7. Time-Domain and Frequency-Domain Analysis

Introduction and Context

Dynamic systems in mechanical engineering are judged not only by their physical construction but also by how they respond to commands, disturbances, and changing operating conditions. A servo-driven positioning stage, for example, must reach a commanded position rapidly, with little overshoot, while rejecting load disturbances and remaining stable despite changes in stiffness, friction, and payload. These requirements cannot be assessed adequately from a static model alone.

Time-domain analysis describes how a system evolves with time. It reveals rise time, settling time, overshoot, oscillation, and steady-state error. Frequency-domain analysis examines how sinusoidal inputs are amplified, attenuated, and phase-shifted over a range of frequencies. It provides practical information about bandwidth, robustness, resonance, noise transmission, and stability margins. The two viewpoints describe the same system from complementary perspectives.

For a linear time-invariant system, the transfer function is commonly written as

G(s) = Y(s)/U(s) = K [(s-z1)(s-z2)...]/[(s-p1)(s-p2)...],

where the zi are zeros, the pi are poles, and K is a gain constant. Although transfer functions are valuable for input-output analysis, state-space models are needed to examine internal dynamics, controllability, and observability. This chapter develops the principal analytical tools used to assess these properties and to support informed controller design.

Learning Objectives

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

1. Interpret poles, zeros, system order, controllability, and observability.

2. Assess stability using time-domain, pole-location, Routh-Hurwitz, root-locus, Bode, and Nyquist methods.

3. Relate pole locations to transient and steady-state response.

4. Calculate or interpret steady-state error for standard inputs.

5. Construct and use root-locus and frequency-response plots.

6. Explain gain margin, phase margin, bandwidth, sensitivity, and disturbance rejection.

7. Select an appropriate analysis method for a mechanical control problem.

8. Recognize modelling assumptions, practical limitations, and common analytical errors.

The Dynamic Model: Poles, Zeros, and Order

A pole is a value of s that makes the denominator of a transfer function zero. Poles determine the natural modes of a system and therefore dominate its free response. A real pole at s = -a produces a term proportional to exp(-at), while a complex conjugate pair at s = -sigma ± j omega_d produces a decaying oscillation. Poles in the left half of the complex plane correspond to decaying modes; poles in the right half-plane correspond to growing modes. Poles on the imaginary axis produce persistent oscillation in the ideal model.

A zero is a value of s that makes the numerator zero. Zeros shape the forced response and may suppress or amplify particular dynamic features. A right-half-plane zero is non-minimum phase: it can produce an initial response in the direction opposite to the final response and imposes limits on achievable speed and bandwidth. Pole-zero cancellation should be treated cautiously. Exact cancellation may conceal an unstable or weakly damped internal mode that remains physically present.

The order of a system is normally the number of independent energy-storage elements, or equivalently the number of independent state variables in a minimal realization. Masses, inertias, springs, capacitors, and inductors are typical sources of stored energy. The degree of the denominator of a minimal transfer function indicates system order. A high-order mechanical system may sometimes be approximated by a dominant second-order model, but the neglected modes must be sufficiently fast, sufficiently damped, and weakly coupled to the controlled output.

For a standard second-order denominator,

s² + 2 zeta omega_n s + omega_n²,

omega_n is the natural frequency and zeta is the damping ratio. For 0 < zeta < 1, the poles are complex and the system is underdamped. The damped natural frequency is omega_d = omega_n sqrt(1 - zeta²). Increasing zeta generally reduces oscillation but may increase response time when excessive damping is introduced.

State-Space Properties: Controllability and Observability

A continuous-time state-space model is

x_dot = A x + B u,

y = C x + D u.

The pair (A, B) is controllable if an input can move the state from any initial state to any desired final state within finite time. For an n-state system, the controllability matrix is

C_o = [B AB A²B ... A^(n-1)B].

The system is controllable if this matrix has rank n. In mechanical terms, an uncontrolled flexible mode or an actuator that does not couple to a particular mode may create an uncontrollable state. Such a mode cannot be assigned arbitrarily by state-feedback control.

The pair (A, C) is observable if the internal state can be reconstructed from measured inputs and outputs. The observability matrix is

O = [C; CA; CA²; ...; CA^(n-1)],

and observability requires rank n. A flexible mode may be physically present but invisible to a sensor located at a node of that mode. An unobservable unstable mode is especially serious because the controller cannot detect its growth.

Controllability and observability are distinct. A system may be controllable but poorly observable, or observable but difficult to control. In numerical work, rank tests should be supplemented by conditioning or singular-value analysis, because a mathematically full-rank matrix may still indicate very weak practical control or measurement authority.

Stability and the Meaning of Response

BIBO stability requires every bounded input to produce a bounded output. For a proper rational transfer function with no pole-zero complications, this requires all poles to lie strictly in the open left half-plane. Internal stability is stronger: all internal modes must be stable, including modes hidden by transfer-function cancellations.

The Routh-Hurwitz criterion provides an algebraic test without explicitly calculating roots. For a characteristic polynomial, the necessary conditions include positive coefficients, while the complete criterion requires all elements in the first column of the Routh array to have the same sign. A sign change indicates a right-half-plane pole; the number of sign changes equals the number of such poles. Special cases, such as a row of zeros, indicate symmetric roots or imaginary-axis roots and require the standard auxiliary-polynomial procedure.

Transient response is commonly measured using rise time, peak time, maximum overshoot, and settling time. For a lightly damped second-order system, the percentage overshoot is approximately

M_p = 100 exp[-pi zeta / sqrt(1-zeta²)] percent,

and the two-percent settling time is approximately

t_s ≈ 4/(zeta omega_n).

These expressions are useful design estimates, not universal formulas for high-order systems. Dominant poles, additional zeros, delays, saturation, friction, and nonlinearities can substantially change the actual response.

Steady-state error is the limiting difference between the reference and output as time tends to infinity. In unity feedback, the error transfer function is E(s) = R(s)/[1 + L(s)], where L(s) is the loop transfer function. The system type is the number of pure integrators in L(s). For a stable unity-feedback system, a type 0 system has finite error to a step, a type 1 system has zero step error but finite ramp error, and a type 2 system has zero step and ramp error but finite parabolic-input error. Increasing type improves tracking of polynomial references but can reduce robustness and increase the risk of instability.

Root Locus as a Design Map

The root locus shows how the closed-loop poles move as a real gain K varies. For unity feedback with open-loop transfer function G(s)H(s), the characteristic equation is

1 + K G(s)H(s) = 0.

A point lies on the root locus when the angle condition is satisfied:

angle[G(s)H(s)] = (2k + 1)180 degrees.

The magnitude condition then gives the gain required at that point. Locus branches begin at open-loop poles and terminate at open-loop zeros or at infinity. Asymptote angles and centroid provide a rapid sketch, while breakaway points and imaginary-axis crossings refine the analysis.

Root locus is particularly useful when a controller gain is the principal design variable. It shows the trade-off between speed and damping: moving poles farther left generally improves speed, but excessive gain may drive branches toward the right half-plane. Compensators add poles and zeros to reshape the locus. A lead compensator can improve damping and speed, whereas a lag compensator can improve low-frequency accuracy with a smaller effect on transient dynamics.

Frequency-Domain Analysis

For a sinusoidal input u(t) = U sin(omega t), a stable linear system reaches a sinusoidal steady state at the same frequency. If G(j omega) is the frequency response, the output amplitude is multiplied by |G(j omega)| and shifted by angle G(j omega). A frequency-response plot therefore describes both magnitude and phase as functions of frequency.

A Bode plot uses logarithmic frequency scales. Magnitude is expressed in decibels as 20 log10 |G(j omega)|, and phase is expressed in degrees. Poles contribute a slope change of approximately -20 dB per decade each, while zeros contribute +20 dB per decade each, subject to their locations and interactions. Resonant peaks reveal lightly damped modes. Low-frequency gain is closely associated with tracking and disturbance rejection; high-frequency gain is associated with noise amplification and unmodelled dynamics.

The gain crossover frequency is where the magnitude of the loop transfer function equals unity, or 0 dB. The phase margin is the additional phase lag required at this frequency to reach -180 degrees. The phase crossover frequency is where loop phase equals -180 degrees. The gain margin is the reciprocal of the magnitude at this frequency, often expressed in decibels. Larger margins generally indicate greater robustness, although margins are not complete guarantees for strongly nonlinear, time-varying, or poorly modelled systems.

Bandwidth is the frequency range over which a closed-loop system tracks its input with acceptable attenuation and phase lag. A wider bandwidth can improve speed and disturbance rejection within the controlled range, but it also increases sensitivity to sensor noise, actuator limitations, sampling effects, and neglected flexible modes. Bandwidth must therefore be selected in relation to mechanical resonances and the available actuator authority.

The Nyquist plot maps L(j omega) in the complex plane as frequency varies. The encirclement of the critical point -1 + j0, together with the number of open-loop right-half-plane poles, determines closed-loop stability through the Nyquist criterion. Unlike a simple pole test, the Nyquist method handles open-loop unstable systems and provides a geometric interpretation of robustness. It is especially valuable when delays or complicated dynamics make algebraic methods inconvenient.

Sensitivity and Disturbance Rejection

For a unity-feedback loop, the sensitivity function is

S(s) = 1/[1 + L(s)],

and the complementary sensitivity function is

T(s) = L(s)/[1 + L(s)] = 1 - S(s).

Low sensitivity at a frequency means that plant uncertainty and disturbances entering before the loop are attenuated there. High complementary sensitivity indicates good reference tracking but also increased transmission of measurement noise and high-frequency unmodelled dynamics. The fundamental trade-off is that reducing S over a broad frequency range tends to increase T elsewhere; this is expressed by the so-called waterbed effect in many stable minimum-phase systems.

A disturbance applied at the plant input is transferred approximately through S(s)P(s), whereas measurement noise is transmitted through T(s), with sign and location depending on the block diagram. Integral action provides strong rejection of constant disturbances but may produce windup when actuators saturate. Anti-windup schemes, feedforward compensation, disturbance observers, and two-degree-of-freedom controllers are practical methods for improving performance without relying exclusively on high loop gain.

An Illustrative Applied Case: Precision Positioning of a Flexible Stage

Background

Consider an illustrative electromechanical positioning stage driven by a motor through a compliant coupling. The stage must follow position commands accurately while rejecting a nearly constant load force. The motor encoder measures rotor position, but the stage contains a flexible mode that is not directly measured.

Situation and Analysis

A rigid-body approximation suggests an integrator in the position plant, and proportional feedback gives nonzero step error to a constant load. Adding integral action removes the steady-state position error. However, frequency-response testing reveals a resonance near the flexible-mode frequency. Increasing integral and proportional gains improves low-frequency tracking but raises the resonant peak and reduces phase margin. The stage begins to overshoot and may oscillate after rapid commands.

A state-space model is formed using motor position, stage position, and their velocities. The controllability test shows that the actuator couples to both rigid and flexible modes, but the observability test using only the motor encoder indicates weak observability of stage motion. This warns the designer that an apparently stable motor response may conceal substantial stage vibration.

Approach and Solution

The design proceeds in stages. First, a root-locus study identifies gain ranges that maintain adequate damping. Next, a lead compensator is introduced to supply phase near the crossover frequency. A notch filter is placed around the measured flexible resonance, but its bandwidth is kept narrow to avoid excessive phase distortion. Integral action is retained at low frequency, with anti-windup protection for actuator saturation. A second-degree-of-freedom reference filter limits command excitation near the resonance while preserving disturbance rejection.

Bode and Nyquist plots are then checked using plant-parameter variations. The selected design has a lower crossover frequency than the fastest nominal design, but it maintains useful phase and gain margins when stiffness and payload vary. Time simulations with step, ramp, and load-disturbance inputs confirm acceptable settling and small steady-state error.

Outcome and Learning Points

The case illustrates that the highest nominal bandwidth is not necessarily the best design. Pole locations explain the observed oscillation, frequency-response plots identify the resonance and robustness limits, and controllability-observability analysis reveals whether the selected actuator and sensor arrangement can control and detect the relevant modes. Good control design balances tracking, disturbance rejection, noise sensitivity, actuator capability, and model uncertainty.

Practical Design Workflow and Common Errors

A sound analysis begins with a physically justified model and clearly defined input, output, disturbance, and measurement locations. The analyst should then determine poles, zeros, order, and possible cancellations; check controllability and observability; assess open- and closed-loop stability; and compare time-domain and frequency-domain results. Validation against experimental data is essential when friction, backlash, saturation, dead zones, or flexible structures are important.

Common mistakes include interpreting open-loop stability as proof of closed-loop stability, using second-order formulas for a strongly high-order response, ignoring right-half-plane zeros, treating a pole-zero cancellation as physically harmless, and quoting gain or phase margins without specifying the loop transfer function. Another frequent error is raising bandwidth without checking sensor noise, sampling frequency, actuator saturation, or unmodelled resonances. Numerical pole calculations should also be checked for scaling and conditioning.

Professional and Emerging Practice

Modern mechanical-control workflows combine symbolic modelling, numerical simulation, frequency-response identification, and hardware-in-the-loop testing. Digital control introduces sampling, computation delay, quantization, and discretization effects. Data-driven frequency-response estimation can identify resonances that are absent from an idealized model, while robust-control tools evaluate structured uncertainty rather than relying on a single nominal plant. Automated tuning and optimization can accelerate design, but engineering judgement remains necessary to define constraints, reject unsafe solutions, and interpret uncertain data.

Professional practice also requires traceability. Stability claims should specify the model and operating point; experimental tests should be conducted within safe actuator and load limits; and automated or adaptive controllers should include supervision, fault detection, and fallback modes. In safety-critical machinery, compliance with applicable machinery, functional-safety, and electromagnetic-compatibility requirements is part of the design problem, not an afterthought.

Best-Practice Guidelines

• Define performance requirements in both time and frequency domains.
• Examine internal modes, not only the input-output transfer function.
• Use controllability and observability tests before selecting actuator and sensor locations.
• Preserve adequate gain and phase margins under realistic parameter variations.
• Keep bandwidth below poorly modelled structural resonances unless those modes are deliberately controlled.
• Use integral action with anti-windup and verify disturbance-rejection requirements.
• Validate analytical predictions through simulation, identification, and controlled experiments.
• Report assumptions, uncertainty, operating conditions, and the exact loop used for margin calculations.

Chapter Summary

Time-domain analysis explains how dynamic systems rise, settle, oscillate, and track commands. Frequency-domain analysis explains how they respond across frequency and provides direct measures of resonance, bandwidth, robustness, and noise transmission. Poles determine natural modes and stability, while zeros shape forced response and may impose fundamental performance limits. State-space controllability and observability establish whether modes can be influenced and detected.

Root locus describes closed-loop pole movement as gain changes. Bode plots provide magnitude and phase information, while Nyquist plots assess stability through encirclement of the critical point. Gain margin, phase margin, sensitivity, and complementary sensitivity connect analytical results with practical robustness and disturbance rejection. Effective mechanical-control design uses these tools together, validates them against physical limitations, and treats actuator saturation, sensor noise, uncertainty, delays, and hidden modes as integral parts of the engineering problem.

Key Terms

Pole: A root of the transfer-function denominator that determines a natural system mode.

Zero: A root of the transfer-function numerator that shapes forced response.

System order: Number of independent dynamic states or energy-storage elements in a minimal model.

Natural frequency: Undamped oscillation frequency associated with a second-order mode.

Damping ratio: Dimensionless measure of modal damping relative to critical damping.

Controllability: Ability of the input to move all system states.

Observability: Ability to reconstruct internal states from measured inputs and outputs.

BIBO stability: Property in which every bounded input produces a bounded output.

Transient response: Time-varying response before steady state is reached.

Steady-state error: Final difference between reference and output.

System type: Number of integrators in the loop transfer function.

Root locus: Path followed by closed-loop poles as a gain parameter varies.

Bode plot: Logarithmic magnitude and phase representation of frequency response.

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

Gain margin: Multiplicative gain increase required to reach the stability boundary.

Phase margin: Additional phase lag required to reach the -180-degree condition at gain crossover.

Bandwidth: Frequency range over which acceptable closed-loop response is maintained.

Sensitivity function: S = 1/(1 + L), measuring closed-loop sensitivity to disturbances and uncertainty.

Complementary sensitivity: T = L/(1 + L), associated with tracking and noise transmission.

Non-minimum-phase zero: A zero in the right half-plane that limits achievable response.

Review Questions

Short Answer Questions

1. Distinguish between a pole and a zero in a transfer function.

2. What do controllability and observability mean physically?

3. Define gain crossover frequency and phase margin.

4. What is meant by system type in a unity-feedback system?

5. State the relationship between sensitivity and complementary sensitivity.

Long Answer Questions

1. Explain how pole locations determine stability and transient response, including the role of dominant poles.

2. Describe the construction and interpretation of a root-locus plot.

3. Explain how Bode plots are used to assess bandwidth, resonance, and robustness.

4. Compare Routh-Hurwitz, root-locus, Bode, and Nyquist methods for stability analysis.

5. Discuss controllability and observability in the design of a mechanical servo system.

Analytical / Application Questions

1. A second-order system has damping ratio 0.5 and natural frequency 10 rad/s. Estimate its percentage overshoot and two-percent settling time.

2. A unity-feedback loop has one integrator in its open-loop transfer function. Determine the expected steady-state error characteristics for step and ramp references, assuming stability.

3. A flexible positioning stage shows a sharp resonance close to its gain crossover frequency. Propose a revised analysis and control strategy, explaining the trade-offs among bandwidth, phase margin, noise, and disturbance rejection.

22. Multiple Choice Questions with Answers

Question 1

Which statement most accurately describes the role of poles in a linear time-invariant system?

A. They determine only the static gain.

B. They determine the natural modes and strongly influence stability.

C. They always eliminate steady-state error.

D. They represent only sensor dynamics.

Correct Answer: B. They determine the natural modes and strongly influence stability.

Question 2

For an n-state system, controllability requires which condition?

A. The observability matrix has rank one.

B. All eigenvalues of A are positive.

C. The controllability matrix has rank n.

D. The transfer function has no zeros.

Correct Answer: C. The controllability matrix has rank n.

Question 3

A stable unity-feedback type-1 system has, ideally, what steady-state error for a step reference?

A. Infinite error

B. A nonzero constant error

C. Zero error

D. An oscillating error of fixed amplitude

Correct Answer: C. Zero error

Question 4

A right-half-plane zero generally causes which characteristic?

A. A non-minimum-phase response and a limitation on achievable speed.

B. Guaranteed closed-loop stability.

C. Removal of all resonances.

D. Infinite low-frequency gain.

Correct Answer: A. A non-minimum-phase response and a limitation on achievable speed.

Question 5

At the gain crossover frequency, the magnitude of the loop transfer function is:

A. -180 dB

B. 0 dB

C. +3 dB

D. Equal to the phase margin.

Correct Answer: B. 0 dB

Question 6

Increasing closed-loop bandwidth without considering unmodelled flexible modes is most likely to:

A. Reduce all sensor noise automatically.

B. Excite neglected resonances and reduce robustness.

C. Guarantee zero actuator effort.

D. Make observability unnecessary.

Correct Answer: B. Excite neglected resonances and reduce robustness.

Question 7

Which function is the sensitivity function for a unity-feedback loop?

A. S(s) = 1 + L(s)

B. S(s) = L(s)

C. S(s) = 1/[1 + L(s)]

D. S(s) = 1/L(s)

Correct Answer: C. S(s) = 1/[1 + L(s)]

Question 8

In a Nyquist analysis, the critical point used for closed-loop stability assessment is:

A. 0 + j0

B. 1 + j0

C. -1 + j0

D. -j1

Correct Answer: C. -1 + j0

Question 9

A lightly damped complex pole pair will usually produce which time-domain feature?

A. A decaying oscillation with possible overshoot.

B. A purely static response.

C. An immediate zero output for every input.

D. A response independent of natural frequency.

Correct Answer: A. A decaying oscillation with possible overshoot.

Question 10

Why is exact pole-zero cancellation potentially dangerous in a physical mechanical system?

A. It always lowers the system order physically.

B. It can hide an unstable or weakly damped internal mode.

C. It guarantees infinite phase margin.

D. It prevents any controller implementation.

Correct Answer: B. It can hide an unstable or weakly damped internal mode.