Budapest University of Technology and Economics
During investigation of dynamical systems, stochastic effects are often neglected, however stochastic excitations many times influence the behaviour of the systems. In case of applying deterministic models to approximate stochastic systems, during the identification process of system parameters usually their mean value is used as the parameter, while the measured variation is used to qualify the measurements. This approach often does not describe well enough the behaviour of the investigated system and leads to false results. An example from mechanical engineering problems is machine tool vibrations, where the goal is to avoid vibration during the material removal process in order to ensure the best surface quality, while maintaining the highest possible material removal rate. The mathematical models, through which the dynamics of the machining is investigated are almost exclusively deterministic delayed differential equations. However, the forces originating from the cutting depend on a large number of factors (e.g. friction, high frequency changes in chip formation, material inhomogeneity), which gives the process a highly stochastic nature. Another possible application is the investigation of traffic dynamics in the presence of vehicles equipped with connected cruise control (CCC), using vehicle-to-vehicle (V2V) communication. One goal of the corresponding researches is to avoid a phenomena called phantom traffic jam, during which spontaneous traffic jam forms without any accident or roadblock, due to the drivers’ large reaction times and limited perception range. The use of autonomous vehicles decreases the reaction times, but to gain beyond line of sight information one has to apply V2V communication. The simplest method for this technology is that each vehicle measures its position and velocity (e.g. with GPS and on board velocity sensors) and broadcasts it to the other vehicles. The broadcasted signal is received by the other vehicles, and they use this data to control their velocity. A challenge is that packet losses occur, which results in a “jerky” behaviour of the vehicle. To investigate the effect of the packet losses, usually deterministic packet loss patterns are used, introducing time-dependent time delay into the system. However, these packet losses occur stochastically, causing the time delay to be stochastic.
To facilitate the investigation of the stationary behaviour of such systems, a Julia package was created. This package is able to handle stochastic delayed linear systems, which can be written in the following structure:
\[\mathrm{d}\mathbf{x}_t = \left(\mathbf{A}(t) \mathbf{x}_t + \mathbf{B}(t) \mathbf{x}_{t-\tau(t)}\right)\mathrm{d}t +\left(\boldsymbol{\sigma}(t)+\boldsymbol{\alpha}(t)\mathbf{x}_t + \boldsymbol{\beta}(t) \mathbf{x}_{t-\tau(t)}\right)\mathrm{d}W_t,\]where $\mathbf{x}(t)\in \mathbb{R}^d$ is the $d$-dimensional state vector, $\mathbf{A}(t),\mathbf{B}(t),\boldsymbol{\alpha}(t),\boldsymbol{\beta}(t) \in\mathbb{R}^{d\times d}$ are deterministic, time dependent coefficient matrices, $\sigma(t)\in\mathbb{R}^d$ is the vector corresponding to the additive noise $\tau(t) \in \mathbb{R}$ is the time delay, which can change between discrete values stochastically. To model the stochastic effects in the parameters, the $W_t$ Wiener process is used. The package is capable to compute moment stability and stationary $\lim_{t\to\infty}\mathbb{E}\left(\mathbf{x}_t\mathbf{x}_t^\top\right)$ second moment of the delayed system. Furthermore, the method is generalised to handle multiple time delays and stochastic excitations. One has to only supply the coefficient matrices and the time delay as functions of time, then the package calculates the mentioned stationary properties of the investigated system.
Machine tool vibrations: Conference proceeding at ASME, Homepage of the ERC SIREN project Connected cruise control V2V communication: Book chapter on Springer.com
Henrik Sykora is a PhD student at the Budapest University of Technology. His main research topic is the investigation of the behaviour of linear stochastic delayed differential equations. After spending some time in Karlsruhe University of Technology and a few months in University of Michigan, he realized that even if there is a good mathematical method, it is worth few, if it cannot be used by others quickly and simply. That’s why he is striving to create a Julia package, with which he can give not only published results, but also a possible tool for others. When he is not glued to a chair gazing at a computer screen, he tries to teach mechanics to mechanical engineering students, reads books about great and not so great (but nevertheless successful) people, and tries not to tap too much during his brazilian Jiu-Jitsu trainings.
