5. Frequency-Domain Analysis and Design

Introduction and Context

Control-system performance is often assessed in the time domain through rise time, settling time, overshoot, and steady-state error. These measures are essential, but they do not by themselves reveal how a system responds across a range of signal frequencies or how close it is to instability. Frequency-domain analysis provides that complementary view. It describes the steady-state response of a system to sinusoidal inputs and uses the resulting magnitude and phase information to evaluate stability, bandwidth, resonance, disturbance rejection, and robustness.

The central object is the frequency response obtained by substituting \(s=j\omega\) into a transfer function. From this response, engineers construct Bode plots, Nyquist diagrams, and Nichols charts. These graphical methods support both analysis and design. They are particularly valuable for systems with uncertain parameters, unmodeled high-frequency dynamics, sensor noise, flexible structures, power-electronic switching effects, and communication delays.

Frequency-domain design is not a substitute for time-domain verification. Rather, it provides a powerful design language in which speed, stability margin, noise sensitivity, and robustness can be considered together.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Derive and interpret the frequency response of a control system.

2. Construct and use Bode, Nyquist, and Nichols plots.

3. Determine gain margin, phase margin, crossover frequencies, bandwidth, and resonance characteristics.

4. Relate loop gain to sensitivity, disturbance rejection, and noise transmission.

5. Apply the Nyquist criterion to assess closed-loop stability.

6. Design lead, lag, lead–lag, and proportional-integral compensators in the frequency domain.

7. Evaluate trade-offs among speed, robustness, noise amplification, and actuator effort.

8. Recognize modeling limitations, nonlinear effects, and practical implementation risks.

The Frequency Response and Loop Description

Consider a linear time-invariant transfer function \(G(s)\). For a sinusoidal input \(r(t)=A\sin\omega t\), the steady-state output is sinusoidal at the same frequency, with amplitude scaled by \(|G(j\omega)|\) and phase shifted by \(\angle G(j\omega)\). Thus,

\[

G(j\omega)=|G(j\omega)|e^{j\angle G(j\omega)}.

\]

The magnitude ratio is commonly expressed in decibels:

\[

M_{dB}=20\log_{10}|G(j\omega)|.

\]

For a unity-feedback system with controller \(C(s)\) and plant \(P(s)\), the open-loop or loop transfer function is

\[

L(s)=C(s)P(s),

\]

and the closed-loop transfer function from reference to output is

\[

T(s)=\frac{L(s)}{1+L(s)}.

\]

The characteristic equation is \(1+L(s)=0\). Consequently, the magnitude and phase of \(L(j\omega)\) determine the proximity of closed-loop poles to instability. Frequency-domain design therefore concentrates on shaping the loop gain rather than directly manipulating closed-loop poles.

A frequency-response plot is also a description of dynamic behavior. Low frequencies are associated with tracking slowly varying references and rejecting slowly varying load disturbances. Intermediate frequencies are associated with useful control bandwidth. High frequencies often contain measurement noise, neglected modes, sampling effects, and actuator limitations.

Bode Plots: A Practical Engineering Representation

A Bode plot consists of two graphs against logarithmic frequency: magnitude in decibels and phase in degrees. Its usefulness arises from the fact that multiplication of transfer-function factors becomes addition of magnitudes in decibels and phase angles.

For a first-order factor \(1/(1+s/\omega_p)\), the magnitude is approximately constant below \(\omega_p\), decreases at approximately \(-20\) dB per decade above \(\omega_p\), and contributes a phase changing from approximately \(0^\circ\) to \(-90^\circ\). A zero factor \(1+s/\omega_z\) contributes a positive slope change of approximately \(+20\) dB per decade and a phase increase toward \(+90^\circ\). An integrator contributes a constant slope of \(-20\) dB per decade and a phase of \(-90^\circ\); a differentiator contributes the opposite.

For a transfer function written as

\[

G(s)=K\frac{\prod_i(1+s/\omega_{z_i})}{s^m\prod_k(1+s/\omega_{p_k})},

\]

the asymptotic magnitude plot can be constructed by starting with \(20\log_{10}K\), adding the effect of integrators, and changing slope at each pole and zero frequency. Exact curves differ from asymptotes near corner frequencies, where a first-order pole or zero contributes approximately 3 dB from its asymptotic value.

The Bode plot supports several important measurements:

• Gain crossover frequency, \(\omega_{gc}\): the frequency at which \(|L(j\omega)|=1\), or 0 dB.
• Phase crossover frequency, \(\omega_{pc}\): the frequency at which \(\angle L(j\omega)=-180^\circ\), for a conventional negative-feedback system.
• Phase margin:

\[

PM=180^\circ+\angle L(j\omega_{gc}).

\]

• Gain margin:

\[

GM=\frac{1}{|L(j\omega_{pc})|},

\]

or in decibels, \(GM_{dB}=-20\log_{10}|L(j\omega_{pc})|\).

Positive margins generally indicate a useful distance from the nominal instability condition, although the precise interpretation depends on system structure and uncertainty.

Stability, Margins, and the Nyquist Criterion

The Nyquist criterion maps the contour in the right-half \(s\)-plane into the complex plane through \(L(s)\). For a system with no open-loop right-half-plane poles, the closed loop is stable if the Nyquist plot does not encircle the critical point \(-1+j0\). If right-half-plane open-loop poles exist, the required number and direction of encirclements must be included.

The critical point has a direct physical interpretation. At \(L=-1\), the loop has unit magnitude and phase of \(-180^\circ\). Negative feedback has effectively become positive feedback at that frequency, satisfying the characteristic equation \(1+L=0\).

Bode margins are local measures near crossover frequencies, whereas the Nyquist plot gives a global geometric view. A system can have apparently acceptable conventional margins but still display problematic behavior because of multiple crossover frequencies, right-half-plane poles or zeros, nonminimum-phase dynamics, or resonant peaks. The Nyquist distance from \(-1\), often called the modulus margin, is a useful robustness indicator:

\[

M_m=\min_\omega |1+L(j\omega)|.

\]

A small value of \(M_m\) means that a relatively small modeling or gain perturbation could move the loop close to instability.

Nichols Charts and Loop-Shaping Insight

A Nichols chart plots open-loop phase on the horizontal axis and magnitude in decibels on the vertical axis. It also contains contours of constant closed-loop magnitude, commonly called M-contours, and sometimes constant sensitivity contours. The chart allows an engineer to see how changes in loop gain affect both open-loop margins and closed-loop resonance.

The Nichols representation is especially useful when the design objective is to place the loop near a desired bandwidth while avoiding excessive closed-loop peaking. A trajectory that passes too close to the high-value M-contours may produce a resonant closed-loop response even if the gain and phase margins appear acceptable. Nichols charts are therefore valuable for loop shaping and for comparing alternative compensators.

Sensitivity, Complementary Sensitivity, and Disturbance Rejection

Two functions organize the relationship between loop gain and robustness:

\[

S(s)=\frac{1}{1+L(s)},\qquad T(s)=\frac{L(s)}{1+L(s)}.

\]

They satisfy \(S(s)+T(s)=1\). The sensitivity function describes how plant perturbations, load disturbances introduced at suitable points, and command tracking errors are transmitted. Where \(|L|\gg1\), \(|S|\approx1/|L|\), so disturbances and modeling errors are attenuated. The complementary sensitivity function describes reference tracking in a unity-feedback arrangement and measurement-noise transmission when noise enters at the sensor output. Where \(|L|\gg1\), \(|T|\approx1\), whereas where \(|L|\ll1\), \(|T|\approx|L|\).

This relationship creates a fundamental compromise. Increasing low-frequency loop gain improves tracking and rejection of low-frequency disturbances, but increasing bandwidth may amplify high-frequency sensor noise and excite unmodeled plant modes. Since \(S+T=1\), both functions cannot be made small at the same frequency. In robust design, the peak values \(M_s=\max|S|\) and \(M_t=\max|T|\) are commonly monitored.

Bandwidth is often defined as the frequency at which the magnitude of the closed-loop transfer function falls by 3 dB from its low-frequency value. It is not identical to gain crossover frequency, although the two may be related in well-damped systems. A large separation between them can indicate unusual phase behavior or resonant peaking.

Resonance and Robustness

A resonant peak occurs when the closed-loop magnitude becomes substantially larger than its low-frequency value. Resonance is usually associated with lightly damped poles and a loop trajectory passing near the critical region of the Nyquist plane. It may cause overshoot, structural vibration, audible noise, or actuator saturation.

Robustness concerns acceptable performance under uncertainty. Sources include uncertain gain, time constants, delays, friction, sensor dynamics, temperature-dependent parameters, flexible modes, and neglected nonlinearities. Gain and phase margins provide simple robustness measures, but they do not capture every uncertainty structure. For multiplicative plant uncertainty \(P_{actual}=P(1+W_m\Delta)\), a useful qualitative requirement is that \(|T(j\omega)W_m(j\omega)|\) remain below unity over frequencies where the uncertainty is significant. This expresses the fact that high complementary sensitivity is dangerous where plant uncertainty is large.

Frequency-Domain Compensator Design

Frequency-domain compensation changes the shape of \(L(s)=C(s)P(s)\). A successful design begins with specifications: desired bandwidth, tracking accuracy, disturbance-rejection range, allowable overshoot, noise level, actuator limits, and robustness margins. The designer then constructs a nominal model, identifies uncertainty, and shapes the loop while checking both frequency- and time-domain behavior.

Lead Compensation

A lead compensator may be written as

\[

C_{lead}(s)=K\frac{1+s/\omega_z}{1+s/\omega_p},\qquad \omega_z<\omega_p.

\]

It provides positive phase over a selected frequency range and generally increases bandwidth. The maximum phase lead is

\[

\phi_{max}=\sin^{-1}\left(\frac{1-\alpha}{1+\alpha}\right),

\]

where \(\alpha=\omega_z/\omega_p<1\). The frequency of maximum phase lead is approximately \(\omega_m=\sqrt{\omega_z\omega_p}\). Lead compensation is appropriate when the system needs more phase margin or faster response, but it can amplify high-frequency measurement noise and demand greater control effort.

Lag Compensation

A lag compensator can be written as

\[

C_{lag}(s)=K\frac{1+s/\omega_z}{1+s/\omega_p},\qquad \omega_p<\omega_z.

\]

It increases low-frequency gain relative to gain near crossover, improving steady-state accuracy with limited change to crossover frequency. Its phase contribution is negative over a transition range, so excessive lag can reduce phase margin. The lag network is commonly placed so that its phase loss occurs well below the desired crossover frequency.

Lead–Lag and PI Compensation

A lead–lag compensator combines the two effects: lag improves low-frequency accuracy, while lead recovers phase margin and supports the desired bandwidth. A PI controller,

\[

C_{PI}(s)=K\frac{1+sT_i}{sT_i},

\]

adds integral action for zero steady-state error to suitable inputs. However, integral action introduces phase lag and can cause windup during actuator saturation. Practical PI controllers therefore require anti-windup logic.

A common design sequence is to establish the required low-frequency gain, select a desired crossover frequency, calculate the phase deficiency at crossover, add lead to supply phase, and then verify gain margin, bandwidth, sensitivity peaks, actuator effort, and time-domain response. The nominal design must also be tested against parameter variations and delays.

Worked Design Illustration

Consider the illustrative plant

\[

P(s)=\frac{1}{s(1+0.2s)}.

\]

The plant contains an integrator and a pole at \(5\) rad/s. With proportional control alone, increasing gain raises the crossover frequency but also moves operation toward a region where the plant contributes additional phase lag. The resulting phase margin may become unsatisfactory.

Suppose the design objective is a moderate bandwidth near 4 rad/s with a phase margin of approximately 50 degrees. At this frequency, the plant phase is approximately

\[

-90^\circ-\tan^{-1}(0.8)\approx-128.7^\circ.

\]

If the loop is near 0 dB at 4 rad/s, the uncompensated phase margin is approximately 51.3 degrees before accounting for any additional implementation lag. If sensor dynamics, computation delay, or a more accurate plant model add phase lag, a lead network may be introduced. For example, a lead compensator centered near the intended crossover can contribute 20–30 degrees of phase. Its gain is then adjusted so that the compensated loop crosses 0 dB at the chosen frequency.

The design is not accepted merely because the target phase margin is obtained. The engineer must inspect the complete Bode plot for additional crossover points, determine the gain margin, calculate or estimate the peak of \(|S|\) and \(|T|\), check control-signal amplitude, and simulate reference, load-disturbance, and measurement-noise responses. This illustrates a central principle: crossover frequency is a design choice, but robustness is a system-level outcome.

Applied Case: Speed Control of an Industrial Drive

Background

The following is an illustrative case involving a digitally controlled permanent-magnet motor drive. The speed loop contains a current-controlled electrical subsystem, mechanical inertia, friction, a speed sensor, and a digital controller.

Situation and Problem

The original proportional speed controller provides adequate low-frequency tracking but exhibits oscillatory behavior when the commanded speed changes rapidly. Measurements show a resonant feature near 70 rad/s caused by mechanical compliance between the motor and load. Increasing proportional gain improves disturbance rejection but moves the loop crossover toward the resonance and reduces phase margin. Sensor quantization and inverter delay also become significant above approximately 150 rad/s.

Analysis

A Bode analysis shows that the plant has high low-frequency gain but a steep magnitude transition near the flexible mode. The Nyquist plot approaches the critical point as gain is increased. The sensitivity peak is large near the resonance, indicating poor robustness and amplification of load disturbances. A purely proportional redesign would therefore trade one problem for another.

Approach and Solution

The design team first places a notch filter around the flexible-mode frequency to reduce the loop gain in a narrow band without substantially reducing low-frequency disturbance rejection. A lead element is then used to restore phase near the selected crossover, which is deliberately kept below the frequency range dominated by sensor delay and unmodeled inverter dynamics. A small integral action is retained for elimination of steady-state speed error, with anti-windup protection for acceleration and braking saturation.

The redesigned loop is evaluated through Bode and Nyquist plots, followed by simulations with inertia variation, load-torque disturbances, sensor noise, computation delay, and actuator saturation. Hardware tests are performed with conservative gain limits before the full operating range is enabled.

Outcome

The frequency-domain redesign reduces resonant peaking, preserves low-frequency disturbance rejection, and provides greater separation between the crossover frequency and uncertain high-frequency dynamics. The remaining performance limitation is not removed by controller design alone; it requires mechanical improvement if still higher speed-loop bandwidth is demanded.

Learning Points

1. High loop gain is useful only where the plant model and actuator remain reliable.

2. A notch filter can control a localized resonance but should not conceal broader modeling errors.

3. Phase margin must be interpreted together with delay, uncertainty, and sensitivity peaks.

4. Frequency-domain analysis guides design, while experiments and nonlinear simulations establish practical acceptability.

Common Errors and Practical Limitations

Several mistakes recur in frequency-domain work. Treating a gain margin or phase margin as an absolute guarantee is unsafe when the model contains delays, nonlinearities, multiple unstable modes, or structured uncertainty. Confusing gain crossover with closed-loop bandwidth can lead to an incorrect speed assessment. Designing for a high crossover frequency without checking sensor noise and actuator limits can produce an apparently fast but unusable controller.

Other errors include ignoring right-half-plane zeros, omitting computational delay, using asymptotic Bode slopes too close to corner frequencies, and assessing only reference tracking while neglecting load disturbance and measurement noise. Saturation can invalidate linear frequency-response predictions, especially when integral action is present. A controller that is stable in a nominal simulation may fail under parameter variation or during mode switching.

Digital Implementation, Computation, and Emerging Practice

In a digital controller, sampling and computation introduce delay and frequency-dependent effects. The continuous compensator must be discretized using an appropriate method, such as the bilinear transformation, and its discrete frequency response should be checked rather than assuming that the continuous design is preserved exactly. Sampling frequency should be sufficiently higher than the intended closed-loop bandwidth, with margins determined by the application and by noise and computational constraints.

Modern computer-aided control tools automate Bode, Nyquist, Nichols, sensitivity, and uncertainty plots. Robust-control software can evaluate structured uncertainty and synthesize controllers using formulations such as \(H_\infty\) methods. Data-driven frequency-response estimation is increasingly used when first-principles models are incomplete. Automated tuning and optimization can be useful, but the engineer remains responsible for selecting meaningful objectives, excluding unsafe operating regions, validating excitation signals, and interpreting results physically.

Professional practice also requires traceability. Assumptions, model versions, uncertainty ranges, test conditions, and acceptance margins should be documented. In safety-critical or regulated systems, controller changes require verification, validation, configuration management, and evidence that failure modes have been considered.

Best-Practice Design Workflow

A disciplined workflow is as follows:

1. Define tracking, disturbance-rejection, bandwidth, robustness, noise, and actuator requirements.

2. Develop and validate a plant model over the intended operating range.

3. Plot the uncompensated loop and identify poles, zeros, delays, resonances, and uncertain regions.

4. Select a target crossover frequency that respects unmodeled dynamics and implementation limits.

5. Shape low-frequency gain for tracking and disturbance rejection.

6. Use lead, lag, PI, lead–lag, notch, or other suitable elements to shape phase and magnitude.

7. Evaluate gain margin, phase margin, Nyquist encirclements, sensitivity peaks, and closed-loop resonance.

8. Test reference, disturbance, noise, saturation, parameter variation, and delay scenarios.

9. Discretize and implement the controller, including filtering and anti-windup where required.

10. Validate through simulation, laboratory tests, and controlled field testing.

Chapter Summary

Frequency-domain analysis represents a control system through its sinusoidal steady-state response. Bode plots provide a practical view of magnitude, phase, crossover frequencies, bandwidth, and classical stability margins. Nyquist plots give a geometric stability test based on encirclement of the critical point, while Nichols charts support loop shaping and visualization of closed-loop resonance.

The sensitivity functions \(S=1/(1+L)\) and \(T=L/(1+L)\) connect loop gain with tracking, disturbance rejection, noise transmission, and robustness. Lead compensation generally increases phase margin and bandwidth; lag and PI compensation improve low-frequency accuracy but add phase lag; lead–lag networks balance these objectives. Successful design requires more than nominal gain and phase margins: uncertainty, delays, resonances, saturation, digital implementation, and actuator and sensor limitations must also be considered.

Key Terms

Frequency response: Steady-state magnitude and phase response to sinusoidal excitation.

Loop transfer function: Product of controller and plant transfer functions around the feedback loop.

Bode plot: Log-frequency plots of magnitude in decibels and phase in degrees.

Nyquist plot: Complex-plane mapping of a loop transfer function as frequency varies.

Nichols chart: Plot of open-loop magnitude against phase with closed-loop magnitude contours.

Gain crossover frequency: Frequency at which loop magnitude equals unity.

Phase crossover frequency: Frequency at which loop phase reaches \(-180^\circ\).

Phase margin: Additional phase lag required at gain crossover to reach the instability condition.

Gain margin: Multiplicative gain increase required at phase crossover to reach instability.

Bandwidth: Frequency range over which closed-loop response remains within a specified magnitude criterion.

Resonance: A pronounced peak in frequency response associated with lightly damped dynamics.

Sensitivity function: \(S=1/(1+L)\), measuring sensitivity to disturbances and plant variation.

Complementary sensitivity: \(T=L/(1+L)\), describing reference transmission and noise-related behavior.

Lead compensator: Network that supplies positive phase and commonly increases bandwidth.

Lag compensator: Network that increases low-frequency gain with relatively small intended crossover change.

Crossover frequency: A frequency at which a specified magnitude or phase crossing occurs.

Modulus margin: Minimum distance of the Nyquist locus from the critical point, expressed through \(|1+L|\).

Robustness: Ability to maintain stability and acceptable performance despite uncertainty and disturbances.

Anti-windup: Technique that prevents excessive integrator accumulation during actuator saturation.

Review Questions

Short Answer Questions

1. What is meant by the frequency response of a transfer function?

2. Define gain crossover frequency and phase crossover frequency.

3. State the relationship between sensitivity and complementary sensitivity.

4. Why can a lead compensator improve phase margin?

5. What information does a Nyquist plot provide that is not directly visible from a simple gain calculation?

Long Answer Questions

1. Explain how Bode plots are constructed from poles, zeros, integrators, and differentiators.

2. Derive and interpret the sensitivity and complementary-sensitivity functions for a unity-feedback system.

3. Explain the Nyquist stability criterion and the significance of the critical point \(-1+j0\).

4. Compare lead, lag, PI, and lead–lag compensation in terms of objectives, benefits, and limitations.

5. Discuss how bandwidth, resonance, measurement noise, actuator limits, and robustness influence one another in frequency-domain design.

Analytical / Application Questions

1. For \(L(s)=K/[s(1+0.2s)]\), explain qualitatively how increasing \(K\) affects crossover frequency, phase margin, and disturbance rejection.

2. A control loop has satisfactory phase margin but a large peak in \(|T(j\omega)|\) near an uncertain flexible mode. Should the design be accepted? Justify your answer and propose a modification.

3. An industrial speed loop requires improved steady-state accuracy but cannot tolerate increased sensor-noise amplification. Select and justify a frequency-domain compensation strategy, including the checks you would perform before implementation.

22. Multiple Choice Questions with Answers

Question 1

For a unity-feedback loop, which condition corresponds to the nominal instability point?

A. Magnitude zero and phase 0 degrees

B. Magnitude unity and phase -180 degrees

C. Magnitude unity and phase -90 degrees

D. Magnitude -3 dB and phase 180 degrees

Correct Answer: B. Magnitude unity and phase -180 degrees

Question 2

At the gain crossover frequency, the phase margin is calculated as:

A. 180 degrees minus the loop phase

B. The loop magnitude in decibels

C. 180 degrees plus the loop phase

D. The reciprocal of the loop magnitude

Correct Answer: C. 180 degrees plus the loop phase

Question 3

If the loop gain is much greater than one at a low frequency, the sensitivity magnitude is approximately:

A. The loop gain

B. One

C. The reciprocal of the loop gain

D. Zero for every frequency

Correct Answer: C. The reciprocal of the loop gain

Question 4

Which compensator is primarily used to add positive phase near a selected crossover frequency?

A. Lag compensator

B. Lead compensator

C. Pure integrator

D. Low-pass measurement filter

Correct Answer: B. Lead compensator

Question 5

A lag compensator is most directly useful for:

A. Increasing low-frequency gain and improving steady-state accuracy

B. Removing all high-frequency noise

C. Adding large positive phase at crossover

D. Eliminating actuator saturation

Correct Answer: A. Increasing low-frequency gain and improving steady-state accuracy

Question 6

The Nyquist criterion is based on the behavior of the loop transfer function relative to:

A. The origin

B. The point +1 on the real axis

C. The point -1+j0

D. The gain crossover frequency only

Correct Answer: C. The point -1+j0

Question 7

Why may increasing crossover frequency reduce robustness?

A. It always removes integral action

B. It can move the loop into regions with delay, resonance, or uncertain dynamics

C. It necessarily reduces low-frequency gain to zero

D. It makes the plant linear only at high frequencies

Correct Answer: B. It can move the loop into regions with delay, resonance, or uncertain dynamics

Question 8

For a unity-feedback system, the complementary sensitivity function is:

A. 1-L

B. 1/(1-L)

C. L/(1+L)

D. 1/L

Correct Answer: C. L/(1+L)

Question 9

A large peak in the sensitivity function generally indicates:

A. Strong robustness at every frequency

B. Potential amplification of disturbances and model uncertainty

C. Zero closed-loop bandwidth

D. Guaranteed absence of resonance

Correct Answer: B. Potential amplification of disturbances and model uncertainty

Question 10

In the illustrative motor-drive case, why was a notch filter used?

A. To add integral action

B. To reduce loop gain near a localized flexible-mode resonance

C. To increase sensor quantization

D. To eliminate the need for validation testing

Correct Answer: B. To reduce loop gain near a localized flexible-mode resonance