero on top of an existing plant pole, which removes that pole from the picture and leaves only the compensator pole to place. It is the simplest choice and the one that gives the cleanest algebra, but it must never be used on an unstable or very lightly damped plant pole, because an imperfect cancellation leaves that pole in the closed-loop response where it is no longer controllable by the gain.

The second is the bisector rule. Draw the line from \(s_d\) to the origin and the horizontal line through \(s_d\); bisect the angle between them, then place the zero and the pole on the real axis at \(\pm\phi_c/2\) from that bisector, as seen from \(s_d\). This construction can be shown to maximise \(\alpha\) — that is, to place the pole and zero as close together as possible for the required angle — which minimises the high-frequency gain \(1/\alpha\) and therefore the noise penalty.

The third is simply to fix the zero somewhere convenient, just to the left of the plant's slowest pole, and solve for the pole from the angle condition. Whichever placement is used, the gain follows from the magnitude condition \(|G_c(s_d)G(s_d)|=1\), and the design is not finished until two checks have been made: that the remaining closed-loop poles are far enough left for \(s_d\) to dominate, and that the error constant has not been ruined — a lead network multiplies \(K_v\) by \(\alpha\) at fixed \(K_c\), which is a reduction.

Reverse the ordering of pole and zero and the network changes character completely.

Now the pole at \(-1/(\beta T)\) is nearer the origin than the zero at \(-1/T\). The magnitude starts at \(K_c\beta\) at low frequency, falls at \(-20\) dB/decade between the two corners, and settles at \(K_c\) — a factor \(\beta\) lower. The phase is negative throughout, with a minimum of \(-\phi_m\) at the geometric mean, where \(\sin\phi_m=(\beta-1)/(\beta+1)\) by the same algebra as before.

Negative phase is exactly what a stability-limited loop does not need, so the lag network is not used for its phase at all. It is used for the ratio \(\beta\) between its low- and high-frequency gains, and the design consists of arranging for the phase penalty to be paid somewhere harmless.

The trick that makes this possible is to place both corner frequencies well below the intended gain crossover. Put the zero one decade below the new crossover \(\omega_2\), so \(1/T=\omega_2/10\). At \(\omega_2\) the zero contributes \(\arctan 10=84.3^\circ\) and the pole contributes very nearly \(90^\circ\) of lag, so the network's net phase there is only about \(-5.7^\circ\) — an amount easily covered by the same safety allowance \(\varepsilon\) used in the lead procedure. Meanwhile the full attenuation \(1/\beta\) is already in effect at \(\omega_2\), because \(\omega_2\) lies a decade above the upper corner.

The frequency-domain procedure mirrors the lead procedure but reads the Bode plot in the opposite order: the phase curve is consulted first to choose the new crossover, and the magnitude curve then tells us how much attenuation is needed there.

On the root locus the lag plays a different-looking but identical role. Suppose the uncompensated locus already passes through an acceptable \(s_d\) — the transient specification is met — but the gain there gives an error constant that is too small by a factor \(\beta\). Add a zero at \(-z\) and a pole at \(-z/\beta\), both very close to the origin, with \(z\) chosen small compared with \(|s_d|\).

The pair is called a dipole. Seen from a point \(s_d\) far away, the pole and the zero are almost at the same place: their angles nearly cancel and their magnitudes nearly cancel, so the locus in the region of interest is essentially unchanged and the gain required at \(s_d\) is essentially unchanged. Seen from the origin, where \(s\) is comparable with \(z\), they are far apart, and the ratio of their distances is exactly \(\beta\). The error constant is multiplied by \(\beta\) for almost nothing.

"Almost" deserves a word. The dipole adds a slow closed-loop pole near \(s=-z\), close to the compensator zero but not on it. Its residue is small, so it contributes a small-amplitude term to the step response — but that term decays with time constant \(1/z\), which is deliberately long. The result is a long, low tail on the settling response, familiar to anyone who has watched a lag-compensated servo creep into its final position. Making \(z\) smaller reduces the tail's amplitude and lengthens its duration; there is no setting that removes it.

Specifications frequently demand both things at once: a large error constant and a fast, well-damped transient. A lag alone would meet the first and slow the system down; a lead alone would meet the second and might even reduce the error constant. Cascading the two gives a network with two zeros and two poles that does both jobs, each in its own frequency band.

The condition on the corner frequencies is what keeps the two halves from interfering: the lag section does all its work one or two decades below crossover, where its \(-6^\circ\) is harmless, while the lead section does all of its work at crossover, where its phase bump is worth the most. Read the magnitude curve from left to right and the compensator lifts the low frequencies by \(\beta\), drops back down, then lifts again by \(\gamma\) around crossover.

The design is done sequentially, and the order matters. Start with the lead half, because it is the lead that determines where the crossover ends up: choose the dominant poles or the phase margin, design the lead section against the plant alone, and compute the error constant the lead-compensated loop achieves. Then take the ratio of the required error constant to the achieved one — that ratio is \(\beta\) — and design a lag dipole to supply it, placed far enough below the new crossover to leave the lead's work undisturbed. A single verification pass at the end confirms both specifications.

A single passive network can realise the whole compensator if the two ratios are linked, \(\gamma=\beta\); this is the classical lag–lead network of the analogue era, and its restriction is the price of using one \(RC\) section instead of two. With an operational amplifier, or in software, the two ratios are independent and the more general form above applies.

Both networks are two-resistor, one-capacitor circuits, which is why they dominated control hardware for forty years before digital implementation made the question moot. Their transfer functions follow from a voltage divider with one frequency-dependent arm.

The lag network is the same exercise with the capacitor moved into the lower arm: \(E_o/E_i=(1+R_2Cs)/\bigl(1+(R_1+R_2)Cs\bigr)\), giving \(T=R_2C\) and \(\beta=(R_1+R_2)/R_2\).

Two practical points follow from these expressions. The passive lead has a DC gain of \(\alpha\), an attenuation that must be made up by an amplifier of gain \(1/\alpha\) somewhere in the loop — the \(K_c=1/\alpha\) of the design procedure is not a bookkeeping device but a real amplifier. The passive lag, by contrast, has unity DC gain and needs no such amplifier, which is one reason lag networks were popular in the era of expensive amplification. And because both are passive dividers, cascading two of them directly makes the second load the first and changes both transfer functions; a buffer between the sections, or a single active realisation, is required.

Decide first which specification the uncompensated system fails — margin or accuracy — because that decides the network. Assume unity feedback throughout. Difficulty rises down the list.