{Outline}

The Compensation Problem

{Compensation Philosophy and Challenges}

Goal

Minimise the degradation of vector control performance caused by motor–controller parameter mismatch by adapting the controller parameters to track the actual motor parameters in real time, without interrupting drive operation.

Fundamental Challenges

Direct parameter measurement during operation is generally impractical (no rotor access in a squirrel-cage motor)
Most adaptation algorithms are themselves sensitive to other machine parameters
Adaptation bandwidth is limited by the signal bandwidth available in the drive signals
Performance near zero speed is a universal limitation of all estimation-based methods

Two Adaptation Strategies

Direct strategy:

Monitor alignment of the flux and torque-producing axes
Use the orientation error directly to update \(\Rr^*\)

Indirect (signal-based) strategy:

Measure a terminal quantity sensitive to \(\Rr\) (e.g., modified reactive power)
Infer parameter variation from observable electrical signals
No flux sensor or observer required

Direct Axis-Alignment Method

{Principle of Axis-Alignment Detection}

Condition for Perfect Field Orientation

Perfect vector control requires:

\[ \lambda_{qr}^e = 0 \quad\Longleftrightarrow\quad \text{\(q\)-axis rotor flux equals zero} \]

If \(\lambda_{qr}^e \ne 0\), the orientation is lost and the flux and torque channels are coupled.

Detection and Adaptation Scheme

Compute \(\lambda_{qr}^e\) from measured stator currents and the estimated flux
A nonzero \(\lambda_{qr}^e\) is the alignment error
Use this error to adapt \(\Rr^*\):
\[ \frac{d\Rr^*}{dt} = -K_{\mathrm{adapt}}\cdot\lambda_{qr}^e\cdot\iqs \]
The update law drives \(\lambda_{qr}^e \to 0\) in steady state

No additional hardware beyond current sensors
Convergence rate set by \(K_{\mathrm{adapt}}\)

Reactive Power Based Method

{Modified Reactive Power — Concept and Adaptation Law}

Why Reactive Power?

The modified reactive power \(Q_m\) of the induction motor, defined as the cross-product of stator voltage and current in the stationary frame:

\[ Q_m = v_{ds}^s\,i_{qs}^s - v_{qs}^s\,i_{ds}^s \]

\(Q_m\) is sensitive to both \(\Lm\) and \(\Rr\)
The sign of \((Q_m^* - Q_m)\) changes when \(\Rr^*\) crosses the true value of \(\Rr\)
Reference \(Q_m^*\) from the controller model is compared to the measured \(Q_m\)
The error \(\Delta Q_m = Q_m^* - Q_m\) drives adaptation

PI Adaptation Law

\[ \Delta Q_m \propto \left(\Rr^* - \Rr\right) \]

When \(\Rr^* > \Rr\): \(\Delta Q_m > 0\) \(\Rightarrow\) reduce \(\Rr^*\) When \(\Rr^* < \Rr\): \(\Delta Q_m < 0\) \(\Rightarrow\) increase \(\Rr^*\) The adaptation is implemented as a PI controller:

\[ \Rr^*(s) = \left(K_p + \frac{K_i}{s}\right)\Delta Q_m(s) \]

Key Advantage

Computed entirely from terminal measurements — no rotor flux observer or shaft position sensor required.

Rotor Flux Deviation Method

{Rotor Flux Deviation as the Error Signal}

Flux Deviation Principle

If the field angle \(\thetaf\) is in error due to \(\Rr\) mismatch:

\(\lambda_{qr}^e \ne 0\) — nonzero quadrature rotor flux
The ratio of \(q\)- to \(d\)-axis flux equals \(\tan(\Delta\theta)\) where \(\Delta\theta\) is the field angle error
In steady state, this ratio is proportional to \((\alpha - 1)\):

\[ \frac{\lambda_{qr}^e}{\lambda_{dr}^e} \approx \frac{(\alpha - 1)\,\omegasl^*\Tr}{1 + \alpha(\omegasl^*\Tr)^2} \]

Implementation

Estimate \(\lambda_{qr}^e\) from measured stator currents and rotor position
Pass through a low-pass filter to remove PWM noise
A PI controller drives \(\lambda_{qr}^e \to 0\) by adjusting \(\omegasl^*\) (equivalently \(\Rr^*\))
Converges in a few rotor time constants

Practical Note

Flux observers require accurate \(\Ls\) and \(\Lm\) — residual errors from these induce a small bias offset but do not cause instability of the adaptation loop.

MRAS Rotor-Resistance Identification

MRAS Framework

{MRAS Concept and Framework}

MRAS Structure

Reference model: uses measured quantities (stator currents, voltages) — has no \(\Rr\) dependence
Adaptive model: uses the controller \(\Rr^*\) — its output depends on \(\Rr^*\)
Error signal: difference between the two model outputs
Adaptation mechanism: adjusts \(\Rr^*\) to drive the error to zero

Choice of Comparison Variable

Rotor flux linkages are used:

\[ \varepsilon = \hat{\lambda}_{qr}^{s\,(\mathrm{ref})} - \hat{\lambda}_{qr}^{s\,(\mathrm{adapt})} \]

Reference model: current-based (stator equations)\\ Adaptive model: voltage-based (rotor equations with \(\Rr^*\))

Popov Stability and Adaptation Law

{MRAS Adaptation Laws — Popov Hyperstability}

Popov's Hyperstability Criterion

The MRAS is asymptotically stable if and only if the adaptation law satisfies Popov's inequality:

\[ \int_0^t \varepsilon^{\mathsf{T}}\cdot w\,\mathrm{d}\tau \ge -\gamma^2, \quad \forall\, t \ge 0 \]

where \(w\) is the adaptive parameter vector and \(\gamma\) is a finite positive constant.

Resulting Adaptation Law for \(\Rr^*\)

\[ \frac{d\Rr^*}{dt} = -K_1\,\varepsilon\cdot\phi_r - K_2\int\!\varepsilon\cdot\phi_r\,\mathrm{d}t \]

\(\phi_r\): state component of the adaptive model proportional to \(\Rr^*\)
\(K_1, K_2 > 0\): design gains (proportional and integral)
Convergence guaranteed by Popov's criterion

Properties of MRAS Adaptation

Global asymptotic stability proven via Lyapunov and Popov methods
Converges to the correct \(\Rr\) regardless of initial error
Adaptation rate tunable independently via \(K_1\), \(K_2\)
Works for both increasing and decreasing \(\Rr\)
Not excitation-dependent at non-zero speed

Fundamental Limitation

The MRAS fails at zero speed where the reference model becomes unobservable — a fundamental limitation shared by all estimation-based methods.

Comparison and Summary

{Comparative Assessment of Compensation Methods}

lccccc@{}} Method	Sensors	Additional param.	Zero speed	Complexity	Stability
Axis alignment	\(i_s\), \(v_s\)	\(\Ls\), \(\Lm\)	No	Moderate	Conditional
Reactive power	\(i_s\), \(v_s\)	\(\Ls\), \(\Lm\)	No	Low	Conditional
Flux deviation	\(i_s\), \(\theta_r\)	\(\Ls\)	No	Moderate	Conditional
MRAS	\(i_s\), \(v_s\)	\(\Ls\), \(\Lm\)	No	High	Global (Popov)

Common Limitations

All methods fail near zero speed (universal observability constraint)
All depend on some additional known parameters
Slow adaptation bandwidth (governed by \(\Tr\))
Cannot simultaneously handle \(\Rr\) and \(\Lm\) changes

Practical Guidance

Parameter compensation is essential for robust vector control in thermally demanding environments
MRAS offers the strongest theoretical stability guarantee
The reactive power method is simplest to implement
Hybrid strategies combining multiple signals improve reliability across the full operating range