8. Advanced Control, Estimation and Optimisation

Introduction and context

Electrical drives have progressed from fixed-gain regulators and measured-speed feedback systems to digitally implemented control architectures capable of prediction, adaptation, estimation and real-time optimisation. This progression has been driven by demanding applications: electric vehicles require rapid torque response over a wide speed range; machine tools require precise position regulation; wind turbines must operate efficiently under changing conditions; and industrial plants increasingly expect drives to diagnose their own faults and coordinate with supervisory control systems.

The control problem is not simply to produce a reference voltage or switching sequence. A modern drive must infer internal electromagnetic states, tolerate uncertain parameters, respect voltage and current limits, reject disturbances, and balance competing objectives such as torque response, efficiency, acoustic noise, thermal stress and switching loss. These requirements are particularly significant in induction motors and permanent-magnet synchronous motors (PMSMs), whose useful variables—rotor flux, electromagnetic torque, rotor position and load torque—are not all directly measurable.

This chapter develops advanced methods for drive control, state and parameter estimation, sensorless operation, constraint handling and optimisation. The discussion assumes familiarity with rotating-machine models, space vectors, coordinate transformations, inverter operation and conventional vector control.

Learning objectives

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

1. formulate state-space models suitable for advanced drive control;

2. distinguish robust, adaptive, predictive and nonlinear control approaches;

3. design and interpret observers for flux, speed, position and load torque;

4. explain parameter identification and sensorless control limitations;

5. incorporate voltage, current, torque and thermal constraints into a controller;

6. formulate single-objective and multi-objective optimisation problems;

7. select an appropriate control architecture for an industrial application;

8. evaluate implementation, safety, ethical and reliability issues in advanced drives.

From machine equations to control-oriented models

A controller requires a model that is sufficiently accurate for prediction but sufficiently simple for real-time computation. For a general electrical drive, a nonlinear state-space representation can be written as

ẋ = f(x,u,p,d), y = h(x,u,p) + n,

where x is the state vector, u is the control input, p is the parameter vector, d represents disturbances such as load torque, y is the measured output and n represents measurement noise. In a PMSM, a useful state vector may contain stator currents, rotor electrical position and mechanical speed. In an induction motor, rotor flux components are often included as states because they determine torque production but are not directly measured.

For a surface-mounted PMSM in the rotor d–q frame, neglecting saturation for a first model,

v_d = R_s i_d + L_d di_d/dt − ω_e L_q i_q,

v_q = R_s i_q + L_q di_q/dt + ω_e(L_d i_d + ψ_f),

T_e = (3/2)p[ψ_f i_q + (L_d−L_q)i_d i_q],

where R_s is stator resistance, L_d and L_q are inductances, ψ_f is permanent-magnet flux linkage, p is the number of pole pairs and ω_e is electrical speed. Mechanical dynamics are represented by

J dω_m/dt = T_e − T_L − Bω_m,

where J is inertia, B is viscous friction and T_L is load torque.

The equations show why advanced control is necessary. Cross-coupling terms depend on speed, resistance changes with temperature, inductances may vary with current because of saturation, and the load torque is usually unknown. A model designed for controller synthesis therefore differs from a detailed finite-element model. The former must expose dominant dynamics and constraints without imposing an impractical computational burden.

Linearisation around an operating point gives

Δẋ = AΔx + BΔu + EΔd,

which supports pole-placement, linear quadratic regulation and local stability analysis. However, a controller based only on one linearisation may perform poorly across the entire speed and torque range. Gain scheduling, nonlinear feedback, adaptive estimation or predictive control can address this variation.

Advanced control architectures

Robust control

Robust control is designed to preserve acceptable stability and performance despite uncertainty in parameters, neglected dynamics and disturbances. In a drive, uncertainty may arise from stator resistance variation, magnetic saturation, dead-time effects, inverter voltage error, friction and changes in load inertia.

A practical robust design begins by identifying bounded uncertainty rather than attempting to model every physical effect. For example,

R_s ∈ [R_min,R_max], L_d ∈ [L_min,L_max].

The controller is then selected so that closed-loop poles, gain margins or specified performance measures remain satisfactory over these ranges. H-infinity methods treat disturbances and modelling errors through a worst-case performance formulation. Sliding-mode control provides another robust approach by forcing the system trajectory toward a selected sliding surface. For a speed loop, a surface may be defined as

s = ė_ω + λe_ω,

where e_ω is speed error and λ is positive. The discontinuous action drives s toward zero, while a boundary layer or continuous approximation is commonly introduced to reduce chattering.

Robustness is not synonymous with aggressive gain. Excessive gains can amplify measurement noise, excite structural resonances and increase switching activity. In practice, robustness must be evaluated together with noise sensitivity, actuator limits and thermal consequences.

Adaptive control and gain scheduling

Adaptive control modifies controller parameters online in response to changing operating conditions. In model-reference adaptive control (MRAC), the actual plant is made to follow a reference model, and an adaptation law adjusts uncertain parameters. In drive applications, adaptation may estimate rotor resistance, stator resistance, inertia or load torque.

A simpler industrial strategy is gain scheduling. Controller gains are selected from speed, torque or flux regions, with interpolation between operating points. Gain scheduling is computationally attractive but is not fully adaptive: it does not necessarily identify the cause of performance degradation. It can also produce discontinuities unless transitions are carefully smoothed.

Parameter adaptation must respect physical observability. At very low frequency, rotor resistance and rotor flux may have similar effects in an induction motor model, making reliable separation difficult. An adaptation law that is mathematically stable under ideal excitation may drift in a real drive with insufficient excitation, noise or inverter nonlinearity.

Nonlinear control

Nonlinear control uses the machine equations directly rather than treating all operating points as small deviations about one equilibrium. Feedback linearisation can cancel selected nonlinear terms, provided the model and state measurements are sufficiently accurate. Input–output linearisation may decouple torque and flux channels, although exact cancellation becomes unreliable when parameters vary.

Backstepping constructs a stabilising controller recursively by treating intermediate variables as virtual control inputs. It is useful where electrical and mechanical dynamics are cascaded, but the resulting expressions may become complicated and sensitive to parameter errors. Passivity-based control exploits the energy structure of the motor and converter. Because electrical machines are physical energy-conversion systems, energy-shaping and damping-injection methods can provide meaningful stability interpretations.

Nonlinear methods are particularly valuable when magnetic saturation, reluctance torque, wide-speed operation or severe disturbances are important. Their disadvantages include model dependence, computational complexity and the need for rigorous implementation checks.

Model predictive control

Finite-control-set model predictive control (FCS-MPC) predicts the future motor state for each admissible inverter switching vector and selects the vector minimising a cost function. A typical cost function is

J = w_i[(i_d^−i_d)^2+(i_q^−i_q)^2] + w_T(T_e^*−T_e)^2 + w_uΔu^2,

where the w terms are weighting factors. Constraints can be included directly by assigning a high penalty or rejecting infeasible candidates.

FCS-MPC has a natural relationship with a two-level inverter because the candidate set is finite. It can provide rapid torque response and explicit constraint handling, but it usually produces variable switching frequency and requires careful weighting-factor selection. Computational delay, parameter mismatch and current measurement timing are significant design concerns.

Continuous-control-set MPC first computes an ideal voltage vector through optimisation and then uses a modulator. This facilitates fixed switching frequency and compatibility with conventional PWM, but adds an optimisation or modulation stage. Long prediction horizons can improve anticipation of constraints but increase computation and sensitivity to modelling errors.

Disturbance-observer-based control

A disturbance observer estimates the combined effect of unknown load torque, friction, parameter error and unmodelled dynamics. The controller then compensates for the estimated disturbance. This approach is attractive because a detailed model of every disturbance is unnecessary. Nevertheless, observer bandwidth must be chosen carefully: a high bandwidth improves disturbance rejection but increases noise sensitivity and may amplify inverter ripple.

State estimation and observers

An observer reconstructs unavailable states from a model and available measurements. The basic Luenberger observer for a linear model is

x̂̇ = Ax̂ + Bu + L(y−Cx̂),

where x̂ is the estimated state and L is the observer gain. The error dynamics are ė = (A−LC)e. The gain is chosen so that the error converges with the desired speed and damping.

For nonlinear drives, an extended Kalman filter (EKF) linearises the model around the current estimate and propagates both the state and its uncertainty covariance. The process-noise covariance Q represents uncertainty in the model and disturbances; the measurement-noise covariance R represents sensor noise. Increasing Q makes the filter respond more readily to measurements and model mismatch, whereas increasing R makes it trust the model more strongly. Poorly selected covariance matrices can produce sluggish estimates or unstable-looking oscillations.

Unscented Kalman filters avoid explicit first-order linearisation by propagating selected sigma points through the nonlinear model. Particle filters can represent non-Gaussian uncertainty but generally demand much greater computation. In industrial drives, EKFs and reduced-order observers remain common because they provide a practical compromise between accuracy and processing requirements.

A phase-locked loop (PLL) is often used to estimate electrical angle from a rotating voltage or flux vector. A second-order PLL can track frequency while filtering noise, but it may suffer phase error under distortion or rapidly changing speed. The observer must be designed together with the angle-generation method; an accurate flux magnitude with a biased angle can still produce severe torque ripple.

Parameter identification and sensorless operation

Parameter identification determines unknown or changing quantities from measured input-output data. Offline tests may estimate stator resistance, inductance, inertia and friction before commissioning. Online identification updates parameters during operation using recursive least squares, gradient adaptation, extended Kalman filtering or test-signal injection.

Recursive least squares, for example, estimates a parameter vector θ from a regression model y = φᵀθ + ε. A forgetting factor allows the estimator to follow slow parameter changes, but excessive forgetting increases sensitivity to noise. Identification requires persistent excitation: if the signals do not sufficiently vary, different parameter values may produce nearly identical outputs.

Sensorless control eliminates the mechanical speed or position sensor. In induction motors, rotor speed can be estimated from stator quantities and the rotor-flux model. At low speed, the back electromotive force becomes small and stator resistance error dominates, so pure model-based estimation becomes unreliable. Signal injection can overcome this limitation by applying a high-frequency voltage or current component and extracting the position-dependent response caused by saliency or magnetic anisotropy.

PMSM sensorless drives often use back-EMF estimation at medium and high speed, while high-frequency injection supports low-speed or standstill position estimation, especially in salient machines. A starting procedure may therefore be divided into regions: initial alignment or injection at zero speed, observer transition at low speed, and back-EMF or flux observation at higher speed. Bumpless transfer is essential; otherwise, the estimated angle can jump and produce a large current transient.

Sensorless operation reduces cabling, cost and mechanical vulnerability, but it does not eliminate sensing. DC-link voltage, phase currents, temperature and protective signals remain necessary. Sensorless algorithms also require fault detection because an erroneous estimate can command unsafe torque.

Constraint handling and safe operation

A drive controller operates within a feasible region. Typical constraints are

|i_s| ≤ I_max, |v_s| ≤ V_max, |T_e| ≤ T_max,

along with DC-link voltage, semiconductor temperature, winding temperature, speed and acceleration limits. The voltage limit becomes especially important in field weakening, where the back-EMF approaches the available inverter voltage. The controller must then reduce flux or adjust current commands to remain feasible.

A current-reference limiter should preserve the most important objective. In a PMSM, the current vector may be limited by a circular current boundary, while the voltage boundary is speed dependent. Maximum-torque-per-ampere (MTPA) control selects the d–q current combination producing required torque with minimum current. At high speed, maximum-torque-per-volt or field-weakening control modifies the flux-producing current to satisfy the voltage ellipse.

Constraint handling can be implemented through saturation, reference governors, anti-windup, barrier functions or predictive optimisation. Simple saturation is easy to certify but can cause integrator windup and poor transient behaviour. Anti-windup feedback prevents the integrator from accumulating an unrealizable command. A reference governor modifies commands before they reach the controller, maintaining feasibility while preserving the requested trajectory as far as possible.

Protection and control should not be confused. A predictive controller may anticipate a current limit, but independent hardware protection is still required for overcurrent, shoot-through, overspeed and insulation faults. Safety-related shutdown paths should not depend exclusively on a complex software estimator.

Optimisation of drive performance

Optimisation begins by defining a decision vector, objective function and constraints. For example, controller parameters k can be selected by

minimize F(k) = αJ_1(k) + βJ_2(k) + γJ_3(k),

subject to g(k) ≤ 0 and h(k) = 0.

Objectives may include speed-tracking error, copper loss, switching loss, torque ripple, acoustic noise, computation time and thermal cycling. Weighting converts multiple objectives into a scalar cost, but the result depends strongly on the selected weights. A Pareto front is often more informative: a solution is Pareto optimal if no objective can be improved without worsening at least one other objective.

Common optimisation methods include gradient-based tuning, sequential quadratic programming, genetic algorithms, particle-swarm optimisation and Bayesian optimisation. Gradient methods are efficient when the cost is smooth and differentiable. Evolutionary methods are useful for discontinuous or mixed-variable problems, such as selecting switching states and controller structures, but they require many evaluations. In engineering practice, optimisation should generally be performed offline or at a supervisory timescale, while the inner current loop remains deterministic.

Loss minimisation can be based on copper, iron, mechanical and inverter losses. For a PMSM, reducing current generally reduces copper loss, but a different current angle may increase iron or inverter loss. Thermal optimisation must include the thermal time constants: a short torque burst may be electrically acceptable while producing excessive long-term winding temperature. Multi-objective optimisation therefore benefits from an electrothermal model rather than an electrical model alone.

Applied case: sensorless, constraint-aware control of an industrial pump drive

Background

Consider an illustrative medium-power PMSM pump drive used in a process plant. The pump operates for long periods at nearly constant speed but experiences frequent flow changes. A mechanical encoder is undesirable because of contamination, cable length and maintenance requirements. The drive must provide reliable starting, limit current during blocked-pipe events and reduce energy consumption during partial-load operation.

Situation and problem

A conventional speed-sensorless back-EMF observer performs adequately above approximately medium speed but fails during starting and produces angle error during rapid flow changes. The original PI current regulators also saturate during acceleration, causing integrator windup. Operating data show unnecessary reactive current at partial load and elevated inverter temperature during repeated transients.

Analysis

The problem has four interacting causes. First, back-EMF is too small at low speed for reliable angle estimation. Second, parameter errors and temperature-dependent resistance bias the flux estimate. Third, voltage and current limits are not explicitly coordinated with the speed controller. Fourth, efficiency and thermal stress are not represented in the tuning objective.

Approach and solution

The revised architecture uses high-frequency signal injection during standstill and low-speed operation, followed by a smoothly blended flux observer at higher speed. Stator resistance is updated using an online adaptation law constrained to a physically reasonable interval. A current-reference governor enforces the current circle and voltage ellipse before commands enter the inner current controller. Anti-windup action is added to both current and speed regulators.

For normal operation, MTPA is used below base speed and field weakening above base speed. A supervisory optimisation routine selects operating points using a cost function containing copper loss, inverter loss, speed error and estimated winding temperature. The inner current loop remains fixed-frequency and deterministic. Independent hardware comparators provide overcurrent protection, while software monitors observer residuals and disables torque production if estimation becomes implausible.

Outcome

The architecture improves starting reliability, reduces transient current peaks and avoids the previous angle discontinuity during observer transition. Partial-load operation uses less unnecessary d-axis current, and the thermal supervisor limits repeated acceleration events when the estimated thermal state approaches its allowable boundary. The example also illustrates that no single advanced algorithm solves the problem: estimator design, constraint handling, optimisation and protection must be integrated.

Learning points

The principal lesson is architectural. Sensorless control is successful only when its operating range and transition logic are explicitly designed. Adaptation must be bounded, optimisation must respect thermal and electrical constraints, and software intelligence must be supported by independent protection.

Challenges, limitations and common mistakes

Advanced control methods can fail for reasons unrelated to their theoretical derivation. A frequent mistake is using an inaccurate motor model and compensating with increasingly complex algorithms. Complexity then conceals rather than solves the modelling error. Another mistake is tuning an observer with an unrealistically high bandwidth, causing switching ripple to be interpreted as physical motion.

Weighting factors in predictive control are often selected by trial and error without normalising the terms. A squared current error measured in amperes can dominate a torque-ripple term expressed in newton-metres simply because of units. Each cost component should be scaled and related to a meaningful engineering priority.

Sensorless schemes may be tested only under steady-state conditions, although the most important failures occur during start-up, reversal, regeneration, voltage sag and rapid load change. Similarly, an optimiser may produce a nominally efficient controller that is too sensitive to parameter variation for production use.

Digital implementation introduces sampling delay, computation time, quantisation, PWM update delay and finite word length. These effects should be included in the design model. Verification should progress from simulation to processor-in-the-loop, hardware-in-the-loop, dynamometer testing and controlled field trials.

Professional, ethical and governance considerations

Drive engineers are responsible for preventing foreseeable hazards, including uncontrolled torque, overspeed, electric shock, thermal fire and mechanical damage. Functional safety requires clearly defined safe states, diagnostic coverage and independence between normal control and protective mechanisms where appropriate. Design should follow applicable electrical, machinery, EMC and functional-safety standards for the intended industry and jurisdiction.

Data-driven optimisation introduces governance questions. Operational data may reveal production rates, machine duty cycles or maintenance conditions and should be protected accordingly. An adaptive controller must have traceable limits and a method for reverting to a validated baseline. Engineers should not present an AI- or optimisation-derived setting as reliable merely because it performs well in simulation. Explainability, test evidence, version control and change approval are essential in safety-critical or highly regulated installations.

Current developments and best practices

Current research and industrial development are moving toward embedded digital twins, physics-informed machine-learning estimators, wide-bandgap inverter control, coordinated multi-drive optimisation and condition-aware control. Machine learning can assist parameter estimation, fault classification or surrogate modelling, but physics-based constraints remain important for extrapolation and safety.

A sound implementation practice is to:

• select the simplest model that captures the performance-limiting phenomena;
• define operating regions, constraints and fallback modes before tuning;
• normalise and document optimisation objectives;
• validate observability and excitation over the full operating envelope;
• use bounded adaptation and bumpless transitions;
• separate performance control from independent protection;
• test parameter variation, sensor faults, communication loss and abnormal loads;
• record software versions, calibration data and acceptance criteria.

Chapter Summary

Advanced electrical-drive control combines dynamic modelling, robust or adaptive feedback, nonlinear and predictive methods, state observation, parameter identification and optimisation. Robust control addresses uncertainty; adaptive control adjusts to changing parameters; nonlinear control exploits machine structure; and model predictive control anticipates future behaviour while handling constraints. Observers and estimators reconstruct flux, speed, position and disturbances, enabling sensorless operation but introducing observability and low-speed limitations. Parameter identification is meaningful only when adequate excitation exists and estimates are physically bounded.

Voltage, current, torque and thermal limits must be incorporated explicitly through saturation management, reference governance, field weakening, predictive constraints or optimisation. Performance optimisation should consider competing objectives such as efficiency, torque ripple, switching loss, thermal stress and response time. The case study demonstrates that reliable advanced drives are systems architectures rather than isolated algorithms. Validation, independent protection, professional governance and carefully managed complexity are as important as mathematical design.

Key Terms

State-space model: A mathematical representation of system dynamics using states, inputs, outputs and disturbances.

Robust control: Control designed to maintain stability and performance despite bounded uncertainty.

Adaptive control: Control that modifies parameters online in response to changing plant behaviour.

Gain scheduling: Selection of controller gains according to operating region or measured conditions.

Sliding-mode control: A nonlinear method that drives system trajectories toward a chosen switching surface.

Model predictive control: Control that predicts future behaviour and selects actions by minimising a cost function.

Observer: An algorithm that estimates unmeasured states from a model and available measurements.

Extended Kalman filter: A recursive nonlinear state estimator based on local linearisation and covariance propagation.

Parameter identification: Estimation of unknown model parameters from measured system data.

Persistent excitation: Sufficient signal variation to make unknown parameters distinguishable.

Sensorless control: Drive control that estimates mechanical speed or position without a mechanical sensor.

Phase-locked loop: A feedback system that estimates phase or frequency by synchronising with a reference signal.

Field weakening: Operation with reduced effective flux to remain within the inverter voltage limit at high speed.

MTPA: Maximum-torque-per-ampere control, which minimises current for a specified torque.

Anti-windup: A technique that prevents an integrator from accumulating error when an actuator is saturated.

Reference governor: A supervisory mechanism that modifies commands to maintain constraint feasibility.

Pareto optimum: A solution for which improvement in one objective requires deterioration in another.

Disturbance observer: An estimator that reconstructs unknown disturbances and model uncertainties for compensation.

Bumpless transfer: A transition between controllers or estimators without an abrupt command or state discontinuity.

Review Questions

Short Answer Questions

1. Why is a control-oriented motor model usually simpler than a finite-element model?

2. What is the main purpose of a disturbance observer in an electrical drive?

3. Define persistent excitation in parameter identification.

4. Why does back-EMF-based sensorless control become difficult at low speed?

5. What function does anti-windup perform in a saturated controller?

Long Answer Questions

1. Explain robust, adaptive, nonlinear and predictive control approaches, comparing their assumptions and practical uses in electrical drives.

2. Derive the role of an observer in reconstructing unmeasured drive states and discuss the influence of observer bandwidth and measurement noise.

3. Discuss sensorless PMSM operation from standstill through high speed, including signal injection, observer transition and failure modes.

4. Explain how current, voltage, torque and thermal constraints can be incorporated into a drive controller.

5. Describe a multi-objective optimisation framework for minimising losses while maintaining torque response and thermal limits.

Analytical / Application Questions

1. A PMSM drive experiences current overshoot during acceleration and integrator windup. Propose a control modification and explain how you would validate it under voltage and current constraints.

2. An induction-motor sensorless estimator gives accurate speed at rated speed but oscillates near zero speed. Analyse the likely causes and recommend an estimation strategy.

3. Design a high-level architecture for a sensorless pump drive that must minimise energy consumption, tolerate parameter variation and enter a safe state when the position estimate becomes unreliable.

22. Multiple Choice Questions with Answers

Question 1

Which feature most directly distinguishes model predictive control from a conventional PI current controller?

A. It requires no motor model

B. It predicts future states and selects an action by minimising a cost function

C. It eliminates all current measurements

D. It operates only at zero speed

Correct Answer: B. It predicts future states and selects an action by minimising a cost function

Question 2

Why is back-EMF-based sensorless position estimation weak at very low speed?

A. Back-EMF magnitude becomes small

B. Rotor inertia becomes zero

C. The DC-link voltage becomes infinite

D. Current sensors stop measuring DC quantities

Correct Answer: A. Back-EMF magnitude becomes small

Question 3

In an observer, the term L(y−Cx̂) primarily serves to:

A. Increase motor resistance

B. Inject switching harmonics into the inverter

C. Correct the model estimate using output error

D. Limit the mechanical speed directly

Correct Answer: C. Correct the model estimate using output error

Question 4

What is the principal purpose of MTPA control?

A. To maximise current for a given torque

B. To minimise current for a specified torque

C. To eliminate the voltage limit

D. To maintain zero electromagnetic torque

Correct Answer: B. To minimise current for a specified torque

Question 5

A forgetting factor in recursive parameter estimation allows the estimator to:

A. Ignore all new measurements

B. Track slowly changing parameters

C. Guarantee perfect identification without excitation

D. Remove the need for a motor model

Correct Answer: B. Track slowly changing parameters

Question 6

Which action best prevents integrator windup during actuator saturation?

A. Increasing the integrator gain without limit

B. Adding anti-windup feedback or limiting integrator accumulation

C. Removing all current feedback

D. Increasing the reference beyond the actuator limit

Correct Answer: B. Adding anti-windup feedback or limiting integrator accumulation

Question 7

In a multi-objective drive optimisation, a Pareto-optimal solution is one for which:

A. Every objective has its absolute maximum

B. No objective can improve without worsening at least one other objective

C. All weighting factors are zero

D. Only switching frequency is considered

Correct Answer: B. No objective can improve without worsening at least one other objective

Question 8

Why is high-frequency signal injection useful in low-speed PMSM sensorless control?

A. It creates a measurable position-dependent saliency response

B. It removes the need for an inverter

C. It prevents all magnetic saturation

D. It directly measures shaft torque mechanically

Correct Answer: A. It creates a measurable position-dependent saliency response

Question 9

Which statement best describes a reference governor?

A. It modifies commands to preserve constraint feasibility

B. It replaces every hardware protection circuit

C. It estimates only stator resistance

D. It increases voltage beyond the DC-link limit

Correct Answer: A. It modifies commands to preserve constraint feasibility

Question 10

In the illustrative pump-drive case, why was independent hardware overcurrent protection retained?

A. Software estimators and predictive controllers cannot replace all safety protection

B. Hardware protection improves energy efficiency only

C. It is required to create permanent sensor error

D. It removes the need for current sensors

Correct Answer: A. Software estimators and predictive controllers cannot replace all safety protection