# Closed-Loop System Identification in Continuous-time Using Parallel PI Controller and Reference Prefiltering: White Noise Case

Jamy Vignaud, Stéphane Victor, Olivier Cahuc, Jean-Yves K'Nevez, Philippe Verlet

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## To cite this version:

Jamy Vignaud, Stéphane Victor, Olivier Cahuc, Jean-Yves K'Nevez, Philippe Verlet. Closed-Loop System Identification in Continuous-time Using Parallel PI Controller and Reference Prefiltering: White Noise Case. Comité Technique Identification/SAGIP, Nov 2024, Paris CNAM, France. hal-05446744

HAL Id: hal-05446744

https://hal.science/hal-05446744v1

Submitted on 7 Jan 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Closed-Loop System Identification Using Parallel PI Controller and Reference Prefiltering
## White Noise Case

J. Vignaud¹,²,³

S. Victor¹

O. Cahuc²

J-Y. K'Nevez²

Ph. Verlet³

¹Univ. Bordeaux, CNRS, Bordeaux INP, IMS, UMR 5218, F-33400 Talence, France

²Univ. Bordeaux, CNRS, Bordeaux INP, I2M, UMR 5295, F-33400, Talence, France

³VLM Robotics, Avenue Mayne Rabbit, F-33114, Le Barp, France

CT Identif, November 28th, 2024

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VLM
Robotics

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INSTITUT DE
MÉCANIQUE
ET D'INGÉNÉRIE

SAGIP
SOUTIEN A L'ÉTUDE
DU LIEU INDUSTRIEL & PRODUITIF

IDENTIF
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# Table of Contents

1. Context

2. Problem formulation

3. System identification

* Least-squares method

* Extended instrumental variable method

* Simplified refined instrumental variable method

4. Application

* Simulated system

* Results

5. Conclusions and Future Works
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# Table of Contents

1. Context

2. Problem formulation

3. System identification

* Least-squares method

* Extended instrumental variable method

* Simplified refined instrumental variable method

4. Application

* Simulated system

* Results

5. Conclusions and Future Works
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# Context

## Thesis objective

Minimizing in real time the mechanical energy of the machining which leads to:

* Improve workpiece durability.

* Reduce residual stresses.

* Enhance ecological sustainability.

<figure><img src="image_20.png" /><figcaption>Figure: Face milling picture</figcaption></figure>
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# Context

## Thesis objective

Minimizing in real time the mechanical energy of the machining which leads to:

* Improve workpiece durability.

* Reduce residual stresses.

* Enhance ecological sustainability.

<figure><img src="image_26.png" /><figcaption>Figure: Milling schema</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Figure: Face milling picture</figcaption></figure>

IDENTIF
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# Context

## Thesis objective

Minimizing in real time the mechanical energy of the machining which leads to:

* Improve workpiece durability.

* Reduce residual stresses.

* Enhance ecological sustainability.

<figure><img src="image_33.png" /><figcaption>Figure: Face milling picture</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Figure: Milling schema</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>Figure: Mechanical energy control</figcaption></figure>

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Context

The aim of this work is to identify the **spindle motor** of a three-axis machine tool.

*A spindle is a motor that drives cutting tools, such as milling cutters or drill bits, at specific speeds.*
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## Context

The aim of this work is to identify the **spindle motor** of a three-axis machine tool.

*A spindle is a motor that drives cutting tools, such as milling cutters or drill bits, at specific speeds.*

## Machine

* Machine tool: DMG Mori CMX600V

<figure><img src="image_47.png" /><figcaption>Figure: Tool machine DMG Mori CMX600V</figcaption></figure>
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# Context

The aim of this work is to identify the **spindle motor** of a three-axis machine tool.
*A spindle is a motor that drives cutting tools, such as milling cutters or drill bits, at specific speeds.*

## Machine

* Machine tool: DMG Mori CMX600V

* Computer Numerical Controller (CNC): Siemens Sinumerik 840DsL

<figure><img src="image_53.png" /><figcaption>Figure: Tool machine DMG Mori CMX600V</figcaption></figure>

<figure><img src="image_54.png" /><figcaption>Figure: CNC software environment</figcaption></figure>

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## Context

The aim of this work is to identify the **spindle motor** of a three-axis machine tool.

A spindle is a motor that drives cutting tools, such as milling cutters or drill bits, at specific speeds.

### Machine

* Machine tool: DMG Mori CMX600V

* Computer Numerical Controller (CNC): Siemens Sinumerik 840DsL

* Spindle motor: inlineMASTER 15000 RPM

<figure><img src="image_57.png" /><figcaption>Figure: inlineMASTER 15000RPM spindle</figcaption></figure>

<figure><img src="image_58.png" /><figcaption>Figure: Tool machine DMG Mori CMX600V</figcaption></figure>

<figure><img src="image_59.png" /><figcaption>Figure: CNC software environment</figcaption></figure>

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# Context

The system to identify:

* is in a closed-loop system,

* is controlled by a parallel PI controller,

* has a reference prefilter set only for the integral controller.

<figure><img src="image_67.png" /><figcaption>Figure: Spindle speed regulation block diagram</figcaption></figure>

System to Identify

IDENTIF
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# Context

The system to identify:

* is in a closed-loop system,

* is controlled by a parallel PI controller,

* has a reference prefilter set only for the integral controller.

## About the external control signal

The system cannot be directly controlled with a specific external control signal. The signal "Anticipated Torque Control" can change but is controlled only internally with Siemens devices. That means the signal $r_1$ cannot be handled.

<figure><img src="image_73.png" /><figcaption>Figure: Spindle speed regulation block diagram</figcaption></figure>

IDENTIF
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# Context

The system to identify consists of:

* the electrical current controller,

* the spindle motor,

* and the encoder.

<figure><img src="image_79.png" /><figcaption>Figure: Spindle speed regulation block diagram</figcaption></figure>

IDENTIF
Hypothesis:

* Linear Time Invariant (LTI) around a working point

* Single-Input Single-Output (SISO)

* stable

* vitiated by a white noise

<figure><img src="image_80.png" /><figcaption>Figure: Spindle speed regulation block diagram</figcaption></figure>

System to Identify
IDENTIF
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# Context

The closed-loop system can be represented by the following block diagram:

<figure><img src="image_86.png" /><figcaption>Figure: Closed-loop system block diagram</figcaption></figure>

## Signals:

* $r_1$: External control signal

* $r_2$: Reference signal

* $u$: Control signal

* $y$: Measured signal

* $e_o$: White noise

* $\epsilon_*$: Error signals

## Systems:

* $G_o$: True system to identify

* $C_p$: Proportional controller

* $C_i$: Integral controller

* $F$: Reference prefilter

IDENTIF
# Context

The closed-loop system can be represented by the following block diagram:

<figure><img src="image_87.png" /><figcaption>Figure: Closed-loop system block diagram</figcaption></figure>

## Signals:

* $r_1$: External control signal

* $r_2$: Reference signal

* $u$: Control signal

* $y$: Measured signal

* $e_o$: White noise

* $\epsilon_*$: Error signals

## Systems:

* $G_o$: True system to identify

* $C_p$: Proportional controller

* $C_i$: Integral controller

* $F$: Reference prefilter

## Remark 1

$r_1$ and/or $r_2$ should be persistently exciting.

## Remark 2

Thanks to a parallel PI controller, $r_2$ is not filtered when passing through the proportional part.

IDENTIF
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# Context

## Aim of the presentation

* Explore various **closed-loop system identification methods** in a **white noise** context by considering such reference prefiltering and parallel controller.

* The system is identified in **discrete-time**.

* A **Monte-Carlo simulation** is then proposed to analyze the statistical properties of the proposed methods.
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# Table of Contents

1. Context

2. Problem formulation

3. System identification
   - Least-squares method

- Extended instrumental variable method

- Simplified refined instrumental variable method

4. Application
   - Simulated system
   - Results

5. Conclusions and Future Works
# Problem formulation

There are two controllers:

* Proportional controller:
  $C_P(q) = K \in \mathbb{R}^*$,

* Integral controller:

$$C_I(q) = \frac{Q(q^{-1})}{P(q^{-1})} = \frac{\sum_{i=1}^{n_q} q_i q^{-i}}{1 + \sum_{j=1}^{n_p} p_j q^{-j}}$$

A reference prefilter is set as input for integral controller:

$$F(q) = \frac{Q_f(q^{-1})}{P_f(q^{-1})} = \frac{\sum_{i=0}^{n_{qf}} q_{fi} q^{-i}}{1 + \sum_{j=1}^{n_{pf}} p_{fj} q^{-j}}$$

The external input signal $r(k)$ is defined as:

$$r(k) = r_1(k) + [C_P(q) + C_I(q)F(q)] r_2(k)$$

<figure><img src="image_98.png" /><figcaption>Figure: Closed-loop system block diagram</figcaption></figure>
# Problem formulation

There are two controllers:

* Proportional controller:

$$C_P(q) = K \in \mathbb{R}^*,$$

* Integral controller:

$$C_I(q) = \frac{Q(q^{-1})}{P(q^{-1})} = \frac{\sum_{i=1}^{n_q} q_i q^{-i}}{1 + \sum_{j=1}^{n_p} p_j q^{-j}}.$$

A reference prefilter is set as input for integral controller:

$$F(q) = \frac{Q_f(q^{-1})}{P_f(q^{-1})} = \frac{\sum_{i=0}^{n_{qf}} q_i f^{-i}}{1 + \sum_{j=1}^{n_{pf}} p_j f^{-j}}.$$

The external input signal $r(k)$ is defined as:

$$r(k) = r_1(k) + [C_P(q) + C_I(q)F(q)] r_2(k)$$

<figure><img src="image_99.png" /><figcaption>Figure: Closed-loop system block diagram</figcaption></figure>

The true closed-loop is expressed by:

$$S: \begin{cases} y(k) = G_o(q)u(k) + e_o(k) \\ u(k) = r(k) - [C_P(q) + C_I(q)]y(k). \end{cases}$$

with the real system to identify

$$G_o(q) = \frac{B(q^{-1})}{A(q^{-1})} = \frac{\sum_{i=0}^{n_b} b_i q^{-i}}{1 + \sum_{j=1}^{n_3} a_j q^{-j}} q^{-n_k}.$$
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# Problem formulation

## System identification objective

From a sufficiently persistent dataset ($r_1, r_2, y$) of $N$ samples and given the system model

$$G(q, \hat{\theta}) = \frac{\hat{B}(q^{-1})}{\hat{A}(q^{-1})} = \frac{\sum_{j=0}^{n_b} \hat{b}_j q^{-i}}{1 + \sum_{j=1}^{n_a} \hat{a}_j q^{-j}} q^{-n_k},$$

the objective is to estimate the parameter vector

$$\hat{\theta} = [\hat{a}_{n_a} \quad \dots \quad \hat{a}_1 \quad \hat{b}_{n_b} \quad \dots \quad \hat{b}_0]^T$$

where

* $n_a = \deg(\hat{A})$,

* $n_b = \deg(\hat{B})$,

* and $n_k$ is the system delay (assumed known).
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# Problem formulation

## System identification objective

From a sufficiently persistent dataset ($r_1, r_2, y$) of $N$ samples and given the system model

$$G(q, \hat{\theta}) = \frac{\hat{B}(q^{-1})}{\hat{A}(q^{-1})} = \frac{\sum_{j=0}^{n_b} \hat{b}_j q^{-j}}{1 + \sum_{j=1}^{n_a} \hat{a}_j q^{-j}} q^{-n_k},$$

the objective is to estimate the parameter vector

$$\hat{\theta} = [\hat{a}_{n_a} \quad \dots \quad \hat{a}_1 \quad \hat{b}_{n_b} \quad \dots \quad \hat{b}_0]^T$$

where

* $n_a = \deg(\hat{A})$,

* $n_b = \deg(\hat{B})$,

* and $n_k$ is the system delay (assumed known).

## Hypothesis

The model system is assumed to be in the same class as the true one.

<figure><img src="image_110.png" /><figcaption>Figure: Closed-loop system block diagram</figcaption></figure>
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# Table of Contents

1. Context

2. Problem formulation

3. System identification

* Least-squares method

* Extended instrumental variable method

* Simplified refined instrumental variable method

4. Application

* Simulated system

* Results

5. Conclusions and Future Works
# Least-squares system method (CLLSSVFFR)

The least-squares algorithm has to minimize the following quadratic criterion:

$$J_{ls} = \frac{1}{N} \sum_{k=1}^{N} \left[ F_{svf}(q) (y(k) - \hat{y}(k, \hat{\theta}_{ls})) \right]^2$$

with:

* the measured output $y(k)$.

* the estimated output $\hat{y}(k, \hat{\theta}_{ls})$ such that:

$$\hat{y}(k, \hat{\theta}_{ls}) = - \hat{a}_{n_a} y(k - n_a) - \dots - \hat{a}_{1} y(k - 1) \\ + \hat{b}_{n_b} u(k - n_b - n_k) + \dots + \hat{b}_0 u(k - n_k).$$

* the control signal $u(k)$ such that:

$$u(k) = r(k) - [C_P(q) + C_I(q)] y(k).$$

* a state-variable filter $F_{svf}(q)$.
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## Least-squares system method (CLLSSVFFR)

The least-squares algorithm has to minimize the following quadratic criterion:

$$J_{ls} = \frac{1}{N} \sum_{k=1}^{N} \left[ F_{svf}(q) (y(k) - \hat{y}(k, \hat{\theta}_{ls})) \right]^2$$

with:

* the measured output $y(k)$.

* the estimated output $\hat{y}(k, \hat{\theta}_{ls})$ such that:

$$\hat{y}(k, \hat{\theta}_{ls}) = -\hat{a}_{n_3}y(k - n_a) - \dots - \hat{a}_{1}y(k - 1) + \hat{b}_{n_b}u(k - n_b - n_k) + \hat{b}_{0}u(k - n_k).$$

* the control signal $u(k)$ such that:

$$u(k) = r(k) - [C_P(q) + C_I(q)]y(k).$$

* a state-variable filter $F_{svf}(q)$.

The chosen state-variable filter $F_{svf}(q)$ is a Poisson filter in discrete-time:

$$F_{svf}(q) = \frac{1}{\left( \frac{2}{T_s \omega_c} \frac{1-q^{-1}}{1+q^{-1}} + 1 \right)^{n_a+1}}$$

with:

* $\omega_c$: the cut-off frequency of the filter,

* $T_s$: the sampling time,

* $n_a = \text{deg}(\hat{A})$

### Remark

This estimation is a direct method.
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## Least-squares method (CLLSSVFFR)

<figure><img src="image_126.png" /><figcaption>Figure: Closed-loop least-squares cllssvffr algorithm schema</figcaption></figure>
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# Extended instrumental variable method (CLIVSVFFR)

The instrumental variable algorithm has to minimize the following criterion:

$$J_{xiv} = \left\| \frac{1}{N} \sum_{k=1}^{N} F_{svf}(q) \left( z(k, \hat{\theta}_{xiv}^{\text{iter}-1}) [y(k) - \hat{y}(k, \hat{\theta}_{xiv}^{\text{iter}})] \right) \right\|_{2}^{2}$$

<figure><img src="image_132.png" /><figcaption>Figure: Auxiliary system block diagram</figcaption></figure>

Defining:

* the instrumental variable control $\hat{v}(k)$ such that:

$$\hat{v}(k) = \frac{r(k)}{1 + [C_P(q) + C_I(q)] \hat{G}(q, \hat{\theta}_{xiv})}$$

* and the instrumental variable output $\hat{x}(k)$ such that:

$$\hat{x}(k) = \hat{G}(q, \hat{\theta}_{xiv}) \hat{v}(k),$$

* the instrumental vector $z(k, \hat{\theta}_{xiv})$ is:

$$z(k, \hat{\theta}_{xiv}) = [-\hat{x}(k - n_a), \dots, -\hat{x}(k - 1), \\ \hat{v}(k - n_b - n_k), \dots, \hat{v}(k - n_k)]^T.$$
Extended instrumental variable method (CLIVSVFFR)

Remark

This estimation is *iterative*.

The stopping criterion is the sum of
relative difference of the parameter
vector.

The initial parameter vector can be
provided by the least-squares method.

<figure><img src="image_133.png" /><figcaption>Figure: Closed-loop extended instrumental variable clivsvffr algorithm schema</figcaption></figure>

IDENTIF
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# Simplified refined instrumental variable method (CLSRIVSVFFR)

The simplified refined instrumental variable algorithm has to minimize the following criterion:

$$J_{sriv} = \left\| \frac{1}{N} \sum_{k=1}^{N} F_{opt}(q) \left( z(k, \hat{\theta}_{sriv}^{iter-1}) [y(k) - \hat{y}(k, \hat{\theta}_{sriv}^{iter})] \right) \right\|_{2}^{2}$$

## Optimal filter explanation

1. From the error function:

$$\epsilon(k) = y(k) - \frac{B(q^{-1})}{A(q^{-1})} u(k)$$

2. After factorization, one gets:

$$\epsilon(k) = A(q^{-1}) \frac{1}{A(q^{-1})} y(k) - B(q^{-1}) \frac{1}{A(q^{-1})} u(k)$$

which comes down to a filtered equation error with

$$y_f(k) = F_{opt}(q)y(k) \text{ and } u_f(k) = F_{opt}(q)u(k).$$

3. From which the “optimal filter” is:

$$F_{opt}(q) = \hat{A}^{-1}(q^{-1}; \hat{\theta}).$$

IDENTIF
The initial parameter vector can be
provided by the extended
instrumental variable method.

<figure><img src="image_139.png" /><figcaption>Figure: Closed-loop simplified refined instrumental variable algorithm clsrivfr schema</figcaption></figure>

IDENTIF
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# Table of Contents

1. Context

2. Problem formulation

3. System identification

* Least-squares method

* Extended instrumental variable method

* Simplified refined instrumental variable method

4. Application

* Simulated system

* Results

5. Conclusions and Future Works
The true discrete-time plantª is

$$G_c(q) = \frac{0.0997 - 0.09q^{-1}}{1 - 1.884q^{-1} + 0.9048q^{-2}} q^{-1}$$

The sampling time is set to 5 ms.

The proportional and integrator controller are respectively:

$$C_P(q) = 10.75 \text{ and } C_I(q) = \frac{1.5q^{-1}}{1 - q^{-1}}$$

The reference prefilter is set as:

$$F(q) = \frac{1.072q^{-1} - 0.9675q^{-2}}{1 - 0.812q^{-1} + 0.0627q^{-2}}$$

<figure><img src="image_145.png" /><figcaption>Figure: Frequency response of the true system</figcaption></figure>

<footnote>ªP.C. Young et al. "Optimal instrumental variable method for closed-loop identification". In: *IET Control Theory & Applications* (2011).</footnote>
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# Simulated system

* Monte-Carlo simulation consists of 500 runs,

* SNR = 10, 15 and 20 dB,

* $r_1$ is not handled ($r_1(k) = 0$),

* $r_2$ is a PRBS,

* $N$ samples: 4088 (simulation time: ≈ 20s),

* The order and system delay are assumed known.

<figure><img src="image_151.png" /><figcaption>Figure: Closed-loop system block diagram</figcaption></figure>

<figure><img src="image_152.png" /><figcaption>Figure: Signals with SNR = 10dB</figcaption></figure>
# Simulated system

* Monte-Carlo simulation consists of 500 runs,

* SNR = 10, 15 and 20 dB,

* $r_1$ is not handled ($r_1(k) = 0$),

* $r_2$ is a PRBS,

* $N$ samples: 4088 (simulation time: ≈ 20s),

* The order and system delay are assumed known.

<figure><img src="image_153.png" /><figcaption>Figure: Closed-loop system block diagram</figcaption></figure>

<figure><img src="image_154.png" /><figcaption>Figure: Signals with SNR = 10dB</figcaption></figure>

## Remark

The external input $r$ will be sufficiently persistent as the reference signal $r_2$ will not be filtered when passing through the Proportional controller.
<figure><img src="image_155.png" /></figure>

<figure><img src="image_156.png" /></figure>

<figure><img src="image_157.png" /></figure>

<figure><img src="image_158.png" /></figure>

<figure><img src="image_159.png" /></figure>

# Results

## Remark

The estimation was made with a Poisson state-variable filter with a cut-off frequency $\omega_c = 60$ rad/s and an order $n_a + 1 = 3$.

## Remark

The classic CLSRIV and CLIV have not converged.
<figure><img src="image_160.png" /></figure>

<figure><img src="image_161.png" /></figure>

<figure><img src="image_162.png" /></figure>

<figure><img src="image_163.png" /></figure>

<figure><img src="image_164.png" /></figure>

# Results

## Remark

The estimation was made with a Poisson state-variable filter with a cut-off frequency $\omega_c = 60$ rad/s and an order $n_a + 1 = 3$.

<figure><img src="image_165.png" /><figcaption>(a) Least-squares</figcaption></figure>

<figure><img src="image_166.png" /><figcaption>(b) Extended instrumental variable</figcaption></figure>

<figure><img src="image_167.png" /><figcaption>(c) Simplified refined instrumental variable</figcaption></figure>

<figure><img src="image_168.png" /></figure>

<figure><img src="image_169.png" /></figure>

<figure><img src="image_170.png" /></figure>

Figure: Frequency responses of the 500 estimated transfer functions for SNR = 10dB

IDENTIF
<figure><img src="image_171.png" /></figure>

<figure><img src="image_172.png" /></figure>

<figure><img src="image_173.png" /></figure>

<figure><img src="image_174.png" /></figure>

<figure><img src="image_175.png" /></figure>

# Results

<figure><img src="image_176.png" /><figcaption>Figure: Comparison of the parameters</figcaption></figure>
<figure><img src="image_177.png" /></figure>

<figure><img src="image_178.png" /></figure>

<figure><img src="image_179.png" /></figure>

<figure><img src="image_180.png" /></figure>

<figure><img src="image_181.png" /></figure>

# Results

<figure><img src="image_182.png" /><figcaption>Figure: Comparison of the parameters</figcaption></figure>

<table><thead><tr><th></th><th colspan="2">a<sub>2</sub></th><th colspan="2">a<sub>1</sub></th></tr></thead><tbody><tr><th scope="row">True</th><td>9.048×10<sup>-1</sup></td><td></td><td>-1.8838</td><td></td></tr><tr><th scope="row">Method</th><td>â<sub>2</sub> × 10<sup>-1</sup></td><td>σ<sub>a2</sub> × 10<sup>-4</sup></td><td>â<sub>1</sub></td><td>σ<sub>a1</sub> × 10<sup>-4</sup></td></tr><tr><th scope="row">cllssvffr</th><td>9.208700</td><td>22.629</td><td>-1.9012902</td><td>23.666</td></tr><tr><th scope="row">clivsvffr</th><td>9.047774</td><td>26.614</td><td>-1.8837456</td><td>29.129</td></tr><tr><th scope="row">clsrivfr</th><td>9.048411</td><td>22.595</td><td>-1.8838148</td><td>24.992</td></tr></tbody></table>

<table><caption>Table: Estimated model coefficients ($\hat{\theta}$: mean; $\sigma_{\theta}$: standard deviation) with SNR = 10 dB</caption><thead><tr><th></th><th colspan="2">b<sub>1</sub></th><th colspan="2">b<sub>0</sub></th></tr></thead><tbody><tr><th scope="row">True</th><td>-9×10<sup>-2</sup></td><td></td><td>9.97×10<sup>-2</sup></td><td></td></tr><tr><th scope="row">Method</th><td>b̂<sub>1</sub> × 10<sup>-2</sup></td><td>σ<sub>b1</sub> × 10<sup>-4</sup></td><td>b̂<sub>0</sub> × 10<sup>-2</sup></td><td>σ<sub>b0</sub> × 10<sup>-4</sup></td></tr><tr><th scope="row">cllssvffr</th><td>-8.66355</td><td>6.905</td><td>9.48120</td><td>6.050</td></tr><tr><th scope="row">clivsvffr</th><td>-8.99825</td><td>7.450</td><td>9.97050</td><td>6.576</td></tr><tr><th scope="row">clsrivfr</th><td>-9.00071</td><td>5.680</td><td>9.97251</td><td>4.727</td></tr></tbody></table>
<figure><img src="image_183.png" /></figure>

<figure><img src="image_184.png" /></figure>

<figure><img src="image_185.png" /></figure>

<figure><img src="image_186.png" /></figure>

<figure><img src="image_187.png" /></figure>

# Table of Contents

1. Context

2. Problem formulation

3. System identification

* Least-squares method

* Extended instrumental variable method

* Simplified refined instrumental variable method

4. Application

* Simulated system

* Results

5. Conclusions and Future Works
# Conclusions and Future Works

## Conclusions

* Three extended algorithms (LS, XIV, and SRIV) were evaluated in a closed-loop context with a parallel PI controller and a prefiltered reference.

* The closed-loop simplified refined instrumental variable method ensures consistent estimates.
<figure><img src="image_188.png" /></figure>

<figure><img src="image_189.png" /></figure>

<figure><img src="image_190.png" /></figure>

<figure><img src="image_191.png" /></figure>

<figure><img src="image_192.png" /></figure>

# Conclusions and Future Works

## Conclusions

* Three extended algorithms (LS, XIV, and SRIV) were evaluated in a closed-loop context with a parallel PI controller and a prefiltered reference.

* The closed-loop simplified refined instrumental variable method ensures consistent estimates.

## Future Works

* Application on a real machine tool to identify the spindle behavior around different operating points.

* Extend system identification to a closed-loop system with colored noise.

* Design a robust control system for the spindle to minimize power consumption during machining.
Thank You!

Questions are welcome!