*Sous réserve de la délivrance de l’autorisation de soutenance :*

**Arthur Perodou** defends his PhD on Dec. 18, 2019 at 10:00 AM.

**Place : ** Ecole Centrale de Lyon, Bâtiment W1, Amphi 203.

**Jury** :

Rapporteurs :

M. Yann Le Gorrec, FEMTO-ST (Besançon)

M. Paolo Rapisarda, University of Southampton (UK)

Autres membres :

Mme. Isabelle Queinnec, LAAS (Toulouse)

M. Eric Kerhervé, IMS (Bordeaux)

M. Ian O’Connor, INL (Lyon), Directeur de thèse

M. Anton Korniienko, Ampère (Lyon), Encadrant

Invités :

M. Gérard Scorletti, Ampère (Lyon), co-Directeur de thèse

M. Mykhailo Zarudniev, CEA-LETI (Grenoble), Encadrant

M. Jean-Baptiste David, CEA-LETI (Grenoble), Encadrant

**Abstract** :

The current explosion of communicating devices (smartphones, drones, IoT ...), along with the ever-growing data to be transmitted, produces an exponential growth of the radiofrequency bands. All solutions devised to handle this increasing demand, such as carrier aggregation, require to synthesise frequency filters with stringent industrial requirements (performance, energy consumption, cost ...). While the technology of acoustic wave (AW) resonators, that seem to be the only passive micro-electronic components available to fulfil these requirements, is mature, the associate design problem becomes dramatically complex. Traditional design methods, based on the intuition of designers and the use of generic optimisation algorithms, appear very limited to face this complexity. Thus, systematic and efficient design methods need to be developed.

The design problem of AW filters happens to be an instance of the more general design problem of passive electronic filters, that played an important role in the early development of Linear Control and System theory. Systematic design methods were developed in particular cases, such as for LC-ladder filters, but do not enable to tackle the case of AW filters. Our aim is then to revisit and generalise these methods using a modern System approach, in order to develop systematic and efficient design methods of passive electronic filters, with a special focus on AW filters.

To achieve this, the paradigm of convex optimisation, and especially the sub-class of Linear Matrix Inequality (LMI) optimisation, appears for us a natural candidate. It is a powerful framework, endowed with efficient solvers, able to optimally solve a large variety of engineering problems in a low computational time. In order to link the design problem with this framework, it is proposed to use modern tools such as the Linear Fractional Transformation (LFT) representation and a mathematical characterisation coming from Dissipative System theory.

Reviewing the different design methods, two design approaches stand out. The first approach consists in directly tuning the characteristic values of the components until the frequency requirements are satisfied. While very flexible and close to the original problem, this typically leads to a complex optimisation problem with important convergence issues. Our first main contribution is to make explicit the sources of this complexity and to significantly reduce it, by introducing an original representation resulting from the combination of the LFT and the Port-Hamiltonian Systems (PHS) formalism. A sequential algorithm based on LMI relaxations is then proposed, having a decent convergence rate when a suitable initial point is available.

The second approach consists of two steps. First, a transfer function is synthesised such that it satisfies the frequency requirements. This step is a classical problem in Control and Signal Processing and can be efficiently solved using LMI optimisation. Second, this transfer function is realised as a passive circuit in a given topology. To this end, the transfer function needs to satisfy some conditions, namely realisation conditions. The issue is to get them with a convex formulation, in order to keep efficient algorithms. As this is generally not possible, an idea is to relax the problem by including common practices of designers. This leads to solve some instances of a general problem denoted as frequency LFT filter synthesis. Our second main contribution is to provide efficient synthesis methods, based on LMI optimisation, for solving these instances. This is achieved by especially generalising the spectral factorisation technique with extended versions of the so-called KYP Lemma.

For particular electronic passive filters, such as bandpass LC-ladder filters, this second approach allows to efficiently solve the design problem. More generally, it provides an initial point to the first approach, as illustrated on the design of a particular AW filter.

**Key Words:**

Frequency Filter Design, LFT systems, Passive elements, Dissipativity, KYP Lemma, Spectral Factorisation, LMI optimisation, Port-Hamiltonian systems, Differential-Algebraic Equations (DAE), Acoustic Wave (AW) resonators