Root-Locus and Classical Controller Design

Introduction and Context

Root-locus methods provide a graphical and analytical means of studying how the closed-loop poles of a feedback system move as a real system parameter, usually the loop gain, varies. Because pole locations determine stability, dominant transient behavior, oscillation, and settling characteristics, root-locus analysis remains one of the most useful classical design methods in control engineering. It is particularly valuable when a designer must connect time-domain requirements—such as settling time, overshoot, and damping ratio—to a controller structure.

Classical controller design also uses frequency-response tools, steady-state error constants, and practical implementation constraints. Proportional, integral, derivative, lead, lag, and combined controllers each modify different aspects of performance. No single controller improves every requirement without trade-offs: integral action improves low-frequency accuracy but may reduce stability margins; derivative or lead action improves transient response but increases sensitivity to measurement noise; lag action improves accuracy with comparatively small changes to transient response but reduces bandwidth.

This chapter develops root-locus construction and interpretation, then applies it to controller design. The discussion assumes linear time-invariant models unless otherwise stated. Real systems may contain saturation, nonlinear friction, sampling, delays, unmodelled modes, and sensor noise; therefore, a mathematically satisfactory design must also be evaluated for implementation robustness.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Formulate the characteristic equation and construct a root locus.

2. Apply angle and magnitude conditions to determine closed-loop pole locations and gain.

3. Relate pole locations to damping ratio, natural frequency, settling time, and overshoot.

4. Design and compare P, PI, PD, PID, lead, lag, and lead–lag controllers.

5. Evaluate steady-state accuracy, disturbance rejection, robustness, and actuator limitations.

6. Use anti-windup and practical tuning procedures in implementable controllers.

7. Critically assess the limitations of classical designs and validate them through simulation and experiment.

1. Closed-Loop Poles and the Root-Locus Concept

Consider a single-input, single-output negative-feedback system with forward transfer function G(s), feedback transfer function H(s), and adjustable gain K. The loop transfer function is

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

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

T(s) = Y(s)/R(s) = K G(s)/[1 + K G(s)H(s)],

when K is included in the forward path. The characteristic equation is therefore

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

If the open-loop transfer function is written as

G(s)H(s) = N(s)/D(s),

the characteristic equation becomes D(s) + K N(s) = 0. The root locus is the set of all values of s satisfying this equation as K varies from zero to positive infinity, with the poles and zeros of G(s)H(s) held fixed.

A point s belongs to the root locus if and only if it satisfies two conditions:

Angle condition: angle[G(s)H(s)] = (2q + 1)180 degrees,

Magnitude condition: K|G(s)H(s)| = 1,

where q is any integer. The angle condition identifies possible pole locations; the magnitude condition determines the gain at a selected location.

At K = 0, the closed-loop poles begin at the open-loop poles. As K tends to infinity, branches terminate at open-loop zeros. If there are more poles than zeros, the remaining branches tend toward infinity along asymptotes.

Important terminology

An open-loop pole is a pole of the loop transfer function before feedback closure. An open-loop zero is a zero of that transfer function. A closed-loop pole is a root of the characteristic equation. A branch is one continuous path of the locus. A breakaway point is where branches leave the real axis; a break-in point is where branches re-enter it. A dominant pole pair is a pair whose dynamics significantly influence the observable transient response because other poles are substantially faster.

2. Rules for Constructing a Root Locus

For a real-coefficient transfer function, the following rules provide a systematic construction.

2.1 Number of branches

The number of root-locus branches equals the number of open-loop poles, counting multiplicity. Each branch starts at an open-loop pole. Branches end at open-loop zeros or at infinity.

2.2 Real-axis segments

A point on the real axis belongs to the positive-gain root locus if the number of real open-loop poles and zeros to its right is odd. This rule assumes the usual negative-feedback characteristic equation 1 + K G(s)H(s) = 0 with K > 0.

2.3 Asymptotes

Let n be the number of open-loop poles and m the number of open-loop zeros, with n > m. The number of asymptotes is n − m. Their angles are

θ_a = [(2q + 1)180 degrees]/(n − m), q = 0, 1, ..., n − m − 1.

The centroid from which the asymptotes emanate is

σ_a = [sum of poles − sum of zeros]/(n − m).

The sums include algebraic values of real poles and the real parts of complex poles and zeros. The asymptote formulas apply to rational transfer functions and describe behavior far from the finite poles and zeros.

2.4 Angles of departure and arrival

For a complex pole, the angle of departure is found by applying the angle condition near that pole. In words, the contribution of the unknown departure angle must compensate for the net angle contributed by all other poles and zeros. Similarly, the angle of arrival at a complex zero is obtained by solving the angle condition near that zero. These calculations are especially useful when branches are crowded and a sketch alone is unreliable.

2.5 Breakaway and break-in points

From the characteristic equation, solve for K as a function of s:

K(s) = −D(s)/N(s).

Candidate real-axis break points satisfy

dK(s)/ds = 0.

Only candidates lying on an actual real-axis root-locus segment and producing an admissible positive gain should be retained. A stationary point outside a root-locus segment is not a valid breakaway or break-in point.

2.6 Imaginary-axis crossings

Imaginary-axis crossings may be found by substituting s = jω into the characteristic equation and equating real and imaginary parts, or by using the Routh–Hurwitz criterion. The crossing gain identifies a stability boundary. For higher-order systems, Routh–Hurwitz analysis is often more reliable than a graphical estimate.

3. Interpreting Pole Locations

For a standard second-order denominator

s² + 2ζω_n s + ω_n²,

the parameters are the natural frequency ω_n in rad/s and damping ratio ζ. The poles are

s = −ζω_n ± jω_n√(1 − ζ²),

for 0 < ζ < 1. The real part is −ζω_n and the damped frequency is ω_d = ω_n√(1 − ζ²).

Useful approximate relationships are:

Peak overshoot: M_p = exp[−πζ/√(1 − ζ²)],

2% settling time: T_s ≈ 4/(ζω_n),

10–90% rise-time approximation for an underdamped second-order system: T_r ≈ 1.8/ω_n.

These formulas apply exactly only to the ideal normalized second-order step response, and approximately to higher-order systems when a dominant complex pair adequately represents the response.

Constant-damping-ratio lines are rays from the origin in the left half-plane. Constant-natural-frequency lines are circles centered at the origin. Thus a transient specification can be translated into a desired region in the s-plane. For example, a maximum overshoot specification determines a minimum ζ, while a settling-time specification determines a minimum ζω_n. The intersection defines an acceptable pole region rather than one unique pole location.

A higher-order system should not be judged solely by its visually nearest poles. A remote pole can have a large residue, a zero can cancel or amplify a mode, and nonminimum-phase zeros can produce inverse response. Dominance must therefore be verified using the complete transfer function and simulated response.

4. Worked Root-Locus Example

Consider the unity-feedback plant

G(s) = K/[s(s + 2)(s + 5)].

The characteristic equation is

s(s + 2)(s + 5) + K = 0,

or

s³ + 7s² + 10s + K = 0.

There are three branches beginning at 0, −2, and −5, and no finite zeros. The real-axis segments are (−∞, −5) and (−2, 0), because those intervals contain an odd number of real poles to their right. The asymptote centroid is

σ_a = (0 − 2 − 5)/3 = −7/3 = −2.333,

and the asymptote angles are 60°, 180°, and 300°.

To determine stability, form the Routh array for

s³ + 7s² + 10s + K:

s³: 1, 10

s²: 7, K

s¹: (70 − K)/7, 0

s⁰: K

For strict stability, the first-column entries must be positive. Hence K > 0 and K < 70. The limiting gain is K = 70. At this value, the auxiliary equation from the s² row is 7s² + 70 = 0, giving s = ±j√10 rad/s. Thus the locus crosses the imaginary axis at approximately ±j3.162 rad/s, and the system is stable for 0 < K < 70.

This example illustrates why a root-locus sketch should be complemented by algebraic stability analysis. The sketch indicates movement and asymptotic behavior, whereas the Routh test gives an exact stability interval for this polynomial.

5. Classical Controller Structures

5.1 Proportional control

A proportional controller is

C(s) = K_p.

It increases loop gain and generally reduces transient error, but it cannot eliminate steady-state error for a type-0 plant subject to a step reference. Increasing K_p may improve speed while decreasing damping and stability margin. Proportional control is often appropriate when actuator simplicity, low noise sensitivity, and moderate accuracy are priorities.

5.2 Integral control and PI control

An ideal PI controller is

C(s) = K_p + K_i/s = K_p(s + z_i)/s,

where K_p is the proportional gain, K_i is the integral gain, and z_i = K_i/K_p when K_p is nonzero. Integral action adds a pole at the origin and raises the system type by one, thereby eliminating steady-state error for a step reference when the closed loop remains stable and the actuator does not saturate.

Integral action also accumulates error during saturation. If the actuator cannot provide the demanded input, the integrator may continue to grow, causing overshoot and delayed recovery when the actuator returns to its linear range. PI design must therefore include anti-windup considerations.

5.3 Derivative and PD control

An ideal PD controller is

C(s) = K_p + K_d s = K_d(s + z_d),

with derivative gain K_d. The zero can reshape the root locus so that selected closed-loop poles have greater damping or faster response. Derivative action responds to the rate of error and does not improve steady-state accuracy by itself. Pure differentiation is physically unrealizable and strongly amplifies high-frequency measurement noise.

A practical derivative term uses a first-order filter:

C_D(s) = K_d s/(1 + s/ω_f),

where ω_f is the derivative-filter corner frequency. The filter should be selected above the intended control bandwidth but below frequencies where sensor noise becomes dominant. Different industrial conventions use equivalent forms such as K_p[1 + T_d s/(1 + T_d s/N)].

5.4 PID control

The ideal parallel PID controller is

C(s) = K_p + K_i/s + K_d s.

A practical filtered form is

C(s) = K_p + K_i/s + K_d s/(1 + s/ω_f).

PID combines low-frequency integral action, proportional feedback, and phase-lead-like derivative action. It is powerful but not automatically superior: excessive integral gain causes oscillation and windup, while excessive derivative gain produces noise-sensitive control. In many applications, PI is preferred because derivative information is unreliable or unnecessary.

5.5 Lead compensation

A lead compensator may be written as

C_lead(s) = K_c (s + z)/(s + p), with 0 < z < p.

Its zero lies closer to the origin than its pole. Around its break frequencies, it contributes positive phase and increases the effective bandwidth. Lead compensation is useful for improving phase margin, damping, rise time, and transient response. The maximum phase lead occurs at

ω_m = √(zp),

and the maximum phase contribution is

φ_max = sin⁻¹[(p − z)/(p + z)],

for positive z and p under the stated pole-zero convention. The phase formula is applicable to the factor (s + z)/(s + p), with 0 < z < p.

5.6 Lag compensation

A lag compensator may be written

C_lag(s) = K_c (s + z)/(s + p), with 0 < p < z.

Its pole lies closer to the origin than its zero. Lag compensation increases low-frequency loop gain relative to high-frequency gain, improving steady-state accuracy while usually reducing bandwidth and adding a small negative phase contribution. It is selected when accuracy is insufficient but the existing transient response is acceptable.

A lag network should not be regarded as a free accuracy improvement. Its low-frequency pole can slow response, and poorly placed lag poles can reduce phase margin. The pole and zero are commonly placed sufficiently below the crossover frequency so that their phase effect near crossover is limited, but the exact placement must be checked through frequency-response analysis.

5.7 Lead–lag compensation

A lead–lag controller combines the two forms:

C(s) = K_c (s + z_l)/(s + p_l) · (s + z_g)/(s + p_g),

where the first factor is selected for lag behavior and the second for lead behavior, subject to the chosen parameter convention. Lead–lag design can simultaneously improve transient response and steady-state accuracy. It is more difficult to tune because it introduces additional parameters and interactions. Pole-zero placement should be validated with root locus, Bode plots, sensitivity functions, and time-domain simulation.

6. Controller Design Through Root Locus

A practical design process begins by translating specifications into constraints:

1. Determine the required steady-state accuracy and disturbance rejection.

2. Determine transient requirements such as overshoot, settling time, and rise time.

3. Identify actuator limits, sensor noise, sampling rate, delays, and safety constraints.

4. Plot the uncompensated root locus and assess whether it passes through an acceptable pole region.

5. If necessary, add controller poles and zeros to reshape the locus.

6. Select a desired dominant pole pair from the damping and speed requirements.

7. Apply the angle condition to place compensator zeros and poles.

8. Apply the magnitude condition to calculate gain.

9. Check all closed-loop poles, not only the selected pair.

10. Validate reference tracking, load disturbance rejection, noise response, saturation behavior, and robustness.

Suppose a desired dominant pole is s_d = −σ + jω_d. For the uncompensated plant, calculate the net phase at s_d. If the angle condition is not satisfied, the compensator must supply the phase deficiency. A lead zero is often placed to contribute positive phase, while a lead pole is placed farther left to limit high-frequency gain. After pole-zero placement, the gain is found from

K = 1/|C_0(s_d)G(s_d)H(s_d)|,

where C_0 excludes the scalar gain K. The resulting closed-loop characteristic polynomial must then be checked for additional poles and stability.

Illustrative design calculation

For a desired second-order response with 10% overshoot, solve

0.10 = exp[−πζ/√(1 − ζ²)].

This gives ζ approximately 0.591. If a 2% settling time of 2 s is required, then

ζω_n ≈ 4/T_s = 2,

so ω_n ≈ 2/0.591 = 3.38 rad/s. The desired dominant poles are approximately

s_d = −2 ± jω_n√(1 − ζ²)

= −2 ± j2.72.

These values define a target region. They do not by themselves guarantee that a higher-order closed loop will have 10% overshoot and 2 s settling time; the complete response must be computed.

7. Accuracy, Disturbance Rejection, and Robustness

For unity feedback, the sensitivity and complementary sensitivity functions are

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

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

where L(s) is the loop transfer function. S describes sensitivity to plant and disturbance effects entering in appropriate locations, while T describes reference transmission and measurement-noise transmission in common feedback configurations. The identity S(s) + T(s) = 1 holds algebraically.

High low-frequency loop gain makes S small near zero frequency, improving tracking and rejection of slowly varying load disturbances. High-frequency loop gain, however, can amplify sensor noise through T and demand excessive actuator effort. Robust design seeks a suitable trade-off rather than maximizing gain everywhere.

Gain margin and phase margin are frequency-domain indicators of robustness. They depend on the loop transfer function and crossover frequencies, and they are meaningful only with a clearly specified feedback convention and adequate model validity. Classical margins do not fully capture time delay, nonlinear saturation, uncertainty with structured frequency dependence, or right-half-plane zeros. For serious applications, uncertainty analysis and worst-case simulation should supplement margin calculations.

Nonminimum-phase zeros impose fundamental limitations. A right-half-plane zero can restrict achievable bandwidth and cause inverse response. A pure time delay contributes phase lag that cannot be removed by simply increasing gain. Flexible structures may contain lightly damped modes that are invisible in a reduced model but are excited by aggressive lead or derivative action.

8. Anti-Windup and Practical Implementation

Actuator saturation changes the closed-loop system from linear to nonlinear. If the commanded signal u_c exceeds the allowable range [u_min, u_max], the actuator applies

u = sat(u_c) = min(max(u_c, u_min), u_max).

With integral action, the integrator state may continue accumulating the error while u is saturated. Two common remedies are conditional integration and back-calculation.

Conditional integration disables or limits integration when saturation is present and the error would drive the actuator further into saturation. Back-calculation adds a correction proportional to the difference between the saturated and unsaturated commands. In a common form,

ẋ_I = K_i e + K_aw(u − u_c),

where x_I is the integrator state, e is the control error, K_aw is an anti-windup gain, and u − u_c is the saturation correction. The exact sign follows the chosen controller and state definitions; implementation should be tested against a known saturation case.

Best practices include bumpless transfer between manual and automatic modes, set-point weighting, derivative action on the measured output rather than on a discontinuous reference, explicit output and rate limits, integral-state initialization, and discrete-time validation. A digital implementation must account for sampling, numerical integration, derivative filtering, quantization, and computational delay. Tuning gains obtained in continuous time should not be transferred blindly to a low-rate controller.

9. Tuning Methods and Their Critical Evaluation

Classical tuning rules based on step tests or ultimate gain can provide initial settings, but they are empirical and may produce excessive overshoot or poor robustness for integrating, delayed, nonlinear, or unstable plants. Relay-feedback tuning estimates an oscillation point without driving the actuator through a full manual search, but safety limits remain essential.

Model-based tuning uses a nominal plant and design objectives. It is more transparent than purely empirical tuning and supports pole placement, frequency shaping, and robustness analysis. However, model error can make a nominally excellent design unsafe. A sound workflow is therefore iterative: obtain an initial model, design conservatively, simulate uncertainty and nonlinearities, test at low amplitude, inspect actuator demand, and increase performance only when margins and safety constraints remain acceptable.

AI-assisted tuning can help identify models, optimize gains, detect operating-mode changes, or recommend gain scheduling. It does not remove the need for stability proofs, constraint handling, explainability, cybersecurity, and human approval. An optimization objective that rewards fast tracking alone may produce actuator chatter, noise amplification, or unsafe operation. AI tools should be treated as design aids subject to engineering validation and governance.

10. Illustrative Case Study: Speed Control of a DC Motor

Background

Consider a laboratory DC motor whose speed must track commanded speed despite load-torque changes. A simplified low-frequency model is used initially, but the real system includes armature saturation, encoder quantization, friction, and a voltage limit.

Problem

The uncompensated proportional loop gives a nonzero speed error under a constant load torque. Increasing proportional gain reduces the error but causes oscillatory behavior near a flexible mechanical resonance. The design requires small step overshoot, acceptable settling time, rejection of a constant load disturbance, and no prolonged recovery after voltage saturation.

Analysis

A PI controller is selected because integral action is required for zero steady-state error to a constant disturbance, while derivative action would amplify encoder noise. Root-locus analysis with the added integrator shows that excessive integral gain moves the dominant poles toward a low-damping region. A lead element is considered to recover phase margin, but its high-frequency gain is limited because the encoder contains quantization noise.

Approach and solution

The design proceeds as follows. First, a conservative proportional gain is selected from the measured plant model. Second, the integral zero is placed below the intended crossover frequency so that low-frequency disturbance rejection is improved without unnecessarily dominating the transient region. Third, the integral gain is increased while monitoring damping and actuator demand. A filtered lead term is added only if the desired crossover and phase margin cannot be reached with PI control. Finally, conditional integration and back-calculation are implemented, together with output voltage and voltage-rate limits.

Validation includes reference steps, positive and negative load-torque steps, speed reversals, sensor-noise tests, supply-voltage variation, and parameter changes in inertia and friction. The controller is tested first at reduced voltage and speed, then under normal operating conditions.

Outcome

The illustrative design achieves zero steady-state error for constant speed commands in the unsaturated region and rejects constant load torque. Anti-windup substantially reduces overshoot after a large speed reversal. The main trade-off is that aggressive lead action improves settling time but increases noise-sensitive voltage activity; consequently, the final design uses a moderate bandwidth rather than the fastest simulated response.

Learning points

The case demonstrates that controller selection follows the disturbance and implementation requirements, not merely the desire for a faster step response. Root locus is valuable for visualizing pole movement, but actuator limits, sensor quality, uncertainty, and nonlinear testing determine whether a design is deployable.

11. Limitations, Safety, and Engineering Governance

Root-locus design is fundamentally a nominal linear method. It does not automatically account for parameter uncertainty, nonlinear saturation, hysteresis, dead zones, time-varying dynamics, or stochastic disturbances. Pole-zero cancellation is especially risky: cancelling a poorly known plant pole can leave a hidden lightly damped mode, and cancelling a right-half-plane pole is not a valid stabilizing strategy.

Safety-critical controllers require independent protection layers, fault detection, bounded commands, and documented verification. A controller should fail to a safe state when sensors, computation, or actuators malfunction. Design records should identify model assumptions, operating envelope, gain schedules, test evidence, and change-control procedures. Cybersecurity is also relevant because malicious modification of controller gains or sensor data can destabilize a physical process.

Chapter Summary

Root locus connects controller parameters to closed-loop pole locations and therefore to stability and transient response. Its angle and magnitude conditions, real-axis rules, asymptotes, break points, and stability crossings provide a systematic design framework. Damping ratio, natural frequency, settling time, and overshoot translate performance specifications into regions of the s-plane.

Proportional control is simple but limited in accuracy; PI adds low-frequency accuracy and disturbance rejection at the cost of possible windup; PD and lead compensation improve phase and transient response but require noise management. Lag improves low-frequency gain with reduced bandwidth, while lead–lag combines both objectives. Practical designs must evaluate all poles, frequency-domain robustness, actuator constraints, saturation, sampling, uncertainty, and sensor noise. Anti-windup, bumpless transfer, filtering, and staged validation are essential for implementation. Classical methods remain powerful when used with realistic models, nonlinear tests, and explicit safety governance.

Key Terms

Root locus: Set of closed-loop pole locations as a parameter, usually gain, varies.

Characteristic equation: Equation whose roots are the closed-loop poles.

Angle condition: Condition that the loop phase equals an odd multiple of 180 degrees.

Magnitude condition: Condition that loop magnitude equals unity after gain selection.

Asymptote: Far-field direction followed by root-locus branches approaching infinity.

Centroid: Point from which root-locus asymptotes originate.

Breakaway point: Real-axis point where branches leave the real axis.

Damping ratio: Dimensionless measure associated with decay and oscillation of a second-order mode.

Natural frequency: Frequency parameter defining the undamped second-order dynamics.

Dominant poles: Poles that primarily determine the observed response.

Steady-state error: Long-term difference between a commanded and achieved output.

Lead compensator: Compensator that adds positive phase and generally increases bandwidth.

Lag compensator: Compensator that increases relative low-frequency gain.

Lead–lag compensator: Controller combining lead and lag effects.

Sensitivity function: S(s) = 1/[1 + L(s)], indicating closed-loop sensitivity to several disturbances and uncertainties.

Integral windup: Unbounded or excessive integrator accumulation during actuator saturation.

Anti-windup: Method that limits or corrects integrator accumulation during saturation.

Phase margin: Frequency-domain measure related to distance from the −180-degree phase condition.

Nonminimum-phase zero: Zero in the right half of the complex plane, imposing performance limitations.

Review Questions

Short Answer Questions

1. State the angle and magnitude conditions for a point to lie on a positive-gain root locus.

2. Why does integral action improve step-reference accuracy?

3. Distinguish the pole-zero placement of lead and lag compensators.

4. What is the purpose of a derivative filter?

5. Define integral windup and name one anti-windup method.

Long Answer Questions

1. Explain the complete procedure for constructing a root locus, including real-axis segments, asymptotes, break points, and imaginary-axis crossings.

2. Derive the relationship between damping ratio, natural frequency, overshoot, and settling time for an ideal second-order system.

3. Compare P, PI, PD, PID, lead, lag, and lead–lag controllers in terms of accuracy, transient response, noise sensitivity, and implementation difficulty.

4. Explain how sensitivity and complementary sensitivity functions support robust controller design.

5. Discuss why nominal root-locus performance may fail in a practical system with saturation, delay, uncertainty, and measurement noise.

Analytical / Application Questions

1. For the unity-feedback plant G(s) = K/[s(s + 2)(s + 5)], determine the real-axis root-locus segments, asymptote centroid and angles, and the gain range for strict stability.

2. A design requires approximately 10% overshoot and a 2% settling time of 2 s. Estimate the desired dominant pole pair and explain why the result must be checked using the complete closed-loop model.

3. Design a controller strategy for a motor-speed system with constant load disturbances, encoder noise, and actuator saturation. Justify the controller structure, anti-windup method, and validation tests.

Multiple Choice Questions with Answers

Question 1

Which condition identifies a point on a positive-gain root locus?

A. The loop angle is an even multiple of 180 degrees

B. The loop angle is an odd multiple of 180 degrees

C. The loop magnitude is zero

D. The closed-loop gain is always unity

Correct Answer: B. The loop angle is an odd multiple of 180 degrees

Question 2

For a root locus with n poles and m zeros, where n > m, how many asymptotes are present?

A. m

B. n

C. n − m

D. n + m

Correct Answer: C. n − m

Question 3

For the characteristic polynomial s³ + 7s² + 10s + K, what is the strict-stability condition for positive K?

A. K > 70

B. 0 < K < 70

C. K = 70 only

D. K < 0

Correct Answer: B. 0 < K < 70

Question 4

Which controller action eliminates steady-state error to a step reference for a stable type-0 plant?

A. Proportional action alone

B. Derivative action alone

C. Integral action

D. Derivative filtering alone

Correct Answer: C. Integral action

Question 5

In a lead compensator C(s) = K(s + z)/(s + p), which relation gives lead behavior under the convention used in the chapter?

A. 0 < z < p

B. 0 < p < z

C. z = p = 0

D. p and z must both be negative

Correct Answer: A. 0 < z < p

Question 6

A 2% settling-time requirement of 2 s implies approximately what value of ζω_n?

A. 0.5 rad/s

B. 1 rad/s

C. 2 rad/s

D. 4 rad/s

Correct Answer: C. 2 rad/s

Question 7

What is the principal danger of integral action during actuator saturation?

A. Loss of low-frequency gain

B. Integrator windup

C. Removal of all feedback

D. Automatic increase in sensor bandwidth

Correct Answer: B. Integrator windup

Question 8

For L(s) as the loop transfer function, which expression is the sensitivity function?

A. S(s) = L(s)

B. S(s) = 1 + 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 9

A lag compensator is primarily used to:

A. Increase relative low-frequency gain and improve steady-state accuracy

B. Eliminate all measurement noise

C. Guarantee stability for every plant

D. Create a right-half-plane zero

Correct Answer: A. Increase relative low-frequency gain and improve steady-state accuracy

Question 10

Why is a practical derivative term filtered?

A. To increase integral accumulation

B. To prevent all actuator saturation

C. To limit high-frequency noise amplification

D. To remove the plant poles

Correct Answer: C. To limit high-frequency noise amplification