---
title: KRE Observer for IPMSM Sensorless Control
url: https://www.emergentmind.com/topics/kreisselmeier-regression-extension-kre-observer
type: topic
---

# KRE Observer for IPMSM Sensorless Control

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 [2210.00190].

## 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 $\hat\theta\to\theta$ [2210.00190].

## 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 [2210.00190].

## 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.

Source: https://www.emergentmind.com/topics/kreisselmeier-regression-extension-kre-observer