# Do you want to discover the beauty of time-varying systems?

## Background

In all engineering disciplines, mathematical models are required to describe the reality surrounding us. The goal of System identification is to compute these models, based on real-life measurements on the system of interest. Instead of making vague assumptions on the physics of the system, one relies on actual observations, which results in a very pragmatic way of working.
For many applications, one assumes that the system of interest is LTI (Linear and Time Invariant). Although this assumption has shown to perform fairly well in many cases, in quite some applications the linearity and time-invariance hypotheses are only approximately true or not valid at all. Think, for example, of

• civil engineering structures (bridges, buildings, …) subject to increasing damage, or to varying loads
• the impedance of metals, subject to (pit) corrosion,
• aging/mortification in biological systems,
• or systems with a varying set point (e.g. flight flutter analysis), or varying system (scheduling) parameters (e.g. pick-and-place robot with extendable gripper arm, a crane lifting a load, see figure).

Figure: Long rope: slow oscillation, Short rope: faster oscillations

The common problem in these examples is that the system dynamics change during the measurements! For example, the oscillation frequency of a swinging mass is varying when it is lifted by a crane.

The challenge lies in extracting models which track the varying dynamics of these systems, from as few experiments as possible. Indeed, measurements are expensive because, time is money, loss in production time …

What will you learn?
At the department ELEC-VUB we have a lot of expertise in system identification in general and, since more recently, in the identification of linear time-varying systems. In this master thesis, you will learn:

• our gathered knowledge on the identification of time-varying systems (understand its spectral response, describe its dynamic behavior mathematically, the state of the art of the identification tools, ...)
• to develop new estimation algorithms, analyze their stochastic properties, and adapt them to be applicable to real-life cases.
• to design and perform experiments on time-varying systems.

## Challenging applications

We have multiple possibilities to apply the developed identification tools to relevant real-life applications:

• the impedance of an electrochemical system, in collaboration with VUB-MACH: the impedance changes due to the evolution of the corrosion
• the determination of the flutter speed of the wings of a plane, in collaboration with VUB-MECH: the damping of the oscillating modes changes as a function of the air speed and flight height
• tracking the mechanical impedance of a human muscle, in collaboration with the Technical University in Delft, The Netherlands: the human adapts its muscle’s stiffness to environmental conditions
• identifying the varying resonance frequency of an adaptable electronic filter: by using variable electronic components, the pass-band of the filter can be changed over time.

## Challenging fundamental problems

The identification of linear time-varying systems faces several fundamental questions that have not been addressed yet. A master thesis can consist of investigating one of these questions, or a combination of multiple of them:

• Most current techniques use basis functions (polynomials, trigonometric functions) to represent the time variation. An important difficulty in this case is to determine the appropriate number of basis functions (model complexity). As an alternative to that, the time variation could be modelled by 'machine learning', which results in a totally different approach to determine the appropriate model complexity. This is a challenging problem, because it has barely been explored in the context of frequency domain system identification.
• In quite some applications the periodic excitation cannot be synchronized exactly with the inherent periodic time variation (e.g. lung impedance measurements, myocardial tissue bio-impedance measurements), resulting in an aperiodic response. The challenge here is to exploit the periodic nature of the excitation taking into account the lack of synchronization with the time variation.
• For slowly time-varying systems the process noise can be approximated very well by a stationary filtered white noise process. This is no longer true for fast time-varying systems (e.g. wind mills). The open problem here is to get nonparametric estimates of the time-varying noise variance in a pre-processing step (periodic and arbitrary time-variations).
• In some applications the system dynamics should be identified from output observations only (e.g. wind turbines). A frequency domain approach is not available yet. The identification of linear parameter varying (LPV) state space models is challenging because the (dynamic) dependency of the state space matrices on the scheduling parameter(s) is - in general - unknown. Therefore, a 3 step procedure will be developed. In a first step a time-varying state space model is identified from the noisy input-output data. Next, the time-varying state space matrices are modeled as a function of the scheduling parameter(s). In a final step the full LPV state space model is estimated.
• As soon as we leave the LTI framework, the dimensions of the data matrices in subspace algorithms usually start to grow exponentially with increasing complexity, like the (unknown) number of non-linearities, or the (known) number of scheduling parameters. A compact representation of the time-varying model, would be beneficial for reducing the computation time and memory usage..
• The frequency domain methods for (non-)parametric modeling of (arbitrary or periodically) time-varying systems have been developed for single input, single output systems (SISO). In quite some applications (modal analysis, process control …) one has multiple inputs and outputs (MIMO). Hence, there is a need for the non-trivial extension of the algorithms from SISO to MIMO.

Since most real life systems behave to some extend nonlinearly, the following fundamental issues should also be addressed:

• Detection and quantification of the nonlinear contribution to the response of the system. In a first step this will be investigated for slowly time-varying, weakly nonlinear systems, using the concept of the best linear time-invariant approximation of a nonlinear time-varying system.
• Study of the influence of the nonlinear distortions on the identified linear time-varying model. The concept of the best linear time-variant approximation of a nonlinear time-varying system should be defined, and its properties should be studied for a certain class of excitations and systems.
• If a linear approximation of the true nonlinear dynamics is not accurate enough for predicting the response, then a full nonlinear time-varying model is needed. A first step in this challenging problem will be set by combining our expertise in block oriented nonlinear modeling with that in linear time-varying system identification.

Note: a challenging master thesis can be the ideal start for a PhD in that research topic at the department ELEC