Frequency-Domain Analysis and Design

Introduction and Context

Frequency-domain methods are central to the analysis and design of feedback control systems. While time-domain methods describe transient responses directly, frequency-domain methods describe how a system responds to sinusoidal signals of different frequencies. This viewpoint is especially valuable for systems with uncertain models, measurement noise, unmodeled dynamics, transport delays, and high-order transfer functions.

The principal tools developed in this chapter are Bode plots, Nyquist diagrams, Nichols charts, gain and phase margins, bandwidth and resonance measures, sensitivity functions, and loop-shaping methods. These tools are closely related: each represents the same open-loop transfer function in a different graphical or analytical form, but each emphasizes different design questions.

For a standard negative-feedback system, let the plant be \(G(s)\), the controller be \(C(s)\), and the sensor or feedback transfer function be \(H(s)\). The loop transfer function is

\[

L(s)=C(s)G(s)H(s).

\]

The closed-loop transfer function from reference input \(R(s)\) to output \(Y(s)\) is

\[

T(s)=\frac{Y(s)}{R(s)}=\frac{C(s)G(s)}{1+C(s)G(s)H(s)},

\]

assuming a conventional negative-feedback connection. When \(H(s)=1\), the loop transfer function is often written \(L(s)=C(s)G(s)\), and

\[

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

\]

Frequency-domain design does not replace time-domain verification. Rather, it provides insight into stability, robustness, disturbance rejection, noise amplification, and achievable speed before final validation using time simulations or experiments.

Learning Objectives

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

1. Construct and interpret Bode, Nyquist, and Nichols representations.

2. Explain the relationship between frequency response and closed-loop stability.

3. Apply the Nyquist criterion, including the effects of right-half-plane poles and zeros.

4. Calculate gain margin, phase margin, crossover frequencies, bandwidth, and resonance measures.

5. Define and interpret sensitivity and complementary sensitivity functions.

6. Design controllers using frequency-domain loop-shaping principles.

7. Evaluate design trade-offs involving robustness, tracking, disturbance rejection, and measurement noise.

8. Recognize practical limitations caused by delays, nonlinearities, actuator constraints, and model uncertainty.

1. Frequency Response and the Frequency-Domain Model

For a linear time-invariant system with transfer function \(G(s)\), the frequency response is obtained by evaluating the transfer function on the imaginary axis:

\[

G(j\omega)=G(s)|_{s=j\omega},

\]

where \(\omega\) is angular frequency in radians per second. If the input is \(u(t)=U\sin(\omega t)\), the steady-state output is sinusoidal at the same frequency, with amplitude multiplied by \(|G(j\omega)|\) and phase shifted by \(\angle G(j\omega)\).

The magnitude is commonly expressed in decibels:

\[

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

\]

The phase is normally expressed in degrees or radians. A frequency-response description is applicable to linear time-invariant systems, or to nonlinear systems after linearization around an operating point. It describes steady-state sinusoidal behavior and does not by itself describe startup transients or large-signal nonlinear behavior.

A transfer function factor illustrates the basic frequency-domain contributions. For a real pole at \(-a\), with \(a>0\),

\[

\frac{1}{1+s/a},

\]

has magnitude

\[

\left|\frac{1}{1+j\omega/a}\right|=\frac{1}{\sqrt{1+(\omega/a)^2}},

\]

and phase

\[

-\tan^{-1}(\omega/a).

\]

Its magnitude is approximately 0 dB for \(\omega\ll a\), decreases at approximately \(-20\) dB per decade for \(\omega\gg a\), and its phase changes from approximately 0° to \(-90°\). A zero contributes the opposite trend. A second-order factor can introduce a resonant peak when its damping ratio is sufficiently small.

2. Bode Representation

A Bode plot consists of two graphs against logarithmic frequency: magnitude in decibels and phase, usually in degrees. The logarithmic frequency axis allows several decades of behavior to be displayed conveniently and makes asymptotic slopes easy to interpret.

For a transfer function written as

\[

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

\]

where the quantities are arranged in normalized form and any complex factors are included in conjugate pairs, the low-frequency slope is determined by the number of poles and zeros at the origin. Each first-order zero increases the high-frequency slope by 20 dB per decade; each first-order pole decreases it by 20 dB per decade. A pole or zero at a break frequency also produces a phase transition over approximately two decades surrounding that frequency.

For a factor \(1+s/a\), the corner frequency is \(a\) rad/s. For a factor \(1/(1+s/a)\), the asymptotic magnitude changes slope by \(-20\) dB/decade. The exact magnitude at the corner is \(-3.01\) dB relative to the low-frequency asymptote for a pole, while the phase is \(-45°\).

For a second-order denominator,

\[

G(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2},

\]

\(\omega_n\) is the undamped natural frequency and \(\zeta\) is the damping ratio. A resonant peak exists when

\[

\zeta<\frac{1}{\sqrt{2}}.

\]

The resonance frequency, when this condition holds, is

\[

\omega_r=\omega_n\sqrt{1-2\zeta^2},

\]

and the resonant magnitude is

\[

M_r=\frac{1}{2\zeta\sqrt{1-\zeta^2}}.

\]

These expressions apply to the normalized low-pass second-order form above. They should not be used without modification for systems containing zeros or additional poles.

Worked Example 1: Bode Characteristics

Consider

\[

L(s)=\frac{10(1+s/10)}{s(1+s/100)}.

\]

There is one integrator, one zero at 10 rad/s, and one pole at 100 rad/s. The low-frequency slope is \(-20\) dB/decade because of the integrator. At 10 rad/s the zero increases the slope to 0 dB/decade. At 100 rad/s the pole decreases it again to \(-20\) dB/decade.

At \(\omega=1\) rad/s,

\[

|L(j1)|=\frac{10\sqrt{1+(1/10)^2}}{1\sqrt{1+(1/100)^2}}\approx 10.05,

\]

which is approximately 20.05 dB. The phase is

\[

\angle L(j1)=-90°+\tan^{-1}(0.1)-\tan^{-1}(0.01)

\approx -84.85°.

\]

The example demonstrates that asymptotic slopes provide a rapid global picture, whereas exact expressions are useful near crossover frequencies.

3. Stability and the Nyquist Criterion

For negative feedback, closed-loop poles are the zeros of

\[

1+L(s)=0.

\]

Thus, stability depends on whether the Nyquist plot of \(L(s)\) encircles the critical point \(-1+j0\). The Nyquist criterion follows from the argument principle. Let:

• \(P\) be the number of right-half-plane poles of \(L(s)\), counted with multiplicity;
• \(N\) be the net number of clockwise encirclements of \(-1+j0\) by the complete Nyquist plot;
• \(Z\) be the number of right-half-plane zeros of \(1+L(s)\), which are the unstable closed-loop poles.

Using the clockwise convention,

\[

Z=N+P.

\]

Closed-loop stability requires \(Z=0\), so the required encirclement condition is

\[

N=-P.

\]

Different textbooks use different signs for clockwise and counterclockwise encirclements. The convention must therefore be stated explicitly.

The Nyquist contour encloses the right half of the complex \(s\)-plane. If \(L(s)\) has poles on the imaginary axis, the contour must be indented around them, and the limiting contribution must be included. A pole on the imaginary axis is not automatically a stable or acceptable closed-loop condition; it requires careful interpretation and usually indicates marginal or structurally fragile behavior.

The Nyquist plot is particularly useful when the open-loop system has right-half-plane poles, time delays, or dynamics for which algebraic characteristic-polynomial tests are inconvenient. For a stable open-loop transfer function, \(P=0\), and closed-loop stability requires that the plot not encircle \(-1\), provided it does not pass through that point.

A time delay contributes the factor

\[

e^{-sT},

\]

whose frequency response has unit magnitude and phase \(-\omega T\). Thus, delay does not change the Bode magnitude but continuously reduces phase margin and can severely limit achievable bandwidth.

4. Gain Margin, Phase Margin, and Relative Stability

Relative stability measures how close a system is to the instability condition. Define the gain crossover frequency \(\omega_{gc}\) by

\[

|L(j\omega_{gc})|=1,

\]

or equivalently, 0 dB magnitude. The phase margin is

\[

PM=180°+\angle L(j\omega_{gc}),

\]

assuming phase is expressed using a continuous or appropriately unwrapped representation.

Define the phase crossover frequency \(\omega_{pc}\) by

\[

\angle L(j\omega_{pc})=-180°.

\]

The gain margin is

\[

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

\]

or in decibels,

\[

GM_{\mathrm{dB}}=-20\log_{10}|L(j\omega_{pc})|.

\]

If there is no phase crossover frequency, the gain margin may be considered infinite under the usual definition. If there are multiple crossover frequencies, the smallest relevant margin generally governs robustness, but the full Nyquist plot should be inspected.

Positive gain and phase margins are useful indicators for many minimum-phase, single-loop systems, but they are not universal guarantees of robust performance. Systems with open-loop unstable poles, multiple loops, nonminimum-phase zeros, or unusual Nyquist geometry require direct Nyquist and closed-loop analysis.

Worked Example 2: Margins from a Local Frequency Point

Suppose measurements of a loop give \(|L(j\omega_{gc})|=1\) at \(\omega_{gc}=20\) rad/s and phase \(-135°\). Then

\[

PM=180°-135°=45°.

\]

At the phase crossover frequency \(\omega_{pc}=35\) rad/s, suppose the magnitude is \(-8\) dB. The gain margin is therefore 8 dB. These values suggest a potentially acceptable but not highly conservative design. Final judgment depends on uncertainty, actuator limitations, and performance requirements.

5. Nichols Charts

A Nichols chart plots open-loop phase on the horizontal axis and magnitude in decibels on the vertical axis. It includes contours of constant closed-loop magnitude, commonly called \(M\)-circles, and sometimes constant closed-loop phase contours, called \(N\)-circles.

For unity feedback,

\[

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

\]

Consequently, each point on a Nichols chart can be mapped to the magnitude and phase of the closed-loop transfer function. The critical point \((-180°,0\text{ dB})\) corresponds to the singular condition \(1+L=0\). Nichols charts are valuable for loop shaping because gain, phase, closed-loop peaking, and bandwidth can be assessed on one diagram.

The chart is especially useful when a designer wishes to shape the open-loop response so that it follows a target region. It provides a direct graphical connection between open-loop design and closed-loop performance, whereas a Bode plot separates magnitude and phase.

6. Bandwidth, Resonance, and Performance Interpretation

For a closed-loop transfer function \(T(s)\), the bandwidth is often defined as the frequency at which \(|T(j\omega)|\) falls to \(1/\sqrt{2}\) of its low-frequency value. In decibels, this is the \(-3\) dB frequency. This definition is appropriate for low-pass responses with a well-defined low-frequency plateau, but other definitions may be needed for band-pass, high-pass, or nonstandard responses.

The closed-loop resonance peak is

\[

M_r=\max_{\omega\ge 0}|T(j\omega)|.

\]

A high resonance peak indicates amplification near a particular frequency and often correlates with poor damping, overshoot, and sensitivity to disturbances. Bandwidth and resonance are not interchangeable: increasing bandwidth may improve speed and tracking but can increase noise transmission and reduce robustness.

For a second-order dominant closed-loop system, phase margin often correlates qualitatively with damping ratio. This relationship is approximate, not a general theorem, because additional poles, zeros, delays, and nonminimum-phase dynamics can invalidate simple second-order interpretations.

7. Sensitivity Functions

For unity feedback, the sensitivity function is

\[

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

\]

and the complementary sensitivity function is

\[

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

\]

They satisfy the identity

\[

S(s)+T(s)=1.

\]

The sensitivity function describes how disturbances, plant perturbations, and command-tracking errors are transmitted through the feedback system. At frequencies where \(|L(j\omega)|\gg1\),

\[

|S(j\omega)|\approx \frac{1}{|L(j\omega)|},

\]

so feedback suppresses low-frequency errors and disturbances. At frequencies where \(|L(j\omega)|\ll1\), \(|T(j\omega)|\approx |L(j\omega)|\), and the system has little closed-loop authority.

The complementary sensitivity function governs reference tracking in unity feedback and transmission of measurement noise to the output. High loop gain improves tracking but can increase high-frequency noise transmission if the loop remains large in that region.

For an additive plant disturbance entering at the plant input, the output response is commonly proportional to \(G(s)S(s)\). For measurement noise entering the feedback measurement, the output response is commonly proportional to \(-T(s)\), depending on the precise signal location and sign convention.

A fundamental robust-control limitation is the waterbed effect: reducing sensitivity over one frequency range generally forces sensitivity to increase elsewhere, especially in systems with unstable poles or nonminimum-phase zeros. Therefore, a design should specify a maximum sensitivity

\[

M_s=\|S\|_\infty=\sup_{\omega}|S(j\omega)|,

\]

rather than relying only on phase margin. Similarly,

\[

M_t=\|T\|_\infty=\sup_{\omega}|T(j\omega)|

\]

quantifies the largest complementary-sensitivity peak. The infinity norm is the supremum over frequency and is not an average value.

8. Loop-Shaping for Controller Design

Loop shaping means selecting the controller so that the loop transfer function has desirable magnitude and phase characteristics. A typical design sequence is:

1. Translate specifications into frequency-domain targets, such as bandwidth, tracking accuracy, noise attenuation, and minimum margins.

2. Select a desired crossover frequency.

3. Shape the low-frequency gain to obtain tracking and disturbance rejection.

4. Shape the phase near crossover to obtain adequate phase margin.

5. Reduce high-frequency gain to limit noise amplification and actuator activity.

6. Verify stability, sensitivity peaks, time response, saturation behavior, and robustness.

A proportional controller,

\[

C(s)=K_p,

\]

changes loop magnitude without adding phase. Increasing \(K_p\) usually increases bandwidth but can reduce phase margin and increase control effort.

An integrator,

\[

C(s)=\frac{K_i}{s},

\]

increases low-frequency loop gain and can eliminate steady-state error for step references in suitable stable systems. It contributes approximately \(-90°\) phase and can reduce stability margins. Practical controllers often use a low-frequency pole with anti-windup protection rather than an unrestricted ideal integrator.

A phase-lead compensator can be written as

\[

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

\]

It contributes positive phase over a finite frequency interval and increases high-frequency magnitude by approximately \(\omega_p/\omega_z\) relative to its low-frequency gain. The maximum phase lead occurs near

\[

\omega_m=\sqrt{\omega_z\omega_p},

\]

with

\[

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

\qquad \alpha=\frac{\omega_p}{\omega_z}>1.

\]

A phase-lag compensator uses \(\omega_z>\omega_p\). It increases low-frequency gain relative to gain near crossover and is useful for improving steady-state accuracy while preserving a selected crossover region. However, it adds negative phase and must be placed carefully.

A lead-lag controller combines both effects. In modern practice, proportional-integral-derivative controllers, notch filters, two-degree-of-freedom controllers, and state-space or robust-control designs may be interpreted using the same frequency-domain principles.

Illustrative Design Calculation

Suppose a loop currently crosses 0 dB at 10 rad/s with phase margin 25°, while the desired phase margin is 50°. A lead compensator is required to add approximately 25° plus a design allowance for magnitude and modeling errors. If 35° maximum phase lead is selected, then

\[

\alpha=\frac{1+\sin35°}{1-\sin35°}\approx 3.69.

\]

The lead network should be centered near the intended crossover, so choose \(\omega_m=10\) rad/s. Then

\[

\omega_z=\frac{\omega_m}{\sqrt{\alpha}}\approx 5.21\text{ rad/s},

\qquad

\omega_p=\omega_m\sqrt{\alpha}\approx 19.2\text{ rad/s}.

\]

The controller gain must then be adjusted so that the compensated loop has 0 dB magnitude near the desired crossover. The calculation is only an initial design: the final phase margin must be obtained from the complete compensated loop, including all plant dynamics and any delay.

9. A Frequency-Domain Design Workflow

A reliable workflow begins with model validation. The plant model should identify relevant poles, zeros, delays, resonant modes, actuator dynamics, sensor dynamics, and uncertainty. A Bode plot of the uncompensated plant reveals low-frequency gain, crossover behavior, and high-frequency limitations.

The designer then specifies targets such as:

• tracking accuracy over a stated frequency range;
• disturbance rejection over a stated frequency range;
• desired bandwidth;
• minimum phase and gain margins;
• maximum sensitivity peak;
• allowable measurement-noise amplification;
• actuator amplitude and rate limits.

Controller parameters are adjusted using Bode or Nichols loop shaping. The Nyquist plot is then checked for encirclements and near crossings of the critical point. Closed-loop functions \(S\), \(T\), and relevant transfer functions from disturbances and noise are evaluated. Finally, time-domain simulations, nonlinear simulations, hardware-in-the-loop tests, and experimental frequency-response measurements are used for validation.

A design should not be accepted solely because it has a nominally positive phase margin. Model uncertainty can move poles and zeros, saturating actuators can invalidate linear analysis, and sampling or computational delay can reduce effective phase margin.

10. Applied Case: Speed Control of a DC Motor

Background

Consider an illustrative DC motor speed-control system. The motor drives a variable mechanical load. The control objective is accurate low-frequency speed regulation, moderate response speed, and limited sensitivity to tachometer noise.

Problem

An initial proportional controller produces acceptable steady-state behavior but exhibits a large speed error under constant load torque. Increasing proportional gain improves regulation but causes oscillatory response because the motor and load introduce an additional lag near the intended crossover frequency.

Analysis

The open-loop Bode plot shows that the proportional loop has a crossover near 18 rad/s and a phase margin of approximately 28°. The sensitivity function has a pronounced peak near crossover, indicating weak robustness. The complementary sensitivity function also transmits significant measurement noise above approximately 100 rad/s.

Approach and Solution

An integral action is added at low frequency to increase loop gain and reduce constant-load error. Because the integrator adds phase lag, a lead network is placed around the desired crossover. The lead network raises phase margin, while a high-frequency roll-off limits noise amplification. An anti-windup mechanism is included because the motor voltage saturates during large speed commands.

The resulting design is evaluated through the following checks:

1. The Nyquist plot does not encircle the critical point and retains a suitable separation from it.

2. The phase margin exceeds the specified minimum under nominal conditions.

3. The sensitivity peak is reduced compared with the proportional design.

4. The low-frequency gain is sufficiently high for load-disturbance rejection.

5. The high-frequency complementary sensitivity is reduced to protect the tachometer channel.

6. Nonlinear simulations with voltage saturation do not produce integrator windup or sustained oscillation.

Outcome

The compensated controller achieves improved load regulation and a faster, better-damped response than the original proportional design. The bandwidth increases moderately rather than aggressively, preserving robustness and reducing noise amplification. Performance remains acceptable when motor inertia and load torque vary within the specified operating range.

Learning Points

The case illustrates that controller design is a compromise. Integral action improves low-frequency performance but consumes phase margin; lead compensation restores phase but may increase high-frequency control effort; and bandwidth must be selected together with noise, uncertainty, and actuator constraints. Frequency-domain plots provide a unified way to see these trade-offs before implementation.

11. Challenges, Limitations, and Contemporary Practice

Frequency-domain analysis assumes linear time-invariant behavior for the model being analyzed. Strong nonlinearities, friction, backlash, hysteresis, saturation, dead zones, and mode switching may invalidate conclusions drawn from a small-signal model. Linearization remains useful, but it must be complemented by nonlinear analysis and testing.

Time delays and sampling are major practical limitations. A digital controller introduces computation and zero-order-hold effects, while communication networks may introduce variable delay and packet loss. The discrete-time frequency response should be analyzed using \(z=e^{j\Omega}\), where \(\Omega\) is digital frequency in radians per sample, and sampling frequency, aliasing, and quantization must be considered.

Nonminimum-phase zeros impose fundamental limitations. A right-half-plane zero can cause inverse-response behavior and restrict achievable bandwidth and sensitivity reduction. It is generally unsafe to cancel such a zero with a controller pole because the cancellation is fragile and may hide unstable internal dynamics.

Robust design uses uncertainty descriptions, structured singular-value analysis, \(H_\infty\) methods, or quantitative feedback theory when classical margins are insufficient. Data-driven frequency-response identification can support loop shaping without relying entirely on a high-order first-principles model. However, identified data require careful excitation, noise treatment, repeatability checks, and safe operating procedures.

AI-assisted control design can help estimate models, tune controllers, detect resonances, and search parameter spaces. Nevertheless, AI-generated designs must be constrained by stability analysis, explainable performance requirements, verification data, cybersecurity controls, and human engineering approval. Safety-critical systems require traceability, validation, fail-safe behavior, and compliance with applicable industry standards. Frequency-domain analysis is therefore a technical method within a broader governance and assurance process, not a substitute for it.

Best Practices

• State the feedback sign convention and frequency units explicitly.
• Use exact frequency-response calculations near crossover rather than relying only on asymptotes.
• Inspect the complete Nyquist plot when open-loop poles are unstable or dynamics are unusual.
• Report gain margin, phase margin, crossover frequencies, bandwidth, and sensitivity peaks together.
• Include delays, sensor dynamics, actuator dynamics, and uncertainty in the loop model.
• Avoid fragile cancellation of unstable or poorly known poles and zeros.
• Check both nominal and worst-case models.
• Validate frequency-domain designs with nonlinear simulation and experimental tests.
• Account for saturation, rate limits, anti-windup, quantization, and sampling.
• Treat margins as indicators of robustness, not as the sole definition of acceptable performance.

Chapter Summary

Frequency-domain analysis represents a control system through its response to sinusoidal signals. Bode plots show magnitude and phase separately and are effective for identifying poles, zeros, crossover frequencies, and controller-shaping opportunities. Nyquist plots provide a stability test based on encirclements of the critical point, with the number of open-loop right-half-plane poles explicitly included. Nichols charts combine open-loop magnitude and phase with closed-loop performance contours.

Gain margin and phase margin quantify distance from instability, while bandwidth and resonance describe speed and peaking. The sensitivity function \(S=1/(1+L)\) characterizes disturbance and uncertainty attenuation, whereas the complementary sensitivity \(T=L/(1+L)\) describes tracking and measurement-noise transmission in common configurations. Their identity \(S+T=1\) expresses a fundamental performance trade-off.

Loop shaping designs the controller by distributing gain and phase across frequency. Integral action improves low-frequency accuracy, lead compensation restores phase near crossover, lag compensation improves low-frequency gain with limited crossover change, and high-frequency roll-off limits noise and actuator demand. Robust design requires consideration of delays, nonminimum-phase dynamics, uncertainty, nonlinearities, saturation, sampling, and safety requirements. Frequency-domain methods are most powerful when combined with time-domain, nonlinear, and experimental validation.

Key Terms

Frequency response: Steady-state sinusoidal input-output relationship as a function of frequency.

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

Nyquist plot: Complex-plane plot of the loop frequency response used for stability analysis.

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

Loop transfer function: Product of controller, plant, and feedback-path transfer functions.

Gain crossover frequency: Frequency at which loop magnitude equals unity or 0 dB.

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

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 satisfies a specified magnitude criterion.

Resonance: Frequency-dependent amplification associated with lightly damped dynamics.

Sensitivity function: \(S=1/(1+L)\), describing sensitivity to disturbances and uncertainty.

Complementary sensitivity: \(T=L/(1+L)\), commonly describing tracking and noise transmission.

Loop shaping: Controller design by shaping loop magnitude and phase across frequency.

Lead compensator: Controller network that supplies positive phase over a selected frequency range.

Lag compensator: Controller network that increases low-frequency gain relative to crossover gain.

Minimum-phase system: System whose stable poles and zeros permit a stable inverse under appropriate conditions.

Nonminimum-phase zero: Right-half-plane zero or analogous discrete-time zero that limits performance.

Waterbed effect: Trade-off in which sensitivity reduction in one frequency region causes increase elsewhere.

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

Review Questions

Short Answer Questions

1. Define gain crossover frequency and phase crossover frequency.

2. What is the relationship between the sensitivity and complementary sensitivity functions?

3. State the Nyquist encirclement condition using \(N\), \(P\), and \(Z\).

4. Why does a time delay reduce phase margin without changing magnitude?

5. State one principal advantage of a Nichols chart over separate Bode plots.

Long Answer Questions

1. Explain how poles and zeros determine the asymptotic magnitude and phase characteristics of a Bode plot.

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

3. Explain the Nyquist criterion for systems with right-half-plane open-loop poles.

4. Compare proportional, integral, lead, lag, and lead-lag compensation in frequency-domain design.

5. Discuss the trade-offs among bandwidth, phase margin, disturbance rejection, measurement noise, and actuator effort.

Analytical / Application Questions

1. For \(L(s)=10(1+s/10)/[s(1+s/100)]\), determine the approximate slope changes in the Bode magnitude plot and calculate the magnitude and phase at \(\omega=1\) rad/s.

2. A loop has gain crossover phase \(-140°\) and phase-crossover magnitude \(-6\) dB. Calculate its phase margin and gain margin, and comment on the meaning of these values.

3. Design an initial lead compensator providing approximately 30° maximum phase lead with its maximum phase contribution centered at a desired crossover of 20 rad/s. State the resulting zero and pole frequencies using the standard lead-network relations, and identify what further checks are required before implementation.

Multiple Choice Questions with Answers

Question 1

At the gain crossover frequency, the loop magnitude is:

A. -180 dB

B. 0 dB

C. +3 dB

D. Equal to the phase margin

Correct Answer: B. 0 dB

Question 2

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

A. L/(1-L)

B. 1+L

C. 1/(1+L)

D. 1/L

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

Question 3

The Nyquist critical point for a conventional negative-feedback loop is:

A. 0+j0

B. 1+j0

C. -1+j0

D. -j1

Correct Answer: C. -1+j0

Question 4

A time delay e^{-sT} contributes which frequency-response effect?

A. Magnitude increase and zero phase

B. Unit magnitude and phase lag -ωT

C. Magnitude decrease and phase lead

D. A fixed phase lead independent of frequency

Correct Answer: B. Unit magnitude and phase lag -ωT

Question 5

A first-order pole contributes, at frequencies well above its break frequency, approximately:

A. +20 dB/decade

B. 0 dB/decade

C. -20 dB/decade

D. -40 dB/decade

Correct Answer: C. -20 dB/decade

Question 6

If the loop phase at gain crossover is -135°, the phase margin is:

A. -45°

B. 45°

C. 135°

D. 315°

Correct Answer: B. 45°

Question 7

For a stable open-loop system with no right-half-plane poles, Nyquist closed-loop stability generally requires:

A. One clockwise encirclement of -1

B. No encirclement of -1, provided the plot does not pass through it

C. Two counterclockwise encirclements of -1

D. The plot to pass through -1

Correct Answer: B. No encirclement of -1, provided the plot does not pass through it

Question 8

Which controller action primarily increases low-frequency loop gain and can remove step steady-state error?

A. Derivative-only action

B. Integral action

C. High-frequency roll-off

D. A measurement filter alone

Correct Answer: B. Integral action

Question 9

A lead compensator is primarily used to:

A. Add positive phase near a selected frequency range

B. Create an integrator at all frequencies

C. Guarantee zero actuator effort

D. Remove every plant pole exactly

Correct Answer: A. Add positive phase near a selected frequency range

Question 10

The identity relating sensitivity S and complementary sensitivity T in unity feedback is:

A. S-T=0

B. ST=1

C. S+T=1

D. S/T=1+L

Correct Answer: C. S+T=1