## A Loewner tensorization technique for MIMO transfer function decoupling

ELEC seminar room (Building K, 7th floor)

Speaker: Philippe Dreesen

Abstract: We address the problem of finding a decoupled representation of a multi-input-multi-output (MIMO) system in the frequency domain. A MIMO frequency response is considered decoupled if it satisfies a description with a number of distinct single-input-single-output (SISO) responses after appropriate linear transformations of inputs and outputs. The procedure we present is based on recent advances in tensor decomposition techniques and their application in signal processing. Indeed, collecting the frequency response matrices over a frequency interval naturally leads to a three-dimensional array. Other essential elements are the fact that the transfer function of a linear time-invariant system is a rational function of the frequency variable, and the low-rank property of the Loewner matrix corresponding to an arbitrary rational function. By imposing the Loewner structure, the proposed method allows for extracting more internal SISO responses than existing methods and exhibits improved noise rejection properties. (Joint work with Dieter Verbeke and Mariya Ishteva.)

## Detection, Classification and Quantification of Nonlinear Distortions in Time-Varying Frequency Response Function Measurements

ELEC seminar room (Building K, 7th floor)

**Speaker**: Noël Hallemans

**Abstract**: In various measurements of real life systems, for instance corrosion processes, one stumbles upon linear time-varying (LTV) behaviour which is corrupted by both noise and nonlinear distortions. The corruption by noise is well known in the literature. However, the corruption by nonlinear distortions is a new problem in the context of time-varying systems. These nonlinear distortions and noise are regarded as disturbances, they could for instance origin from the measurement equipment or unwanted behaviour of the system. The problem consists of extracting a linear time-varying model for the system from the measurements. Since the noise and the nonlinearities are disturbances, we can compute the associated uncertainty on the model. More speciﬁcally, one would like to distinguish between the uncertainty caused by, on the one hand, the noise and on the other hand the nonlinear distortions. Solutions for these problems are proposed in this presentation.

## A Novel LPV/LTV Method for Nonlinear System Identification

ELEC seminar room, Building K, 6th floor

**Speaker**: Mehrad Sharabiany

**Abstract**: The identification and control of LTI systems is well established among engineers. So, not surprisingly, LPV modeling approaches have received a lot of attention from the community. For LPV systems, there is a (set of) scheduling parameter(s) (p) which is constant for a local LTI model. There are two major LPV identification approaches: local and global. In this presentation, we propose a novel Global LPV modeling paradigm for nonlinear systems. Then, we model a pendulum which is followed by the simulation results. The proposed identification method is as follows: the system is excited with a combined large-amplitude slow signal and a small-amplitude rapid signal. After de-trending, it will be shown that it is possible to approximate the system with an LTV model. This LTV model is estimated by a Gaussian kernel-based estimator. It is known that any LPV model is inherently an LTV model too. Therefore, the coefficients of that linear time-varying system can be considered as the coefficients of the corresponding LPV model. The main idea is that the large signals in input and output themselves can be used as scheduling parameters. So, the LTV coefficients must somehow be related to the scheduling trajectory (i.e. the large signals in input and outputs).the presentation will be followed by some advantages of this proposed modeling method. The underlying maths come from a Taylor series expansion of a nonlinear function around a trajectory. In this case, the coefficients of the LPV/LTV system are obtained via linearization of the plant around the large signal trajectory.

## Every unfalsified model must include the most powerful unfalsified model

ELEC seminar room, Building K, 6th floor

**Speaker**: Vikas Mishra

**Abstract**: Exact model identification refers to the problem of identifying the true data generating model from an observed trajectory of the model. For an infinite noise-free time series, the problem has been tackled by Jan C. Willems by introducing the notion of the most powerful unfalsified model (MPUM). The MPUM is defined as the least complex linear time-invariant model explaining a given time series, wherein the complexity is defined by the pair (number of inputs, model order) in lexicographic manner. It is known that the MPUM always exists and is unique, and under certain assumptions it coincides with the data generating model. However, for a finite time series, there are two issues: (i) The MPUM may not exist or be non-unique, and (ii) The MPUM always coincides with a finite dimensional space corresponding to an autonomous model (model with no inputs) due to the above definition of complexity. In this talk, these issues are resolved by assuming that the number of inputs is a priori known and minimizing the complexity by minimizing the order of the model. First, necessary and sufficient conditions are established for the existence and uniqueness of the MPUM in this setting. Then, in case of non-uniqueness of unfalsified models, it is shown that the set of unfalsified models admits an affine structure. More specifically, every unfalsified model is a sum of the most powerful unfalsified model and an autonomous model of bounded order. Finally, numerical examples are discussed to illustrate the results.

## Best Linear Approximation of Nonlinear Continuous-Time Systems Subject to Process Noise

ELEC seminar room

**Speaker:**** **Rik Pintelon

**Abstract:** In many engineering applications the level of nonlinear distortions in frequency response function (FRF) measurements is quantiﬁed using specially designed periodic excitation signals called random phase multisines and periodic noise. The technique is based on the concept of the best linear approximation (BLA) and it allows one to check the validity of the linear framework with a simple experiment. Although the classical BLA theory can handle measurement noise only, in most applications the noise generated by the system – called process noise – is the dominant noise source. Therefore, there is a need to extend the existing BLA theory to the process noise case. In this presentation we study in detail the impact of the process noise on the BLA of nonlinear continuous-time systems. It is shown that the existing nonparametric estimation methods for detecting and quantifying the level of nonlinear distortions in FRF measurements are still applicable in the presence of process noise. All results are also valid for discrete-time systems and for systems operating in closed loop.