Simulation Of Robust Algebraic Control Of A Delayed Heat Exhanger With Controller Rationalization
Concise complete controller design with simulations for a laboratory circuit heating plant with feedback delays is the aim of this contribution. The particular steps of the design include a mathematical modelling summary of then plant, introduction of relevant algebraic tools, robust controller design and tuning, its discretization and rational simplification. The plant model contains internal delays, thus, it is infinite dimensional. A special ring of stable and proper quasipolynomial functions represent the main algebraic mean for controller structure design. A possible range of adjustable controller parameters is then limited by using the robust stability and performance analysis. Their eventual values are consequently obtained by simulations of control responses. A discretization of the resulting controller is suggested and a method for its rational simplification is verified by simulations in MATLAB and Simulink. All the obtained results are promising for a practical implementation of the presented approach. INTRODUCTION Dynamical plants and processes, even linear ones, with internal (state) delays constitute a family of systems that are not trivial to be controlled by the use of many traditional control design approaches, or it is even impossible to implement them (Loiseau et al. 2009; Richard 2003; Vyhlidal et al. 2014) Some possible approaches, e.g. Lyapunov Krasovskii methods, are too complex and troublesome to be practically implemented by engineers. Algebraic structures, in the contrary, proved to be effective tools for delayed system dynamics description and control design (Dostal et al. 2002; Loiseau 2000; Kucera 1991) Namely, the ring of proper and stable quasipolynomial meromorphic functions (RMS) (Zitek and Kucera, 2003; Pekar 2012) as the main algebraic tool is used in this paper to determine the controller structure. The solution of the Bezout equation followed by the Youla Kucera parameterization satisfy the controller properness and control feedback stability. In this contribution, a circuit laboratory heat exchanger (Dostalek et al. 2008) serves as a controlled plant since its mathematical model (Pekar et al. 2009) evinces internal feedback delays. The laboratory appliance can be considered as a small scaled model of a real word system, e.g. as the cooling system in cars. Recently, it has been utilized to perform the verification of solutions of various engineering and scientific tasks, see e.g. (Bobal et al. 2013; Prokop et al. 2012) however, usually plant models with only input output delays have been used. Once the controller structure by means of the RMS ring is designed, its free and adjustable controller parameters ought to be set properly. To solve this task, possibilities are plentiful and many various ideas can be adopted. We decided to satisfy robust stability and robust performance under unstructured multiplicative uncertainty (Doyle et al. 1992) that is based on the wellknown Nyquist criterion, the validity of which for delayed systems was proved e.g. in (Pekar et al. 2011) As the outcome of robust analysis, only possible controller parameters' ranges are obtained; therefore, their values are precised via simulations in MATLAB and Simulink with some integral criteria evaluation. Presented algebraic design usually yields controllers with delayed dynamics that is linear but far from the conventional proportional integral derivative (PID) control law. Since up to 95 of control loops in industry are equipped by PID controllers (Desborough and Miller 2001) it is desirable to approximate the eventual controller structure by the PID one. The idea of the Pade expansion is adopted in this contribution where the whole controller transfer function is rationalized instead of the traditional approximation of separate delay elements. A simple controller discretization for digital implementation based on delta models (Middleton and Goodwin 1990) in input output space finalizes the controller design procedure. The method is easy to handle for practitioners in contrast to some other more sophisticated methods operating in the state space domain, see e.g. pseudospectral methods (Breda et al. 2005) Proceedings 29th European Conference on Modelling and Simulation (c)ECMS Valeri M. Mladenov, Petia Georgieva, Grisha Spasov, Galidiya Petrova (Editors) ISBN: 978 0 9932440 0 1 ISBN: 978 0 9932440 1 8 (CD) HEAT EXCHANGER MODEL A photo and a rough scheme of the laboratory heat exchanger to be controlled is provided to the reader in Figures 1 and 2 (Dostalek et al. 2008) The model works as follows: Distilled water inside the piping is driven by a pump {6} continuously controllable via the voltage t uP through a flow heater {1} with maximum power t PH of 750 W. The heater output temperature, t HO th is measured by a platinum thermometer. Warmed liquid then goes through a 15 meters long insulated coiled pipeline {2} which causes the significant delay in the system. A heat consuming appliance is represented by the air water heat exchanger (cooler) {3} equipped with a continuously adjustable (by means of the voltage t uC and an on/off cooling fans {4, 5} Input and output temperatures of the cooler, t CI th and t CO th respectively, are measured by platinum thermometers as well. The expansion tank {7} compensates for the expansion effect of the heat fluid. Let the ambient temperature be t A th Figure 1: Laboratory Heat Exchanger A Photo Figure 2: Laboratory Heat Exchanger A Sketch The complete mathematical model was published in (Pekar et al. 2009) For the sake of this paper, however, it is sufficient to consider the relationship between t uP D and t CO th D governed by the transfer function s a a s a s a s s b s b s P s s G D D H CO th t t th D D exp exp exp
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