Parametric dynamical models allows us to have a mathematical description of a dynamical system. Such models play a key role in many fields of science and engineer-ing: Chemistry, Mechanics, Electronic, Biology and many others. Without them, we cannot make predictions on the evolution of an epidemic outbreak, we cannot design the control system of a resonator, monitor the health of a wind turbine in order to plan its maintenance, and the list may continue.
How do we obtain a dynamical model ? We can build a model for a dynamical system starting from first principles, the laws of physics governing its behaviour. We refer to this method as white-box modelling or first principle modelling. The model so obtained has an immediate physical interpretation, but it may rely on constants or parameters that are uncertain. These parameters can then be calibrated/determined using input-output data collected on the true system (grey-box modeling).
Determining mathematical models based on data is the objective of system iden-tification . As mentioned in the previous paragraph, the parameters that will be calibrated/identified using the data collected on the system may be physical parame-ters. However, these parameters can also be just the coeﬃcients of a transfer function. In the latter case, the parameters do not have necessarily a physical meaning and we then speak of black-box modeling.
In this thesis, we will restrict attention to the modeling of linear systems using black-box models (transfer functions). We can divide a system identification procedure in the following four steps:
1. Model structure selection. In this step, we determine the set of transfer functions in which the identified model will be determined1. This set of transfer functions is generally the set of all transfer functions having a certain order. These transfer functions are thus entirely determined by the coeﬃcients of their numerator and denominator, coeﬃcients that we regroup in a parameter vector θ. In this thesis, we will generally make the assumption that the to-be-identified system (the so-called true system) can be described by one (unknown) element of the chosen set of transfer functions i.e., there exists a true parameter vector θ0. Under this full-order model structure assumption, the goal of system identification will therefore be to determine an accurate estimate θ of θ0. Determining such a full-order model structure is generally obtained in an iterative manner .
2. Experiment Design. The estimate θ of θ0 will be determined based on the input-output data collected on the system. These input-output data are obtained during an identification experiment where the system is excited using an excitation signal during a certain amount of time. When the system is operated in open loop, the excitation signal is the input signal itself. When the system is operated in closed loop, the excitation signal is an external signal that is added to the loop. It is important to note that deciding all the details of the experiment is not a merely formal step. Indeed, many of these experimental choices (such as the properties of the excitation signal and the experiment duration) have an important eﬀect on the quality of the final model.
3. Parameter estimation. Starting from the data collected we have then to obtain ˆ ∈ R k of θ0 . This is done by minimizing a certain criterion which an estimate θ aims at maximizing the capability of the model to explain data. A widely used criterion is the Prediction Error criterion .
4. Model Validation. In this validation step, the uncertainty of the identified model is assessed by e.g., inspecting the covariance matrix Pθ of the identified parameter vector θ. Other model validation tests verify the ability of the identified model to explain not only the data used for the identification, but also other input-output data (that have been reserved for this validation step).
This thesis will be made up of a theoretical part and an applied part. In the theo-retical part, we will make new contributions to the optimal design of an identification experiment. In the second part, we will show that system identification can be used as an eﬃcient tool to solve a problem that arises in the Next4MEMS project, the project that helped fund this PhD thesis.
Optimal identification experiment design
Optimal identification experiment design consists in optimally designing the exci-tation signal of an identification experiment to balance two contradictory objectives:
obtaining the most accurate estimate θ of i.e., obtaining an estimate θ with the smallest possible covariance matrix Pθ
• perturbing the to-be-identified system with the excitation signal as little as pos-sible i.e., the cost of the experiment is the smallest possible.
These objectives are contradictory since the larger the spectrum of the excitation signal, the smaller Pθ, but, at the same time, the larger , the larger the cost of the experiment. To balance these contradictory objectives, one can e.g. maximize the accuracy of θ given a certain bound on the cost of the experiment. This is the classical formulation of the optimal experiment design problem that dates back from the seven-ties (see e.g., ). More recently, a second (dual) formulation (the so-called least costly formulation) has been considered where the objective is to determine the spectrum of the excitation signal in such a way that the cost of the experiment is minimized while guaranteeing a given accuracy for the identified model . Mathematically speaking, the least costly experiment design problem consists in determining the spectrum of the excitation signal which minimizes the cost J ( ) of the experiment while guaran-teeing that Pθ−1( ) ≥ Radm where Radm is a matrix reflecting the desired accuracy and where the cost J ( ) is measured by a linear combination of the power of the perturbation induced by the excitation signal onto the input and output signal of the to-be-identified system [3, 4].
The least costly formulation for optimal experiment design is closer to the require-ments in robust control where the robust performance of a controller designed with the identified model can be guaranteed if the model uncertainty is smaller than a given limit (or, equivalently, if the accuracy of the identified model is larger than a certain limit). This has led to a number of contributions on least costly experiment design for robust control. In , a methodology is developed to design the excitation spectrum in such a way that, with the smallest identification cost, the identified model is guaran-teed to have a suﬃciently small uncertainty to enable robust control design. In , an H∞ control framework is used, but other control frameworks can also be considered as shown in [4, 5]. The results in  and  are based on the technical results in [6, 7, 8]. Indeed, the papers [6, 7, 8] show that, using smart parametrizations of the to-be-determined spectrum , both J ( ) and Pθ−1( ) can be expressed as aﬃne functions of the (to-be-determined) spectrum coeﬃcients. Two parametrizations of are in fact considered in [6, 7, 8]: one corresponding to the spectrum of a multisine excitation signal and one corresponding to the spectrum of a filtered white noise where the filter is restricted to have a finite impulse response. Based on these technical results, the least costly experiment design problem introduced in the previous paragraph can be turned to an LMI2 optimization problem3 . It is however important to note that this
2 Linear Matrix Inequality.
3An LMI optimization problem is an optimization problem with a scalar objective function that is aﬃne in the decision variables and with constraints that are also aﬃne in the decision variables and that are under the form of matricial inequalities.
LMI optimization problem depends on the unknown true parameter vector θ0 since both the covariance matrix Pθ and the cost J of the experiment not only depends on , but also on θ0 (we will therefore use the notations J (θ0, ) and Pθ(θ0, ) in the sequel). A classical (but not fully satisfactory) approach to circumvent this so-called chicken-and-egg issue is to replace, in the LMI optimization problem, the unknown θ0 by an initial estimate θinit (see e.g. [8, 3]). One of the contribution of this thesis is to derive a method to deal with this chicken-and-egg issue via convex optimization.
The results in  and  pertain to experiments to identify linear time-invariant (LTI) single-input single-output (SISO) systems that can be described by ordinary dif-ferential equations and that are operated either in open loop or in closed loop. The results have been extended to multivariable systems in  and to systems that can be described by partial diﬀerential equations in [11, 12]. First attempts to deal with optimal experiment design for nonlinear systems can be found in [13, 14, 15, 16]. A first contribution towards optimal experiment design in another operating configura-tion than the open or closed loop can be found in  where a network configuration is considered. A second contribution of this thesis will be to extend further the results presented in .
Robust experiment design
The least costly experiment design problem consists in determining the spectrum of the excitation signal that minimizes the cost J (θ0, ) of the experiment under the constraint that Pθ−1(θ0, ) ≥ Radm. As mentioned in the previous section, this optimization problem can be expressend as an LMI optimization problem provided a solution is found to the fact that the true parameter vector θ0 is unknown. As already mentioned, this chicked-and-egg issue is generally circumvented by replacing θ0 in the optimization problem by an initial estimate θinit of θ0 (see e.g. [8, 3]). How- ever, this approach has the drawback that the optimal spectrum θoptinit obtained in this way is not guaranteed to yield the desired accuracy and that the experiment cost ˆ ˆ ˆ ˆ can underestimate the actual experiment θinit θinit J (θinit, opt ) computed with θinit and opt cost J (θ0, θoptinit ). These observations are at the root of the research areas on adaptive experiment design [18, 19] and on robust optimal experiment design (see  for a good survey). In adaptive experiment design, both the estimate and the optimal spectrum are obtained in a recursive manner (see [18, 19] for more details).
In robust experiment design, diﬀerent lines of research have been considered. In , a spectrum that yields good accuracy for a very broad set of systems (also of diﬀer-ent orders) is discussed. However, in the engineering literature, the most widely used approach is the one that consists in considering an uncertainty set U containing the unknown true parameter vector θ0 (the so-called min-max design [20, 22]). The opti-mal experiment design problem can then be formulated as determining the spectrum minimizing the value of a scalar γ under the constraints that J (θ, ) ≤ γ ∀θ ∈ U and that Pθ−1(θ, ) ≥ Radm ∀θ ∈ U. If we denote by γopt and opt the solution of this optimization problem, we have the guarantee that Pθ−1(θ0, opt) ≥ Radm and that γopt is an upper bound for the actual experiment cost J (θ0, opt). However, finding a tractable approach to deal with such a robustified optimal experiment design problem is still an open research question. While, for very particular or simple situations, this optimization problem can be either exactly solved (see e.g. [20, 23]) or tackled via an Sum-Of-Squares approach (see ), the general approach when facing more complex systems is to replace the initial uncertainty set (containing an infinite number of el-ements) by a number ng of grid points of this uncertainty set U (see e.g. [8, 3, 20, 25]). Consequently, the cost constraint and the accuracy constraint over the set U in the robustified optimal experiment design problem are replaced each by ng constraints (one for each grid point). Even though it is obviously better from a robustification point-of-view than just replacing θ0 by one grid point i.e. θinit, this relaxation of the original robustified optimal experiment design problem cannot yield the guarantees linked to the original problem.
In [26, 27, 28], approaches are presented to uniquely tackle the robustified cost constraint J ( ) ≤ ∀ ∈ (i.e. in the accuracy constraint, 0 is replaced by ˆinit). θ,γθU θ θ However, these approaches all entail some approximation: a first-order approximation in , a second-order approximation in  and an approximation based on the un-scented transform in .
In this thesis, our contribution is to present an approach in order to tackle the ro-bust optimal experiment design problem without approximation4. For this purpose, we observe that, except for its dependence on the to-be-determined spectrum, the robus-tified cost constraint and the robustified accuracy constraint are similar to constraints that are treated in robustness analysis. Based on this observation and on the sepa-ration of graph framework [29, 30, 31], we derive constraints that are linear in the decision variables of the optimal experiment design problem and that imply the orig-inal robustified cost and accuracy constraints. As we will see, the proposed approach will nevertheless not cover all possible situations. These restrictions will be detailed in Chapter 3, together with our robust experiment design approach.
Least costly experiment design for a network of locally controlled systems
Until recently, system identification has only considered systems operated in open loop or in a closed loop. Due to the increasing importance of the concept of network in control engineering, we have however recently seen major eﬀorts to develop techniques for the identification of large-scale or interconnected systems. In many papers, the problem is seen as a multivariable identification problem and structural properties of the system are then used to simplify this complex problem (see e.g. ). The iden-tifiability of the multivariable structure is studied in a prediction error context in  while this multivariable structure is exploited in other papers to reduce the variance of a given module in the network (see [34, 35, 36]). In other contributions, conditions are derived for the consistent estimation of a given module in a dynamic network (see e.g. [37, 38, 39]).
While many diﬀerent problems have thus been studied in the dynamical network context, this is not the case for optimal experiment design. In the very limited lit-erature on this particular subject, the contribution  presents first steps towards optimal experiment design in a dynamical network context.
The paper  considers the case of a network made up of the interconnection of locally controlled systems (also called modules or nodes). In these networks, the inter-connection is realized by the fact that neighbouring modules are allowed to exchange their measured output. In particular, the reference signal of each module in the net-work will be computed based on the measured outputs transmitted by its neighbouring modules. This type of networks is usual in the literature on multiagent systems (see e.g. [40, 41])) and can e.g. be used to ensure that all modules of the network track a given reference known only to one of these modules. For this particular type of dy-namical networks, it is showed in  how to design the excitation signals that have to be added to each module in order to identify models of these diﬀerent modules that are suﬃciently accurate to enhance the network performance by a redesign of the local controllers. The accuracy of each model can be measured by the inverse of the covari-ance matrix of the identified parameter vector of each module. In , an expression for the inverse of this covariance matrix as an aﬃne function of the excitation signal spectra is derived. It is important to note that the inverse of the covariance matrix of a given module l is obviously a function of the excitation signal applied to this particular module, but also, though in a lesser extent, a function of the excitation signals applied to all modules k having a path to l. Consequently, the excitation signal applied to such a module k contributes to the accuracy of the model of l. In other words, the prop-agation of the excitation signals due to the interconnection is a positive feature when we want to obtain suﬃciently accurate estimates of every module with the smallest excitation power (see also [42, 43, 44]).
Table of contents :
1.1 System Identification
1.2 Optimal identification experiment design
1.3 Resonance frequency tracking of a MEMS gyroscope
1.4 Content of the thesis
2 Prediction Error Identification and Optimal Experiment Design
2.1 Introduction and description of the to-be-identified system
2.2 Prediction-error identification in open loop
2.3 Prediction-error identification in closed loop
2.4 Optimal experiment design: least costly experiment design
3 Robust Optimal Identification Experiment Design for Multisine Excitation
3.2 Open-loop identification in a BJ model structure
3.3 Robust optimal experiment design
3.4 Tackling the robustified cost constraint using robustness analysis tools
3.5 Tackling the robustified accuracy constraint
3.6 Convex formulation of the optimal experiment design problem
3.7 Possible extensions of the framework
3.8 Dealing with the numerical complexity
3.9 Numerical illustrations
4 Least Costly Identification Experiment for the Identification of One Module in a Dynamic Network
4.2 Description of the network configuration
4.3 Identification of one module in the network and cost of the experiment
4.4 Optimal experiment design problem
4.5 Tackling the robust cost constraint in a convex way
4.6 Computation of (!) using the hierarchical approach
4.7 Numerical illustrations
5 Resonance Frequency Tracking of a MEMS Gyroscope
5.2 The drive mass closed-loop system
5.3 Recursive Identification
5.4 Extremum Seeking
5.5 Comparison of the RLS and the ES algorithms: Tracking of !r,x(t)
6.2 Suggestions for future works
A Appendix for Chapter 2
B Appendices for Chapter 3
B.1 Example for Observation 3.1
B.2 Proof of Proposition 3.1
B.3 LFT representation of Fu,(z, )
B.4 Useful lemma for the proof of Proposition 3.3
C Appendices for Chapter 4
C.1 Derivation of the expression for Ru(z, 0) and Ry(z, 0)
C.2 Proof of Proposition 4.1
C.3 Consistency and accuracy of (4.23)
C.4 Proof of Proposition 4.3
C.5 Computation of the quantities ˜ci(!) and i(!)
C.6 Proof of Proposition 4.5
D Appendix for Chapter 5
E Résumée de la Thèse
E.2 Conception Optimale Robuste de l’Expérience d’Identification avec une Excitation Multisinusoïdal
E.3 Expérience least costly pour l’identification d’un module dans un réseau dynamique
E.4 Suivi de la fréquence de résonance d’un gyroscope MEMS
E.5 Conclusion .