KRE Observer for IPMSM Sensorless Control

• KRE Observer is a state estimation framework for sensorless control of IPMSMs, providing a globally exponentially stable flux observer.
• It uses dynamic regressor extension, LTI filtering, and a virtual invariant manifold to rapidly reconstruct rotor position from stator measurements.
• Empirical validations show that KRE outperforms gradient-based observers with faster settling times and improved transient performance.

The Kreisselmeier Regression Extension (KRE) Observer is a state estimation framework applied to the sensorless control of interior permanent magnet synchronous motors (IPMSMs). It provides a globally exponentially stable (GES) flux observer that avoids limitations inherent to earlier gradient-based methods, particularly the requirement for small adaptation gains that compromise transient behavior. The KRE observer achieves rapid, stable estimation for real-time applications, with rigorous theoretical backing based on a virtual invariant manifold construction and dynamic regressor extension. It enables high-performance estimation of the IPMSM flux and rotor position from available stator current and voltage measurements (Yi et al., 2022).

1. Mathematical Model and Problem Statement

The IPMSM electrical subsystem in the stationary $\alpha\beta$ frame is described by:

• $\dot\lambda = -R\,i + v$
• $i = L(\theta)^{-1}[\lambda - \psi_m(\theta)]$

where $\lambda\in\mathbb{R}^2$ is the stator flux linkage, $i\in\mathbb{R}^2$ is stator current, $v\in\mathbb{R}^2$ is stator voltage, $R>0$ is stator resistance, $\psi_m(\theta)$ defines the permanent magnet flux vector, and $L(\theta)$ is the position-dependent inductance matrix. The "active flux" is given by:

$\phi = \lambda - L_q i$

The mechanical angle $\theta$ can be recovered from the active flux via:

$\tan\theta = \frac{\phi_2}{\phi_1}$

The observer is designed to reconstruct $\phi(t)$ (and thus $\theta(t)$) from $i(t)$ and $v(t)$.

2. Regression Structure and Virtual Invariant Manifold

To establish a regression form suitable for observer design:

• Two LTI filters, $H_1(p) = \frac{\alpha p}{p+\alpha}$ and $H_2(p) = \frac{\alpha}{p+\alpha}$, with $\alpha > 0$, are applied to the plant equations.
• This filtering generates measurable signals $y$ and $\Phi \in \mathbb{R}^2$ and introduces a small perturbation $d(t)$, producing the regression:

$y = \Phi^\top \phi + d$

Traditional gradient observers (e.g., [Choi et al.’19]) employ $E = \gamma \Phi (y - \Phi^\top \hat\phi)$, yielding only practical convergence unless the adaptation gain $\gamma$ is sufficiently small. The KRE observer replaces this approach via the construction of a dynamic regressor extension and a virtual manifold.

Define KRE system states:

• $Q(t) \in \mathbb{R}^{2 \times 2}$ (dynamic regressor covariance)
• $Y(t) \in \mathbb{R}^2$
• $\xi(t) \in \mathbb{R}^2$

and errors:

• $\tilde\phi = \hat\phi - \phi$
• $\tilde d = \hat d - d$

where $\hat d$ is a certainty-equivalent estimate of $d$.

The set:

$$\mathcal{M} = \{(Y, Q, \xi)\;|\; Y = Q\,\tilde\phi + \xi\}$$

is forward-invariant if $\dot\xi = -a(\xi - \Phi \tilde d)$, $\xi(0) = 0$.

3. KRE Observer Design Equations

The core of the KRE observer consists of the following system:

KRE extension:

$$\begin{aligned} \dot Q & = -a(Q - \Phi\Phi^\top), \quad Q(0) = 0 \ \dot Y & = -a(Y - \Phi e) + Q E, \quad Y(0) = 0 \ E & = -\gamma Y \end{aligned}$$

Prediction error:

$e = \Phi^\top\hat\phi + \hat d - y$

Flux-position observer:

$$\begin{aligned} \dot{\hat\lambda} & = -R\,i + E \ \hat\phi & = \hat\lambda - L_q i \ \hat\theta & = \mathrm{atan2}(\hat\phi_2, \hat\phi_1) \end{aligned}$$

Disturbance estimation:

$$\hat d = -\ell H_1[\Phi^\top \sigma(\hat\phi)], \quad \sigma(\hat\phi) = \begin{cases} \hat\phi/|\hat\phi|, & |\hat\phi| \ge \epsilon \ 0, & \text{otherwise} \end{cases}$$

with $\ell = \psi_m L_0$, $\epsilon \in (0, x_{\min})$, and tuning gains $a, \alpha, \gamma > 0$.

4. Stability Analysis and Global Exponential Convergence

Under persistency of excitation (Assumption 1), after a finite time $T$, the dynamic gain satisfies $Q(t) \ge q I_2 > 0$. On the invariant manifold $Y = Q\,\tilde\phi + \xi$, the closed error dynamics are:

$$\dot{\tilde\phi} = -\gamma Q \tilde\phi - \gamma \xi$$

Stacking all errors as $\chi = [\tilde\phi; \xi; z]$ and casting the system as a linear time-varying (LTV) system yields:

$\dot\chi = [A(t) + \alpha\Delta(t)]\,\chi$

with $A(t)$ uniformly negative-definite for $t \ge T$ and $\Delta(t)$ bounded. A quadratic Lyapunov function

$V(\chi) = \chi^\top P \chi$

with block-diagonal $$P = \mathrm{diag}(1/\gamma,\,1/aq,\,\mu_\Phi^2/(q\alpha))$$ demonstrates that

$\dot V \le -(\beta - \alpha\rho)\|\chi\|^2$

where $\beta > 0$ and $\rho > 0$ do not depend on $\alpha$. For $0 < \alpha < \alpha_{\max} := \beta/\rho$, this ensures global exponential convergence ($\chi \to 0$), thus $\tilde\phi\to 0$ and $\hat\theta\to\theta$ (Yi et al., 2022).

5. Tuning Guidelines

The KRE observer offers multiple tuning parameters affecting convergence and transient quality:

Parameter	Effect	Considerations
$\gamma > 0$	Scaling speed of $-\gamma Q$ in $\dot{\tilde\phi}$	No upper stability bound; noise sensitivity increases with $\gamma$
$\alpha$	Filter bandwidth	Must satisfy $\alpha < \alpha_{\max}$; too large destabilizes, too small slows filter
$a$	KRE "forgetting factor"	Small $a$: long memory; large $a$: tracks current value quickly; moderate $a$ (lowest electrical frequency) common

This suggests that with proper tuning, the transient performance can be arbitrarily improved without sacrificing stability, a departure from the small-gain limitation in gradient-based observers.

6. Comparative Performance: Simulations and Experiments

Simulation and experimental validation at 1000 rpm with initial flux–angle error of $\pi/2$, $\gamma=\{1,5\}$, and $(\alpha,a)=(200\pi,20\pi)$, demonstrate the following:

• KRE observer achieves settling in approximately $0.08$ s with $\gamma=1$ and $0.02$ s with $\gamma=5$
• Standard gradient observer as in [Ort et al.’21] requires $\gamma \ll 1$ for GES, failing with $\gamma=5$ (manifesting bias and oscillation)
• Real-time experiments on a SiC-inverter bench show:
  • KRE observer ($\gamma=5$): flux error $<0.02$ Wb in $40$ ms; angle error $<0.5^\circ$ after $50$ ms
  • Gradient observer at same $\gamma$: settling $\approx 120$ ms, residual fluctuations $\pm 0.05$ Wb, $\pm 2^\circ$

Key insight: By employing the dynamic regressor extension $Q(t)$ and auxiliary variable $Y(t)$, the standard $\Phi\Phi^\top$ gain is replaced with a memory-rich gain, scalable arbitrarily fast via $\gamma$ yet provably stabilizing for all $\gamma > 0$ if $\alpha$ is sufficiently small (Yi et al., 2022).

7. Impact and Theoretical Significance

The KRE observer, underpinned by the construction of a virtual invariant manifold and regressor extension, streamlines stability analysis, decouples filter-induced perturbation from the main estimation error, and provides provable GES for all $\gamma>0$. The observer framework enables high-performance sensorless control of IPMSMs, overcoming longstanding adaptation gain limitations. A plausible implication is that this methodology can be extensible to broader classes of nonlinear observer design where similar regression structures are present.