Feedback Control Fundamentals

Introduction and Context

Feedback control is the foundation of modern robotics. A robot must move accurately despite changing payloads, friction, modeling errors, sensor noise, and external disturbances. Feedback compares a measured output with a desired reference and uses the error to modify the actuator command. This closed-loop arrangement is fundamentally different from open-loop control, in which the controller does not measure the result of its action.

For a standard negative-feedback system, let the reference be r(t), output be y(t), error be e(t), controller be C(s), and plant be P(s). With sensor transfer function H(s),

and

Therefore, the closed-loop transfer function is

The product L(s) = C(s)P(s)H(s) is the loop transfer function, and 1 + L(s) = 0 is the characteristic equation for closed-loop poles. These poles determine stability and strongly influence transient response.

Learning Objectives

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

1. Derive closed-loop transfer functions and characteristic equations.
2. Assess stability using pole locations and the Routh–Hurwitz criterion.
3. Relate pole locations to rise time, overshoot, settling time, and damping.
4. Calculate steady-state error for standard reference inputs.
5. Explain sensitivity and disturbance rejection in feedback systems.
6. Use root-locus, Bode, and Nyquist methods to analyze and design control systems.
7. Select practical control strategies for robotic mechanisms.

1. Fundamental Feedback Concepts

A unity-feedback system has H(s) = 1. Its closed-loop function is

where L(s) = C(s)P(s). The sensitivity function is

and the complementary sensitivity function is

These expressions assume linear time-invariant models and zero initial conditions. They are normally derived in the Laplace domain, where s has units of inverse seconds for physical dynamic systems.

A feedback loop has several important signals:

Feedback provides three major benefits: tracking of commands, rejection of disturbances, and reduced sensitivity to plant uncertainty. It also introduces risks, particularly instability, actuator saturation, sensor noise amplification, and excessive control effort.

2. Stability in the Time Domain

For a continuous-time linear system, asymptotic stability requires every closed-loop pole to lie strictly in the left half of the complex s-plane. A pole at s = σ + jω contributes a term proportional to e^(σt). Thus, σ < 0 produces decay, σ = 0 produces sustained oscillation or nondecaying behavior, and σ > 0 produces divergence.

For a characteristic polynomial

with real coefficients and a_n > 0, the Routh–Hurwitz test determines the number of right-half-plane roots without explicitly solving the polynomial. The first column of the Routh array is inspected: the number of sign changes equals the number of right-half-plane poles. A necessary condition for stability is that all coefficients be positive, although positive coefficients alone are not sufficient for order three and higher.

For a cubic polynomial

stability requires

These conditions apply to a third-order polynomial with no special degeneracies. A zero row or zero first-column element requires the standard special procedures of the Routh algorithm.

Worked Example 1: Routh Stability

Consider the characteristic equation

where K is a dimensionless positive gain. The cubic conditions give K > 0 and

Therefore, the stable range is

At K = 20, a pair of roots lies on the imaginary axis and the system is marginal at the stability boundary. For K > 20, two roots enter the right half-plane and the system is unstable.

3. Transient Response

The transient response describes the behavior immediately after a change in command. The dominant poles are those closest to the imaginary axis because their associated modes decay most slowly.

A common second-order model is

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

for 0 < ζ < 1. The damped natural frequency is ω_d = ω_n√(1 − ζ²).

For a unit-step input and an underdamped second-order system, useful approximations are:

Here M_p is percent overshoot, t_s(2%) is the 2% settling time, and t_r is the 0–100% rise time. These formulas apply primarily to a dominant second-order response with 0 < ζ < 1; higher-order zeros and nondominant poles can substantially alter the results.

For ζ = 0.5 and ω_n = 4 rad/s,

and

Increasing ζ generally reduces overshoot but can increase rise time. Increasing ω_n usually makes the response faster, but requires greater actuator bandwidth and often greater control effort.

A pole at −2 ± j3 has natural frequency

and damping ratio

The pole is stable and underdamped.

4. Steady-State Error

The steady-state error is the limiting tracking error after transients have decayed:

Using the final-value theorem,

provided all poles of sE(s) lie in the left half-plane, except possibly a simple pole at the origin as permitted by the theorem.

For unity feedback,

The system type is the number of pure integrators in L(s). For a stable unity-feedback system, the static error constants are

For step, ramp, and parabolic inputs respectively:

The input amplitudes must be included when they are not unit inputs. These results apply to stable systems and unity feedback; nonunity feedback requires using the complete error transfer function.

Worked Example 2: Steady-State Error

Let

This is a type-1 system. For a unit step, e_ss = 0. For a unit ramp,

so

The integrator eliminates constant position error but does not eliminate ramp error.

5. Disturbance Rejection and Sensitivity

Suppose a disturbance d enters additively at the plant input and measurement noise n enters the sensor. For unity feedback, the output can be expressed conceptually as

Thus, low-frequency loop gain gives S(s) ≈ 0 and T(s) ≈ 1. This improves reference tracking and rejects plant-input disturbances. However, at high frequency, T(s) should generally decrease so that sensor noise is not transmitted strongly to the output.

For a multiplicative plant uncertainty P_actual(s) = P(s)[1 + Δ(s)], the approximate relative effect of uncertainty is governed by T(s)Δ(s). Robust performance therefore requires low sensitivity in the frequency range where uncertainty is important. A fundamental trade-off is that sensitivity cannot be made small at every frequency; reducing sensitivity in one region generally increases it elsewhere. This is often called the waterbed effect.

The sensitivity of closed-loop output to plant gain changes can be written approximately as

for small perturbations under compatible loop definitions. This demonstrates why high loop gain reduces the effect of plant variation at frequencies where |L| is large.

Integral action is valuable for rejecting constant disturbances, but it can cause windup when an actuator saturates. Anti-windup schemes commonly freeze, back-calculate, or condition the integrator during saturation.

6. Root-Locus Analysis

The root locus shows how closed-loop poles move as a real gain K varies. For unity feedback with L(s) = KG(s), the characteristic equation is

A point s lies on the root locus when the angle condition is

and the corresponding gain is obtained from the magnitude condition

The locus begins at open-loop poles when K = 0 and ends at open-loop zeros or infinity as K becomes large. If there are n poles and m finite zeros, n − m branches go to infinity along asymptotes. Their centroid is

where pole and zero locations are algebraic sums in the s-plane. Asymptote angles are

for q = 0, 1, …, n − m − 1.

Root locus is especially useful for choosing gain to meet damping-ratio or settling-time requirements. It assumes a parameterized characteristic equation and is less convenient when several independent controller parameters must be selected.

7. Frequency Response

The frequency response is obtained by evaluating the transfer function at s = jω. The magnitude |L(jω)| describes amplification, while the phase ∠L(jω) describes phase shift. Frequency methods are valuable because they expose bandwidth, resonance, noise sensitivity, and stability margins.

The gain crossover frequency ω_gc satisfies

The phase margin is

The phase crossover frequency ω_pc satisfies ∠L(jω_pc) = −180°. The gain margin is

or, in decibels,

These definitions assume a conventional negative-feedback loop and a stable open-loop transfer function when interpreting classical margins. Positive phase and gain margins generally indicate increased relative stability, but margins alone do not guarantee satisfactory performance for all plant uncertainties.

Bode Plot Rules

A first-order pole 1/(1 + s/ω_p) contributes approximately 0 dB below ω_p, a −20 dB/decade slope above ω_p, and phase approaching −90°. A first-order zero contributes +20 dB/decade above its break frequency and phase approaching +90°. An integrator contributes −20 dB/decade and −90° phase at all positive frequencies. Exact plots use arctangent phase expressions rather than only asymptotic approximations.

Bandwidth is often associated with the frequency at which the closed-loop magnitude falls by 3 dB from its low-frequency value. Greater bandwidth can improve speed but may transmit more noise and demand more actuator power.

8. Nyquist Stability Criterion

The Nyquist criterion determines closed-loop stability from the mapping of the open-loop frequency response around a contour enclosing the right-half s-plane. For L(s), the critical point is −1 + j0. If P is the number of right-half-plane poles of L(s), N is the net number of clockwise encirclements of −1, and Z is the number of right-half-plane zeros of 1 + L(s), then, under a stated contour and sign convention,

Closed-loop stability requires Z = 0. When L(s) has no right-half-plane poles, the Nyquist plot must not encircle −1 for stability, provided there are no problematic poles on the contour. Poles on the imaginary axis require indented contours and careful interpretation.

Nyquist is powerful for systems with delays, high order, or unusual frequency behavior. Unlike a simple phase-margin statement, it can account explicitly for open-loop unstable poles.

9. Illustrative Robotic Application: Joint Position Control

Background

Consider one revolute robot joint driven by a motor and gearbox. The objective is to track a commanded angle while resisting load torque and friction changes. A simplified plant is

where J is reflected rotational inertia in kg·m² and b is viscous damping in N·m·s/rad. The factor 1/s represents position integration.

Problem

Suppose a proportional controller C(s) = K_p is used with unity feedback. The closed-loop characteristic equation is

With J = 0.01 kg·m² and b = 0.1 N·m·s/rad, increasing K_p raises the natural frequency, but the damping ratio is

Thus, excessive proportional gain can make the joint more oscillatory even while increasing speed.

Analysis and Approach

For K_p = 1 N·m/rad,

The estimated overshoot is approximately 16.3% and the 2% settling time is approximately 0.8 s. A proportional controller gives zero step position error because the plant contains one integrator, but it cannot eliminate constant load-torque error in every practical configuration and may produce unacceptable transient behavior.

A velocity-feedback or derivative term can increase effective damping. Integral action can reject constant torque disturbances, but it must be protected against saturation using anti-windup. A practical design should also include actuator limits, encoder quantization, sample delay, gearbox backlash, friction, and safety limits on position and torque.

Outcome and Learning Points

The example illustrates a central design trade-off: gain improves stiffness and speed but can reduce damping margin. A successful robotic controller is therefore selected using both time-domain requirements and frequency-domain margins. Simulation should be followed by hardware testing with torque limits and emergency stopping procedures.

10. Design Practice, Limitations, and Contemporary Tools

A sound workflow is to establish a physical model, identify dominant dynamics, specify tracking and disturbance requirements, select a controller structure, check pole locations, evaluate time response, inspect Bode or Nyquist margins, and validate actuator and sensor constraints. Model uncertainty should be tested through parameter sweeps and, where appropriate, Monte Carlo simulation.

Linear analysis is local: it may not predict behavior under friction dead zones, backlash, saturation, contact transitions, flexible links, or large-angle nonlinearities. Sampling introduces discrete-time poles and computational delay; continuous-time margins should not be used blindly for a digital controller. Discretization, sensor filtering, and real-time execution must be included in the final model.

Modern robotic systems increasingly combine classical feedback with state estimation, model-based control, adaptive methods, and learning-based components. Machine learning may assist plant identification or gain scheduling, but safety-critical feedback should retain verifiable bounds, fault detection, human override, and traceable validation. Engineers must prevent unsafe autonomous motion, protect recorded sensor data, and comply with relevant machinery and functional-safety standards.

Chapter Summary

Feedback control compares desired and measured behavior to regulate a system. The closed-loop denominator 1 + L(s) determines poles and therefore stability and transient behavior. Stable continuous-time poles lie in the left half-plane; the Routh–Hurwitz criterion provides an algebraic stability test. Second-order parameters connect pole locations with damping, natural frequency, overshoot, and settling time. System type and static error constants predict steady-state tracking errors for step, ramp, and parabolic inputs. Sensitivity explains robustness, while complementary sensitivity describes command transmission and noise behavior. Root locus shows pole movement with gain, Bode plots provide gain and phase information, and Nyquist analysis determines stability through encirclements of −1. Robotic controller design must combine these analytical tools with actuator limits, uncertainty, sampling, safety, and experimental validation.

Key Terms

Feedback: Using measured output information to modify a control input.

Open-loop system: A system without output measurement in the control decision.

Closed-loop system: A system whose control action depends on measured output.

Loop transfer function: The product of forward-path and feedback-path transfer functions.

Characteristic equation: The equation whose roots are closed-loop poles.

Asymptotic stability: Decay of all natural-response modes to zero.

Dominant poles: Poles whose modes most strongly influence the response.

Damping ratio: A dimensionless measure of oscillatory decay.

Natural frequency: The characteristic frequency of a second-order mode.

Overshoot: The amount by which a response exceeds its final value.

Settling time: Time required for a response to remain within a specified tolerance band.

Steady-state error: The limiting tracking error after transients vanish.

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

Sensitivity function: S(s) = 1/[1 + L(s)].

Complementary sensitivity: T(s) = L(s)/[1 + L(s)].

Root locus: Locus of closed-loop poles as a gain varies.

Gain margin: Gain increase required to reach instability at phase crossover.

Phase margin: Additional phase lag required to reach instability at gain crossover.

Nyquist criterion: A frequency-domain test based on encirclements of −1.

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

Review Questions

Short Answer Questions

1. What is the characteristic equation of a negative-feedback system?
2. What condition on continuous-time pole locations indicates asymptotic stability?
3. Define damping ratio and natural frequency for a second-order system.
4. What is meant by system type?
5. Why can integral action cause windup?

Long Answer Questions

1. Derive the closed-loop transfer function for a nonunity feedback system and explain each signal.
2. Explain the Routh–Hurwitz procedure and its interpretation of sign changes.
3. Discuss the relationship between pole location and transient response.
4. Compare root-locus, Bode, and Nyquist methods, including their advantages and limitations.
5. Explain how sensitivity and complementary sensitivity affect robustness, tracking, disturbance rejection, and noise transmission.

Analytical / Application Questions

1. For L(s) = K/[s(s + 3)], determine the system type and the unit-step and unit-ramp steady-state errors for a stable positive K.
2. A second-order design requires less than 10% overshoot. Estimate the minimum damping ratio using the standard overshoot formula and discuss what additional information is required to select the natural frequency.
3. A robot joint becomes oscillatory after its proportional gain is doubled. Use pole and frequency-response concepts to explain two possible causes and propose two corrective actions.

Multiple Choice Questions with Answers

Question 1

For a negative-feedback system, which equation determines the closed-loop poles?

A. 1 − L(s) = 0

B. 1 + L(s) = 0

C. L(s) = 0

D. C(s) − P(s) = 0

Correct Answer: B. 1 + L(s) = 0

Question 2

A continuous-time linear system is asymptotically stable when all closed-loop poles are located in the:

A. Right half of the s-plane

B. Imaginary axis only

C. Left half of the s-plane

D. Origin

Correct Answer: C. Left half of the s-plane

Question 3

For a stable unity-feedback type-1 system subjected to a unit-step input, the steady-state error is:

A. Zero

B. One

C. Infinite

D. Equal to the phase margin

Correct Answer: A. Zero

Question 4

The approximate 2% settling time of a dominant second-order system is:

A. ζω_n/4

B. 4ζω_n

C. 4/(ζω_n)

D. ω_n/ζ

Correct Answer: C. 4/(ζω_n)

Question 5

The sensitivity function for a negative-feedback loop is:

A. L(s)

B. 1 + L(s)

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

D. 1 − L(s)

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

Question 6

On a root locus, the magnitude condition for L(s) = KG(s) is:

A. K = |G(s)|

B. K = 1/|G(s)|

C. K = ∠G(s)

D. K = 1 + |G(s)|

Correct Answer: B. K = 1/|G(s)|

Question 7

The gain crossover frequency is the frequency at which:

A. The phase is zero

B. The magnitude of the loop transfer function is one

C. The closed-loop output is zero

D. The gain margin is infinite

Correct Answer: B. The magnitude of the loop transfer function is one

Question 8

A phase margin is measured at the frequency where:

A. |L(jω)| = 1

B. The phase is 0°

C. The phase is −90°

D. The output reaches its peak value

Correct Answer: A. |L(jω)| = 1

Question 9

For L(s) = 10/[s(s + 2)], what is the velocity error constant?

A. 0.2

B. 2

C. 5

D. 10

Correct Answer: C. 5

Question 10

The principal critical point in the Nyquist criterion for conventional negative feedback is:

A. 0 + j0

B. 1 + j0

C. −1 + j0

D. −j1

Correct Answer: C. −1 + j0