MODELLING, SIMULATION AND ROBUST ANALYSIS OF THE TEMPERATURE PROCESS CONTROL

The 6 th edition of the Interdiscilinarity in Engineering International Conference Petru Maior University of Tîrgu Mureş, Romania, 2012 MODELLING, SIMULATION AND ROBUST ANALYSIS OF THE TEMPERATURE PROCESS CONTROL Mircea DULĂU #1, Adrian GLIGOR #2, Horaţiu-Ştefan GRIF #3 # Deartment of Electrical Engineering and Comuter Science, Petru Maior University of Tg. Mureş No.1 N.Iorga St., Tg. Mureş Romania 1 mircea.dulau@ing.um.ro 2 adrian.gligor@ing.um.ro 3 horatiu.grif@ing.um.ro ABSTRACT Processes characterized by large time constant, such as those with temerature control systems, are found in a large number of industrial alications. To control such systems, the aroriate mathematical models identification is needed, considering the uncertainties modeling, handling in dynamic oerating regime and roosing control solutions. The aer resents a control system used in a art of a comlex lant where is needed to maintain temerature at a set oint. The controlled installation consists from two connected tanks, for which a simlified mathematical model based on transfer functions are determined, considering additive uncertainties and classical cascade PI control structures are roosed. Develoed models and solutions are analyzed by simulation in Matlab environment. Keywords: control engineering, mathematical modeling, temerature rocess, PID controllers, robust control 1. Problem formulation In mixing tank V1 is inserted a steam flow rate and cold water in a roortion to rovide heating temerature of the mixture from the shell of the tank V2, where, by thermal transfer the roduct is heated and maintained at a constant temerature (Fig. 1). Fig. 1 Plant diagram The temeratures from oututs of the tanks are measured considering homogeneous mixtures of thermal agents (steam and cold water) and roduct. For resented installation are determined the followings: the mathematical model of the rocess based on balance equations; transfer functions for each block and the equivalent transfer function; temerature control scheme; controller tye and its tuning arameters; obtained erformances. A control scheme with two controllers connected in cascade is chosen, and tuning arameters are determined knowing that (Fig.2): in main loo - roduct temerature at the exit of the tank V2 is measured, conversion temerature - unified voltage signal is erformed, the result is comared with reference T ref value, the resulting error value is inut into the controller TIC 2; in secondary loo - the heating mixture temerature at the outut of the tank V1 is measured, a temerature-unified voltage signal conversion is rovided, and the resulting value is comared with the outut value of the controller TIC 2; the value of the resulting error is the inut value for controller TIC 1; Fig. 2 Block diagram of the cascade control system 312

the flow rates of thermal agents (cold water and steam) from the entering of tank V1 is controlled by two valves controlled via the controller; 2. Determination of mathematical models In rocess of mathematical modeling are considered the terms with the meaning from Table 1: Table 1. The meaning of values used in modeling rocess Value Meaning F ar +F ab = 300 l steam and cold water flow rates that maintain a constant amount of mixture in tank V1. c ar = 4191 J/kgK incoming cold water secific heat at the average temerature T ar = 10 o C h ab = 2769 kj/kg enthaly of saturated steam at a temerature of 185 o C and a ressure of 8 bar c i = 4208 J/kgK secific heat of the mixture at the outut, considered at the temerature T i = 90 o C; M 1 = 300 kg the liquid weight in the tank V1 C i = c i M 1 heat caacity of the mixture from V1 tank F i mixture flow from outut of the tank V1 and the inut of tank V2 c 2 = 4179 J/kgK secific heat of the roduct at outut when outut temerature of tank is T2 = 60 o C M 2 = 500 kg weight of roduct from the tank V2 C 2 = c 2 M 2 heat caacity of the roduct from tank V2 V 2 = 500 l c e = 4179 J/kgK F e F = 21 kg/min 1 = 999,7 kg/m 3 F 1 c 1 V2 tank volume secific heat of the mixture at the outut of the shell, at the outut temerature T e = 60 o C heat flow rate at the outut of V2 equal to the outut um the outut um roduct density at inut, at average temerature TP 1 = 10 o C roduct flow at the inut of the tank V2 roduct secific heat at inut into V2 In mixing tank V1 is introduced and mixed cold water and steam, resulting hot water at outut (see Fig. 1). Analytical descrition of the mixing rocess is based on the following heat balance equation [7,12,20]: qar qab qi (1) - q ar reresents the heat introduced by cold water; - a ab heat introduced by steam; - q i the heat of the outut mix. We can write the equation (1) in the new form: dti Far car Tar Fab hab Fi ci Ti Ci (2) the terms have the meanings in Table 1. Density and secific heat of the water deends on its temerature, and steam enthaly deends on the ressure and temerature [19]. Considering the mixture flow rate equal with the outut um, i.e. F i = F, equation (2) becomes: M 1 dti car Tar hab Ti Far Fab F F c F c (3) 1 T1 F i M reresents the time constant for rocess from tank V1. In V2 tank is introduced the roduct to be heated by heat transfer between the shell and the tank and maintained at a constant temerature at the outut (Fig. 1). Analytical descrition of the heat transfer rocess is based on heat balance equation [7,20]: ( q q C dt q (4) 1 i ) 2 2 - q 1 heat sulied by the roduct in the tank; - q i heat sulied by the mixture; - q e heat of the outut mix. Considering Te = T2 and neglecting the heat sulied by the roduct, equation (4) becomes: M 2 dt2 T 2 Ti (5) F M 2 reresents the time constant for 2 T A F rocess from tank V2. Based on data given in Table 1, the mathematical models that describe the mixing rocess in the V1 tank, resectively heat transfer in the V2 tank, are: dti 857,1 Ti 805,6 Far 1, 9 Fab (6) dt2 1428,5 T2 Ti (7) After alying Lalace transform, and given the condition F ar +F ab =300, from equations (6) and (7), we obtain the transfer functions for the two tanks: kv1 803,7 HV 1s (8) T s 1 857,1 s 1 V1 e i 313

kv 2 1 HV 2s (9) T s 1 1428,5 s 1 V 2 T V1 =857.114 min. and T V2 =857.124 min. are time constants. 3. Controller selection and tuning Figure 3 resents the block diagram of the cascade control system, with the secification of the variation ranges of the values. Fig. 5 Ste resonse of closed-loo (inner loo) Analysis of nominal values shows a hase margin of arox. 67 degrees (Fig. 6). For nominal rocess and affected by the uncertainties is resented the mode of disturbance rejection aeared on the inner loo (Fig. 7) [10,11,16]. Fig. 3 Block diagram of the cascade control system Let's consider the transfer function of the mixture temerature inner control loo deending on the inut flow rate of cold water (at a constant steam flow rate), obtained by series connection of the temerature sensor, the actuator and the rocess [9]: k k T1 kv1 f 1 H f 1s ke1 TT 1 TV 1 TT 1 TV 1 (10) Process arameters secified by the equation (8) are considered as nominal values and the nominal rocess. For modeling of the model uncertainties are considered deviations of arameters from nominal values, exressed as a ercentage, amounting to 25%. Thus [1,3,6], kv1inc kv 1 kv1 (11) TV 1inc TV 1 TV 1 The roosed control algorithm, for the nominal rocess, is PI tye [2,21,22]: kr H R 1s Ti s 1 (12) Ti s The equivalent second order system in closed loo is adoted as design method, resulting [17,18]: nti T i = T V1 ; kr ((13) 2k the dynamic erformance of the system are secified by arameters n,. Ste resonses of the oen and closed inner loo, for nominal rocess, affected by uncertainties are resented in Fig. 4 and Fig. 5 [4,8,24]. Fig. 4 Ste resonse of oen-loo (inner loo) f Fig. 6 Phase margin analysis (inner loo) Fig. 7 Robustness on disturbance rejection Let's consider the transfer function of the fixed art of the roduct temerature main control loo deending on the water temerature from shell, obtained by series connection of the inner loo (equivalent circuit) and the rocess described by nominal transfer function (9). Comared to the nominal values are considered deviations (uncertainties) exressed as a ercentage, amounting to 25%. Similar as in case of the inner loo, is adoted a PI control law for the controller structure, given by equations (12), with the tuning arameters: Ti T V 2, k R = 55. The main loo ste resonses in oen and closed configuration, for nominal rocess and rocess affected by uncertainties are resented in Fig. 8 and Fig. 9. To analyze the robustness to disturbance rejection in the frequency domain is considered the sensitivity function as measure of the system erformance [13,14,15]. Thus, the sensitivity function is comuted and lotted for the nominal situation and the situation with uncertainties (Fig. 10). At the time domain, the sensitivity function 314

indicates the degree of disturbance rejection. In Fig.11 is lotted the sensitivity function over the entire range of uncertainties for nominal and worst case [4,24]. Fig. 8 Ste resonse of oen-loo (main loo) Fig. 9 Ste resonse of closed-loo (main loo) Fig. 10 Frequency resonse sensitivity functions (main loo) Fig. 11 Disturbance rejection (main loo) 4. Conclusions Temerature control is usually of stabilization tye and resents issues more comlex than other control tyes (flow, level, ressure) due to comlex heat transfer henomena, at that are also associated large time constants. The temerature control roblems are, in fact, roblems of heat transfer, whether based on radiation, conduction or convection. Temerature control systems are imlemented as control structures after error, but in most cases as develoed cascade structures and control after disturbance. A cascade control scheme of temerature in a tank with the main variable the temerature inside the tank and intermediate variable the ambient temerature around the tank is resented in this aer. The roosed scheme reresents an alicable solution because heat exchange between thermal agent and useful substance inside the tank is made with a relatively large time constant, of the order of minutes or tens of minutes. Comared to the main loo, secondary loo has a faster dynamic, so the whole control system is invariant to unwanted fluctuations of the mixing flow rate. A major disturbance remains however, the flow rate of the roduct that will follow to make heat exchange with the mixture. Changing the inut flow rate of V2 tank or the suly fluid temerature is a temerature erturbation that directly affects the temerature at the outut of V2. Also, changing arameters of feedwater and steam are disturbances that occur on the tank V1, resectively on the temerature of the mixture. Overall, the roosed control scheme ensures a good behavior at both ste inut signals and the disturbances due to flow rates. References [1] Astrom, K.J. (2001), Model Uncertainty and Robust Control, Deartment of Automatic Control, Lund University, Sweden. [2] Astrom, K.J., Hagglund, T. (2004), Revisiting the Ziegler-Nichols ste resonse method for PID control, Journal of Process Control, Vol. 14, Issue 6,. 635-650. [3] Balas, G.J. et al (2004), Next generation of tools for robust control, American Control Conference, Boston, Vol. 1-6,. 5612-5615. [4] Balas, G.J. et al (2006), Robust Control Toolbox. For Use with Matlab, Version 3, The MathWorks. [5] Dittmar, R. et al (2009), Robust otimizationbased multi-loo PID controller tuning: A new tool and its industrial alication, IFAC Symosium on Advanced Control of Chemical Processes, Istanbul, Vol. 20, Issue 4,. 355-370. [6] Doyle, J. (1996), Robust and Otimal Control, IEEE Conference on Decision and Control, Kobe, Vol. 1-4,. 1595-1598. [7] Dulău, M. (2004), Automation of continuous rocesses. Thermal and chemical rocesses, Petru Maior University of Tîrgu-Mureş Publishing House, Romania. [8] Dulău, M. et al, Behavioural Study of a Thermal Process Control Under Uncertainties, IEEE International Conference on Automation, Quality and Testing, Robotics, Tome I, Cluj- Naoca,. 198-201, 2010. [9] Dumitrache, I. et al (2010), Automatics, Romanian Academy Publishing House. [10] Gene, F. et al (2006), Feedback Control of Dynamic systems, 5 th ed., Pearson Prentice Hall. 315

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