![]() For analysis of the casting process dynamics as DPS, especially temperature fields in the casting mould and control synthesis purposes, the benchmark casting plant with steel mould of complex-shape was designed at Faculty of Mechanical Engineering STU in Bratislava. Modelling, simulation and evaluation of real-time experiments in this area is now widely accepted as an important tool in product design and process development to improve productivity and casting quality. There in order to obtain the desired solidification structure, the casting process requires a specific temperature field of the mould, which is defined on complex-shape 3D definition domain. The casting technology is a typical case of the DPS. In the time domain, synthesis of control is performed by lumped parameter control loops, where robust controllers are used. In the space domain, approximation problems are solved. ![]() ![]() Based on this decomposition a methodical framework of control synthesis decomposition into space and time tasks will be presented. In this chapter the decomposition of dynamics of controlled LDS into time and space components is introduced. In this sense the actuators and the controlled plant as a DPS create a controlled lumped-input and distributed-parameter-output system (LDS). In the field of lumped parameters system (LPS) control, where the state/output quantities x(t)/y(t) – parameters are given as finite dimensional vectors, the actuator together with the controlled plant make up a controlled LPS. Well-known reduction methods based on finite difference method (FDM), or finite element method (FEM), spectral method require an accurate nominal PDE model and usually lead to a high-order model, which requires unpractical high-order controller.Īn engineering approach for the control of DPS is being developed since the eighties of the last century ( Hulkó et al., 1981, 1987, 1998, 2009a, 2009b). Variety of transfer functions for systems described by PDE are illustrated by means of several examples in ( Curtain & Morris, 2009). Methodical approach from the view of time-space separation with model reduction is presented in ( Li & Qi, 2010). Continuous and approximation theories aimed to control of parabolic systems presents monograph ( Lasiecka & Triggiani, 2000). That is the decomposition of dynamics into time and space components based on the eigenfunctions of the PDE. In the first mathematical foundations of DPS control, analytical solutions of the underlying PDE have been used ( Butkovskij, 1965 Lions, 1971 Wang, 1964). There are many dimension reduction methods, which can be used to solve this problem. ![]() ![]() However, from point of view of implementation of DPS control in technological practice, where a finite number of sensors and actuators for practical sensing and control is at disposal, such infinite-dimensional systems need to be approximated by finite-dimensional systems. The time-space coupled nature of the DPS is usually mathematically described by partial differential equations (PDE) as infinite-dimensional systems. Such systems are classified as distributed parameter systems (DPS). Most of the dynamical systems analysed in engineering practice have the dynamics, which depends on both position and time. ![]()
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