Robust controller design for a heat exchanger using H 2, H, H 2 /H, and n-synthesis approaches

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1 Robust controller design for a heat exchanger using H, H, H /H, and nsynthesis approaches Anna Vasičkaninová, Monika Bakošová Slovak University of echnology in Bratislava, Faculty of Chemical and Food echnology, Radlinského 9, 8 37 Bratislava, Slovakia anna.vasickaninova@stuba.sk, monika.bakosova@stuba.sk Abstract: Possibilities of using robust controllers for a shellandtube heat exchanger control were studied, tested and compared by simulations and obtained results are presented in this paper. he heat exchanger was used to preheat petroleum by hot water; the controlled output was the measured output temperature of the heated fluid petroleum, and the control input was the volumetric flow rate of the heating fluid water. Robust controllers were designed using H, H, H /H strategies and msynthesis. A comparison with the classical PID control demonstrated the superiority of the proposed robust control especially in case when the controlled process is affected by disturbances. Keywords: heat exchanger, PID control, H control, H control, msynthesis Introduction Heat exchangers are nonlinear processes (Janna, 9) and their control is a complicated problem due to their nonlinear behaviour and com plexity caused by many phenomena such as leakage, friction, temperaturedependent flow properties, contact resistance, unknown fluid properties, etc. (Serth and Lestina, 4). Many factors enter into the design of heat exchangers, including thermal analysis, weight, size, structural strength, pressure drop and cost. he heat exchanger modelling framework has been described and demonstrated previously (Skaugen et al., 3). he main purpose of Daróczy et al. (4) was to illustrate and analyse a heat exchanger arrangement problem in its most general form. In Manassaldi et al. (4), a mathematical model for optimal design of an air cooled heat exchanger is described. Owing to the wide utilisation of heat exchangers in industrial processes, their cost minimisation is an important task for both, designers and users (Pan et al., ). Cost evaluation is obviously an optimisation pro cess dependent upon other design parameters. A method capable of utilising the maximum allowable stream pressure drops has been described by Panjeshahi et al. (). his approach can result in minimum surface area requirements. Economics plays a key role in the design and selection of heat exchanger equipment. he weight and size of heat exchangers are significant parameters in the overall application and thus may be considered as economic variables (Holman, 9). Effective control of heat exchangers requires the application of some advanced methods, e.g. robust control (Gerhard et al., 4; lacuahuac et al., 5). Robust control has emerged as one of the most important areas in the modern control design since Doyle (98), Zames (983), and many others. In Veselý (3), a survey of robust control design procedures is given. H and H control theories have been the area of active research for years and they have been successfully implemented in many engineering applications. While the H norm represents the mean energy of the system, the H norm represents the maximum energy. If there are uncertainties in the system model, some quantity combining the H norm and the H norm can be a desirable mea sure of a system s robust performance (Bansal and Sharma, 3). hus, the mixed H /H performance criterion provides an interesting measure for the controller evaluation. heoretic motivation for mixed H /H control has been discussed in literature (Doyle, 984; Kwakernaak, ). he goal of Zarabadipour et al. () was to design a reduced order robust controller based on the balanced realisation technique. he simulation showed that the reduced order controller design based on the H /H approach has good results in the frequency and time domains. In Ganji et al. (3), different conventional and intelligent controllers were implemented with a clear objective to control the outlet fluid temperature of a shellandtube heat exchanger system. For the dynamic system with time varying characteristics and parametric uncertainties, a sliding mode controller was developed and an optimal H controller was designed based on msynthesis with the DKiteration algorithm by Moradi et al. (). In De Souza et al. (4), the problems of robust stability analysis and robust control of linear 84 Acta Chimica Slovaca, Vol. 9, No., 6, pp , DOI:.55/acs63

2 discretetime periodic systems was investigated. In Ahmad et al. (3), the design of H and H control for a twin rotor system is described. Dlapa (5) presented an application of algebraic msynthesis to the airheating system, where the temperature of bulb was controlled by its voltage. Zhu and Yang (6) solved a simultaneous H /H stabilisation problem for a chemical reaction system that can be modelled as a finite collection of sub systems. Mahitthimahawong et al. (6) develo ped a control system for a split range control in a heat exchanger network. Vasičkaninová and Bakošová (3) dealt with the design and application of robust H and H controllers for a heat exchanger. In this study, robust controllers are designed using H, H, H /H and msynthesis approaches. he H control technique is based on the minimisation of the quadratic norm of the transfer function between the input disturbance signal and the plant s output signal. he disadvantage is that the H control does not show guaranteed robustness a priori, when unstructured uncertainties are present. he H control has the basic objective to minimise the effect of disturbances on the plant output, but, on the other hand, it presents limitations of the system s performance. he H /H mixed control joins the ability of the H control to minimise the effect of the input disturbances on the plant output employing the H control. In the msynthesis design, a perturbation matrix Δ is chosen and through an optimisation procedure, a stabilising controller for the worst perturbation is obtained. As all these approaches have been implemented in the Robust Control oolbox of Matlab, the robust controllers have been designed using this toolbox. he controllers were tested in set point tracking and disturbance rejection of the shellandtube heat exchanger and compared with conventional PID controllers. he presented simulation results confirm that robust controllers are able to provide better results especially in case of disturbance rejection. Robust control he basic feature of robust control is that it guarantees closedloop stability and performance even though the controlled process has uncertain parameters or is affected by disturbances or measurement noise. H, H and H /H control Consider a plant, P(s), that has to be controlled (Bosgra et al., 7). he goal of the H, H and H /H control strategies is to design an internally stabilising linear timeinvariant (LI) dynamic feed back controller, K(s), that minimises the H and the H norm of the performance transfer function, zw, of the closedloop system (see Fig. ). Fig.. Standard feedback control loop. zw is described as follows: = F( P, K) = P + P K( I P K) P () zw l where F l (P, K) is the lower linear fractional transformation of P and K (Zhou et al., 998). he transfer matrix, P(s), and the signals in the feedback control loop (see Fig. ) can be expressed as: ézù ép P ùé wù = () ê y ú ê P P úê u ú û û û Here, z are performance outputs, w the exogenous inputs, y the measurements, and u are the control inputs: z = P w+ P u, y = Pw+ Pu, u = Ky. Statespace realisation of the controlled process is x ( t) = Ax( t) + Bw( t) + B u( t) z( t) = C x( t) + D u( t) y( t) = C x( t) + D w( t) (3) and P(s) = C(s I A) B + D. Assumptions A to A8 are typically made for H or H or H /H control design problems (Skogestad and Postlethwaite, 5): A. system (A, B, C ) is stabilisable and detectable, A. matrices D and D have full rank, A3. matrix éa jwi B ù êc D ú has full column rank for û all w, where w Î é, ) is the frequency, A4. matrix éa jwi B ù êc D ú û has full row rank for all values of w, A5. D =, D =. A is required for the existence of a stabilising controller, K, and A is required to achieve proper value of K. A3 and A4 ensure that the controller does not cancel the poles or zeros on the imaginary axis, which leads to instability. A5 is required for the H control. Condition D = results in strictly proper P which is required because H is a set of Vasičkaninová A et al., Robust controller design for a heat exchanger using H, H, H /H 85

3 proper transfer matrices. Condition D = provides a strictly proper P and simplifies the H design problem. In the H control, none of the assumptions A A5 is required but if met, they simplify the problem. Sometimes, additional assumptions can be made to simplify the solution significantly (Skogestad and Postlethwaite, 5): é ù, êi ú û A6. D = ê ú D = [ I ] D C =, BD =, A7. A8. (A, B ) is stabilisable and (A, C ) is detectable. Assumption A6 is required for the H problem and it means that there is no cross term in the cost function. A7 means common LQG control with no crosscoupling. If A7 is true, A8 replaces A3 and A4. For a plant, P(s), satisfying assumptions A A8 and the closedloop configuration according to Fig., the H optimal controller, K(s), is found to minimise the H norm min zw l K K, g = min = min F( P, K) (4) he suboptimal H design problem is defined as: find all internally stabilising controllers that assure = F( P, K) < g (5) zw l where g > g min. A unique optimal controller is obtained as follows. Find positive definite X X = ³ and Y = Y ³ that are the solutions of A X + X A+ C C X B B X = (6) AY + Y A + B B Y C C Y = (7) such that Â{ l é i A BBX ù û} < " i and { l é ù}  û < " i A YC C i (stabilising solutions), where  (.) is the real part of the argument. hen, with F = BX, L = YC, Aˆ = A+ BF + LC, the central controller is éaˆ L B ù K c () s = F I C I ê úû (8) and the family of all stabilising controllers such that zw g holds, is obtained from K ( s) = F( K, Q) (9) l c where Q(s) is stable, strictly proper, and meets min Q g g. he optimal controller is obtained from Eq. (9) by setting Q(s) =. H suboptimal controller design Consider a plant, P(s), with configuration as in Fig., which satisfies assumptions A A8. All stabilising controllers, K(s), have to be found that satisfy Fl ( P, K) < g () for a suboptimal bound g > g min. hey can be obtained as follows (Glover and Doyle, 988; Doyle et al., 989): search for a positive definite X = X ³ and Y = Y ³ solving A X X A CC ( g ) + X BB B B X = AY Y A BB ( g ) + Y C C C C Y = such that { (  l é i A g BB BB) X ù ê + ú} < " i û (stabilising solution of ()), { (  l é i A Y g CC CC) ù ê + ú} < " i û (stabilising solution of ()) and r( X Y ) < g. All controllers, K(s), are given by K = F( K, Q) l c () () where the central controller, K c, is of the same order as P and is given by where é A Z L Z B ù K = c F I C I ê ûú A A g BB X BF Z L C = + + +, ( g ) Z = I Y X, L = Y C, F =B X (3) and Q(s) are stable proper transfer functions for which Q < g holds. he central controller can be, similarly as the LQG controller, separated into the estimator 86 Vasičkaninová A et al., Robust controller design for a heat exchanger using H, H, H /H

4 xˆ = Axˆ+ B B X xˆ+ B u+ Z L ( C xˆy) (4) g and the state feedback u = F xˆ (5) If the stated conditions in the Riccati equations listed above are not met, the tested value of g is too low and hence infeasible. his is used to formulate a bisection algorithm, called giteration, to search for a feasible value g g min close to the optimum within tolerance g such that g g min < f. Mixed sensitivity control Mixed sensitivity stands for the transfer function shaping problems in which the sensitivity function S = (I + GK) is shaped along with one or more other closedloop transfer functions such as KS or the complementary sensitivity function = I S in a typical one degreeoffreedom configuration, where G denotes the plant and K the (sub)optimal controller to be found (Skogestad and Postlethwaite, 5). Shaping of multivariable transfer functions is based on the idea that a satisfactory definition of gain (range of gains) for a matrix transfer function is given by its singular values, s. Hence, the classical loopshaping ideas of feedback design can be generalised to multivariable systems. In addition to the requirement that K stabilises G, the closedloop objectives are as follows (Skogestad and Postlethwaite, 5): for disturbance rejection: low s ( S), for noise attenuation: low s ( ), for reference tracking: s( )» s( )», for input usage (control energy) reduction: low s ( KS), for robust stability in the presence of an additive perturbation, GP = G+ D : low s ( KS), for robust stability in the presence of a multiplicative output perturbation, GP = ( I + D) G : low s ( ), where s ( A) is the maximum and s ( A) is the minimum singular value of A. It is known that robust controllers are designed to minimise the H norm of the plant. hree weight functions were added to the control system for loop shaping (Bansal and Sharma, 3). he classical feedback control system structure with weighting is shown in Fig.. For this problem, the cost function given by Eq. (6) with weighting functions W, W, and W 3 penalising the error signal, control signal and output signal, respectively, can be used. F = F ( P, K ) = W KS < g (6) zw l WS W 3 Fig.. Mixedsensitivity configuration. For this structure it is possible to use the generalised plant, P, as follows éw WGù W P( s) = WG 3 I G ê úû (7) he returned values of S, KS, and satisfy the following loop shaping inequalities: s( S( jw) gs( W ( jw)) s( KS( jw) gs( W ( jw)), " w s( ( jw) gs( W ( jw)) 3 (8) It is necessary to choose low W inside the desired control bandwidth to achieve good disturbance attenuation (i.e. performance), and to choose low W 3 outside the control bandwidth to ensure good stability margin (i.e. robustness). msynthesis with DKiteration he DK iteration controller design method combines the H synthesis that provides robustness with the manalysis that considers parametric uncertainties and yields a good performance of the robust controller. he basic idea is to find a controller that minimises the maximum value of the upper bound on m over the whole range of frequency. At present, no analytical method to calculate a moptimal controller is available. However, a numerical method for complex perturbations known as DKiteration can be used (Balas et al., 998). he generalised openloop representation of the configuration in Fig. 3 is given by Eq. (9) (Griffin and Fleming, 3). he input and output vectors for this configuration contain inputs and outputs related to the input perturbations u and y, respectively. Uncertainty at the plant input and performance requirements at the system output are described by weights W u and W p, respectively, K is the controller, and G d is the disturbance model: Vasičkaninová A et al., Robust controller design for a heat exchanger using H, H, H /H 87

5 éy ù é W ù é u u ù éu ù D D D z WpG WpGd WpG d P d = = êv ú G G u u d G ê ú ê ú û ê úû û û (9) he closedloop interconnection structure N which includes P and K is given as: éw ukgs WuKG dsù N = ê WpGS WpGdS ú () û msynthesis with DKiteration is based on the minimisation of the upper bound on m specified in terms of the scaled singular value m( N ) mins DÎD ( DND ) () A scaling matrix, D, was chosen so as to commute with the plant perturbation, i.e. D = D. hen, the synthesis problem is to find controller K that minimises the peak value of this upper bound over the given frequency range, namely: ( ) min min DN ( K ) D K DÎD () by alternating the minimisation with respect to K and D (keeping the other ones fixed). he iteration proceeds as follows (Skogestad and Postlethwaite, 5):. Kstep. Synthesise an H controller for the scaled problem with fixed D(s). ( ) min DN ( K ) D K. Dstep. For N fixed, find D(jw) to minimise ( D( j ) ND ( j )) s w w (3) (4) at each frequency. 3. Fit the magnitude of each element of D(jw) to a stable and minimumphase transfer function, D(s). Go to step. Continue the iteration until the norm no longer decreases. Simulations and results DND <, or until Process description Consider a shellandtube heat exchanger (Vasičkaninová and Bakošová, 5), where petroleum is heated by hot water passed through a copper tube (Fig. 4). Among the input variables, the hot water flow rate, q 3 (t), was selected as the control input. he controlled variable was the outlet petroleum Fig. 3. Closedloop system with uncertainty and performance weighting. Fig. 4. Scheme of the countercurrent heat exchanger. 88 Vasičkaninová A et al., Robust controller design for a heat exchanger using H, H, H /H

6 temperature, out. he mathematical model of the heat exchanger was derived under several simplifying assumptions. Coordinate z measures the distance of a modelled section from the inlet. he fluids move in a plug velocity profile and the petroleum, tube and water temperatures: (z,t), (z,t) and 3 (z,t), are functions of the axial coordinate, z, and time, t. he petroleum, water and tube material densities, i, as well as the specific heat capacities, c pi, i =,, 3, were assumed to be constant. he simplified nonlinear dynamic mathematical model of the countercurrent heat exchanger is described by three partial differential equations: ( z,t ) + ( z,t) = d rc ( z,t) d rc q ( z,t) = h t h d z dh dh+dh p p + 4 p d h ( z,t) ( z,t ) + ( z,t) = 3 dh+dh ( d d) rcp ( z,t) = 4( dh+dh) t ( z,t) ( z,t) = 3 ( d d ) r c ( z,t) 3 3 p3 3 = 4d h t ( d d ) r c q ( t) 3 ( z,t) d h ( d d ) z 3 3 p3 3 p 3 (5) (6) (7) Here, l is the length of the tube, d is the tube diameter, the density, c p the specific heat capacity, h the heat transfer coefficient, q the volumetric flow rate, is petroleum, is copper, and 3 is hot water. Parameters and steadystate inputs of the heat exchanger are listed in able, where the subscript in denotes the inlet, and the superscript s denotes the steady state. ab.. Heat exchanger parameters and inputs. Variable Unit Value d m.5 d m.8 d 3 m.5 h W m K 75 h W m K 48 kg m 3 8 kg m c p J kg K c p J kg K 48 c p3 J kg K 486 l m q m 3 s q 3in s m 3 s. 4 in s K 39.6 in s K in s K 34.8 For identification, ± %, ±3 %, ±5 % step changes of the inlet volumetric flow rate of heating water were generated. Step responses of the outlet temperature are shown in Fig. 5. Here, the response to the input change of ± % is represented by a blue solid line, ±3 % by a red dashed line, ±5 % by a brown dotted line. According to the step responses, the heat exchanger is a nonlinear system with asymmetric dyna mics. he nominal model was identified using the Strejc method (Mikleš and Fikar, 7) from the step responses in form of the n th order transfer function in Eq. (8). As for various step responses, intervals of the values of gain, K, time constant, t, and timedelay, D, were obtained and the heat exchanger is represented as a system with interval parametric uncertainty, where [ 37, 7 ], t Î,3, K Î [ ] Fig. 5. Step responses of the outlet temperature to the step changes of the volumetric flowrate of heating water. Vasičkaninová A et al., Robust controller design for a heat exchanger using H, H, H /H 89

7 [.,.8] D Î, n =. he nominal values of the parameters are the mean values of the identified parameter intervals. G =.5 e 535 = e n ( ts + ) ( 5s + ) K Ds s Control of the heat exchanger (8) PID control Conventional PID controllers described by the transfer function k = + + (9) s i C kp kds were tuned using the ZieglerNichols and the RiveraMorari methods (Bequette, 99). he PID controller parameters tuned using the ZieglerNichols formulas for the nominal values of parameters are: k p =.9 4, k i = 9.7 5, k d =.9 4, and those obtained using the Rivera Morari formulas are: k p = 7.7 5, k i =. 6, k d = Robust controllers he transfer function describing the nominal system was assumed in the form of Eq. (8). he H controller was found in the form:.s +.6s+.46 C = (3) 3 s s +.9s +.5 the H controller was found in the form: C = s.35s.7 3 s + s + s+ the H /H controller was found in the form: (3).5s +.5s+.3 C( s) = (3) 3 s + 7.s + 5s +.3 and finally, the msynthesis controller was found in the form in Eq. (33). In Figs. 6 and 7, the reference trajectory, w, and the responses of the outlet petroleum temperature are shown as obtained using the designed PID and H, H, H /H and msynthesis controllers. he simulations were performed employing the nonlinear model characterised by Eqs. (5) (7). Fig. 6 shows the simulation results of setpoint tracking, while Fig. 7 presents the simulation results of setpoint tracking and disturbance rejection. Changes of the inlet petroleum temperature from 33.5 K to 38.5 K at s, from 38.5 K to 34.5 K at 6 s and to 37.5 K at s represent the disturbances. he proposed controllers were compared using the IAE integral performance index described as follows: IAE = ò edt (34) he obtained IAE values are given in able. he lowest IAE values were obtained using the m synthesis controller, which attenuates disturbances in the shortest time and the overshoots caused by the disturbances are also minimal. However, the best results were obtained applying the most complicated structure of the controller. Among the robust controllers with a simpler structure, the H /H controller has been shown to be the best. All robust controllers provided better results in terms of IAE than the conventional PID controllers; again, a more complicated structure of the controllers and a more complicated design were needed. Stability is often investigated by analysing the Nyquist curve. o achieve stability, the Nyquist curve must be sufficiently far away from the critical point at. he sensitivity function, S, expresses how the closedloop properties are influenced by small variations in the process and disturbances. he complementary sensitivity measures the influence of feedback on the measurement noise. he maximum magnitude of sensitivity and complementary sensitivity, S = max ( w), = max ( w), (35) M S j M j w are also used as robustness measures. Variable /M S can be interpreted as the shortest distance between the Nyquist curve and the critical point at. Recommended values for M S are typically within the range of.4. (Åström and Hägglund, 6). A good compromise is M S =.7 (oivonen, 998). he use of the maximum sensitivity as a robustness measure has the advantage that lower bounds on the gain, A m, and phase, j m, margins (Åström and Hägglund, 6) can be assured according to: j m A m MS > M S S w (36) æ ö > sin (37) èç M ø herefore, M S = provides the minimum robustness requirement, i.e. A m > and j m > 9. For M S =.4, A m > 3.5 and j m > 4. M is the largest magnitude of the complementary sensitivity. ypi s s s s s +.37s +.7s+. C = (33) s s + 46s + 479s + 54s + 333s + 339s + s +. 9 Vasičkaninová A et al., Robust controller design for a heat exchanger using H, H, H /H

8 ab.. Values of IAE, M S, M. set point tracking set point tracking and disturbance rejection controller IAE IAE M S M PID ZieglerNichols PID RiveraMorari H control H control H /H control msynthesis control Fig. 6. emperature response in set point tracking. Fig. 7. emperature response in set point tracking and disturbance rejection. cally, it is required that M is below.5 (Skogestad and Postlethwaite, 5). In able, values of M S and M are given. M is below.5 in all cases besides the PID ZieglerNichols controller. he condition MS Î.4; is not met for the PID RiveraMorari controller and is almost assured by the H controller. According to the values of M S and M it can be stated that all controllers designed using the described advanced control strategies are robust, while the best performance was shown by the msynthesis and H /H controllers, as they are closest to the recommended value M S =.7. aking into account the controller structure, the H /H controller is recommended. Vasičkaninová A et al., Robust controller design for a heat exchanger using H, H, H /H 9

9 Conclusions Robust controllers designed using H, H, H /H and msynthesis approaches were used to control the heat exchanger. he presented results show satisfactory control responses for the set point tracking as well as for the disturbance attenuation. Design of conventional PID controllers is the most simple one, but these controllers are not sufficiently robust which leads to the worst performance. he H, H, H /H and msynthesis controller design requires the choice of weighting functions and of a reference model, but the designed controllers are robust and assure good performance. he msynthesis controller design is the most difficult and leads to the most complicated controller. However, such a controller provides the best performance. aking into account the controller design, controller structure and performance, the H /H controller is a reasonable compromise for the control of nonlinear processes with asymmetric dynamics. Acknowledgement he authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grants //6. References Ahmad U, Anjum W, Bukhari S (3) H and H Controller Design of win Rotor System (RS), Intelligent Control and Automation, 4(): doi:.436/ica Åström KJ, Hägglund (6) Advanced PID control. nd Edition. ISA: he Instrumentation, Systems, and Automation Society. ISBN: Balas GJ, Doyle JC, Glover K, Packard A, Smith R (998) μanalysis and Synthesis oolbox. Mathworks Inc, Natick, MA, USA. Bansal A, Sharma V (3) Design and Analysis of Robust Hinfinity Controller. Control heory and Informatics. 3(): 7 4. ISSN 549. Bequette BW (99) Nonlinear control of chemical processes: a review. Industrial Engineering Chemical Research 3: Bosgra OH, Kwakernaak H, Meinsma G (7) Design Methods for Control Systems. Notes for a course of the Dutch Institute of Systems and Control. Daróczy L, Janiga G, hévenin D (4) Systematic analysis of the heat exchanger arrangement problem using multiobjective genetic optimization. Energy. 65: doi:.6/j.energy Dlapa M (5) emperature control in airheating set via direct search method and structured singular value. Procedia Engineering : doi:.6/j. proeng Doyle JC, Stein G (98) Multivariable Feedback Design Concepts for a Classical Modern Synthesis. IEEE ransactions on Automatic Control 6: 4 6. Doyle JC (984) Lecture Notes, ONR/HoneywellWorkshop on Advances in Multivariable Control, Minneapolis. Doyle JC, Glover K, Khargonekar PP, Francis BA (989) Statespace solutions to standard H and H control problems. IEEE ransactions on Automatic Control, 34(8): Ganji DD, Zahmatkesh A, Barouz A, Zahedi H, Sajadifar MJ (3) A survey study on different methods of controlling temperature of shell and tube heat exchangers. International Research Journal of Applied and Basic Sciences. 4(3): ISSN 5838X. Gerhard J, Monningmann M, Marquardt W (4) Robust stable nonlinear control and design of a CSR in a large operating range. In Proceedings of the 7 th International Symposium on Dynamics and Control of Process Systems. Massachusetts. Glover K, Doyle JC (988) Statespace formulae for all stabilizing controllers that satisfy an H norm bound and relations to risk sensitivity. Systems and Control letters. : Griffin IA, Fleming PJ (3) A Multiobjective Optimization Approach to DKIteration. Proceedings of the European Control Conference (ECC), , ISBN Holman JP (9) Heat ransfer. McGrawHill New York USA. Janna WS (9) Engineering Heat ransfer. 3 rd ed. ennessee: he University of Memphis. ennessee USA. Kwakernaak H () H Optimization heory and Applications to Robust Control Design. Annual Reviews in Control 6: Mahitthimahawong S, arapoom N, Skogestad S, Srinophakun, Lertbumrungsuk V (6) Application of Passivity Concept for Split Range Control of Heat Exchanger Networks. Journal of Chemical Engineering & Process echnology 7(): 74. doi:.47/ Manassaldi JI, Scenna NJ, Mussati SF (4) Optimization mathematical model for the detailed design of air cooled heat exchangers. Energy. 64: doi:.6/j.energy Mikleš J, Fikar M (7) Process Modeling, Identification, and Control. Springer Verlag Berlin Germany. ISBN Moradi H, SaffarAvval M, BakhtiariNejad F () Sliding mode control of drum water level in an industrial boiler unit with time varying parameters: A comparison with H robust control approach. Journal of Process Control : doi:.6/j. jprocont...3. Pan M, Bulatov I, Smith R, Kim JK () Improving energy recovery in heat exchanger network with intensified tubeside heat transfer. Chemical Engi neering ransactions. 5: Panjeshahi MH, Joda F, ahouni N () Pressure drop optimization in multistream heat exchanger using genetic algorithms. Chemical Engineering ransactions. : Serth RW, Lestina G (4) Process Heat ransfer: Principles. Applications and Rules of humb. nd Ed. Academic Press. ISBN: Skaugen G, Kolsaker K, Walnum H, Wilhelmsen Ø (3) A flexible and robust modelling framework for 9 Vasičkaninová A et al., Robust controller design for a heat exchanger using H, H, H /H

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