Feedback Control and Motion Control

Introduction and Context

Mechatronic machines combine mechanical structures, sensors, actuators, power electronics, computation, and control software. In an open-loop system, a command is converted into actuator action without measuring whether the desired result has occurred. Such systems are simple, but they cannot automatically compensate for load changes, friction, disturbances, component variation, or modelling error. Feedback control closes this gap by measuring the controlled variable, comparing it with a reference, and using the error to modify the actuator command.

Motion control is a particularly important application of feedback control. Industrial robots, computer numerical control (CNC) machines, servo presses, packaging equipment, autonomous vehicles, and semiconductor stages must regulate position, velocity, and torque accurately while meeting requirements for speed, smoothness, stability, and safety. A motion controller therefore does more than make a motor move: it shapes the dynamic response of a complete electromechanical system.

This chapter develops the analysis and design of feedback systems, including stability, transient and steady-state response, PID control, feedforward, cascade control, state-space methods, trajectory generation, and coordinated motion. The presentation assumes knowledge of basic differential equations, Laplace transforms, linear algebra, and electrical and mechanical fundamentals.

Learning Objectives

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

1. Explain the structure and purpose of a feedback control loop.
2. Derive closed-loop transfer functions and assess stability.
3. Analyse transient and steady-state response using standard performance measures.
4. Design and implement PID, feedforward, cascade, and state-feedback controllers.
5. Select practical tuning methods while considering saturation, noise, sampling, and actuator limits.
6. Generate feasible motion trajectories and coordinate multiple axes.
7. Evaluate an industrial motion-control architecture and identify safety and implementation risks.

1. Feedback-Control Fundamentals

A typical single-input, single-output loop contains a reference signal r(t), a comparator, a controller, an actuator, a plant, and a sensor. The measured output is fed back and compared with the reference. For negative feedback,

where e(t) is the error, r(t) is the reference, and y_m(t) is the measured output. If the sensor transfer function is H(s), the measured signal is Y_m(s) = H(s)Y(s). The controller is represented by C(s), and the plant by G(s). With zero initial conditions, the closed-loop transfer function is

The formula applies to a linear time-invariant loop with negative feedback. The product L(s) = C(s)G(s)H(s) is the loop transfer function, and 1 + L(s) is the characteristic equation. Units must be interpreted carefully: the loop gain must be dimensionless after the signal scaling and sensor units are included.

A disturbance may enter at several locations. For example, a load torque disturbance acting on a motor is reduced by the feedback loop, provided that the loop has sufficient gain at the disturbance frequency. Feedback does not eliminate disturbances instantaneously; it reacts after the disturbance affects the measured output.

Open-loop and closed-loop control

Open-loop control is appropriate when the plant is predictable, disturbances are small, and feedback would be expensive or impractical. A timed toaster is a simple example. Closed-loop control is preferred when accuracy, disturbance rejection, or automatic adaptation is important. A servo axis that must reach a commanded position despite changing payload is a closed-loop application.

Feedback introduces several trade-offs:

• Higher low-frequency gain generally reduces tracking error and disturbance sensitivity.
• Excessive gain can produce oscillation, actuator saturation, or instability.
• Greater bandwidth improves response speed but increases sensitivity to sensor noise and unmodelled dynamics.
• Measurement delay and computation delay reduce stability margins.

2. Modelling Electromechanical Motion

For a rotational drive, Newton's law gives

where J is the combined inertia in kg·m², ω is angular velocity in rad/s, T_m is motor torque in N·m, T_L is load torque in N·m, and B is viscous friction in N·m·s/rad. Since radian is dimensionless in SI units, both J(dω/dt) and torque have units of N·m.

Position θ is related to speed by

For a simplified torque-controlled motor with negligible friction and no flexible transmission,

This is a double-integrator model. Real drives also include electrical dynamics, torque-loop dynamics, backlash, Coulomb friction, compliance, sampling delay, and saturation. A model is useful only within the range where its assumptions are reasonable.

For a translational axis,

where m is mass, v is velocity, F is applied force, F_L is load force, and b is viscous damping. Position x satisfies dx/dt = v.

3. Stability and Dynamic Response

Stability concept

A feedback system is stable when every bounded input produces a bounded output and natural responses decay with time. For a continuous-time rational system, asymptotic stability requires all closed-loop poles to lie strictly in the left half of the complex s-plane. Poles on the imaginary axis generally indicate marginal or sustained oscillation, while any right-half-plane pole causes an exponentially growing response.

The characteristic polynomial is obtained from

For low-order systems, pole locations can be found directly. The Routh–Hurwitz criterion provides an algebraic test without explicitly calculating roots. For a second-order polynomial

with real coefficients, asymptotic stability requires a₁ > 0 and a₀ > 0. These conditions are necessary and sufficient for this second-order monic polynomial.

In sampled-data control, stability is assessed in the z-plane. All discrete-time poles must lie inside the unit circle, meaning |z| < 1. A continuous pole s maps approximately to z = e^(sT_s), where T_s is the sampling period, under zero-order-hold discretisation. This relation explains why a left-half-plane pole maps inside the unit circle.

Standard second-order response

A common closed-loop model is

where ω_n is the undamped natural frequency in rad/s and ζ is the damping ratio. For 0 < ζ < 1, the poles are complex and the response is underdamped. The damped natural frequency is

For a unit-step response, the percent overshoot is

where M_p is expressed as a percentage. This equation applies to the standard second-order form with 0 < ζ < 1 and no additional zeros that materially alter the response.

The approximate 2% settling time is

and the approximate 5% settling time is T_s ≈ 3/(ζω_n). These are approximations based on the exponential envelope; actual systems with zeros, higher-order poles, saturation, or nonlinear friction may differ.

For example, if ζ = 0.5 and ω_n = 10 rad/s, the overshoot is

M_p = 100 exp[−0.5π/√(0.75)] ≈ 16.3%.

The 2% settling-time estimate is T_s ≈ 4/(0.5 × 10) = 0.8 s.

Steady-state error

The final-value theorem states

provided all poles of sE(s) are in the left half-plane, except possibly a simple pole at the origin as permitted by the theorem's conditions. It must not be applied to an unstable system.

For unity feedback, the open-loop system type is the number of pure integrators in C(s)G(s). 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 for ideal conditions. Increasing type improves low-frequency tracking but can reduce stability and increase windup risk.

For a unity-feedback loop, the static error constants are

for step, ramp, and parabolic inputs respectively. These constants are useful for ideal linear systems, but practical performance must also account for limits, friction, quantisation, and reference filtering.

4. PID Control

The ideal continuous-time PID controller is

where u(t) is the controller output, e(t) is error, K_p is proportional gain, K_i is integral gain in units consistent with u/(error·s), and K_d is derivative gain in units consistent with u·s/error. In the Laplace domain,

The proportional term responds to present error, the integral term removes persistent error, and the derivative term anticipates error change. A pure derivative is sensitive to high-frequency noise, so practical implementations use a filtered derivative:

where N is a dimensionless filter factor, usually selected as a moderate multiple of the desired derivative corner frequency. The exact choice depends on sampling, sensor noise, and bandwidth.

Practical PID implementation

A digital controller with sample period T_s approximates the integral using a numerical sum:

A backward-difference derivative is

These approximations are not interchangeable with the continuous controller at very high frequencies. The sample period should be sufficiently smaller than the dominant closed-loop time scale, and the resulting discrete controller should be tested for delay and numerical effects.

A major practical issue is integral windup. If the actuator saturates at u_max while the integral continues to accumulate, the controller may take a long time to recover after the error changes sign. Anti-windup methods include conditional integration and back-calculation. In back-calculation, the integrator is corrected using the difference between saturated and unsaturated commands:

where K_aw is an anti-windup gain, u_unsat is the calculated command, and u_sat is the limited command. Sign conventions must be implemented consistently.

Derivative kick occurs when the reference changes abruptly because differentiating error produces a large impulse-like action. A common remedy is to apply derivative action to the measured output rather than to the error, or to filter the reference. Proportional-on-measurement is another implementation choice that reduces command-induced kick.

Tuning approaches

Manual tuning begins with a conservative proportional gain, followed by integral and derivative adjustment. A process-reaction or ultimate-gain method can provide initial settings for suitable plants, but it should not be applied blindly to unstable, highly nonlinear, delayed, or safety-critical systems. Model-based tuning uses an identified plant and desired bandwidth or pole locations. In motion systems, inner current and velocity loops are normally tuned before the outer position loop.

5. Feedforward and Cascade Control

Feedback corrects error after it appears. Feedforward uses a model of the plant to predict the actuator effort required for a command or disturbance. For a translational axis with desired acceleration a_d and velocity v_d, a basic command is

where m is mass, b is viscous damping, and F_friction is an estimated friction force. For a rotary axis,

where α_d is desired angular acceleration. Feedforward can greatly reduce tracking error, but it cannot by itself correct modelling error, payload variation, or unexpected disturbances. Feedback remains necessary.

Cascade control uses nested loops. A common servo architecture has an outer position loop, a middle velocity loop, and an inner current or torque loop. The inner loop should be substantially faster than the outer loop so that the outer controller sees a predictable effective plant. If the inner loop is slow or poorly tuned, the assumed separation of time scales fails and the overall system may oscillate.

A disturbance should be measured and fed forward only when the measurement is reliable and the path from disturbance to output is understood. Excessive feedforward can cause actuator saturation and may amplify model errors.

6. State-Space Control

Transfer functions are convenient for single-input, single-output linear systems. State-space models are more suitable for multivariable systems and systems with internal states. A continuous linear model is

where x is the state vector, u is the input vector, y is the output vector, and A, B, C, and D are system matrices of compatible dimensions.

With state feedback,

where K is the feedback-gain matrix, r is the reference, and N is a precompensator or reference gain. The closed-loop state matrix is A − BK. Pole placement selects K so that A − BK has desired eigenvalues, provided the pair (A, B) is controllable. A controllability matrix is

For an n-state single-input system, full rank n is the standard controllability test.

If all states cannot be measured, an observer can estimate them:

where x̂ is the estimated state and L is the observer gain. The estimation error dynamics are governed by A − LC. The observer must be designed with regard to sensor noise and model uncertainty; placing observer poles extremely fast can amplify noise.

Linear quadratic regulation (LQR) selects K by minimizing a quadratic cost such as

where Q is positive semidefinite and R is positive definite. Larger entries in Q penalize state deviations more strongly, while larger entries in R penalize control effort. LQR is powerful, but its result depends on model quality, weighting choices, constraints, and operating-point linearisation.

7. Motion Trajectory Generation

A position command should be a physically feasible trajectory rather than an instantaneous step. A trajectory generator limits position, velocity, acceleration, and sometimes jerk. A trapezoidal velocity profile consists of acceleration, constant velocity, and deceleration phases. For a move of distance D, acceleration magnitude a, and maximum velocity v_max, the triangular or trapezoidal form depends on whether the required distance permits reaching v_max.

If the profile is triangular and acceleration and deceleration magnitudes are equal, the peak velocity is

The total time is

These equations assume zero initial and final velocity, constant acceleration magnitude, and no cruise interval. If √(aD) > v_max, the axis reaches the velocity limit and a trapezoidal profile is required. The acceleration distance is v_max²/(2a), so the cruise distance is D − v_max²/a, provided that this quantity is nonnegative.

S-curve profiles limit jerk, defined as

Reducing jerk improves mechanical smoothness, decreases excitation of flexible structures, and can reduce acoustic noise. The trade-off is longer motion time or more complex trajectory generation.

Interpolation must also respect actuator limits, coordinate transformations, workspace constraints, and collision boundaries. A controller should not be expected to compensate for an infeasible trajectory.

8. Coordinated Multi-Axis Motion

In a multi-axis machine, coordination means that axes follow a common time base and maintain the required geometric relationship. In point-to-point motion, each axis may reach its endpoint simultaneously by scaling its profile duration. In contouring, the relevant error is deviation from the desired path, not merely the individual axis errors.

For a planar commanded path x_d(t), y_d(t), a tangent direction is related to the velocity vector [ẋ_d, ẏ_d]. Cross-coupling control can use the error in one axis to compensate for error in the other, but it requires a reliable kinematic model and careful bandwidth design. Kinematic transformations convert task-space commands into joint-space commands. Near singular configurations, small Cartesian demands may require very large joint velocities; a trajectory planner must detect or avoid these conditions.

Time synchronisation, deterministic communication, encoder resolution, interpolation rate, and mechanical compliance all influence coordinated accuracy. EtherCAT, PROFINET IRT, and similar industrial networks can support deterministic updates, but network capability does not remove the need for correct control design.

9. Worked Design Example: Servo Position Loop

Consider an ideal torque-controlled rotary axis with J = 0.01 kg·m². Use a proportional position controller and assume the torque command is u = K_p(θ_r − θ). The plant is Θ(s)/T(s) = 1/(Js²). The closed-loop characteristic equation is

This produces undamped imaginary-axis poles, so proportional position control alone is not asymptotically stable in this ideal model. Add velocity feedback, producing torque

Since ω = dθ/dt, the characteristic equation becomes

Dividing by J and comparing with the standard second-order form gives

Suppose the design target is ω_n = 20 rad/s and ζ = 0.7. Then

K_p = Jω_n² = 0.01 × 400 = 4 N·m/rad.

The required velocity gain is

K_v = 2ζ√(JK_p) = 2(0.7)√(0.01 × 4) = 0.28 N·m·s/rad.

The approximate 2% settling time is

T_s ≈ 4/(ζω_n) = 4/(0.7 × 20) ≈ 0.286 s.

The expected overshoot is

M_p = 100 exp[−0.7π/√(1 − 0.7²)] ≈ 4.6%.

This calculation is an ideal design starting point. Real implementation must check torque limits, encoder noise, sampling delay, flexible modes, friction, velocity limits, and whether the commanded acceleration is feasible.

10. Applied Case: Illustrative Packaging-Axis Upgrade

Background

A fictional packaging machine uses a servo-driven indexing axis to position cartons beneath a filling head. The original system uses a position PID controller with step-like commands. Operators report overshoot, settling delay, and occasional carton misalignment when product mass changes.

Problem analysis

The step command excites mechanical compliance, while the integral term accumulates during torque saturation. The velocity loop is not sufficiently faster than the position loop, and the trajectory generator does not limit acceleration. The machine therefore experiences both control-structure and motion-profile problems.

Approach and solution

The engineering team first identifies inertia at two representative payloads and measures the frequency response. The current loop is tuned first, followed by the velocity loop. The position loop bandwidth is then selected below the dominant flexible resonance. A jerk-limited S-curve replaces the step command. Velocity and acceleration feedforward are added using identified inertia and viscous-friction values. Anti-windup is enabled, and the derivative signal is filtered.

A safety-rated motion controller supervises the drive-enable circuit, emergency stop, safe torque off, guarding, and speed limits. The controller logs following error, saturation, peak torque, and drive faults. Acceptance tests include unloaded and loaded operation, repeated indexing, loss-of-feedback simulation, and controlled power interruption.

Outcome

In this illustrative case, overshoot is reduced, settling becomes repeatable across payloads, and peak torque decreases because the feedforward term supplies much of the predictable effort. The remaining feedback action compensates for uncertainty and disturbances. The improvement is not attributed to PID gains alone; it results from coordinated plant modelling, trajectory planning, nested-loop tuning, and safety verification.

Learning points

1. A feasible trajectory can prevent problems that cannot be solved robustly by feedback alone.
2. Inner-loop performance establishes the foundation for outer-loop tuning.
3. Feedforward improves nominal tracking but does not replace feedback.
4. Saturation and windup must be treated as design conditions, not exceptional events.
5. Performance claims require tests over payload, temperature, speed, and fault conditions.

11. Limitations, Safety, and Contemporary Practice

Linear controllers are local approximations. Friction, backlash, saturation, dead zones, changing payload, thermal drift, and flexible modes create nonlinear or time-varying behaviour. Robust design therefore includes gain and phase margins, disturbance testing, parameter variation, and conservative operating boundaries.

Motion systems can cause crushing, impact, cutting, or unexpected restart hazards. Functional safety must be designed independently of ordinary performance control where required. Emergency stop, guard interlocking, safe torque off, safe limited speed, and safe limited position may involve standards such as ISO 13849, IEC 62061, IEC 61800-5-2, or applicable national regulations. The exact compliance route depends on machine type, jurisdiction, risk assessment, and certification responsibilities.

Data-driven and AI-based methods are increasingly used for system identification, friction compensation, predictive maintenance, and anomaly detection. They should not be allowed to bypass deterministic safety functions without a validated safety architecture. A learned controller must be tested against distribution shift, sensor failure, adversarial or corrupted data, and unmodelled operating conditions. Explainability and traceability are especially important when control decisions affect personnel safety or expensive equipment.

Best practice is to maintain a documented signal-flow diagram, version-controlled parameters, identified sampling rates, actuator limits, test procedures, and fault-response requirements. Commissioning should progress from simulation to low-energy tests, unloaded motion, loaded motion, and validated production operation. Every tuning change should be linked to measured evidence.

Chapter Summary

Feedback control compares a reference with a measured output and modifies actuator action to reduce error and reject disturbances. The closed-loop characteristic equation determines stability, while pole locations influence speed, damping, and oscillation. Standard second-order relationships provide useful estimates of overshoot and settling time, but their assumptions must be respected.

PID control combines proportional, integral, and derivative action. Practical implementations require derivative filtering, anti-windup, output limits, sampling awareness, and protection against noise and derivative kick. Feedforward uses a plant model to improve nominal tracking, while cascade control places faster current and velocity loops inside a slower position loop.

State-space methods support multivariable control, pole placement, observers, and LQR design. Motion control also requires trajectory generation: velocity, acceleration, and jerk limits prevent infeasible commands and reduce mechanical excitation. Coordinated axes require common timing, kinematic awareness, contour-error analysis, and attention to singularities and communication determinism.

Industrial success depends on the complete architecture rather than one controller formula. Modelling, tuning, trajectory planning, actuator limits, safety functions, validation, and lifecycle documentation must be treated as an integrated engineering process.

Key Terms

Feedback control: Control in which measured output is compared with a reference to form corrective action.

Open-loop control: Control performed without using measured output for correction.

Closed-loop transfer function: Mathematical relationship between reference and output in a feedback system.

Loop transfer function: Product of controller, plant, and feedback-path transfer functions.

Stability: Property by which bounded inputs produce bounded outputs and natural responses do not diverge.

Damping ratio: Dimensionless parameter describing the damping of a standard second-order system.

Natural frequency: Characteristic oscillation frequency of a second-order system.

Steady-state error: Difference between desired and actual output after transients have decayed.

PID controller: Controller using proportional, integral, and derivative actions.

Integral windup: Excessive accumulation of integral action during actuator saturation.

Feedforward: Control action calculated from a model or measured disturbance before error develops.

Cascade control: Nested control structure with inner and outer loops.

State: Variable needed, with the input, to describe future system behaviour.

Controllability: Ability to move system states through suitable inputs.

Observer: Dynamic estimator that reconstructs unmeasured states.

Trajectory generation: Creation of time-based position, velocity, acceleration, or jerk commands.

Jerk: Time derivative of acceleration.

Contouring error: Deviation of a multi-axis motion from its desired geometric path.

Bandwidth: Frequency range over which a control system responds effectively.

Safe torque off: Safety function that prevents torque-producing energy from being applied by a drive.

Review Questions

Short Answer Questions

1. What is the purpose of negative feedback in a motion-control system?
2. State the closed-loop transfer function for a plant G(s), controller C(s), and feedback H(s).
3. Why is integral windup undesirable in a PID controller?
4. What is the purpose of a cascade velocity loop inside a position loop?
5. Define jerk and explain why it matters in machine motion.

Long Answer Questions

1. Explain the relationship between pole location, stability, damping, overshoot, and settling time.
2. Describe a practical procedure for tuning a nested servo system from the inner current loop to the outer position loop.
3. Compare feedback and feedforward control, including their benefits, limitations, and appropriate applications.
4. Explain state feedback, controllability, and observer-based control using state-space notation.
5. Discuss the engineering and safety considerations involved in deploying a coordinated multi-axis motion system.

Analytical / Application Questions

1. A rotary axis has J = 0.02 kg·m². Determine K_p and K_v for a second-order position response with ω_n = 15 rad/s and ζ = 0.8. State the assumptions of the calculation.
2. A move has D = 0.5 m, acceleration limit a = 2 m/s², and velocity limit v_max = 1.5 m/s. Determine whether the ideal zero-velocity move is triangular or trapezoidal, and justify the result.
3. A packaging axis exhibits good performance without load but oscillates with a heavy payload. Propose a structured diagnostic and redesign process involving modelling, trajectory limits, cascade tuning, feedforward, saturation handling, and validation.

Multiple Choice Questions with Answers

Question 1

For a negative-feedback system with controller C(s), plant G(s), and feedback H(s), which expression is the closed-loop transfer function?

A. C(s)G(s)H(s)

B. C(s)G(s) / [1 + C(s)G(s)H(s)]

C. C(s)G(s) / [1 − C(s)G(s)H(s)]

D. [1 + C(s)G(s)] / H(s)

Correct Answer: B. C(s)G(s) / [1 + C(s)G(s)H(s)]

Question 2

For the standard second-order system, what condition gives asymptotic stability for s² + a₁s + a₀?

A. a₁ < 0 and a₀ < 0

B. a₁ = 0 and a₀ > 0

C. a₁ > 0 and a₀ > 0

D. a₁ > 0 and a₀ < 0

Correct Answer: C. a₁ > 0 and a₀ > 0

Question 3

A standard second-order system has ζ = 0.5 and ω_n = 10 rad/s. What is its approximate 2% settling time?

A. 0.08 s

B. 0.20 s

C. 0.80 s

D. 5.00 s

Correct Answer: C. 0.80 s

Question 4

Which PID action is primarily responsible for eliminating persistent steady-state error in an ideal linear system?

A. Proportional action

B. Integral action

C. Derivative action

D. Measurement filtering

Correct Answer: B. Integral action

Question 5

What is the main purpose of anti-windup logic?

A. To increase sensor quantisation

B. To prevent excessive integral accumulation during actuator saturation

C. To remove all mechanical friction

D. To make the plant open loop

Correct Answer: B. To prevent excessive integral accumulation during actuator saturation

Question 6

Which expression represents basic inertial and viscous feedforward torque for a rotary axis?

A. T_ff = Jα_d + Bω_d

B. T_ff = J/α_d + B/ω_d

C. T_ff = α_d/(J + B)

D. T_ff = Jω_d + Bα_d

Correct Answer: A. T_ff = Jα_d + Bω_d

Question 7

In a conventional servo cascade, which loop is normally designed to be fastest?

A. Position loop

B. Velocity loop

C. Current or torque loop

D. Trajectory-planning loop

Correct Answer: C. Current or torque loop

Question 8

For a state-space model, what does full rank of the controllability matrix indicate?

A. All outputs are noise-free

B. The system is controllable under the stated model

C. The system is necessarily stable

D. The actuator has unlimited power

Correct Answer: B. The system is controllable under the stated model

Question 9

For a triangular move beginning and ending at zero velocity, with distance D and acceleration magnitude a, what is the peak velocity?

A. aD

B. D/a

C. √(aD)

D. 2D/a

Correct Answer: C. √(aD)

Question 10

What is the principal purpose of limiting jerk in an S-curve trajectory?

A. To increase encoder resolution

B. To reduce abrupt changes in acceleration and mechanical excitation

C. To eliminate the need for feedback

D. To guarantee zero contouring error

Correct Answer: B. To reduce abrupt changes in acceleration and mechanical excitation