# Closed-loop continuous-time model identification with noisy input-output

Stéphane Victor, Arnold Diudichi, Pierre Melchior

► To cite this version:

Stéphane Victor, Arnold Diudichi, Pierre Melchior. Closed-loop continuous-time model identification with noisy input-output. 20th World Congress of the International Federation of Automatic Control (IFAC WC 2017), Jul 2017, Toulouse, France. pp.12853 - 12858, 10.1016/j.ifacol.2017.08.1936. hal-01703951

HAL Id: hal-01703951

https://hal.science/hal-01703951v1

Submitted on 13 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Closed-loop continuous-time model identification with
noisy input-output

Stéphane VICTOR, Arnold DIUDICHI, Pierre MELCHIOR

IMS, UMR 5218 CNRS - Université de Bordeaux - Bordeaux INP
351 cours de la Libération, 33405 TALENCE cedex, FRANCE
(e-mail: firstname.lastname@ims-bordeaux.fr)

Abstract

In this paper, system identification is proposed for closed-loop systems in input/output noisy context. Although instrumental variable techniques exist in the literature, the proposed filtering in input/output noisy context is not optimal. Also, three algorithms are proposed, and the closed-loop simplified refined instrumental variable for continuous-time system gives optimal results: estimates are consistent and with minimum variance. The main contributions relies on the filtering of the instruments and signals considered.

Key words: System identification, closed-loop system, noisy input-output systems, continuous-time models, errors-in-variables, linear estimation

# 1 Introduction

For safety, complexity and production reasons, a lot of industrial processes can not be identified in open-loop. Data measurements can then only be obtained in closed-loop (CL), and therefore this loop induces correlation between the disturbances and the control signal. Closed-loop system identification is already widely studied but not very often applied. Several algorithms have been proposed with direct, indirect and joint input-output approaches (Ljung (1999); Söderström and Stoica (1989); Gilson et al. (2008)).

The model under consideration is referred to continuous-time errors-in-variables (EIV) model. Discrete-time EIV system identification have been widely studied such as bias compensating least squares methods (Zheng (1998)), prediction error and maximum likelihood methods (Karlsson et al. (2000)), and instrumental variable based approaches (Söderström and Mahata (2002)). Although EIV system identification have started in continuous-time (CT) (Mahata and Garnier (2006)), the closed-loop case has not been studied so far and especially in the instrumental variable case. Instrumental variable (IV) techniques presents many assets: algorithms are simple and iterative with linear regression formulation and estimates are consistent (asymptotically unbiased). IV methods for system identification have been developed for decades (see e.g. Young (1981, 2011); Söderström and Stoica (1983); Söderström (2012)). In open-loop context, **Refined Instrumental Variables for Continuous system (rivc)**, and its simplified version (**srivc**) when the additive measurement noise is white, were developed in Young and Jakeman (1980); Young (1981, 2002) and extended to fractional systems in Victor et al. (2013). When the model lies in the same class as the true system, the *rivc* method and consequently the *srivc* can be interpreted in optimal statistical terms yielding consistent estimates with minimum variance. Both methods use an iterative-adaptive prefiltering based on a quasi-optimal statistical solution and provide an estimation of the parametric error covariance matrix. Recent developments aimed at extending the *rivc* method to handle multi-input transfer function model identification (Garnier et al. (2007)), hybrid Box-Jenkins model identification (Young et al. (2008)), closed-loop identification (Gilson et al. (2008)), irregularly sampled data (Wang et al. (2009)), non-uniformly sampled data (Chen et al. (2014)) and linear parameter varying model identification (Laurain et al. (2011)).
Section 2 relates the system identification context, then algorithms for closed-loop system identification are proposed in section 3. Then, statistical properties are deduced from Monte Carlo simulation in section 4, to finally conclude in section 5.

## 2 Problem formulation

Consider a stable, linear, SISO, closed-loop system as shown in Figure 1:

* $p = \frac{d}{dt}$ denotes the differential operator;

* the process input $u(t)$ and the noise-free output $y^*(t)$ are related by the following model of the form

$$G(p, \theta) = \frac{B(p)}{A(p)} = \frac{\sum_{i=0}^{n_B} b_i p^i}{1 + \sum_{j=1}^{n_A} a_j p^j} \quad (1)$$

where $B(p)$ and $A(p)$ are assumed coprime and $\theta$ is the parameter vector defined as

$$\theta = [b_0, b_1, \dots, b_{n_B}, a_1, \dots, a_{n_A}]^T, \quad (2)$$

* the controller is assumed known and given by

$$C(p) = \frac{Q(p)}{P(p)} \quad (3)$$

with $P(p)$ and $Q(p)$ assumed coprime;

* $r_1(t)$ and $r_2(t)$ are external signals that are collected into a single signal for simplicity sake

$$r(t) = r_1(t) + C(p)r_2^*(t); \quad (4)$$

* $\tilde{r}_2(t_k)$ and $\tilde{y}(t_k)$ are mutually independent, zero-mean, discrete-time white noise sequences with respectively, $\sigma_{\tilde{r}_2}^2$ and $\sigma_{\tilde{y}}^2$ variances.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Closed-loop configuration</figcaption></figure>

$K$ data samples are collected at regular time-instants from $t = 0$ to $t = T_{final}$ with a sampling period $T_s$. $K$ is assumed large enough to guarantee convergence of the estimated parameters to the true ones.

The CL system is assumed asymptotically stable and the quasi-stationary input signal $\{r(t), 0 \le t \le K\}$ applied to the system is persistently exciting giving rise to an output signal $\{y^*(t), 0 \le t \le K\}$.

Discrete-time (DT) noises are associated with the sampled data (see e.g. Young et al. (2008)) as it is easier to consider white noise in the discrete-time case (continuous-time white noise being a non-trivial extension of the DT case).
The complete data-generating closed-loop system equations can be written in the form:

$$ 
\left\{
\begin{aligned}
v(t_k) &= S(p)r(t_k) \\
y(t_k) &= G(p)v(t_k) \\
y^*(t_k) &= y(t_k) + S(p)\tilde{y}(t_k) \\
u(t_k) &= r(t_k) - C(p)y^*(t_k),
\end{aligned}
\right.
\qquad (5) $$

where the sensitivity function is defined as

$$ S(p) = \frac{1}{1 + C(p)G(p)}. \qquad (6) $$

Note that in (5), $u(t_k)$ can also be expressed as

$$ u(t_k) = v(t_k) - C(p)S(p)\tilde{y}^*(t_k). $$

The output noise-free signals $v$ and $y^*$ are necessary for instrumental variable algorithms (see Å§3.3) and are not available from measurements.

The CT model identification problem is to find estimates of the parameter vector $\theta$ in (2) from finite input/output (I/O) sequences $Z\{r(t_k), y^*(t_k)\}_{k=1}^K$ of respectively the external signal and output signal DT data.

Even though there are two approaches to estimate CT models (the indirect one where an initial discrete-time model is estimated from sampled data and is converted into a CT model, and the direct one where a CT model is directly estimated from sampled data), the direct approach is considered here.

# 3 Closed-loop Identification methods

A basic IV estimator can be used to estimate the parameter vector $\theta$ assuming that the time derivatives of the output $y^*$ and external $r$ signals are available. There are various ways of considering IV estimation.

## 3.1 Optimal Instrumental Variable estimator in open-loop

In an open-loop context, the iterative SRIVC (Simplified Refined Instrumental Variable Continuous-time) algorithm, known to deliver optimal estimates, when the additive measurement noise is white, is recalled in order to extend it to the closed-loop context.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Open-loop configuration</figcaption></figure>

From the SRIV algorithm for DT transfer function model identification, the SRIVC algorithm was developed by converting the maximum likelihood estimation equations to a pseudo-linear form by means of optimal prefilters (see e.g. Young (2011)). Using the model system in Fig. 2 and the prediction error method, a suitable error function $\epsilon$ is given by the output error (OE)

$$ \epsilon(t, \theta) = y^*(t) - \frac{B(p, \theta)}{A(p, \theta)}u(t), \qquad (7) $$
which can be rewritten as

$$
\begin{align}
\varepsilon(t, \theta) &= A(p, \theta) \left( \frac{1}{A(p, \theta)} y^*(t) \right) - B(p, \theta) \left( \frac{1}{A(p, \theta)} u(t) \right) \nonumber \\
&= A(p, \theta) y_f^*(t) - B(p, \theta) u_f(t) \tag{8}
\end{align}
$$

where $y_f^*(t) = \frac{1}{A(p, \theta)} y^*(t)$ and $u_f(t) = \frac{1}{A(p, \theta)} u(t)$.

Direct differentiation being avoided, a linear transformation (low-pass filter) is applied to input and output signals.
As shown in (8), the required filter

$$
F^{\text{opt}}(p) = \frac{1}{A(p, \theta)} \tag{9}
$$

turns out to lead to optimal Instrumental Variable (IV) estimates in open-loop context (see e.g. Young (2011)). In
practical cases, A(p, θ) is unknown. Also, an estimation ˆA(p, ˆθ^iter) is iteratively calculated:

$$
F^{\text{iter}}(p) = \frac{1}{\hat{A}(p, \hat{\theta}^{\text{iter}})} = \frac{1}{1 + \sum_{j=1}^{N} \hat{a}_j p^{\alpha_j}},
$$

where iter = 1, 2, . . . is the iteration number and ˆa_j the estimated a_j coefficient at iteration iter.

An auxiliary model, defined with transfer function coefficients computed iteratively, gives the noise-free output
estimates

$$
y_{\text{iv}}^{\text{iter}}(t) = \frac{\hat{B}(p, \hat{\theta}^{\text{iter}})}{\hat{A}(p, \hat{\theta}^{\text{iter}})} u(t)
$$

and the instruments can be built. The filtered derivatives of the input, the output, and the instruments are calculated
with:

$$
\begin{align*}
p^i u_f(t) &= F_i^{\text{iter}}(p) u(t), & 0 \le i \le n_B, \\
p^j y_f^*(t) &= F_j^{\text{iter}}(p) y^*(t), & 1 \le j \le n_A, \\
p^j y_{\text{iv},f}^{\text{iter}}(t) &= F_j^{\text{iter}}(p) y_{\text{iv}}^{\text{iter}}(t), & 1 \le j \le n_A,
\end{align*}
$$

where

$$
F_{\sigma}^{\text{iter}}(p) = p^{\sigma} F^{\text{iter}}(p) = \frac{p^{\sigma}}{\hat{A}(p, \hat{\theta}^{\text{iter}})}. \quad (10)
$$

The regression $\phi_f(t)$ and the instrumental variable $\phi_{iv,f}^{\text{iter}}(t)$ vectors are then defined as:

$$
\begin{align*}
\varphi_f(t) &= \begin{bmatrix} p^0 u_f(t), \dots, p^{n_B} u_f(t), \\ -p^1 y_f^*(t), \dots, -p^{n_A} y_f^*(t) \end{bmatrix}^T, \\
\varphi_{iv,f}^{\text{iter}}(t) &= \begin{bmatrix} p^0 u_f(t), \dots, p^{n_B} u_f(t), \\ -p^1 y_{iv,f}^{\text{iter}}(t), \dots, -p^{n_A} y_{iv,f}^{\text{iter}}(t) \end{bmatrix}^T.
\end{align*}
$$

The optimization problem of the SRIVC algorithm being formulated as the following

$$
\hat{\theta}^{\text{iter}+1} = \arg \min_{\theta} \left\| \left[ \mathbf{\Phi}_{\text{iv},f}^{\text{iter}} \mathbf{\Phi}_f^{\text{T}} \right] \theta - \left[ \mathbf{\Phi}_{\text{iv},f}^{\text{iter}} \mathbf{Y}_f^{*} \right] \right\|^2,
$$
with

$$
\begin{align*}
\mathbf{\Phi}_{\text{iv},f}^{\text{iter}} &= [\varphi_{\text{iv},f}^{\text{iter}}(0) \quad \dots \quad \varphi_{\text{iv},f}^{\text{iter}}(K)]^{\text{T}}, \\
\mathbf{\Phi}_{f} &= [\varphi_{f}(0) \quad \dots \quad \varphi_{f}(K)]^{\text{T}}, \\
\mathbf{Y}_{f}^{*} &= [y_{f}^{*}(0) \quad \dots \quad y_{f}^{*}(K)]^{\text{T}},
\end{align*}
$$

a solution is obtained by:

$$
\hat{\theta}^{\text{iter}+1} = (\mathbf{\Phi}_{\text{iv},f}^{\text{iter}} \mathbf{\Phi}_{f}^{\text{T}})^{-1} \mathbf{\Phi}_{\text{iv},f}^{\text{iter}} \mathbf{Y}_{f}^{*}.
$$

This algorithm is iteratively repeated until convergence, when

$$
\max_j \left| \frac{\hat{\theta}_j^{\text{iter}+1} - \hat{\theta}_j^{\text{iter}}}{\theta_j^{\text{iter}+1}} \right| < \epsilon, \quad (11)
$$

where, at the iteration iter, $\hat{\theta}_j^{\text{iter}}$ defines the j-th element of the estimated parameter vector.

The initialization of the SRIVC algorithm can be done by using the least squares-based state-variable filter (LSSVF) or the instrumental variable SVF (IVSVF) method (see Garnier (2006)). In the latter methods a non-optimal state variable filter is used:

$$
F_{\sigma}^{svf}(p) = \frac{p^{\sigma}}{\left(1 + \frac{p}{\omega_c}\right)^{N_c}}, \qquad (12)
$$

where two parameters should be predefined: the integer order $N_c > n_A$ (generally set to the integer above the system order) and the cut-off frequency $\omega_c$ (generally chosen below the sampling frequency inside the frequency bandwidth (Garnier et al. (2003)). State variable filters enable at the same time to have a differentiator behavior in the frequency band of interest and to filter noise in high frequencies.

Under persistently exciting input signal and the white zero-mean noise signal, the IV scheme gives consistent esti-
mates under these two conditions

$$
\begin{cases}
E \left[ \varphi_{\text{iv},f}^{\text{iter}}(t_k) \varphi_f^{\text{T}}(t_k) \right] & \text{is non-singular,} \\
E \left[ \varphi_{\text{iv},f}^{\text{iter}}(t_k) \xi(t_k) \right] & = 0.
\end{cases}
$$

When the noise signal is normally distributed, an estimation of the asymptotic covariance matrix (Young et al. (2008)) is computed:

$$
\hat{\mathbf{P}}_{\theta} = \hat{\sigma}^2 (\mathbf{\Phi}_{\text{iv},f} \mathbf{\Phi}_{\text{iv},f}^{\text{T}})^{-1}, \qquad (13)
$$

where $\hat{\sigma}^2$ defines the estimation of the noise variance, and $\mathbf{\Phi}_{\text{iv},f}$ is $\mathbf{\Phi}_{\text{iv},f}^{\text{iter}}$ computed at the last iteration.

Under a white Gaussian noise and sufficient persistency of the input signal assumptions, the Cramér-Rao lower-
bound is computable (Young et al. (2008)), and an IV estimator can asymptotically reach this bound (Wellstead
(1978)):

$$
\mathbf{P}_{\theta}^{\text{opt}} = \frac{\sigma_{\tilde{y}}^2}{N} \left( \text{E} \left[ \varphi_{\text{iv},f}^{\text{opt}}(t_k) \varphi_{\text{iv},f}^{\text{opt}}(t_k)^{\text{T}} \right] \right)^{-1}, \quad (14)
$$

which can be obtained with the true signals and the true noise variance $\sigma_{\tilde{y}}^2$. The instrumental vector $\varphi_{\text{iv},f}^{\text{opt}}(t_k)$ is the noise-free pre-filtered IV vector, with $F^{\text{opt}}(p)$ defined in (9) and

$$
\varphi_{\text{iv},f}^{\text{opt}}(t_k) = F^{\text{opt}}(p) \left[ \begin{array}{c} p^0 u(t_k), \dots, p^{n_B} u(t_k), \\ -p^1 y(t_k), \dots, -p^{n_A} y(t_k) \end{array} \right]^T . \quad (15)
$$

If the SRIVC algorithm is assumed to converge in the following sense $\hat{\theta} \to \theta$, then the covariance matrix (13) tends in probability to (14).
## 3.2 Closed-loop instrumental variable estimator

In the closed-loop context, Gilson et al. (2008) have proposed several IV algorithms that are effectively optimal when considering only (white or colored) output noise $\tilde{y}$ ($\hat{r}_2 = 0$). In the simplified refined context, the additive measurement noise $\tilde{y}$ is assumed white with a noise model $H(q^{-1}) = 1$. In this case, the model structure is a CT hybrid OE model described by:

$$ 
\begin{aligned}
y^*(t_k) &= \frac{B(p, \theta)}{A(p, \theta)} u(t_k) + \tilde{y}(t_k) \\
u(t_k) &= r(t_k) - C(p)y(t_k).
\end{aligned}
 $$

The CLSRIVC algorithm<fnref n="1" /> proposed in Gilson et al. (2008) follows the SRIVC algorithm approach in terms of prediction error minimisation in open-loop context (see equation (8)). Therefore, the CT filter is defined such as the one used in the SRIVC algorithm (see (9)).

Unfortunately, this filter is well-adapted in output white noise context, and consequently, it is not the optimal filter in input noise context. Therefore, an improved filter is proposed in the following paragraph.

## 3.3 Closed-loop instrumental variable estimator for errors- in-variables context

The CL configuration such as defined in Fig. 1 can be rewritten in an open-loop manner such as represented in Fig. 3. The output is unchanged and can be written as:

$$ y^*(t_k) = G(p)S(p)r_2(t_k) + w(t_k), $$

where the sensitivity function is given in (6) and the noise output is defined as

$$ w(t_k) = S(p)\tilde{y}(t_k) $$

or even:

$$ w(t_k) = S(p)y^*(t_k) - S(p)G(p)u(t_k). $$

Therefore, a suitable error function $\varepsilon$ can be given by:

$$ \varepsilon(t_k, \theta) = S(p, \theta)y^*(t_k) - S(p, \theta)G(p, \theta)u(t_k). $$

This error function can be rewritten in an equation error manner such as (8):

$$ 
\begin{aligned}
\varepsilon(t_k, \theta) &= S(p, \theta)y^*(t_k) - S(p, \theta)G(p, \theta)u(t_k) \\
&= A(p, \theta) \left( \frac{S(p, \theta)}{A(p, \theta)} y^*(t_k) \right) - B(p, \theta) \left( \frac{S(p, \theta)}{A(p, \theta)} u(t_k) \right) \\
&= A(p, \theta)y_f^*(t_k) - B(p, \theta)u_f(t_k)
\end{aligned}
 $$

where $y_f^* = \frac{S(p, \theta)}{A(p, \theta)} y^*(t)$ and $u_f(t) = \frac{S(p, \theta)}{A(p, \theta)} u(t)$. Now, the adequate filter $F(p)$ can be expressed as:

$$ F(p) = \frac{S(p, \theta)}{A(p, \theta)} = \frac{P(p)}{A(p)P(p) + B(p)Q(p)}. \qquad (16) $$

In practice, $G(p, \theta) = \frac{B(p, \theta)}{A(p, \theta)}$ is unknown, also an estimation of $\hat{B}(p, \theta^{\text{iter}})$ and $\hat{A}(p, \theta^{\text{iter}})$ is iteratively computed.

<footnote n="1">¹ The CLSRIVC algorithm is a simplified CLRIVC algorithm where two step of filtering are necessary: a CT filter and a DT filter.</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 3. Closed-loop EIV configuration reformulated in open-loop configuration</figcaption></figure>

### 3.3.1 Improved closed-loop SRIVC method: ICLSRIVC

In this paragraph, thanks to the IV mechanism, estimate will become consistent and the optimal solution (ICLSRIVC) is proposed in the following algorithm.

#### Algorithm 1 Improved closed-loop SRIVC algorithm

**Inputs:** The I/O signals $Z\{r(t_k), y^*(t_k)\}_{k=1}^K$ and the controller $C(p)$

**Outputs:** Model $\hat{G}(p, \hat{\theta})$ with covariance matrix $\hat{\mathbf{P}}_{\theta}$

**Procedure:**

(1) Initialization:

Compute the input signal $u(t_k)$ with:

$$u(t_k) = r(t_k) - C(p)y(t_k)$$

and generate an initial estimate $\hat{\theta}^0$ from data $Z\{r(t_k), y(t_k)\}_{k=1}^K$ and controller $C(p)$ using e.g. ICLLSSVF Algorithm 2 or ICLIVSVF Algorithm 3.

(2) Iterative estimation:

for iter = 1 : convergence

(a) Set the auxiliary model $G(p, \hat{\theta}^{iter-1})$

(b) Compute the output noise-free signals:

$$v(t_k) = S(p, \hat{\theta}^{iter-1})r(t_k)$$

$$y(t_k) = G(p, \hat{\theta}^{iter-1})v(t_k)$$

$$\text{with } S(p, \theta^{iter-1}) = \frac{1}{1+C(p)G(p, \hat{\theta}^{iter-1})}.$$

(c) Set the optimal filter as (16):

$$F^{\text{opt}}(p, \hat{\theta}^{\text{iter}-1}) = \frac{S(p, \hat{\theta}^{\text{iter}-1})}{A(p, \hat{\theta}^{\text{iter}-1})} \qquad (17)$$

and compute the I/O successive derivatives:

$$p^j y_f^*(t_k) = F^{\text{opt}}(p, \hat{\theta}^{\text{iter}-1}) p^j y^*(t_k), 0 \le j \le n_A$$

$$p^j y_f(t_k) = F^{\text{opt}}(p, \hat{\theta}^{\text{iter}-1}) p^j y(t_k), 0 \le j \le n_A$$

$$p^i u_f(t_k) = F^{\text{opt}}(p, \hat{\theta}^{\text{iter}-1}) p^i u(t_k), 0 \le i \le n_B$$

$$p^i v_f(t_k) = F^{\text{opt}}(p, \hat{\theta}^{\text{iter}-1}) p^i v(t_k), 0 \le i \le n_B$$

(d) Generate the regression vector $\varphi_f(t)$ and the instrumental vector:

$$\varphi_f(t) = \begin{bmatrix} p^0 u_f(t), \dots, p^{n_B} u_f(t), \\ -p^1 y_f^*(t), \dots, -p^{n_A} y_f^*(t) \end{bmatrix}^T$$

$$\varphi_{iv,f}^{iter}(t) = \begin{bmatrix} p^0 v_f(t), \dots, p^{n_B} v_f(t), \\ -p^1 y_f(t), \dots, -p^{n_A} y_f(t) \end{bmatrix}^T$$
(e) Compute the improved closed-loop SRIVC estimate:

$$ \hat{\theta}^{iter} = (\mathbf{\Phi}_{iv,f}^{iter} \mathbf{\Phi}_{f}^{T})^{-1} \mathbf{\Phi}_{iv,f}^{iter} \mathbf{Y}_{f}^{*}, $$

with

$$ \mathbf{\Phi}_{iv,f}^{iter} = [\varphi_{iv,f}^{iter}(0) \quad \dots \quad \varphi_{iv,f}^{iter}(K)]^{\mathrm{T}}, $$

$$ \mathbf{\Phi}_{f} = [\varphi_{iv,f}^{iter}(0) \quad \dots \quad \varphi_{iv,f}^{iter}(K)]^{\mathrm{T}}, $$

$$ \mathbf{Y}_{f}^{*} = [y_{f}^{*}(0) \quad \dots \quad y_{f}^{*}(K)]^{\mathrm{T}}. $$

(3) Compute the asymptotic covariance matrix of estimation errors associated with the estimate $\hat{\theta}_{ICLLSSVF}$:

$$ \hat{\mathbf{P}}_{\theta} = \hat{\sigma}_{e}^{2} (\mathbf{\Phi}_{iv,f}^{iter} \mathbf{\Phi}_{iv,f}^{iter T})^{-1}, $$

where $\mathbf{\Phi}_{iv,f}^{iter}$ is the IV matrix obtained at convergence and $\hat{\sigma}_{e}^{2}$ is the estimated residual variance.

Convergence is achieved when the estimate do not evolve, also, the iterative algorithm can be stopped with condition (11).

### 3.3.2 Improved suboptimal closed-loop methods: LSSVF and IVSVF methods

Suboptimal algorithms can be used to initialize the latter algorithm by using state variable filters. As in the open-loop identification methods, a filter is still needed to construct the derivatives of input-output data. Poisson filters as in (12) can be used to define these state variable filters. Knowing that the numerator order and denominator order of filter (16) are respectively $n_P + n_A$ and $2n_A + n_Q$, the filter used for SVF becomes:

$$ F^{\mathrm{svf}}(p) = \frac{p^{n_A+n_P}}{\left(1 + \frac{p}{\omega_c}\right)^{2n_A+n_P}}, \qquad (18) $$

where the cut-off frequency $\omega_c$ is chosen in the frequency bandwidth of the system.

#### Algorithm 2 Improved CLLSSVF algorithm

**Inputs:** The I/O signals $Z\{r(t_k), y^*(t_k)\}_{k=1}^K$ and the controller $C(p)$

**Outputs:** Model $\hat{G}(p, \hat{\theta})$ with covariance matrix $\hat{\mathbf{P}}_{\theta}$

**Procedure:**

(1) Compute the input signal $u(t_k)$ with:

$$ u(t_k) = r(t_k) - C(p)y^*(t_k) $$

(2) Set the state variable filter as in (18) and compute the filtered derivatives of input and output:

$$ p^i u_f(t) = F^{\mathrm{svf}}(p) p^i u(t), \qquad 0 \le i \le n_B $$

$$ p^j y_f^*(t) = F^{\mathrm{svf}}(p) p^j y^*(t), \qquad 1 \le j \le n_A $$

(3) Generate the regression vector $\varphi_f(t)$:

$$ \varphi_f(t) = \begin{bmatrix} p^0 u_f(t), \dots, p^{n_B} u_f(t), \\ -p^1 y_f^*(t), \dots, -p^{n_A} y_f^*(t) \end{bmatrix}^T $$

and consequently the regression matrix:

$$ \mathbf{\Phi}_{f} = [\varphi_{f}(0) \quad \dots \quad \varphi_{f}(K)]^{\mathrm{T}}, $$
(4) Compute the improved CL least-squares estimate:

$$ \hat{\theta}_{ICLLSSVF} = (\mathbf{\Phi}_f \mathbf{\Phi}_f^T)^{-1} \mathbf{\Phi}_f \mathbf{Y}_f^*, $$

with

$$ \mathbf{Y}_f^* = [y_f^*(0) \quad \dots y_f^*(K)]^T. $$

(5) Compute the asymptotic covariance matrix of estimation errors associated with the estimate $\hat{\theta}_{ICLLSSVF}$:

$$ \hat{P}_{\theta} = \hat{\sigma}_{e}^{2} (\mathbf{\Phi}_{f} \mathbf{\Phi}_{f}^{\mathrm{T}})^{-1}, $$

where $\hat{\sigma}_{e}^{2}$ is the estimated residual variance.

In order to improve the consistency of the estimates, an IV technique can also be used.

**Algorithm 3 Improved CLIVSVF algorithm**

The algorithm follows exactly the same procedure as in Algorithm 1 except that the filter (17) is replaced by filter (18) used for SVF.

## 4 Numerical example

The following numerical example is used to compare the performances of the different algorithms proposed above and to the performances of the “classic” algorithms proposed by Gilson et al. (2008).

The process to be identified is under feedback control as described in Fig. 1, where

$$ G(p) = \frac{p+1}{1+0.5p+p^2} $$

$$ C(p) = \frac{10p+15}{p}. $$

The external signal $r_2$ is chosen to be a pseudo-random binary signal of maximum length generated from a shift register with 4 stages and a clock period of 500 (N=7500 data points). The sampling period $T_s$ is fixed to 5 ms.

Note that in output noise context, the classic algorithms proposed in Gilson et al. (2008) give satisfying results for mean values of the estimates thanks to the IV mechanism, however the estimates are not optimal in EIV context and especially in reference input noise context.

Monte Carlo simulation (MCS) of 300 experiments is performed with white noise disturbance $\tilde{r}_2$ applied with a signal to noise ratio $SNR = 20$dB and white noise disturbance $\tilde{y}$ with $SNR = 20$dB. The Monte Carlo simulation results are summarized in Table 1.

The 3 first methods are algorithms proposed in Gilson et al. (2008) with SVF stemming from Poisson’s filter (12) (algorithms CLLSSVF and CLIVSVF) and with filter stemming from optimal filtering in open-loop context (9). These algorithms are effectively adapted in output noise context but the filtering is not appropriate in input noise context. Thanks to the IV mechanism, CLIVSVF and CLSRIVC give estimates with less bias, but the estimates are still far from the real true parameters. Moreover, the estimate variance is not minimal.

On the contrary, the three improved algorithms (3 last line in Table 1) give better result as compared to the classic methods. Even with the LS approach (CLLSSVF), the bias is reduced as compared to the classic one. Furthermore, the IV mechanism helps the estimation and cancels the bias. Finally, the improved closed-loop SRIVC (ICLSRIVC) algorithm gives the best results in input noisy context with consistent estimates and minimum variance.

Fig. 4 presents the summary of the MCS where only the magnitude Bode diagrams of the estimated models are plot. Fig a. gives the models stemming from the closed-loop LSSVF methods, Fig. b the models stemming from the closed-loop IVSVF methods, and Fig. c of the models stemming from the closed-loop SRIVC methods. Clearly, the proposed improved algorithms give the best results in closed loop EIV context given Table 1 and Fig. 4.
<figure><img src="image_5.png" /><figcaption>Fig. b</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. c</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Magnitude Bode diagrams of the estimated model for the 300 Monte Carlo simulation runs: true system (-), CL methods (-), improved CL methods (-)</figcaption></figure>

<table><caption>Table 1</caption><thead><tr><th>Method</th><th>Filter</th><th>ā<sub>0</sub> ± σā<sub>0</sub></th><th>ā<sub>1</sub> ± σā<sub>1</sub></th><th>b̂<sub>0</sub> ± σb̂<sub>0</sub></th><th>b̂<sub>1</sub> ± σb̂<sub>1</sub></th></tr></thead><tbody><tr><td>True value</td><td></td><td>1</td><td>0.5</td><td>1</td><td>1</td></tr><tr><td>CLLSSVF</td><td><sup>p<sup>σ</sup></sup>&frasl;<sub>(1 + <sup>p</sup>&frasl;<sub>ω<sub>c</sub></sub>)<sup>N<sub>c</sub></sup></sub></td><td>0.348 ± 0.096</td><td>0.250 ± 0.037</td><td>0.077 ± 0.128</td><td>1.052 ± 0.039</td></tr><tr><td>CLIVSVF</td><td><sup>p<sup>σ</sup></sup>&frasl;<sub>(1 + <sup>p</sup>&frasl;<sub>ω<sub>c</sub></sub>)<sup>N<sub>c</sub></sup></sub></td><td>0.553 ± 0.210</td><td>0.113 ± 0.166</td><td>0.244 ± 0.353</td><td>1.073 ± 0.039</td></tr><tr><td>CLSRIVC</td><td><sup>1</sup>&frasl;<sub>A(p,θ̂)</sub></td><td>0.606 ± 0.365</td><td>0.660 ± 0.233</td><td>0.488 ± 0.569</td><td>1.645 ± 0.473</td></tr><tr><td>ICLLSSVF</td><td><sup>p<sup>n<sub>A</sub>+n<sub>P</sub></sup></sup>&frasl;<sub>(1 + <sup>p</sup>&frasl;<sub>ω<sub>c</sub></sub>)<sup>2n<sub>A</sub>+n<sub>P</sub></sup></sub></td><td>0.977 ± 0.046</td><td>0.345 ± 0.050</td><td>0.775 ± 0.070</td><td>0.977 ± 0.011</td></tr><tr><td>ICLIVSVF</td><td><sup>p<sup>n<sub>A</sub>+n<sub>P</sub></sup></sup>&frasl;<sub>(1 + <sup>p</sup>&frasl;<sub>ω<sub>c</sub></sub>)<sup>2n<sub>A</sub>+n<sub>P</sub></sup></sub></td><td>0.974 ± 0.048</td><td>0.390 ± 0.055</td><td>0.858 ± 0.081</td><td>1.004 ± 0.012</td></tr><tr><td>ICLSRIVC</td><td><sup>S(p,θ̂)</sup>&frasl;<sub>A(p,θ̂)</sub></td><td>0.985 ± 0.045</td><td>0.416 ± 0.052</td><td>0.898 ± 0.069</td><td>0.993 ± 0.009</td></tr></tbody></table>

Monte-Carlo simulation results: 300 runs with input white noise (SNR = 20 dB) and output white noise (SNR = 20 dB) ($\bar{\theta}$ is the mean and $\sigma$ is the standard deviation)

## 5 Conclusion

Finally, system identification methods are proposed for closed-loop systems and in input/output noisy context. Instrumental variable algorithms exist in the literature however the filtering proposed in input noisy context is not optimal. Also, three algorithms are proposed, and the closed-loop simplified refined instrumental variable for continuous-time system gives consistent estimates with minimum variance. The main contributions relies on the filtering of the instruments and signals considered.
Errors-in-variables models under closed-loop configuration may be identifiable and estimable by applying a relative simple method, even if the reference is disturbed. Monte Carlo simulation show that the proposed CLSRIVC algorithm gives consistent estimates with minimum variance.

Several perspectives are still open such as the extension to colored noise context in input and output noise context. Furthermore, a direct extension to fractional system identification in closed-loop context can also be studied.

## References

Chen, F., Gilson, M., Aguero, J.C., Garnier, H., and Schorsch, J. (2014). Closed-loop identification of continuous-time systems from non-uniformly sampled data. In *13th European Control Conference, ECC'14*. Strasbourg, France.

Garnier, H. (2006). Identification de modèles paramétriques à temps continu. Méthodes, outil logiciel et avantages. *Journées Identification et Modélisation Expérimentale (JIME'2006)*. Poitiers (France).

Garnier, H., Gilson, M., Young, P., and Huselstein, E. (2007). An optimal IV technique for identifying continuous-time transfer function model of multiple input systems. *Control Engineering Practice*, 46(15), 471–486.

Garnier, H., Mensler, M., and Richard, A. (2003). Continuous-time model identification from sampled data. Implementation issues and performance evaluation. *International Journal of Control*, 76(13), 1337–1357.

Gilson, M., Garnier, H., Young, P., and Van den Hof, P. (2008). Instrumental Variable Methods for Closed-Loop Continuous-time Model Identification. In H. Garnier and L. Wang (eds.), *Identification of continuous-time models from sampled data*, 133–160. Springer-Verlag.

Karlsson, E., Söderström, T., and Stoica, P. (2000). The cramér-rao lower bound for noisy input-output systems. *Signal Processing*, 80(11), 2421–2447. doi:10.1016/S0165-1684(00)00126-2.

Laurain, V., Toth, R., Gilson, M., and Garnier, H. (2011). Direct identification of continuous-time LPV input/output models. *IET Control Theory & Applications*, 5(7), 878–888.

Ljung, L. (1999). *System identification – Theory for the user*. Prentice-Hall, Upper Saddle River, N.J., USA, 2 edition.

Mahata, K. and Garnier, H. (2006). Identification of continuous-time errors-in-variables models. *Automatica*, 42(19), 1477–1490. doi:10.1016/j.automatica.2006.04.012.

Söderström, T. (2012). How accurate can instrumental variable models become? In L. Wang and H. Garnier (eds.), *System Identification, Environmental Modelling, and Control System Design*, 3–25. Springer-Verlag.

Söderström, T. and Mahata, K. (2002). On instrumental variable and total least squares approaches for identification of noisy systems. *International Journal of Control*, 75(6), 381–389. doi:10.1080/00207170110112278.

Söderström, T. and Stoica, P. (1983). *Instrumental variable methods for system identification*. Springer Verlag, New York.

Söderström, T. and Stoica, P. (1989). *System Identification. Series in Systems and Control Engineering*. Prentice-Hall.

Victor, S., Malti, R., Garnier, H., and Oustaloup, A. (2013). Parameter and differentiation order estimation in fractional models. *Automatica*, 49(4), 926–935. doi:10.1016/j.automatica.2013.01.026.

Wang, J., Zheng, W.X., and Chen, T. (2009). Identification of linear dynamic systems operating in a networked environment. *Automatica*, 45(12), 2763–2772.

Wellstead, P.E. (1978). An instrumental product moment test for model order estimation. *Automatica*, 14, 89–91.

Young, P. (1981). Parameter estimation for continuous-time models – a survey. *Automatica*, 17(1), 23–29.

Young, P. (2002). Optimal IV identification and estimation of continuous-time TF models. In *15th World IFAC Congress*. IFAC, Elsevier, Barcelona (Spain).

Young, P. (2011). *Recursive estimation and Time-Series Analysis*. Springer-Verlag, Berlin, 2nd edition.

Young, P., Garnier, H., and Gilson, M. (2008). Identification of continuous-time models from sampled data. In H. Garnier and L. Wang (eds.), *Identification of continuous-time models from sampled data*, chapter Refined Instrumental Variable Identification of Continuous-time Hybrid Box-Jenkins Models, 91–131. Springer-Verlag.

Young, P. and Jakeman, A. (1980). Refined instrumental variable methods of time-series analysis: Part III, extensions. *International Journal of Control*, 31, 741–764. doi:10.1080/00207178008961080.

Zheng, W.X. (1998). Transfer function estimation from noisy input and output data. *International Journal of Adaptive Control and Signal Processing*, 12, 365–380. doi:10.1002/(SICI)1099-1115(199806)12:4<365::AID-ACS496>3.0.CO;2-A.