Drive Modeling and Feedback Control

Introduction

An electrical drive is an energy-conversion system that regulates mechanical motion. It normally contains a power converter, an electric motor, a mechanical load, sensors, signal-conditioning circuits, and a controller. The purpose of modeling is to express the relationships among electrical variables, electromagnetic torque, speed, and position in a form suitable for analysis and design.

Drive models support four important activities:

1. predicting steady-state and transient performance;
2. assessing stability and damping;
3. selecting controller gains and bandwidths; and
4. implementing reliable digital control.

This chapter develops models for a separately excited DC motor and then generalizes the modeling approach to converter-fed drives, mechanical loads, sensors, and cascaded feedback loops. The same principles apply, with suitable changes, to induction-motor and permanent-magnet synchronous-motor drives.

Learning Objectives

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

• formulate state-space and transfer-function models of converter-fed motors;
• combine electrical, mechanical, sensor, and load models into a block diagram;
• identify poles, zeros, damping, bandwidth, and stability limitations;
• design current, torque, speed, and position loops using PI or PID controllers;
• explain why cascaded loops are normally tuned from the innermost loop outward;
• recognize practical limitations such as saturation, delay, noise, friction, and parameter variation.

1. Modeling Structure of an Electrical Drive

A useful drive model separates the system into five subsystems:

A model is always an approximation. It is valid only over a stated operating range and under stated assumptions. For example, a linear model may be accurate near one operating point but inadequate during magnetic saturation, inverter overmodulation, or large speed reversal.

A standard workflow is:

1. choose variables and reference directions;
2. write physical laws for each subsystem;
3. select state variables;
4. linearize nonlinear equations if necessary;
5. obtain state-space and transfer-function forms;
6. interconnect blocks and analyze the closed loop;
7. validate the model against measurements.

2. Separately Excited DC Motor Model

Consider a separately excited DC motor with constant field flux. Let [[M:v_a]] be armature voltage, [[M:i_a]] armature current, [[M:R_a]] armature resistance, [[M:L_a]] armature inductance, [[M:e_a]] back electromotive force, [[M:T_e]] electromagnetic torque, [[M:J]] total inertia, [[M:B]] viscous friction coefficient, [[M:T_L]] load torque, [[M:ω_m]] mechanical angular speed, and [[M:θ_m]] angular position.

2.1 Electrical equation

Kirchhoff’s voltage law gives

For constant flux,

where [[M:K_e]] is the back-emf constant. In SI units, [[M:K_e]] has units V·s/rad, numerically equivalent to N·m/A when consistent motor conventions are used.

2.2 Torque equation

The electromagnetic torque is

where [[M:K_t]] is the torque constant in N·m/A. With SI units and the same flux definition, [[M:K_t]] and [[M:K_e]] have equal numerical values.

The mechanical equation is

and position satisfies

These equations assume a rigid shaft, constant parameters, linear magnetic operation, and viscous friction. Coulomb friction, backlash, elastic shafts, and magnetic saturation require additional terms.

2.3 State-space representation

Choose the state vector

with inputs [[M:v_a]] and [[M:T_L]]. The state equations are

Thus,

where

and, for [[M:u = [v_a T_L]ᵀ]],

The output equation [[M:y = Cx + Du]] is selected according to the measured variable. For speed measurement, [[M:C = [0 1 0]]]. For position measurement, [[M:C = [0 0 1]]].

2.4 Transfer functions

Taking Laplace transforms with zero initial conditions gives

For the voltage-to-speed transfer function with load torque set to zero,

The voltage-to-position transfer function is

The load-torque disturbance-to-speed transfer function is

The negative sign indicates that an increased opposing load torque reduces speed.

3. Converter and Motor Interconnection

A converter is often represented by an average model rather than individual switching waveforms. For a PWM voltage-source converter operating below significant saturation, a first-order approximation is

where [[M:V_c]] is the controller command, [[M:V_d]] is average motor-side voltage, [[M:K_c]] is converter gain, [[M:τ_c]] is an effective converter time constant, and [[M:T_d]] is computational, sampling, and switching delay.

For a fast inner-loop design, the delay is frequently approximated by a first-order factor using

This approximation is useful for preliminary design but does not reproduce the exact phase behavior at high frequency.

A complete forward path may therefore be written as

where [[M:G_e]] represents electrical motor dynamics, [[M:G_m]] mechanical dynamics, and [[M:G_s]] sensor dynamics. Block diagrams are not merely illustrations: they expose signal paths, disturbance entry points, loop gains, and possible algebraic errors.

4. Mechanical Loads and Equivalent Parameters

For a rigidly coupled load,

where [[M:J_m]] and [[M:J_L]] are motor and load inertias referred to the motor shaft. If a gearbox has speed ratio [[M:n = ω_m/ω_L]], then load inertia referred to the motor is

and, for ideal power transmission, load torque referred to the motor is

The squared ratio in the inertia expression is essential. It follows from equating kinetic energy or virtual work. Gear efficiency, elasticity, and backlash must be included when their effects are significant.

A two-inertia drive includes motor inertia, load inertia, shaft stiffness, and shaft damping. Such a model can produce torsional resonance and antiresonance. A rigid-load approximation should therefore be avoided when the control bandwidth approaches the first shaft resonance.

5. Sensors and Measurement Dynamics

A speed sensor can be modeled as

where [[M:K_ω]] converts speed to measurement units and [[M:τ_ω]] is sensor time constant. An encoder may instead be modeled as a position measurement with quantization and sample delay. A current sensor is commonly approximated by a gain plus a low-pass filter.

Measurement filtering reduces noise but adds phase lag. Excessive filtering decreases achievable control bandwidth and may destabilize a loop that was tuned using an ideal sensor model. Sensor scaling must be included consistently: a controller designed for volts of feedback cannot be connected directly to a signal expressed in revolutions per minute without a conversion factor.

6. Stability and Transient Response

A continuous-time linear system is internally stable when all poles of its characteristic equation have negative real parts. Poles on the imaginary axis require special qualification and generally do not provide asymptotic stability. For a closed-loop transfer function

the characteristic equation is [[M:1 + G(s)H(s) = 0]].

Important transient measures include rise time, peak overshoot, settling time, and steady-state error. For a standard second-order denominator

[[M:ω_n]] is the undamped natural frequency and [[M:ζ]] is the damping ratio. For 0 < [[M:ζ]] < 1, the approximate percent overshoot to a unit step is

and the 2% settling-time approximation is

These formulas apply primarily when higher-order poles are sufficiently faster than the dominant pair. They should not be used uncritically for systems with delays, saturation, resonance, or poorly separated poles.

The final-value theorem gives the steady-state value when the relevant stability conditions hold:

where [[M:F(s)]] is the Laplace transform of [[M:f(t)]].

7. PI and PID Controllers

The ideal PID controller is

where [[M:K_p]] is proportional gain, [[M:K_i]] integral gain, and [[M:K_d]] derivative gain. The integral term removes steady-state error to a constant reference or disturbance when the closed loop is stable. The derivative term anticipates error change and can improve damping, but it amplifies measurement noise.

A practical derivative controller uses filtering:

where [[M:T_f]] is a small derivative-filter time constant. In many electrical drives, PI control is preferred for current and speed loops because derivative action is unnecessary or noisy.

Integral action creates windup when the actuator saturates. Anti-windup methods include conditional integration, back-calculation, and integrator clamping. A controller should also include output limits, reference limits, and bumpless transfer between operating modes.

8. Cascaded Drive Control

The conventional hierarchy is:

1. current loop;
2. torque loop, if separately required;
3. speed loop; and
4. position loop.

The inner loop must be substantially faster than the outer loop. A common engineering rule is to make the speed-loop bandwidth roughly 5–10 times lower than the current-loop bandwidth, and the position-loop bandwidth lower than the speed-loop bandwidth. Exact ratios depend on sampling, noise, resonances, and performance requirements.

8.1 Current loop

Neglecting back-emf during the fast current transient, the armature plant is approximately

With a PI controller

choosing the controller zero to cancel the electrical pole gives

and therefore

This cancellation is a design approximation; resistance variation and converter delay prevent exact cancellation in practice. If the desired current-loop crossover frequency is [[M:ω_ci]] and converter gain is approximately [[M:K_c]], a preliminary proportional gain is often selected from the desired loop gain, then verified using the complete converter and delay model. The final gains should be checked by frequency response or simulation rather than by cancellation alone.

8.2 Torque loop

For a constant-flux DC motor, torque is proportional to current. Thus a current loop is effectively a torque loop after scaling by [[M:K_t]]. In AC drives, a torque loop may be implemented through current components such as [[M:i_q]], with motor-specific decoupling and parameter compensation.

8.3 Speed loop

When the current loop is much faster than the speed loop, it may be approximated as unity. The speed plant is then approximately

A speed PI controller provides zero steady-state error to a constant speed command, subject to current and voltage limits. The speed-loop design must account for load-torque disturbance, speed-sensor filtering, and the finite response of the current loop.

8.4 Position loop

Since position is the integral of speed,

A position controller may be proportional for a sufficiently fast speed loop. Integral action can remove position bias but may cause sluggishness or windup. Derivative action is often unnecessary because the speed signal already supplies a physically meaningful rate signal.

9. Worked Example: Preliminary Current-Loop Design

Consider a separately excited DC motor with

• [[M:R_a = 2 Ω]];
• [[M:L_a = 10 mH]];
• [[M:K_t = K_e = 0.8]] in SI units.

The electrical pole is

The electrical time constant is

A PI controller whose zero approximately cancels the motor electrical pole must satisfy

Suppose the desired proportional gain is selected as [[M:K_p = 4]]. Then

The numerical units of [[M:K_i]] depend on the voltage, current, and error scaling used in the implementation. This design is only preliminary: converter delay, current-sensor filtering, voltage saturation, and back-emf must be included before commissioning. At high speed, back-emf reduces the voltage available to force current changes, so the current loop may become voltage limited even if its small-signal design is satisfactory.

10. Applied Case: Servo Axis with Cascaded Control

Background

Consider an illustrative packaging-machine axis driven by a permanent-magnet servo motor. The axis must move a carriage accurately, reject a disturbance caused by changing product mass, and stop without excessive overshoot.

Problem

An initial single speed loop produced oscillation when the carriage inertia changed. Current peaks also exceeded the motor rating during rapid acceleration. Measurements showed that the speed sensor was filtered heavily and that the controller integrator continued to accumulate while the inverter was voltage limited.

Analysis

The original model treated the motor and load as a constant inertia and omitted converter delay. The changing product mass altered the mechanical pole. Heavy sensor filtering added phase lag, reducing stability margin. The absence of anti-windup caused a large stored integral command, leading to overshoot after saturation ended.

Approach and solution

The engineering team used a cascaded structure. A sampled current loop was tuned first, including converter delay and current-sensor filtering. Its output was limited to the safe motor-current range. A speed PI loop was then tuned at a lower bandwidth, with load torque treated as a disturbance. Finally, a position loop was added outside the speed loop. The position reference was passed through a trajectory generator that limited velocity and acceleration.

Anti-windup back-calculation was added to both relevant integrators. The mechanical model was updated to include the measured carriage inertia, gearbox ratio, and a friction estimate. A commissioning test measured step response, frequency response, current peaks, and settling time at minimum and maximum payload.

Outcome

The cascaded design reduced interaction between electrical and mechanical dynamics. Current limiting protected the motor, while trajectory shaping reduced position overshoot. Retuning across the payload range maintained acceptable damping, although the maximum achievable acceleration decreased at the heaviest load.

Learning points

• Inner-loop bandwidth determines how well outer-loop assumptions hold.
• Saturation is part of the plant, not merely a software detail.
• Sensor filtering and computation delay must be included in stability analysis.
• Model validation should cover the full operating range, not only nominal load.
• Safe operation requires independent hardware limits in addition to software control.

11. Practical Limitations and Professional Practice

Real drives differ from ideal models because of:

• parameter variation with temperature and frequency;
• magnetic saturation and inverter dead time;
• Coulomb friction, backlash, and elastic couplings;
• sampling, quantization, computation, and communication delays;
• actuator voltage and current limits;
• sensor noise and loss of feedback;
• regenerative energy and DC-link overvoltage.

Good practice includes using SI units consistently, documenting sign conventions, measuring motor parameters, checking every saturation limit, and validating both reference tracking and disturbance rejection. Simulation should include nonlinearities after the linear design is complete. Hardware-in-the-loop testing can expose timing and saturation problems before full-power commissioning.

Drive controllers are safety-relevant systems. Emergency stopping, overspeed protection, insulation coordination, fault detection, and safe torque-off functions should not depend solely on a software PI algorithm. Applicable machinery, electrical-safety, EMC, and functional-safety standards must be identified for the installation. Data-driven and AI-based tuning can assist parameter identification or fault diagnosis, but any learned controller must be bounded, validated, and supervised by deterministic protection functions.

Chapter Summary

Drive modeling converts physical electrical and mechanical laws into state-space, transfer-function, and block-diagram representations. The separately excited DC motor provides a transparent example: armature voltage determines current, current produces electromagnetic torque, and torque determines acceleration against inertia, friction, and load torque. Converter, sensor, and mechanical-load dynamics must be included when their time scales affect the desired bandwidth.

State-space models are suitable for multivariable analysis, simulation, and digital implementation, while transfer functions clarify poles, zeros, disturbance paths, and loop-shaping requirements. Stability depends on the closed-loop characteristic equation, and transient specifications are linked to damping ratio and natural frequency only when the dominant second-order approximation is justified.

PI control is central to drive applications. Integral action removes steady-state error but requires anti-windup. Cascaded current, torque, speed, and position loops are tuned from the inside outward, with progressively lower bandwidths. Successful industrial design combines mathematical analysis with parameter measurement, nonlinear simulation, saturation management, safety provisions, and experimental validation.

Key Terms

State variable: A variable that, with the inputs, describes the future behavior of a dynamic system.

State-space model: First-order vector differential equations with an output equation.

Transfer function: Ratio of an output transform to an input transform under zero initial conditions.

Back emf: Motor voltage induced by rotation and opposing applied armature voltage.

Torque constant: Proportionality constant between motor current and electromagnetic torque.

Electrical time constant: Ratio [[M:L_a/R_a]] for a basic armature circuit.

Mechanical time constant: A characteristic speed-response time determined by inertia, damping, and electromagnetic coupling.

Converter gain: Ratio between a commanded converter signal and average motor-side voltage.

Load torque: Mechanical torque opposing or assisting motor motion.

Equivalent inertia: Inertia of motor and load expressed about a common shaft.

Sensor dynamics: Gain, filtering, delay, and quantization associated with measurement.

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

Damping ratio: Dimensionless measure describing the decay and oscillation of a second-order response.

PI controller: Controller containing proportional and integral actions.

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

Cascaded control: Nested feedback loops in which an inner loop regulates a faster variable.

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

Windup: Excessive integrator accumulation while an actuator is saturated.

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

Review Questions

Short Answer Questions

1. What is the purpose of representing a drive in state-space form?
2. State the electrical and mechanical equations of a separately excited DC motor.
3. Why is the load inertia divided by the square of the gear ratio when referred to the motor shaft?
4. What is the function of integral action in a PI controller?
5. Why are current loops normally faster than speed loops?

Long Answer Questions

1. Derive the voltage-to-speed transfer function of a separately excited DC motor and state the assumptions used.
2. Explain how converter delay and sensor filtering influence closed-loop stability.
3. Describe a systematic procedure for designing current, speed, and position loops.
4. Explain the causes and consequences of integral windup and compare two anti-windup methods.
5. Discuss the limitations of a rigid-load model when a drive contains an elastic shaft.

Analytical / Application Questions

1. A DC motor has [[M:R_a = 1.5 Ω]] and [[M:L_a = 6 mH]]. Calculate its electrical time constant and electrical pole frequency in rad/s. Explain the relevance of each result to current-loop design.
2. A speed-controlled drive becomes oscillatory after a low-pass filter is added to the speed sensor. Using phase-lag reasoning, explain why this occurs and propose two corrective actions.
3. Design a block-level control strategy for a conveyor drive that must reject load-torque changes, limit motor current, and achieve accurate stopping position. Identify the controlled variable, actuator limit, sensor, and likely disturbance at each loop.

Multiple Choice Questions with Answers

Question 1

For a separately excited DC motor, which equation relates electromagnetic torque to armature current under constant flux?

A. [[M:T_e = K_t i_a]]

B. [[M:T_e = K_e ω_m]]

C. [[M:T_e = J dω_m/dt]]

D. [[M:T_e = R_a i_a]]

Correct Answer: A. [[M:T_e = K_t i_a]]

Question 2

Which state vector is suitable for the basic DC motor model developed in this chapter?

A. [[M:[v_a, e_a, T_L]ᵀ]]

B. [[M:[i_a, ω_m, θ_m]ᵀ]]

C. [[M:[R_a, L_a, B]ᵀ]]

D. [[M:[K_t, K_e, J]ᵀ]]

Correct Answer: B. [[M:[i_a, ω_m, θ_m]ᵀ]]

Question 3

If a gearbox speed ratio is [[M:n = ω_m/ω_L]], how is load inertia referred to the motor shaft?

A. [[M:J_L n²]]

B. [[M:J_L/n]]

C. [[M:J_L/n²]]

D. [[M:J_L + n]]

Correct Answer: C. [[M:J_L/n²]]

Question 4

What is the principal benefit of integral action in a stable speed-control loop?

A. It removes measurement noise.

B. It eliminates steady-state error to a constant command or disturbance.

C. It guarantees no actuator saturation.

D. It always increases phase margin.

Correct Answer: B. It eliminates steady-state error to a constant command or disturbance.

Question 5

For the armature circuit, [[M:R_a = 2 Ω]] and [[M:L_a = 0.01 H]]. What is the electrical time constant?

A. 0.005 s

B. 0.02 s

C. 5 s

D. 200 s

Correct Answer: A. 0.005 s

Question 6

Which loop is normally tuned first in a cascaded electrical-drive controller?

A. Position loop

B. Speed loop

C. Current loop

D. Supervisory loop

Correct Answer: C. Current loop

Question 7

What is the main danger of derivative action when applied directly to a noisy speed measurement?

A. It removes all damping.

B. It amplifies high-frequency measurement noise.

C. It prevents current control.

D. It changes motor resistance.

Correct Answer: B. It amplifies high-frequency measurement noise.

Question 8

The characteristic equation of a negative-feedback system is obtained from which expression?

A. [[M:1 − G(s)H(s) = 0]]

B. [[M:G(s) + H(s) = 0]]

C. [[M:1 + G(s)H(s) = 0]]

D. [[M:G(s)/H(s) = 0]]

Correct Answer: C. [[M:1 + G(s)H(s) = 0]]

Question 9

Why is anti-windup needed in a PI-controlled drive?

A. To increase sensor quantization.

B. To prevent excessive integral accumulation during actuator saturation.

C. To cancel motor back emf permanently.

D. To eliminate mechanical inertia.

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

Question 10

A rigid-load approximation is most questionable when which condition applies?

A. The shaft is stiff and resonance is far above control bandwidth.

B. The load torque is constant and small.

C. The shaft has significant elasticity and control bandwidth approaches torsional resonance.

D. The motor operates at zero speed only.

Correct Answer: C. The shaft has significant elasticity and control bandwidth approaches torsional resonance.