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This report deals with rational approximation of any speci ed order n of transfer functions. Transfer functions are assumed to be matrices whose entries belong to the Hardy space for the complement of the closed unit disk endowed with the L 2-norm. A new approach is developed leading to an original algorithm, the rst one to our knowledge which concerns matrix transfer functions. This approach generalizes the ideas developed in the scalar case, but involves substantial new di culties. The inner-unstable factorization of transfer functions allows to express the criterion in terms of inner matrices of Mac-Millan degree n. These matrices form a di erential manifold. Based on a tangential Schur algorithm, an atlas of this manifold is given for which the coordinates vary in n copies of the unit ball. Then a gradient algorithm can be used to solve this problem. The di erent cases which can arise while processing the algorithm are studied : how to switch to another chart of the atlas, what has to be done when a boundary point is reached. In the neighbourhood of a boundary point, the criterion can be smoothly extended. Moreover, such a point can be considered as an initial point for the research of a lower degree approximant. It is explained how to cope with the decrease and the increase of the degree. The convergence of the algorithm to a local minimum of appropriate degree is proved and demonstrated on a simple example.

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The choice of a reference model in data-driven control techniques is a critical step. Indeed, it should represent the desired closed-loop performances and be achievable by the plant at the same time. In this paper, we propose a method to build such a reference model, both reproducible by the system and having a desired behaviour. It is applicable to Linear Time-Invariant (LTI) monovariable systems and relies on the estimation of the plant's instabilities through a data-driven stability analysis technique. The L-DDC (Loewner Data Driven Control) algorithm is used to illustrate the impact of the choice of the reference model on the control design process. Finally, the proposed choice of specifications allows to use a controller validation technique based on the small-gain theorem.

Performing a stability analysis during the design of any electronic circuit is critical to guarantee its correct operation. A closed-loop stability analysis can be performed by analysing the impedance presented by the circuit at a well-chosen node without internal access to the simulator. If any of the poles of this impedance lie in the complex right half-plane, the circuit is unstable. The classic way to detect unstable poles is to fit a rational model on the impedance. In this paper, a projection-based method is proposed which splits the impedance into a stable and an unstable part by projecting on an orthogonal basis of stable and unstable functions. When the unstable part lies significantly above the interpolation error of the method, the circuit is considered unstable. Working with a projection provides one, at small cost, with a first appraisal of the unstable part of the system. Both small-signal and large-signal stability analysis can be performed with this projection-based method. In the small-signal case, a low-order rational approximation can be fitted on the unstable part to find the location of the unstable poles.Comment: Longer version of the paper published in IEEE Transactions on Microwave Theory and Technique