Complex Engineering Systems

Open Access Research Article

Correspondence to: Prof. José Ragot, Université de Lorraine, Centre de Recherche en Automatique de Nancy, CNRS UMR 7039, 2, Avenue de la forêt de Haye, Vandœuvre-les-Nancy Cedex 54518, France. E-mail: José.Ragot@univ-lorraine.fr

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The estimation of the parameters of a system by a set membership approach consists in characterizing the set of parameters completely compatible with all the measurements made on the system, the model of this system and the characteristics of the errors and uncertainties that affect the measurements and the system. In this context, it is assumed that the error affecting the measurements is bounded and belongs to a set that is realizable a priori. The estimation problem to be solved then consists in finding the set of admissible values of the model parameters in adequacy with the measurements, the errors and the uncertainties. These uncertainties are handled by an approach that takes into account the unknowns that are the structural error of the model and the values of these parameters. From a practical point of view, the result obtained is a domain of parameters varying in time, domain which is characterized by its bounds. The volume of this domain is minimized, the proposed model explaining the measurements made at each time by optimizing a criterion of precision of the volume in consideration.

Parameter estimation, bounded error, uncertain model

Obtaining a representative image of the functioning of a system remains an important step in the management of such a system. Indeed, even the techniques based on data use in one way or another the synthesis of a model capable of representing the functioning of a system. In this approach, several difficulties make this synthesis difficult, namely: the uncertainties on the data and measurements, the variations of the parameters which characterize the functioning of the system, the assumptions made on the structure of the model. If we consider each difficulty separately, many solutions have been proposed to deal with them. However, taking them into account at the same time remains a major difficulty. In what follows, we are interested in the identification of a system from uncertain data and taking into account the variability over time of the parameters of the system. More precisely, the aim is to estimate the bounds of the variable parameters simultaneously with those of the measurement errors.

From a historical point of view, the first works on parametric estimation taking into account bounds were published in the 80's and the proposed strategies were interested in circumscribing the domain describing the model uncertainties by a simple form. This estimation problem then leads to the determination of the set of admissible parameters known as the Feasible Parameter Set (FPS). This approach was initially designed to deal with a linear model with uncertain parameters and bounded errors. The estimation procedure amounts to determining the set of parameter values explaining all available observations. In this way we can guarantee that these observations are consistent with the bounds of the errors and the structure of the model. Among these sets are, for example, polytopes, zonotopes and ellipsoids. Generally, the ellipsoidal set is privileged in literature for its simplicity. More recently, constrained zonotopes have been introduced ^{[1-3]} which provide a new representation of sets allowing to combine the flexibility of convex polytopes with the efficiency of zonotopes. For that reason they have been extensively used in several fields in automatic control that include in particular state estimation, reachability analysis, identification, fault detection and isolation, diagnosis ^{[3-8]}.

For models that are linear with respect to their parameters, the FPS is characterized by a convex polytope that can be easily approximated by an ellipsoid ^{[9]} or by an orthotope ^{[10]} containing it in the least pessimistic way. One can in particular refer to the works by Walter *et al*. ^{[11]} and Mo *et al*. ^{[12]} which made use of polytopic domain for an exact and recursive characterization. In 2014, a recursive approach has been developed to define an approximation by an orthotope containing a set of parameters, the latter belonging to a polytope ^{[13]}. The main idea is to select a restricted number of constraints providing a quantified approximation of the exact set.

Many results have been published by Milanese *et al*. ^{[14]}. In this paper, which is a historical reference, the authors' idea was to get rid of the representation of uncertainties by Gaussian stochastic variables and to substitute them with a set of possible values whose bounds are only known. In another works ^{[15, 16]} the authors use the same idea of interval representation and develops it for the estimation of parameters of autoregressive-moving-average-exogenous (ARMAX) models. Of course this approach has been extended to non linear models with respect to their parameters. In order to reduce complexity, various methods have been proposed to determine an approximation of the FPS and some linear techniques have been extended to the nonlinear case using a succession of linearizations of the model ^{[17]}.

To solve the problem of nonlinear estimation with bounded error, the study by Jaulin *et al*. ^{[18]} proposed to use set inversion techniques and based on interval analysis, the idea being to characterize the FPS by means of boxes enclosing it externally and internally. We can also refer to the study by Bravo *et al*. ^{[19]} which uses a bounded description of the measurement noise and considers a representation of the set of parameters by a zonotope. The dimension of the monotype is adapted recursively as the measurements are acquired. The article ^{[20]} proposes a minimax estimation of parameters of nonlinear parametric models using experimental data. After choosing model structures, it is then possible to exhibit sets of linear inequalities to describe a domain approximating the FPS. The proposed algorithm effectively combines a local search procedure to decrease the upper bound of the solution with a pruning procedure based on the propagation of interval constraints.

Of course, identification is not an end in itself and many works use bounded parameter models for control, observer synthesis and diagnosis. For example, the article ^{[21]} is located in a Bayesian framework to address the problem of identification and detection of fault. A new approach to estimating fault by interval is proposed in the study by Zhou *et al*. ^{[8]}. A zonotope is used to represent a discrete linear time system whose parameters vary in the presence of bounded parametric uncertainties, measured disturbances and system disturbances. In general, many articles widely use polytope representation for the synthesis of state observers. For example in the study by Valero *et al*. ^{[22]}, the authors present an alternative state estimation method using convex polyhedra. The Kalman filter condition estimate was also discussed in light of the zonotope representation ^{[6]} and ^{[4]} in which, based on a new zonotope dimension criterion and combining observer gain design according to optimality and robustness criteria, a zonotopic Kalman filter is proposed with a robust convergence proof. The recent paper ^{[23]} also exploits the interval representation of stochastic uncertainties affecting a system in order to synthetize a Kalman filter dedicated to sensor fault detection.

Despite a possible resemblance, the problem considered in the study by Ploix *et al*. ^{[24]} is significantly different in that the uncertain parameters depend on time; more precisely, they are defined by random variables whose realizations have limited amplitudes. In addition, the proposed method does not use probabilistic formalism to determine the imprecision with which each model parameter is known. Only a class of linear structured and static models in uncertain parameters is considered. As already mentioned in citerag the measurement errors are bounded while the system parameters fluctuate within a limited domain invariant in time represented by a convex domain.

Thus, the proposed paper deals with parameter estimation in a bounded error context for linear models in parameters. Parameters may vary in a bounded volume domain, measurement errors also belong to a bounded domain, but the two domains are not known a priori. The objective of the proposed method is to determine the geometric characteristics of these domains (centre and radii for example). The idea is to determine the nominal value of the parameter vector and some time-variant uncertainties making it possible to explain the current observation. Maximal magnitudes of these uncertainties make it possible to deduce the characteristics of the considered domain. By fluctuating inside this one, parameters can explain all measurements. Moreover, in order to obtain the most precise model, the estimation problem is then to find the smallest domain.

In the following, section 2 formulates the variable parameter estimation problem and section 3 defines an accuracy criterion to obtain the smallest possible parametric uncertainty domain while guaranteeing the adequacy of the data to the system model. Sections 4 and 5 are related to the implementation of the proposed approach and to the presentation and discussion of numerical results.

What are the contributions of the proposed paper? As indicated by the previous bibliographic references, the bounded error approach to identify the parameters of a system is not new. Therefore, the contribution of our proposal lies in the following points:

● The representation of the uncertainties in additive form but also in multiplicative form, according to whether they affect the measurements of the outputs or the parameters of the considered system;

● The taking into account of the coupling between outputs of the system due to the presence of parametric uncertainties. Therefore the parametric domain has a minimal volume;

● The taking into account of a matrix characterizing the direction of the parametric uncertainties, this matrix can be a priori known or obtained in an experimental way;

● The joint identification of the parameters of the system and the bounds of the uncertainties, this estimation guaranteeing the membership of all the measurements to the identified model.

We describe in subsection 2.1 the structure of an uncertain system and its modelisation. Subsection 2.2 gives the principle of estimating the parameters of the model.

Let us consider an uncertain model of a system with several outputs, linear in parameters and observations, and represented by the following structure at each time instant

where

This vector allows to represent the uncertain nature of model parameters. Theses uncertainties are distributed on the various components of the vector

where

In the numerical applications of section 5, without affecting generality, the matrix

This type of model includes the particular case of multi input single output (MISO) systems and that of multi input multi output (MIMO) systems. However, in the MIMO case, according to the presence of uncertain parameters

**Remark 1.***The reader will note that the model describing $$ \theta(k) $$ in (1) is that of a zonotope usually defined as a Minkowski sum, but formulated here from its center and its generating vectors ^{[7]}. The values of $$ M $$ and $$ \lambda $$, but also the number of faces of the zonotope linked to $$ q $$, impact the shape of the domain $$ \mathcal{P}_\theta(\lambda, \theta_c) $$, which allows to describe a large number of situations. As an example, with $$ M=\begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\end{bmatrix} $$ and $$ \theta_c = \begin{bmatrix} 0 \\ 0\end{bmatrix} $$, the Figure 1 visualizes the shapes obtained for two values of $$ \lambda $$. The left part visualises the $$ 3 $$ generating vectors $$ m_1, m_2, m_3 $$ (the columns of the $$ M $$ matrix) and the left part shows the deformation of the zonotope according to a modification of its pareameters.*

**Remark 2.***Obviously, the formulation (1) of the system output also applies to a dynamic model. Indeed, without affecting generality, let us consider the stable system with a completely measured state:*

*By restricting the estimation problem to the coefficients of the matrix *

*with *

*the structure of this equation being then consistent with (1).*

The problem involved with parameter estimation is to characterize the unknown parameters

In the case of time-invariant parameters, Milanese and Belforte ^{[10]} suggest approximating the parameter domain with an orthotope aligned with the parameter coordinate axes and finding the minimal and maximal values of the ^{[9]} propose an ellipsoidal outer-bounding recursive algorithm: the centre of the ellipsoid and the positively defined symmetrical matrix which definies it are considered, respectively, as the central value of the parameter and its measurement of uncertainty.

In our formulation, the parameter estimation problem consists in finding the values of the vectors

with:

In this paper, it is assumed that a certain level of knowledge about the parameter domain is available in the sense that

● *In a first case*, if we consider that the parameters of the model are constant (

● *In a second case*, if the measurements are not affected by errors (the

● *In the other cases*, it will be possible to define a criterion for adjusting the bounds which is representative of the accuracy of the model, the latter being linked to the size of the domain. Indeed, increasing "arbitrarily" the values of the bounds

In view of the above remarks, it is necessary to define an indicator that is sensitive to the difference between the actual measurements and their model-generated estimates, and this indicator should depend explicitly on the model parameters. Ragot ^{[25]}, defined a criterion based on interval arithmetic ^{[26]} for a model with only one output. In this paper, a MIMO model is studied and the aim is to characterise uncertainties while minimising a criterion of precision related to the dimension of the output domain

The extension of this domain characterisation procedure, when ^{[27]}, which are unusable to find a solution and make the calculation very delicate. It is therefore necessary to establish a criterion which, at the same time, is representative of the precision of the model and does not lead to major difficulties in calculation.

The aim of this section is to define a mathematical criterion which provides a solution

Now, we are interested in the computation of all vertices of

with:

The expression (10) clearly shows that the components of

with

the time

These equations being only coupled with respect to

whose structure is in accordance with (12). Given the bounds on the uncertainties

In the space

Figure 2. Domain

In the aforementioned, the reader's attention has been drawn to the problem of the coupling of the equations by the parametric uncertainties. Of course, this coupling could also be taken into account depending on how the noise

Following the previous observation, in what follows all the intersections of the hyperplanes defining the half spaces will be considered without distinguishing between vertices and pseudo vertices of the domain

In fact, to characterize the domain, three steps are considered, the first for analyzing separately the components of

● *For the first step*, knowing that

where

Then

and highlights the influence matrix of the

● *Thus, the second step*, consists in reducing the coupling effect between the components of the

For that purpose, we try to eliminate, as much as possible, subsets of the components of the uncertainties

where

It goes without saying that the dimension of this co-kernel cannot be given because it depends on the rank of the matrix

The elimination of a part of the uncertain parameters

Taking into account the bounds of

By iterating this procedure for of all possible sets

where

● In the *last step* gathering inequalities (19) and (25) allows to describe the domain

with:

Note that this domain can also be defined by the hyperplanes of equation :

which depend linearly on the parameters

The main result of section 3.2 provides the bounded domain to which the measurements

For that purpose, we propose to compute the distances between the centre

* First*, it is necessary to determine the intersections of the hyperplanes (28) defining the domain

which consists of

Then, using (28a) in which

We then have the coordinates of the intersections of hyperplanes:

which is a linear expression in respect to the parameters

* Second*, the quadratic distance

Given (31), the distance is directly explained in terms of the parameters

with

Consequently, the quadratic mean of

* Third*, taking into account all the available data, the criterion of precision may be expressed:

This criterion is clearly a quadratic function of the magnitude of the uncertainties

Given the previous formulations, the characterization of the uncertain model takes into account two objectives.

The first one concerns the consistency of the data with the bounds of the model domain model, i.e., the parameter domain must be designed in order to explain all the available data. The second is an accuracy constraint, as the model parameters must control the volume of the domain.

For the first objective the principle of parameter estimation is to explain all the measurements. Thus, the vector

Thus, from (36), all the measurements

where

Then, for the second objective, the procedure of parameter estimation is reduced to a convex optimisation problem that consists to minimize the criterion (35) under linear inequality constraints (37) which define a domain in

under the constraint (37). The search for the solution ^{[28, 29]}. Finally, the proposed procedure is summarized by the two algorithms 1 and 2, the first dedicated to the synthesis of the model and the second to its validation.

Algorithm 1 Model Identification |

1: Collect a set of 2: Define the model structure (1): choose the structure (2) of the matrices 3: Define matrix 4: Define co-kernels 5: Define matrices 6: Define matrix 7: Define matrix 8: Solve problem (39) and obtain model parameters |

The implementation of the proposed procedure does not present any particular difficulty, except for the choice of the initial values of the parameters to be estimated. However, the reader's attention should be drawn to the possible numerical complexity which may be due, on the one hand, to the dimension of the matrices involved in the generation of the vertices of the domain

Algorithm 2 Identified model validation |

1: Adapt model (1) with identified parameters 2: Construct domain 3: Using data 4: Verify that |

We also mentioned the dimension problem in the second step of section 3.2 about the coupling of the

Regarding the search for the intersections

The above dimensions can be partly explained by the fact that the polytopic domains are accurate, since on the one hand no approximations were made for their evaluation and on the other hand the dependencies between variables due to uncertainties were taken into account. In the end, it is still possible to use a quantified simplification technique for polytopic domains. For standard zonotopes, this problem is addressed by applying reduction techniques that overapproximate a given zonotope by another with fewer generators ^{[30, 31]}.

The first three examples that follow are from the same system, but different by the nature of the data that have been generated : a pseudo-static system, i.e., where the matrix

The system (1) is used with the definitions

It is considered as static because the coefficients of the matrix

The reader will have noted that there are five orthogonal vectors to the left of the

The matrices

The variables

The Figure 3, in the

The Figure 4, in the ^{[27]} :

In conclusion, independently of the values of the identified parameters, we can see that the domains

Table 1

True and identified parameters

Identified | 0.91 | 1.24 | 1.77 | 0.07 | 0.26 | 5.02 | 5.02 |

True | 1.00 | 1.50 | 2.00 | 0.10 | 0.10 | 5.00 | 5.00 |

**Remark 3.***In the same spirit, a priori knowledge about the magnitudes of the parameters *

*it is then sufficient to modify the system (37) accordingly (43).*

The formulation (1) uses the matrix

The other data of the previous example have been kept. The previous proposed procedure applied to determine the center

Figure 6 visualizes the identified domain of the outputs. The blue markers locate the measured outputs, the solid and dashed contours correspond respectively to the

With regard to the identification of the parameters, we obtained

This third example concerns a system simulated with a matrix

The output

and the vectors

Table 2

True and identified parameters

Identified | 1.07 | 1.32 | 1.84 | 4.97 | 4.99 |

True | 1.00 | 1.50 | 2.00 | 5.00 | 5.00 |

In conclusion, we can see that the domains

The following continuous system

describes in a conventional way the simplified dynamics of a

where the uncertainties affect the two parameters

It is now necessary to structure this model in the form defined in (1). Following remark (2), after eliminating the states according to the measurements, we obtain :

the terms

The

The last graph (12) compares the results of our approach with those of a simpler approach which does not take into account the couplings between model parameters due to uncertainties. The implementation of the latter is limited to taking into account only the inequalities (19) to define the domain of the parameters. The comparison of Figures 11 and 12 shows the interest of taking into account the coupling of the parameters.

An approach, consisting in explaining the set of measurements while optimizing an accuracy criterion, is proposed in the most general case where the parameters are variable in time without considering the notion of speed of variation of the parameters. Moreover, the characterization of the uncertainties of a MIMO model highlights the dependencies between the model outputs, these dependencies being created by the parameters to be estimated. A technique taking into account these dependencies, combined with the calculation of an accuracy criterion is proposed. It provides an optimal solution (via the accuracy criterion) in the form of a set of parameters, its central value and the limits of the equation error.

In the future, this approach could be extended to other model structures. Moreover, it seems relevant to us to take up and deepen a remark that has been made about the richness of the excitation signals, namely: how to choose, when possible, the nature of the excitations to be applied to a system with uncertain parameters so as to best characterize the bounds of these parameters? A second area to explore concerns the presence of non-linearities. When these nonlinearities bring a bounded contribution to the evolution of the state variables of the system, to what extent can we also model their effects by bounded variable parameters?

Wrote and reviewed the manuscript: Ragot J

Not applicable.

None.

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

© The Author(s) 2022.

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* Complex Eng Syst* 2022;2:10. http://dx.doi.org/10.20517/ces.2022.13

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