Control System

Chapter 3
Time-Domain Analysis and Stability

Post Graduate Program in Electrical Engineering
Control System • Post Graduate

3. Time-Domain Analysis and Stability

Introduction and Context

A control system is ultimately judged by how its variables evolve with time. A reference command may be applied suddenly, a load may change unexpectedly, or a disturbance may enter while the system is operating. The resulting output is not determined only by the final equilibrium value; the speed, overshoot, oscillation, settling behavior, and residual error are equally important. Time-domain analysis provides the principal framework for describing these characteristics.

In earlier studies of control systems, transfer functions and block diagrams establish relationships between input and output. Time-domain analysis extends that description by asking how the system responds to standard inputs and whether the response remains bounded. These questions lead naturally to the ideas of transient response, steady-state response, system type, error constants, dominant poles, performance indices, and stability.

For linear time-invariant systems, stability can be investigated directly from pole locations, without explicitly solving the complete time response. The Routh–Hurwitz criterion offers an algebraic method for determining whether poles lie in the left half of the complex plane. Lyapunov theory provides a more general energy-based perspective and remains important for nonlinear, multivariable, and modern control systems.

Learning Objectives

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

1. Distinguish transient and steady-state components of a control response.

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

3. Determine system type and calculate static error constants.

4. Evaluate steady-state error for standard reference inputs.

5. Identify dominant poles and estimate important time-domain specifications.

6. Apply performance indices such as IAE, ISE, ITAE, and ITSE.

7. Use the Routh–Hurwitz criterion to assess absolute stability and determine stability ranges.

8. Explain the direct and indirect Lyapunov methods.

9. Interpret stability results in practical engineering contexts.

The Time Response of a Feedback System

Consider a standard negative-feedback system with forward transfer function G(s), feedback function H(s), and closed-loop transfer function

T(s) = C(s)/R(s) = G(s)/[1 + G(s)H(s)].

The characteristic equation is

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

Its roots are the closed-loop poles. If the reference input is R(s), the output is C(s) = T(s)R(s). Similarly, for a disturbance input, the appropriate disturbance-to-output transfer function must be used. The poles of the relevant transfer function determine the natural modes appearing in the response.

For a stable linear system, the response is commonly 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. The transient component is associated with system poles and decays with time when the system is asymptotically stable. The steady-state component is the behavior that remains as t approaches infinity.

A first-order mode has the form e^(-at), where a > 0. Its time constant is tau = 1/a. A second-order pair of complex poles produces terms such as e^(-zeta omega_n t) sin(omega_d t), where omega_n is the natural frequency, zeta is the damping ratio, and omega_d = omega_n square root of (1 - zeta^2) is the damped natural frequency for 0 < zeta < 1.

The transient response is often described using rise time, peak time, maximum overshoot, and settling time. These quantities are not independent. Increasing damping generally reduces overshoot but may increase rise time. Increasing natural frequency generally makes a response faster, although actuator limitations and noise sensitivity may then become significant.

First- and Second-Order Response Characteristics

For the standard first-order transfer function

T(s) = 1/(tau s + 1),

a unit-step input produces

c(t) = 1 - e^(-t/tau).

The response reaches approximately 63.2 percent of its final value at t = tau, about 95 percent at 3tau, and about 98.2 percent at 4tau. A practical two-percent settling time is approximately 4tau.

The standard underdamped second-order form is

T(s) = omega_n^2/[s^2 + 2zeta omega_n s + omega_n^2].

For 0 < zeta < 1, the unit-step response oscillates before reaching its final value. Important approximations are:

Maximum overshoot: M_p = exp[-pi zeta/square root of (1 - zeta^2)] times 100 percent.
Peak time: t_p = pi/omega_d.
Two-percent settling time: t_s approximately 4/(zeta omega_n).
Ten-percent settling time: t_s approximately 2.3/(zeta omega_n).

These approximations apply to a dominant second-order response. They should not be used uncritically for systems with nearby additional poles, zeros, delays, saturation, or nonlinearities.

A damping ratio greater than one gives an overdamped response with two real negative poles and no oscillation. A damping ratio equal to one gives critical damping, which is the boundary between oscillatory and non-oscillatory second-order behavior. An undamped system has poles on the imaginary axis and does not settle after a step input.

Zeros also shape the response. A left-half-plane zero can accelerate the initial response but may increase overshoot. A right-half-plane zero produces nonminimum-phase behavior: the output may initially move in the direction opposite to the final response. Such inverse response imposes fundamental limitations on achievable speed and robustness.

Steady-State Error, System Type, and Error Constants

For a unity-feedback system, the error is

E(s) = R(s)/[1 + G(s)].

The final value theorem gives

e_ss = lim(t to infinity) e(t) = lim(s to 0) sE(s),

provided that the final-value theorem is applicable; in particular, the relevant closed-loop response must be stable. Applying the theorem to an unstable system can produce a misleading result.

The system type is the number of pure integrations, or poles at the origin, in the open-loop transfer function G(s)H(s), after pole-zero cancellations are treated carefully. For a unity-feedback system:

Type 0: no pole at the origin.
Type 1: one pole at the origin.
Type 2: two poles at the origin.

The static error constants for unity feedback are

K_p = lim(s to 0) G(s),

K_v = lim(s to 0) sG(s),

K_a = lim(s to 0) s^2G(s).

For a unit-step input, a Type 0 system has finite error e_ss = 1/(1 + K_p), whereas Type 1 and higher systems have zero step error if stable. For a unit-ramp input, a Type 0 system has infinite error, a Type 1 system has finite error e_ss = 1/K_v, and a Type 2 system has zero error. For a unit-parabolic input, a Type 0 or Type 1 system has infinite error, while a Type 2 system has finite error e_ss = 1/K_a.

These results reveal an important design trade-off. Adding integrators improves low-frequency tracking and disturbance rejection, but it also introduces phase lag and can reduce stability margins. Consequently, system type alone does not guarantee acceptable performance; stability and transient specifications must be checked together.

For non-unity feedback, the error signal and the controlled output must be identified carefully. The simple unity-feedback formulas may not apply directly, although the final value theorem remains valid when used with the correct E(s).

Poles, Zeros, and Dominant Modes

The natural response of a rational system is a sum of modal terms determined by its poles. A pole at s = -a contributes a mode e^(-at). Poles closer to the imaginary axis decay more slowly and therefore tend to dominate the response. A complex pair at

s = -sigma plus or minus j omega_d

produces a decaying oscillation. Its decay rate is controlled by sigma, while its oscillation frequency is related to omega_d.

A pole pair is called dominant when its contribution is substantially slower than that of the remaining poles and zeros do not significantly cancel or distort it. A commonly used engineering approximation is to regard other poles as sufficiently remote when their real parts are several times farther from the imaginary axis than those of the dominant pair. This is a rule of thumb, not a universal theorem. Near cancellations, lightly damped non-dominant poles, delays, and right-half-plane zeros can invalidate it.

Dominant-pole approximation is valuable because it reduces a high-order design problem to an interpretable second-order model. Nevertheless, the approximation must be verified by comparing the actual and reduced responses, especially when tight performance or safety requirements are involved.

Performance Indices and Design Interpretation

Time-domain design can be expressed as an optimization problem. Common integral performance indices include:

IAE = integral from 0 to infinity of |e(t)| dt,

ISE = integral from 0 to infinity of e^2(t) dt,

ITAE = integral from 0 to infinity of t|e(t)| dt,

ITSE = integral from 0 to infinity of te^2(t) dt.

IAE treats errors linearly and is less dominated by large initial errors than ISE. ISE penalizes large errors strongly and often leads to responses with relatively small peak deviations but possibly longer low-amplitude tails. ITAE and ITSE assign increasing importance to errors that persist for a long time, thereby encouraging rapid settling.

No single index is universally best. A control designer must decide whether overshoot, energy expenditure, tracking duration, actuator movement, or robustness has greatest importance. In an industrial process, excessive actuation may reduce equipment life; in a positioning system, residual error may be more damaging than a brief initial deviation. Performance indices are therefore engineering decision tools rather than purely mathematical objectives.

Stability Through Pole Locations

For a continuous-time linear time-invariant system, asymptotic stability requires every closed-loop pole to lie strictly in the open left half of the complex s-plane. A pole in the right half-plane produces an exponentially growing mode and makes the system unstable. Simple poles on the imaginary axis result in sustained oscillations and are associated with marginal stability under restricted conditions. Repeated poles on the imaginary axis are unstable because their time responses contain growing polynomial factors such as t sin(omega t).

The distinction between internal and input-output stability is important in advanced analysis. A transfer function may appear to have a pole-zero cancellation, but the hidden internal mode can remain unstable. Exact cancellation is also physically fragile because model parameters are never known perfectly. Stability should therefore be assessed from the complete state-space realization or characteristic equation whenever internal behavior matters.

Discrete-time systems have a corresponding condition: all poles must lie inside the unit circle in the z-plane. The continuous-time results in this chapter apply directly to continuous systems; transformations such as the bilinear transformation may be used when connecting continuous and discrete analyses.

The Routh–Hurwitz Criterion

Suppose the characteristic polynomial is

P(s) = a_n s^n + a_(n-1)s^(n-1) + ... + a_1s + a_0,

with real coefficients and a_n positive. The Routh array is constructed as follows:

s^n: a_n, a_(n-2), a_(n-4), ...

s^(n-1): a_(n-1), a_(n-3), a_(n-5), ...

Each subsequent element is formed from the preceding two rows. For example, the first element in the s^(n-2) row is

b_1 = [a_(n-1)a_(n-2) - a_n a_(n-3)]/a_(n-1).

The first column is the principal result. If all entries in the first column are positive, with no sign changes and no exceptional cases, all roots lie in the left half-plane. The number of sign changes in the first column equals the number of roots in the right half-plane.

Illustrative stability-range example

Consider the characteristic polynomial

P(s) = s^3 + 2s^2 + 3s + K.

The Routh array is

s^3: 1, 3

s^2: 2, K

s^1: (6 - K)/2, 0

s^0: K

For stability, the first-column entries must be positive:

1 > 0, 2 > 0, (6 - K)/2 > 0, K > 0.

Hence the stable range is

0 < K < 6.

At K = 6, the s^1 element becomes zero and the system is at a stability boundary. Substitution or an auxiliary polynomial can be used to identify the imaginary-axis roots. For K outside the range, the number of sign changes indicates how many roots have entered the right half-plane.

Special Routh cases

If a first-column element is zero while the rest of its row is not zero, a small positive quantity epsilon is substituted for the zero and the limit as epsilon approaches zero is interpreted. If an entire row becomes zero, the coefficients of an auxiliary polynomial formed from the row above are differentiated and used to continue the array. The all-zero row usually indicates symmetric roots, often including a pair on the imaginary axis.

The Routh criterion determines stability without calculating roots, but it does not directly provide settling time, overshoot, or robustness margins. It is therefore complementary to pole computation, root-locus analysis, frequency-domain methods, and simulation.

Lyapunov Stability: An Energy-Based View

Lyapunov theory analyzes stability without requiring an explicit solution of the differential equations. For a system

x_dot = f(x),

an equilibrium point x_e satisfies f(x_e) = 0. After shifting coordinates, the equilibrium may be taken as the origin.

The equilibrium is stable in the sense of Lyapunov if every trajectory starting sufficiently close to the origin remains close. It is asymptotically stable if it is stable and all sufficiently close trajectories converge to the origin. It is globally asymptotically stable if this convergence holds for every initial state. Exponential stability is a stronger property involving a bounded exponential rate of convergence.

A scalar function V(x) is a Lyapunov function if it behaves like stored energy: V(0) = 0, V(x) > 0 for x not equal to zero, and its derivative along system trajectories satisfies

V_dot(x) = partial V/partial x multiplied by f(x).

If V is positive definite and V_dot is negative definite, the origin is asymptotically stable. If V_dot is only negative semidefinite, stability may still be established, but asymptotic convergence generally requires an additional argument such as LaSalle's invariance principle.

For a linear system x_dot = Ax, choose

V(x) = x^T P x,

where P is symmetric positive definite. Then

V_dot = -x^T Qx

provided that

A^T P + PA = -Q.

If Q is positive definite, the system is asymptotically stable. A fundamental result states that A is Hurwitz—that is, all eigenvalues have negative real parts—if and only if, for every positive-definite Q, the Lyapunov equation has a unique positive-definite solution P. This establishes the equivalence between pole-based stability and the energy-based Lyapunov approach for linear systems.

Lyapunov analysis is especially valuable for nonlinear systems, adaptive control, robot control, power electronics, and interconnected systems, where a transfer-function polynomial may not adequately represent the problem. A limitation is that finding a useful Lyapunov function can be difficult. Computational methods, including linear matrix inequalities and sum-of-squares techniques, help automate this task for important classes of systems.

Applied Scenario: Speed Control of a Motor Drive

Background

Consider a separately excited motor drive required to track speed commands while rejecting load-torque disturbances. The drive contains a power converter, motor dynamics, a speed sensor, and a feedback controller. A sudden speed command is a standard test, while a step increase in load torque tests disturbance rejection.

Situation and problem

An initial proportional controller provides acceptable rise time but leaves a persistent speed error under constant load torque. Increasing proportional gain reduces this error but produces excessive overshoot and oscillation because the electrical and mechanical dynamics introduce phase lag.

Analysis

The proportional design is effectively Type 0 with respect to speed-reference tracking. Its finite position error under a step disturbance or reference is a consequence of limited low-frequency loop gain. Introducing integral action changes the system type and can eliminate steady-state error for a constant command or load disturbance. However, the added pole reduces phase margin and may move a closed-loop pole toward the imaginary axis.

The design team first specifies an allowable overshoot and settling time. A desired dominant pole pair is selected from the corresponding damping ratio and natural frequency. The remaining closed-loop poles are placed sufficiently farther into the left half-plane. The Routh criterion is then used to verify stability over the expected range of motor inertia, converter gain, and controller parameters.

Approach and solution

A proportional-integral controller is introduced with anti-windup protection. The integral gain is selected to meet the steady-state requirement, while the proportional gain is adjusted to obtain the desired dominant-pole damping. The full characteristic polynomial, rather than only its reduced second-order approximation, is checked. Simulations include command changes, load steps, sensor noise, converter saturation, and parameter variation.

Outcome

The final design has negligible steady-state speed error for constant load torque and a faster response than the original proportional controller. Anti-windup prevents a large recovery delay when the converter saturates. Robustness testing shows that the nominal dominant-pole estimates are useful but not sufficient by themselves; actuator limits and parameter uncertainty materially affect the measured response.

Learning points

The scenario illustrates that system type addresses low-frequency accuracy, whereas pole locations determine transient behavior and stability. A design that satisfies one requirement may violate another. Controller tuning must therefore combine error-constant analysis, pole analysis, stability tests, simulation, and practical hardware constraints.

Engineering Limitations, Risks, and Professional Practice

Several mistakes recur in time-domain analysis. The final value theorem is sometimes applied to unstable systems. System type is sometimes inferred from the closed-loop transfer function rather than the appropriate open-loop loop gain. Dominant-pole approximations are accepted without checking neglected poles and zeros. A Routh array is constructed with an algebraic sign error, or a zero row is treated as an ordinary numerical anomaly rather than a stability-boundary indication.

Real systems also include saturation, dead zones, backlash, quantization, friction, delays, sensor noise, and parameter uncertainty. These effects can invalidate a linear model outside a limited operating region. Integral action may cause windup during saturation. High controller gains can amplify measurement noise and demand unrealistic actuator effort. Stability must therefore be considered together with robustness, safety, maintainability, and fault behavior.

In regulated industries, control-system decisions should be traceable: assumptions, models, test conditions, parameter ranges, and acceptance criteria should be documented. Safety-related systems require independent verification, fault analysis, and conservative validation. It is not professionally adequate to report only a nominal pole calculation when failure could endanger people, equipment, or the environment.

Computational Tools and Emerging Directions

Software packages can calculate poles, construct Routh arrays, simulate nonlinear responses, estimate performance indices, and solve Lyapunov equations. These tools improve productivity but do not replace engineering judgment. Numerical pole locations may be sensitive to scaling, nearly cancelling factors, or poorly conditioned polynomial coefficients. Results should be cross-checked using independent formulations and physically meaningful tests.

Current control practice increasingly combines classical time-domain specifications with optimization, robust control, model predictive control, and data-driven methods. Automated tuning algorithms may minimize ITAE or other objectives, while formal verification tools use Lyapunov inequalities to certify stability over parameter ranges. Machine-learning components can assist modeling or tuning, but their use in safety-critical loops requires explainability, validation outside training data, and safeguards against unverified adaptation. The fundamental concepts of pole location, bounded response, and energy dissipation remain central even when the controller is computationally sophisticated.

Best-Practice Workflow

A disciplined analysis generally follows this sequence:

1. Define the input, output, error signal, operating point, and performance requirements.

2. Derive the correct open-loop and closed-loop models, noting assumptions and possible cancellations.

3. Determine characteristic poles and establish nominal stability.

4. Evaluate system type, static error constants, and steady-state errors.

5. Estimate transient specifications using dominant poles, then verify them with the complete model.

6. Apply Routh–Hurwitz or an equivalent algebraic test over uncertain parameter ranges.

7. Use simulation to investigate saturation, disturbances, noise, nonlinearities, and initial conditions.

8. Where appropriate, construct a Lyapunov certificate for nonlinear or state-space behavior.

9. Validate the design experimentally and document limitations, margins, and safety provisions.

Chapter Summary

Time-domain analysis describes how a control system evolves after commands and disturbances. A response consists of transient and steady-state components, with transient behavior governed primarily by closed-loop poles and shaped by zeros. First- and second-order models provide useful relations among time constant, damping ratio, natural frequency, overshoot, peak time, and settling time.

System type, static error constants, and the final value theorem quantify steady-state tracking accuracy, although these tools are valid only when applied to the correct error transfer function and a stable system. Dominant poles permit useful reduced-order approximations, but the complete model must be checked when poles, zeros, delays, or nonlinearities are significant. Integral performance indices provide alternative ways to express design objectives.

Continuous-time asymptotic stability requires all closed-loop poles to lie in the open left half-plane. The Routh–Hurwitz criterion determines stability and the number of right-half-plane roots without explicitly solving for them. Lyapunov theory gives an energy-based stability framework and generalizes naturally to state-space and nonlinear systems. Sound professional design combines these analytical methods with simulation, robustness assessment, actuator considerations, and documented validation.

Key Terms

Transient response: The portion of a response that decays or changes before steady operation is reached.

Steady-state response: The limiting behavior of a system as time approaches infinity.

Time constant: The reciprocal of the positive decay rate of a first-order mode.

Natural frequency: The frequency parameter associated with a standard second-order system.

Damping ratio: A dimensionless measure of damping relative to critical damping.

Overshoot: The amount by which a response exceeds its final value, usually expressed as a percentage.

Settling time: The time required for a response to remain within a specified band around its final value.

System type: The number of open-loop poles at the origin in the loop transfer function.

Static error constant: A low-frequency gain measure used to calculate steady-state error.

Position constant K_p: The error constant associated with a step input.

Velocity constant K_v: The error constant associated with a ramp input.

Acceleration constant K_a: The error constant associated with a parabolic input.

Dominant poles: Poles whose modes contribute most significantly to the observed response.

Performance index: A numerical measure used to evaluate or optimize control-system behavior.

Routh–Hurwitz criterion: An algebraic test for determining the number of right-half-plane roots.

Asymptotic stability: Stability in which trajectories starting sufficiently close converge to the equilibrium.

Lyapunov function: A positive-definite scalar function used to assess stability through its derivative.

Hurwitz matrix/system: A system whose characteristic roots all have negative real parts.

Anti-windup: A method that limits undesirable integral accumulation during actuator saturation.

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

Review Questions

Short Answer Questions

1. What distinguishes transient response from steady-state response?

2. Define system type for a unity-feedback system.

3. State the condition for asymptotic stability of a continuous-time LTI system in terms of pole locations.

4. What is meant by a dominant pole pair?

5. Why must the final value theorem not be applied indiscriminately to unstable systems?

Long Answer Questions

1. Derive the steady-state errors for step, ramp, and parabolic inputs for Type 0, Type 1, and Type 2 unity-feedback systems.

2. Explain the construction and interpretation of the Routh array, including the significance of sign changes.

3. Discuss the relationships among damping ratio, natural frequency, overshoot, peak time, and settling time.

4. Explain the direct Lyapunov method and its application to the linear system x_dot = Ax.

5. Compare IAE, ISE, ITAE, and ITSE as alternative performance criteria.

Analytical / Application Questions

1. Use the Routh–Hurwitz criterion to determine the range of K for which s^4 + 3s^3 + 5s^2 + 2s + K = 0 is stable.

2. A speed-control loop has one integrator in its open-loop transfer function. Determine its expected steady-state error for a stable unit-step and unit-ramp reference, and explain what additional design trade-off may result from the integrator.

3. A measured response has approximately 12 percent overshoot and a two-percent settling time of 2 seconds. Estimate the damping ratio and the product zeta omega_n using the standard second-order approximations, and comment on the limitations of this estimate.

22. Multiple Choice Questions with Answers

Question 1

For a stable unity-feedback Type 1 system, what is the steady-state error to a unit-step input?

A. Zero

B. One

C. 1/Kp

D. Infinite

Correct Answer: A. Zero

Question 2

The number of sign changes in the first column of a Routh array equals the number of:

A. Left-half-plane poles

B. Right-half-plane poles

C. Imaginary-axis zeros

D. System zeros at the origin

Correct Answer: B. Right-half-plane poles

Question 3

For the standard underdamped second-order system, increasing the damping ratio generally:

A. Increases overshoot

B. Reduces overshoot

C. Moves poles into the right half-plane

D. Eliminates the natural frequency

Correct Answer: B. Reduces overshoot

Question 4

Which condition is required for asymptotic stability of a continuous-time LTI system?

A. All poles lie on the imaginary axis

B. All poles lie in the open left half-plane

C. At least one pole lies in the right half-plane

D. All zeros lie in the left half-plane

Correct Answer: B. All poles lie in the open left half-plane

Question 5

For the characteristic polynomial s^3 + 2s^2 + 3s + K, the Routh criterion gives stability when:

A. K < 0

B. K > 6

C. 0 < K < 6

D. K = 6 only

Correct Answer: C. 0 < K < 6

Question 6

Which performance index gives increasing weight to errors that persist for a long time while using absolute error?

A. ISE

B. IAE

C. ITAE

D. ITSE

Correct Answer: C. ITAE

Question 7

A pole at s = -0.5 generally contributes a slower-decaying mode than a pole at s = -5 because it:

A. Is farther from the imaginary axis

B. Is closer to the imaginary axis

C. Is necessarily a zero

D. Creates a right-half-plane mode

Correct Answer: B. Is closer to the imaginary axis

Question 8

In Lyapunov analysis, a positive-definite V(x) with negative-definite V_dot indicates:

A. Asymptotic stability

B. Guaranteed instability

C. Marginal stability only

D. A noncausal system

Correct Answer: A. Asymptotic stability

Question 9

For a unity-feedback Type 1 system with finite velocity constant Kv, the steady-state error to a unit ramp is:

A. 0

B. 1/(1 + Kv)

C. 1/Kv