Simulation of Distributed Parameter Systems by Transfer Function Models - Schäfer, Maximilian (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2020)
Most real-world systems are distributed parameter systems. This means that their dynamics depend not only on their temporal, but also on their spatial behaviour. Particularly, their parameters are not concentrated, but distributed over their spatial volume. Well-investigated examples include the sound of a guitar string, which depends on its spatial oscillation on the guitar, or an electrical transmission line, whose resistance is dependent on its length. Distributed parameter systems also occur naturally in the human body, where the transport of particles through a blood vessel is influenced by its spatial and temporal properties. An abstraction of the real world delivers a comprehensive mathematical description of distributed parameter systems in terms of initial-boundary value problems, which are derived by the first principles of physics. Initial-boundary value problems describe the dynamics of a distributed parameter system by partial differential equations, where its temporal initial state and its spatial boundary behaviour is modelled by suitable initial and boundary conditions. The analysis of natural existing distributed parameter systems as well as the design of synthetic distributed parameter systems are very important tools that may be employed to analyse the dynamics of particle transport in the human body from a communications point of view, or to create a digital guitar synthesizer. Therefore, it is indispensable to obtain suitable models to simulate the spatio-temporal dynamics of distributed parameter systems. The main challenge is to choose a suitable modelling technique which leads to a model that meets the predefined requirements. In the literature, a considerable number of different modelling techniques is found, each with its own advantages and disadvantages. These modelling techniques may be roughly divided into two different categories: numerical methods and analytical methods. Most numerical methods apply a suitable discretization rule to a set of partial differential equations. They lead to powerful simulation algorithms which are capable of simulating spatially complex physical problems in a very accurate way. However, these methods often have a very high computational complexity and provide only little insight into the influence of parameters on the output signal. In contrast to that, analytical methods try to find an explicit solution of an initial-boundary value problem before a discrete algorithm is established. Most analytical methods are based on well-investigated techniques from mathematics and systems theory. The derived models allow to establish a relationship between input and output variables of a system in terms of its parameters. Furthermore, they can lead to low-complexity algorithms with real-time capability by the application of a convenient discretization method. However, the elegance of analytical methods decreases for non-linear systems. Therefore, they are mostly applied to distributed parameter systems which can be described mathematically by linear initial-boundary value problems. The choice of a suitable modelling technique thus depends strongly on the distributed parameter system to be modelled and the requirements on the simulation model. If exact numerical results of a spatially complex distributed parameter system are required, numerical methods are preferable. However, if an exact closed-form description is necessary – for the analysis of a distributed parameter system, for example – analytical methods are advantageous. The modelling procedure used in this thesis is the Functional Transformation Method. This procedure contains diverse functional transformations, i.e., a Laplace and a Sturm-Liouville transformation. Finally, a model is formulated in terms of multidimensional transfer functions. The functional transformation method belongs to the class of analytical modelling techniques. But as most other analytical methods, the functional transformation method is not suitable for complex spatial shapes, non-linear distributed parameter systems and for complex boundary behaviour. Then the method loses its elegance and no explicit solution is obtained as numerical evaluations have to be involved. Nevertheless, analytical methods are a desired approach for the modelling of distributed parameter systems. Therefore, this thesis marks a starting point in overcoming some of the previously mentioned problems of analytical modelling techniques, i.e., of the functional transformation method. By developing suitable extensions it is possible to derive an explicit model of a distributed parameter system which includes the influence of complex boundary behaviour. Although the procedure of the functional transformation method is already formalised, the first goal of this dissertation is to improve its formulation. As an extension, an operator-based version of the involved Sturm-Liouville transformation is incorporated into the functional transformation method. Applying this extension variant to an initial-boundary value problem, a multidimensional state space description is obtained as a solution, which constitutes the model of the underlying distributed parameter system. This formulation as a state space description exhibits several advantages: it constitutes a unified solution of the functional transformation method and allows its analysis and modification by concepts from control and systems theory. The formulation of the simulation model in terms of a state space description is the basis for the second goal of this dissertation. The functional transformation method is extended by adapting concepts from control theory to incorporate the influence of complex boundary behaviour by the design of feedback loops. First, the complex boundary behaviour is separated from the system and it is modelled with a generic simple boundary behaviour which defines the open loop system. The complex boundary behaviour is used to design a feedback matrix which is attached to the simple model to form the closed loop system that fulfils the desired complex boundary behaviour. In particular, the feedback matrix shifts the eigenvalues of the open loop system into a position where they fulfil the complex boundary behaviour. With the developed concept it is possible to model distributed parameter systems with complex boundary behaviour in an explicit form. The same concept can be used to incorporate other physical effects into the model of a distributed parameter system. Furthermore, the concept allows to model interconnected systems, which builds the basis for a block-based modelling approach of interconnected distributed parameter systems. Applying the developed techniques to specific problems from different fields of application, their validity is confirmed in the third part of this dissertation. Specifically, the techniques are employed to model musical systems, electrical transmission lines and biological systems in the context of molecular communications. Within these applications, the developed methods are used to incorporate complex boundary behaviour and physical effects. In addition, general system modifications to change the timbre of a musical system are shown. Furthermore, two biological systems are interconnected by the design of a connection matrix. Where possible, the modelling results are compared to numerical simulations or measurements. All considered problems show that the developed concepts are suitable for the modelling of distributed parameter systems and constitute a meaningful extension to the functional transformation method.<br