Lag Compensator Design

A lag compensator is a frequency domain controller whose primary purpose is to improve the steady-state accuracy of a control system without significantly affecting its transient response. Unlike the lead compensator which boosts phase near the crossover frequency, the lag compensator works by increasing the open-loop low-frequency gain, thereby reducing steady-state error, while attenuating the gain at higher frequencies to preserve the existing stability margins.

Core Concept of Phase Lag Compensation

A lag compensator has a zero at s = -z and a pole at s = -p, where z is greater than p. This means the zero is farther from the origin than the pole, which is the opposite arrangement to a lead compensator. The ratio beta = z/p is greater than 1, and this ratio determines the DC gain increase provided by the compensator. The lag compensator increases the gain at low frequencies by a factor of beta, reducing the steady-state error proportionally.

The phase lag contributed by the compensator is negative (lagging), which appears between the pole corner frequency p and the zero corner frequency z. To prevent this phase lag from degrading the phase margin at the gain crossover frequency, both corner frequencies (p and z) must be placed well below the existing gain crossover frequency. Typically, the zero is placed at one-tenth of the gain crossover frequency, ensuring the phase lag from the compensator at crossover is less than 5 degrees.

The key insight is that the lag compensator works not by adding phase (as the lead compensator does) but by using its high-frequency attenuation to shift the gain crossover frequency to a lower value where the uncompensated system already has sufficient phase margin. This is fundamentally different from lead compensation and is the reason why the design goals for the two compensators differ.

Mathematical Expressions for Lag Compensator Design

The transfer function of the lag compensator is C(s) = (s + z)/(s + p) with beta = z/p greater than 1, or equivalently in normalized form C(s) = (1 + tau*s)/(1 + beta*tau*s), where tau = 1/z. The DC gain of the compensator is C(0) = z/p = beta. When the uncompensated system gain K is set for adequate steady-state accuracy but the resulting phase margin is insufficient, the lag compensator is not the right choice. Lag compensation is used when the gain K is lower than what is needed for steady-state requirements.

The design procedure is as follows. First, determine the required beta from the steady-state error specification. If the current Kv needs to be increased by a factor of beta, then beta is that factor. Second, identify the frequency w_c_new where the uncompensated system G(jw) has a phase of -180 + PM_desired + 5 degrees (the 5-degree buffer compensates for the small residual lag from the compensator). This w_c_new becomes the new desired gain crossover frequency. Third, place the zero at z = w_c_new / 10 and the pole at p = z / beta = w_c_new / (10*beta).

Effect on System Performance

Lag compensation improves steady-state error by increasing the low-frequency gain by a factor of beta. The gain crossover frequency decreases (the system becomes slower), which means the bandwidth reduces and the settling time increases. The transient response is essentially unchanged in character but becomes slower in absolute time. This is the fundamental trade-off of lag compensation: better steady-state accuracy at the cost of slower transient response.

The phase margin is preserved approximately because the compensator's phase lag at the new crossover frequency is small (by design, less than 5 degrees). The gain margin typically improves as well because the gain rolls off more steeply after crossover. For systems where steady-state accuracy is the primary concern and speed of response is secondary, lag compensation is the preferred choice over lag-lead or PID designs.

Solved Numerical Example

A unity feedback system has G(s) = K / (s(s+1)(s+4)). Design a lag compensator such that Kv = 5 s^-1 and phase margin is at least 40 degrees.

Given:
G(s) = K / [s(s+1)(s+4)]
Desired Kv = 5, Desired PM >= 40 degrees

Step 1: Kv = lim(s->0) s*G(s) = K/4 = 5 => K = 20
So G(s) = 20 / [s(s+1)(s+4)]

Step 2: Find phase of G(jw) at frequencies
Need phase = -180 + 40 + 5 = -135 degrees
Phase of G(jw) = -90 - arctan(w) - arctan(w/4)

Trial w = 0.5 rad/s:
Phase = -90 - arctan(0.5) - arctan(0.125)
= -90 - 26.6 - 7.1 = -123.7 deg (too little lag, move higher)

Trial w = 0.7 rad/s:
Phase = -90 - arctan(0.7) - arctan(0.175)
= -90 - 34.9 - 9.9 = -134.8 deg ≈ -135 deg  ✓

New gain crossover w_c_new = 0.7 rad/s

Step 3: Check magnitude at w_c_new
|G(j0.7)| = 20 / [0.7 * sqrt(1+0.49) * sqrt(16+0.49)]
= 20 / [0.7 * 1.22 * 4.06]
= 20 / 3.47 = 5.76

beta = |G(jw_c_new)| = 5.76
(Compensator must attenuate by factor 5.76 at crossover)

Step 4: Compensator zeros and poles
z = w_c_new / 10 = 0.07 rad/s
p = z / beta = 0.07 / 5.76 = 0.012 rad/s

Final Compensator:
C(s) = (s + 0.07) / (s + 0.012)
DC gain of C = z/p = 0.07/0.012 = 5.83 ≈ beta ✓

Final Answer: C(s) = (s+0.07)/(s+0.012)
Kv = 20*5.83/4 ≈ 29 (exceeds target due to rounding; K can be reduced)
Phase margin ≈ 40 degrees as designed.

Exam Tip: For a lag compensator, remember that beta = z/p is always greater than 1. The DC gain of the compensator equals beta, not 1/beta. The compensator pole p is always closer to the origin than the zero z (unlike a lead compensator where the pole is farther). In GATE, if you see a compensator with zero to the right of the pole in the s-plane, it is a lag compensator.

Mechanism Summary

• Lag compensator: C(s) = beta*(s+z)/(s+p) with z greater than p. Pole is closer to origin than zero. beta = z/p greater than 1.
• Mechanism: the compensator attenuates high-frequency gain, shifting gain crossover to a lower frequency where the phase is naturally better.
• DC gain of compensator = beta. This directly increases Kv, Kp, or Ka depending on the system type.
• Both corner frequencies must be placed well below the desired gain crossover frequency (z at about w_c/10) to prevent significant phase lag at crossover.
• Lag compensation reduces bandwidth and slows transient response but significantly improves steady-state accuracy.
• It does not improve phase margin directly; rather, it exploits the existing phase margin at lower frequencies of the uncompensated system.

Quick Revision

• Lag compensator: C(s) = (s+z)/(s+p), z > p. beta = z/p > 1. Pole is to the right of zero on the negative real axis.
• Primary effect: DC gain increase by factor beta, reducing steady-state error by factor beta.
• Design rule: Place zero at z = w_c_new/10. Place pole at p = z/beta. This keeps phase lag at crossover below 5 degrees.
• Find w_c_new as the frequency where uncompensated phase = -(180 - PM_desired - 5) degrees.
• Effect on response: slower bandwidth, better steady-state, approximately preserved phase margin.
• GATE trap: Confusing lag and lead compensators. In a lag compensator, the zero is farther from origin (larger value) than the pole. In lead, the pole is farther from origin than the zero.
• beta = z/p > 1 for lag. alpha = z/p < 1 for lead. Both are called pole-to-zero ratios but in different senses depending on which text is used.