Format And Type Fonts CHEMICAL ENGINEERING TRANSACTIONS VOL. 45, 2015 A publication of The Italian Association of Chemical Engineering www.aidic.it/cet Guest Editors: Petar Sabev Varbanov, Jiří Jaromír Klemeš, Sharifah Rafidah Wan Alwi, Jun Yow Yong, Xia Liu Copyright © 2015, AIDIC Servizi S.r.l., ISBN 978-88-95608-36-5; ISSN 2283-9216 DOI: 10.3303/CET1545290 Please cite this article as: Mészáros A., Čirka Ľ., Bakošová M., Vasičkaninová A., 2015, On stability and controllability of processes with internal recycle, Chemical Engineering Transactions, 45, 1735-1740 DOI:10.3303/CET1545290 1735 On Stability and Controllability of Processes with Internal Recycle Alajos Mészáros, Ľuboš Čirka, Monika Bakošová, Anna Vasičkaninová Slovak University of Technology, Faculty of Chemical and Food Technology, Institute of Information Engineering, Automation and Mathematics, Radlinského 9, 81237 Bratislava, Slovak Republic alajos.meszaros@stuba.sk In the presented paper dynamic state analysis, stability and controllability aspects of recycle processes are discussed, in general. A comparison of process dynamic behaviour and investigation of influence of process recycle loop parameters are carried out, applying simulation experiments on linear transfer function models. Stability and controllability considerations are provided in both, open and closed loop mode. 1. Introduction In most cases, recycle leads to positive feedback effects. For example, increasing the concentration of a chemical species in a process stream will normally increase the amount of this species in the recycle stream, and, thus, lead to a reinforcement of the original increase. It refers to a self-reinforcing mechanism associated with the recycle. This positive feedback will usually increase the plant time constant, and also increase the process sensitivity to slow disturbances. This is because recycle will tend to “store” material or energy within some part of the plant. An example is a high purity distillation column where extremely long time constants can be observed. The overall dynamics of chemical processing plants with material recycle or heat integration can be very different from the dynamics of the individual processing units. Material recycle and heat integration may dramatically alter the overall gain and time constants of the plant, and may give rise to oscillatory or instable behaviour, even when the individual processing units are stable by themselves. Moreover, plant interconnections may introduce fundamental limitations in the achievable performance of any control system. The knowledge of such phenomena is important for controller design, and their effects may even pose a threat to plant safety if not foreseen. There has been lot of works devoted to the problem of investigating and handling chemical processes with recycle. Luyben (1993) shows that, by changing the gain of the recycle process independently from the other process parameters, the open loop response can become slow, oscillating and even unstable. Morud and Skogestad (1994) analyse the effects of different elements on the dynamics. Scali and Antonelli (1995) investigate the performance of different regulators (PI, IMC) for plants with recycle. Taiwo and Krebs (1996) show how a robust control system can be designed and cope with recycle plant problems. The aim of this paper is to give a comprehensive picture on dynamic behaviour of processes with internal recycle (PIR) in both, open loop and closed loop configuration. A comparison of process dynamic behaviour and investigation of influence of process recycle loop parameters is carried out, applying simulation experiments on linear transfer function models. Stability and controllability conditions are derived and influence of recycle parameters are shown for open loop and closed loop cases. 2. Open-loop system with internal recycle Let us consider a heating plant process identified in the form of simple linear system consisting of two forward paths and a recycle unit as depicted in Figure 1. GM, GD and GR stand for plant forward unit, 1736 disturbance and recycle unit u, d and y are the input, load disturbance and output. Assuming GM = GD, we get the simplified structure of the recycle process, shown as Figure 2. Figure 1: Open-loop process with recycle Figure 2: Simple open-loop process with recycle Let the first object of our study be the recycle system in Figure 2, assuming d = 0. This can be described by the open loop transfer function (s being the derivation operation) RM M S GG G su sy sG   1)( )( )( (1) In dynamic analysis of the recycle system Eq(1), we will focus on investigation of influence of the recycle path (GR) parameters onto the overall plant behaviour. For open-loop and closed-loop stability considerations and simulation experiments, GM and GR will be considered as units of first order dynamics. It can be done without significant loss of generality, as to the results expected, because lot of real plants dynamics can be identified, in a close neighbourhood of the operation regime, as a first order lag with steady-state gain and dead time (if necessary). Simulation experiments could prove that increasing order of either GM or GR does not basically alter the tendency of recycle parameters influence onto overall process dynamics. Then, the forward path is described by a simple linear transfer function consisting of a steady-state gain, KM, and a first order lag with time constant, M. The unit in the recycle path also has a simple gain and lag transfer function (with KR and R), as follows 1  s K G M M M  , 1  s K G R R R  (2) Inserting (2) into (1), we get the overall transfer function RMRMRM RM R R M M M M S KKss sK s K s K s K G        1)( )1( 11 1 1 2     (3) From characteristic equation of system (3), 01)( 2  RMRMRM KKss  (4) the necessary and sufficient condition for recycle plant stability is 1 RM KK or M R K K 1  (5) 2.1 Influence of recycle loop gain and time constant In order to show the effect of recycle parameters, KR and R, we express the overall process steady-state gain, KS, and time constant, S, using Eq(3) RM M S KK K K   1 (6) 1737 RM RM S KK  1   (7) and calculate the limits of GS for boundary values of KR and R, from the same equation M K S GG R  0 lim , 0lim  RK S G (8) 1 1 1 lim 0      s KK KK K G RM M RM M S R  , MS GG R   lim (9) From Eq(8) and Eg(9) it is obvious that the recycle effect is diminishing for “very small” values of the gain as well as “very large” values of the time constant. Large values of KR cause instability and “very large” values stop the whole process operation while “very small” values of R reduce the order of process. To give a complete picture on recycle effect, step-wise response dynamic simulations have been carried out and evaluated with respect to recycle loop parameter changes. In Figure 3 and Figure 4, a comparison of the plant output responses is given for various values of KR and R. The other process parameters during the two courses of simulation kept the following values: KM = M = R = 1 and KM = M = 1, KR = 0.8. It can be concluded that the effect of KR is more straightforward because it influences both, overall gain and dynamics, while R changes system behaviour in transient stage only. From theoretical and simulation results it is obvious that one of the most important effects of recycle is to slow down the response of the process, i.e. increase the process overall time constant. Figure 3: Influence of recycle loop gain onto process dynamics Figure 4: Influence of recycle loop time constant onto process dynamics 3. Closed-loop system with recycle effect Considering the recycle process with closed loop control as depicted in Figure 5, the system characteristic equation takes the form 0 1 1    RM MC GG GG (10) 1738 Figure 5: Recycle system with closed loop control 3.1 Dynamic Analysis and Stability Considerations Case I. Proportional controller The controller transfer function is CC KG  (11) With respect to (1), (2) and (3), characteristic eq. (10) results in 01)( 2  MCRMRMCRMRM KKKKsKKs  (12) The necessary and sufficient condition of closed loop system stability is M RM C K KK K 1  (13) For boundary values holds 0 C K : M R K K 1  (14)  C K : CR KK  (15) where Eq(14) corresponds to the open loop case Eq(5). To give a picture on recycle effect, set-point step-wise response dynamic simulations have been carried out and evaluated with respect to recycle loop parameter changes. In Figure 6 and Figure 7, a comparison of the controlled plant output responses is given for various values of KR and R. The other process parameters during the two courses of simulation kept the following values: d = 0, KM = τM = R =1, KC = 5 and d = 0, KM = M = 1, KR = 0.8, KC = 5. Figure 6: Influence of recycle loop gain onto controlled process dynamics Figure 7: Influence of recycle loop time constant onto controlled process dynamics 1739 Simulation results confirmed the theoretical stability analysis. It is straightforward from (13) that the closed loop system remains stable only for the values KR  6. It can be concluded that the effect of KR is more straightforward because it influences both, overall gain and dynamics, while R changes system behaviour in transient stage only. Case II. Proportional-integral controller The controller transfer function is s sK G i iC C   )1(   (16) Characteristic eq. (10) takes the form 0)1)(()1()( 223  ssKKsKKss iRiRMCRMiRMiiRM  (17) Applying the Hurwitz criterion, the necessary and sufficient condition of closed loop system stability is )( )1( iRM RMi C K KK K      (18) For boundary values of τi holds 0 i  : 0 C K (19)  i  : M RM C K KK K 1  (20) where the latter corresponds to the case of P-control - Eq (13). As above, set-point step-wise response dynamic simulations have been carried out and evaluated with respect to recycle loop parameter changes. In Figure 8 and Figure 9, a comparison of the controlled plant output responses is given for various values of KR and R. The other process parameters during the two courses of simulation kept the following values: d = 0, KM = M = R = 1, KC = 5, i = 0.2 and KM = M = 1, KR = 0.8, KC = 5, i = 0.2. Figure 8: Influence of recycle loop gain onto controlled process dynamics Figure 9: Influence of recycle loop time constant onto controlled process dynamics It can be concluded that the effect of both, KR and τR is well compensated by PI controller and it is in accordance with the theoretical stability results. The closed loop system for given parameters remains stable for values KR  31. However, larger values of the recycle parameters may cause more oscillatory behaviour of the overall system. 1740 4. Controllability of PIR For controllability analysis, let us consider the PIR identified in the form of transfer functions (1), (2) and (3). Then, the input-output process model takes the form a differential equation of second order )()()()1()()()( tuKtuKtyKKtyty MRMRMRMRM   (21) Introducing state variables y=x1, y'=x2, u1=u, u2=u', the PIR state space model )( )()( )( txCy tuBtxA dt txd   (22) is composed by the following matrices            RM RM RM RM KKA    1 10 ,        RMM KK B  00 , )01(C Applying the Controllability Theorem in state space context (Mikleš and Fikar, 2000), the Controllability Matrix of the system can be constructed as )( ABBQ C  (23) The PIR system is then completely controllable if the rank of (23) is equal to two. This leads to the following condition of controllability 0 RM K  (24) The above results refer to a significant condition: a PIR system can be completely controlled only if the recycle loop contains a first or higher order lag with non-zero time constants. A static positive feedback can make the overall system uncontrollable. 5. Conclusion In this paper, dynamic state properties and control aspects of recycle processes, in general, have been studied. Assuming first order dynamics, comparison of process dynamic behaviour and investigations of influence of process recycle loop parameters were carried out, applying simulation experiments on linear transfer function models. A comprehensive picture of dynamic behaviour of processes with internal recycle (PIR) in both, open loop and closed loop configuration was given. Stability and controllability conditions were derived and influence of recycle parameters were shown for open loop and closed loop cases. Theoretical as well as simulation results have shown serious limitations as to PIR system stability and controllability conditions. Acknowledgement The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grants 1/0973/12. References Luyben W.L., 1993, Dynamics and control of recycle systems 1. Ind. Eng. Chem. Res, 32, 466-475. Morud J., Skogestad S., 1994, Effects of recycle on dynamics and control of chemical processing plants, Computers Chem. Engng, 18, Suppl., S529-S534. Scali C., Antonelli R., 1995, Performance of different regulators for plants with recycle, Computers Chem. Engng, 19, Suppl., S409-S414. Taiwo O., Krebs V. 1996, Robust control system design for plants with recycle, Chem. Engng. J., 61, 1-6. Mikleš J., Fikar M., 2000, Process Modelling, Identification and Control. STU Press, Bratislava, Slovakia.