Time-Domain Analysis and Performance

Introduction

Time-domain analysis examines how a control system responds as time progresses after an input is applied. It is especially useful when specifications are expressed in terms of speed, oscillation, overshoot, settling, and accuracy. Although frequency-domain methods are essential for robustness and bandwidth analysis, time-domain measures often correspond more directly to engineering requirements: a positioning system must reach its commanded location quickly, a flight-control surface must not oscillate excessively, and a communication tracking loop must maintain small steady-state error.

This chapter develops the response of first- and second-order systems, establishes performance measures, and extends the analysis to dominant poles, zeros, higher-order modes, steady-state error, and integral performance indices. Particular attention is given to the assumptions behind commonly used formulas. Approximate expressions are valuable for design, but they must not be applied outside their validity range.

Learning Objectives

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

1. Derive and interpret first- and second-order time responses.

2. Relate pole locations to damping, speed, oscillation, and overshoot.

3. Calculate rise time, peak time, maximum overshoot, and settling time.

4. Distinguish exact settling-time calculations from engineering approximations.

5. Explain the effects of zeros, dominant poles, and neglected higher-order modes.

6. Evaluate steady-state error using the final-value theorem under correct stability conditions.

7. Calculate integral performance indices such as IAE, ISE, and ITAE.

8. Select time-domain specifications appropriate to practical control applications.

1. Time-Domain Models and Response Components

For a linear time-invariant system with zero initial conditions,

\[

C(s)=G(s)R(s),

\]

where \(R(s)\) is the input transform, \(G(s)\) is the transfer function, and \(C(s)\) is the output transform. The unit-step input is \(R(s)=1/s\), the unit-ramp input is \(R(s)=1/s^2\), and the unit-impulse input is \(R(s)=1\).

The response may be written as

\[

c(t)=c_{tr}(t)+c_{ss}(t),

\]

where \(c_{tr}(t)\) is the transient component and \(c_{ss}(t)\) is the steady-state component. For a stable system, the transient component tends to zero as \(t\to\infty\). In a rational transfer function, transient behavior is governed mainly by poles, while zeros shape amplitudes, initial slopes, overshoot, and sometimes the apparent rise time.

A pole at \(p=\sigma+j\omega\) contributes a mode proportional to \(e^{\sigma t}\cos(\omega t+\phi)\). Thus, poles in the open left half-plane decay; poles on the imaginary axis produce sustained modes; and poles in the right half-plane produce growing modes.

2. First-Order Systems

A standard first-order transfer function is

\[

G(s)=\frac{K}{\tau s+1}, \qquad \tau>0,

\]

where \(K\) is the static gain and \(\tau\) is the time constant. For a unit step,

\[

C(s)=\frac{K}{s(\tau s+1)},

\]

and inversion gives

\[

c(t)=K\left(1-e^{-t/\tau}\right), \qquad t\geq0.

\]

The normalized response reaches approximately 63.2% of its final value at \(t=\tau\), 86.5% at \(2\tau\), 95.0% at \(3\tau\), and 98.2% at \(4\tau\). The exact time to reach a fraction \(q\), with \(0<q<1\), is

\[

t_q=-\tau\ln(1-q).

\]

For a 10–90% rise time,

\[

t_r=t_{0.9}-t_{0.1}=\tau\ln 9\approx2.197\tau.

\]

The exact 2% settling time is obtained from \(e^{-t/\tau}\leq0.02\):

\[

t_s^{(2\%)}= -\tau\ln(0.02)=3.912\tau.

\]

The 5% value is \(t_s^{(5\%)}=2.996\tau\). These values assume a positive monotonic step response and a final value of \(K\).

Example 1. For \(G(s)=5/(0.4s+1)\), \(K=5\), \(\tau=0.4\) s. The 10–90% rise time is \(2.197(0.4)=0.879\) s, and the 2% settling time is \(3.912(0.4)=1.565\) s. The output at one second is \(5(1-e^{-2.5})=4.590\), approximately 91.8% of the final value.

3. Standard Second-Order Systems

The normalized second-order model is

\[

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

\]

where:

• \(K\) is the DC gain;
• \(\omega_n>0\) is the undamped natural frequency in rad/s;
• \(\zeta\) is the damping ratio.

The poles are

\[

s_{1,2}=-\zeta\omega_n\pm\omega_n\sqrt{\zeta^2-1}.

\]

For \(0<\zeta<1\), define the damped natural frequency

\[

\omega_d=\omega_n\sqrt{1-\zeta^2}.

\]

The poles are \(-\zeta\omega_n\pm j\omega_d\). The real part, \(-\zeta\omega_n\), controls the exponential envelope; \(\omega_d\) controls oscillation frequency.

For a unit-step input and zero initial conditions, the underdamped response is

\[

c(t)=K\left[1-e^{-\zeta\omega_n t}\left(\cos\omega_d t+\frac{\zeta}{\sqrt{1-\zeta^2}}\sin\omega_d t\right)\right].

\]

Equivalently, using \(\phi=\tan^{-1}(\sqrt{1-\zeta^2}/\zeta)\),

\[

c(t)=K\left[1-\frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}}\sin(\omega_d t+\phi)\right].

\]

These formulas apply only to the standard second-order form with no additional influential zeros.

Damping classifications

• \(\zeta=0\): undamped; poles lie on the imaginary axis and the step response does not settle.
• \(0<\zeta<1\): underdamped; oscillatory response with overshoot.
• \(\zeta=1\): critically damped; repeated real pole and no oscillation.
• \(\zeta>1\): overdamped; two distinct negative real poles and no oscillation.

Critical damping does not mean the fastest response under every possible criterion or actuator constraint. It means the boundary between oscillatory and nonoscillatory standard second-order responses.

4. Time-Domain Specifications

For an underdamped standard second-order step response, common specifications are as follows.

Peak time

The first maximum occurs at

\[

t_p=\frac{\pi}{\omega_d}=\frac{\pi}{\omega_n\sqrt{1-\zeta^2}}.

\]

Maximum overshoot

The fractional maximum overshoot is

\[

M_p=\exp\left(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\right),

\]

and the percentage overshoot is \(100M_p\) percent. This formula is valid for \(0<\zeta<1\), for the standard second-order model and a step input.

Solving for damping ratio from a specified fractional overshoot gives

\[

\zeta=\frac{-\ln M_p}{\sqrt{\pi^2+(\ln M_p)^2}}.

\]

Settling time

The frequently used estimates

\[

t_s\approx\frac{4}{\zeta\omega_n}\quad(2\%), \qquad

t_s\approx\frac{3}{\zeta\omega_n}\quad(5\%)

\]

come from the exponential envelope. They are generally useful for underdamped systems when the oscillatory envelope is representative of the actual error. They are not exact universal formulas, and the 2% approximation is not appropriate for critical or overdamped responses without checking the exact response.

Rise time

Rise time must be defined. For an underdamped standard second-order system, the 0–100% first-crossing rise time is

\[

t_r=\frac{\pi-\phi}{\omega_d},

\qquad

\phi=\tan^{-1}\left(\frac{\sqrt{1-\zeta^2}}{\zeta}\right).

\]

The 10–90% rise time generally requires solving \(c(t)/K=0.1\) and \(0.9\) numerically because the response is not a single exponential. Approximate expressions should therefore identify their range of validity.

Exact settling-time computation

The exact settling time is the smallest time \(T\) such that

\[

|c(t)-c(\infty)|\leq \delta |c(\infty)| \quad \text{for every }t\geq T,

\]

where \(\delta=0.02\) or \(0.05\). For an underdamped response, this requires checking the oscillatory waveform or its successive extrema. An envelope crossing is conservative in some cases but is not necessarily the exact first time after which all later values remain within the band.

For critical damping, the normalized unit-step response is

\[

\frac{c(t)}{K}=1-(1+\omega_n t)e^{-\omega_n t}.

\]

The normalized error is therefore \(e(t)=(1+x)e^{-x}\), where \(x=\omega_n t\). Exact 2% settling satisfies

\[

(1+x)e^{-x}=0.02.

\]

The solution is \(x\approx5.834\), so

\[

t_s^{(2\%)}\approx\frac{5.834}{\omega_n},

\]

not \(4/\omega_n\). The exact solution can also be expressed using the Lambert W function:

\[

x=-1-W_{-1}(-0.02/e).

\]

The \(W_{-1}\) branch is required because the physically relevant solution has \(x>0\).

For overdamped systems, the response is a sum of two decaying exponentials. Exact settling is found by solving the response inequality; replacing the system with its slow pole is justified only when the fast mode has become negligible and its coefficient does not cause an early-band violation.

5. Worked Second-Order Design Example

Consider a standard second-order system with \(\zeta=0.5\) and \(\omega_n=4\) rad/s. Then

\[

\omega_d=4\sqrt{1-0.25}=3.464\text{ rad/s}.

\]

The peak time is

\[

t_p=\pi/3.464=0.907\text{ s}.

\]

The fractional overshoot is

\[

M_p=e^{-\pi(0.5)/\sqrt{0.75}}=0.163,

\]

or 16.3%. The 2% envelope estimate is

\[

t_s\approx4/(0.5\times4)=2.00\text{ s}.

\]

This is a design estimate, not a claim of exact settling. The exact boundary must be checked from the complete response if a strict specification is imposed.

Suppose instead that the overshoot must not exceed 10%. With \(M_p=0.10\),

\[

\zeta=\frac{2.3026}{\sqrt{\pi^2+2.3026^2}}\approx0.591.

\]

If a 2% envelope-based settling requirement is 1.5 s, then

\[

\omega_n\geq\frac{4}{\zeta t_s}

=\frac{4}{0.591(1.5)}\approx4.51\text{ rad/s}.

\]

The resulting pole pair is approximately \(-2.67\pm j3.63\) rad/s. The inequality is a preliminary design calculation; simulation or exact evaluation is required before certification.

6. Critical Case Study: Precision Servo

Illustrative case. A precision positioning servo is modeled by a critically damped standard second-order closed-loop transfer function with \(\omega_n=50\) rad/s. The design requirement is a 2% settling time no greater than 0.10 s.

Analysis

Because \(\zeta=1\), the normalized error is

\[

e(t)=(1+50t)e^{-50t}.

\]

The exact 2% condition is

\[

(1+50t_s)e^{-50t_s}=0.02.

\]

Using \(x=50t_s\), the positive solution is \(x\approx5.834\). Therefore,

\[

t_s=5.834/50=0.1167\text{ s}.

\]

Outcome

The servo does not meet the 0.10 s requirement, although the rough expression \(4/(\zeta\omega_n)=0.08\) s might incorrectly suggest that it does. The correct conclusion is that \(\omega_n\) must satisfy

\[

\omega_n\geq5.834/0.10=58.34\text{ rad/s}

\]

for the ideal critically damped model. In practice, actuator saturation, sensor noise, unmodeled flexible modes, computational delay, and parameter uncertainty may require additional design margin.

Learning points

1. The exponential-envelope approximation is not an exact critical-damping settling formula.

2. A strict requirement should be checked against the actual response inequality.

3. Increasing natural frequency improves speed but may increase control effort and sensitivity to unmodeled dynamics.

7. Zeros, Higher-Order Systems, and Dominant Poles

Pole-only intuition is incomplete. Consider

\[

G(s)=K\frac{1+\tau_zs}{(s+p_1)(s+p_2)}.

\]

A left-half-plane zero can increase the initial slope and reduce apparent rise time, but it can also increase overshoot. A right-half-plane zero produces nonminimum-phase behavior: the response may initially move opposite to its final direction, producing undershoot and limiting achievable speed. A zero near a dominant pole can substantially reshape the response, invalidating standard second-order formulas even when the pole pair itself appears familiar.

A higher-order closed-loop transfer function may be approximated by a dominant second-order model when the remaining poles are sufficiently faster. A practical, but not universal, guideline is that additional poles should have real parts several times farther left than the dominant pole pair. The required separation depends on residues, zeros, performance tolerance, and input type. A fast pole with a large residue, or a zero near the dominant poles, may remain important.

Model reduction should therefore compare:

• pole locations;
• residues or modal participation;
• step-response error between full and reduced models;
• peak and settling specifications;
• robustness under parameter variation.

For systems with lightly damped flexible modes, neglecting a non-dominant mode can be unsafe because resonance may be excited by command shaping or disturbances. Time-domain simulation, modal testing, and uncertainty analysis are appropriate postgraduate-level validation tools.

8. Steady-State Error and the Final-Value Theorem

For a unity-feedback system with forward path \(G(s)H(s)\), the error transform is

\[

E(s)=\frac{R(s)}{1+G(s)H(s)}.

\]

The system type is the number of pure integrators in the loop transfer function \(G(s)H(s)\). For unity feedback and a stable closed loop, a type-0 system has finite step error, 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 error, subject to gain and stability conditions.

The final-value theorem states

\[

\lim_{t\to\infty}e(t)=\lim_{s\to0}sE(s),

\]

provided the theorem applies. The precise condition is that all poles of \(sE(s)\) lie strictly in the open left half-plane. Equivalently, \(sE(s)\) must have no poles in the right half-plane or on the imaginary axis. A simple pole at the origin may occur in \(E(s)\) before multiplication by \(s\), for example because of a step input; it must not remain as a pole of \(sE(s)\). This condition is stronger and more precise than merely requiring closed-loop stability in an informal sense.

For a unity-feedback type-0 system with position constant

\[

K_p=\lim_{s\to0}G(s),

\]

unit-step error is

\[

e_{ss}=\frac{1}{1+K_p}.

\]

For a type-1 system, the velocity constant is

\[

K_v=\lim_{s\to0}sG(s),

\]

and unit-ramp error is \(e_{ss}=1/K_v\), when the final-value theorem conditions hold. For a type-2 system, the acceleration constant is

\[

K_a=\lim_{s\to0}s^2G(s),

\]

and unit-parabolic error is \(e_{ss}=1/K_a\).

These classifications concern steady-state tracking, not transient quality. Adding integral action may reduce step error but can reduce damping, increase overshoot, and create actuator windup.

9. Integral Performance Indices

For error signal \(e(t)\), common indices are:

\[

IAE=\int_0^\infty |e(t)|dt,

\]

\[

ISE=\int_0^\infty e^2(t)dt,

\]

\[

ITAE=\int_0^\infty t|e(t)|dt,

\]

and sometimes

\[

ITSE=\int_0^\infty te^2(t)dt.

\]

IAE weights all absolute error uniformly; ISE penalizes large errors strongly; ITAE and ITSE penalize errors that persist for a long time. The integrals must converge, so the error must decay sufficiently and the system must be stable for the selected input.

Example 2. For a first-order error \(e(t)=e^{-2t}\),

\[

IAE=\int_0^\infty e^{-2t}dt=0.5,

\]

\[

ISE=\int_0^\infty e^{-4t}dt=0.25,

\]

and

\[

ITAE=\int_0^\infty te^{-2t}dt=\frac{1}{2^2}=0.25.

\]

The units differ: IAE has error-times-time units, ISE has error-squared-times-time units, and ITAE has error-times-time-squared units. Consequently, indices should not be compared across differently scaled variables without normalization.

In optimization, minimizing ISE can produce a fast response with substantial initial control action, whereas minimizing ITAE often favors rapid elimination of long-lasting tails. No single index universally represents engineering quality; actuator limits, noise, robustness, and safety constraints must also be included.

10. Practical Design and Validation Considerations

A disciplined time-domain workflow is:

1. Define the input, output, operating point, and performance-band convention.

2. Obtain a validated model, including significant delays, zeros, and flexible modes.

3. Determine closed-loop poles and check stability.

4. Use standard formulas only when the model assumptions apply.

5. Compute exact or numerical settling when specifications are strict.

6. Examine sensitivity to parameter uncertainty and unmodeled dynamics.

7. Check actuator saturation, sensor noise, quantization, and sampling effects.

8. Validate using simulation and, where possible, experimental data.

Digital control introduces sampling and computation delay. A continuous-time approximation may be misleading if the sampling interval is not small relative to the dominant time constants. Saturation can create nonlinear behavior, including integrator windup and recovery delays. Safety-critical applications require traceable requirements, independent verification, documented model assumptions, and conservative margins.

AI-based tuning tools can search controller parameters or estimate models, but their recommendations require stability analysis, explainability of constraints, data-quality assessment, and human engineering approval. Automated optimization does not replace theorem conditions or physical validation.

Challenges and Limitations

Time-domain specifications are useful but incomplete. Two systems can have identical rise and settling times yet differ greatly in control effort, robustness, noise amplification, or disturbance rejection. Percent overshoot depends on the selected final value and can be misleading for responses with undershoot or multiple modes. The phrase “settling time” is ambiguous unless the tolerance band and permanence condition are specified. Finally, linear models cannot predict saturation, friction, backlash, hysteresis, or rate limits without additional nonlinear analysis.

Chapter Summary

Time-domain analysis connects pole locations and system structure to transient and steady-state performance. First-order responses are characterized by a time constant, while standard second-order responses depend on natural frequency and damping ratio. For underdamped systems, peak time and overshoot have useful closed-form expressions, whereas settling and rise-time formulas require attention to definitions and assumptions. The familiar \(4/(\zeta\omega_n)\) settling estimate is an envelope approximation, not a universal law; critical and overdamped systems should be checked from their exact responses. Zeros and higher-order modes can substantially alter performance, so dominant-pole approximations require validation. The final-value theorem is valid only when all poles of the multiplied transform, such as \(sE(s)\), lie strictly in the open left half-plane. Steady-state error constants describe tracking accuracy, while IAE, ISE, ITAE, and ITSE provide alternative measures of transient quality. Sound control design combines analytical formulas with simulation, uncertainty analysis, actuator consideration, and experimental validation.

Key Terms

Time constant: Parameter \(\tau\) determining the exponential speed of a first-order response.

Natural frequency: Undamped frequency \(\omega_n\) of a standard second-order model.

Damping ratio: Dimensionless parameter \(\zeta\) describing relative damping.

Damped natural frequency: Oscillation frequency \(\omega_d=\omega_n\sqrt{1-\zeta^2}\) for an underdamped system.

Rise time: Time required to move between specified response levels, commonly 10% and 90%.

Peak time: Time at which the first response maximum occurs.

Settling time: Time after which the response remains within a specified tolerance band.

Maximum overshoot: Maximum excursion above the final value, usually expressed as a percentage.

Transient response: Portion of a response that decays or changes before steady operation.

Steady-state error: Limiting difference between reference and output as time tends to infinity.

Final-value theorem: Transform-domain rule for obtaining a time-domain final value under pole conditions.

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

Position constant: Low-frequency gain \(K_p\) used for step-error analysis.

Velocity constant: Low-frequency gain \(K_v\) used for ramp-error analysis.

Dominant poles: Poles whose modes chiefly determine observed system behavior.

Nonminimum-phase zero: Right-half-plane zero that can cause inverse response and performance limitations.

IAE: Integral of the absolute error.

ISE: Integral of squared error.

ITAE: Integral of time-weighted absolute error.

Model reduction: Approximation of a higher-order model by a lower-order model while controlling approximation error.

Review Questions

Short Answer Questions

1. Define damping ratio and explain the physical significance of the real part of a second-order pole.

2. What is the difference between peak time and settling time?

3. State the exact 2% settling condition for a critically damped standard second-order system.

4. Why can a zero invalidate a pole-only prediction of overshoot?

5. Define IAE and ISE and identify one difference in their weighting of error.

Long Answer Questions

1. Derive the unit-step response of an underdamped standard second-order system and obtain its peak-time and overshoot formulas.

2. Explain the final-value theorem conditions and apply them to steady-state error analysis.

3. Compare underdamped, critically damped, and overdamped responses, including the limitations of approximate settling-time formulas.

4. Discuss dominant-pole approximation for higher-order systems and explain how residues and zeros affect its validity.

5. Compare IAE, ISE, ITAE, and ITSE as controller-performance criteria, including their engineering trade-offs.

Analytical / Application Questions

1. A critically damped system has \(\omega_n=40\) rad/s. Calculate its exact ideal 2% settling time using the critical-response equation. Would it satisfy a 0.12 s requirement?

2. For a standard second-order system with 12% maximum overshoot and a 2% envelope settling-time target of 1.8 s, calculate the required damping ratio and the approximate minimum natural frequency.

3. A fourth-order servo has a dominant pole pair at \(-3\pm j4\), but also has a right-half-plane zero at \(+2\). Explain why a standard second-order step formula may be unreliable and identify simulations or measurements needed before approving the design.

Multiple Choice Questions with Answers

Question 1

For a first-order system with time constant 0.5 s, what is its exact 2% settling time?

A. 0.50 s

B. 1.50 s

C. 1.96 s

D. 2.50 s

Correct Answer: C. 1.96 s

Question 2

For a standard underdamped second-order system, which parameter determines the exponential decay rate of the envelope?

A. \(\omega_n\) alone

B. \(\zeta\omega_n\)

C. \(\omega_d/\zeta\)

D. \(1/\omega_n\)

Correct Answer: B. \(\zeta\omega_n\)

Question 3

A standard second-order system has \(\zeta=0.5\). Its fractional maximum overshoot is closest to:

A. 0.016

B. 0.050

C. 0.163

D. 0.500

Correct Answer: C. 0.163

Question 4

For a critically damped standard second-order system, the normalized step-response error is:

A. \(e^{-\omega_nt}\)

B. \(\cos(\omega_nt)\)

C. \((1+\omega_nt)e^{-\omega_nt}\)

D. \(1-e^{-\omega_nt}\)

Correct Answer: C. \((1+\omega_nt)e^{-\omega_nt}\)

Question 5

The exact ideal 2% settling time of a critically damped system with \(\omega_n=50\) rad/s is approximately:

A. 0.040 s

B. 0.080 s

C. 0.100 s

D. 0.117 s

Correct Answer: D. 0.117 s

Question 6

The final-value theorem for \(\lim_{t\to\infty}c(t)=\lim_{s\to0}sC(s)\) requires:

A. All poles of \(C(s)\) to be at the origin

B. All poles of \(sC(s)\) to lie strictly in the open left half-plane

C. At least one right-half-plane pole in \(sC(s)\)

D. A zero of \(C(s)\) at the origin

Correct Answer: B. All poles of \(sC(s)\) to lie strictly in the open left half-plane

Question 7

For a unity-feedback type-1 system with finite velocity constant \(K_v\), the steady-state error to a unit ramp is:

A. \(1+K_v\)

B. \(K_v\)

C. \(1/K_v\)

D. Zero for every input

Correct Answer: C. \(1/K_v\)

Question 8

Which index gives greater weight to errors occurring later in time while using absolute error?

A. IAE

B. ISE

C. ITAE

D. Instantaneous error

Correct Answer: C. ITAE

Question 9

A right-half-plane zero commonly causes which behavior?

A. Guaranteed monotonic response

B. Inverse response or initial motion opposite to the final direction

C. Removal of all overshoot

D. Automatic closed-loop stability

Correct Answer: B. Inverse response or initial motion opposite to the final direction

Question 10

For \(e(t)=e^{-2t}\), the ISE is:

A. 0.125

B. 0.250

C. 0.500

D. 1.000

Correct Answer: B. 0.250