Model based numerical state feedback control of jet impingement cooling of a steel plate by pole placement technique
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1 Internatonal Journal of Computatonal Engneerng Research Vol, 03 Issue, 8 Model based numercal state feedback control of jet mpngement coolng of a steel plate by pole placement technque Purna Chandra Mshra 1, Swarup Kumar Nayak 1, Santosh K. Nayak 1, Premananda Pradhan 1, Durga P. Ghosh 1 1 School of Mechancal Engneerng, KIIT Unversty, Pata, Bhubaneswar , Odsha, Inda ABSTRACT Ths artcle presents the applcaton of a state feedback, Pole-Placement temperature-trackng control formulaton leadng to a comprehensve Sngle-nput, multple-output (SIMO) structure for coolng of a steel plate by an mpngng ar jet.. Followng a sem-dscrete control volume formulaton of the 1-D transent heat conducton; the state space model of nodal varaton of metal temperature wth tme has been arrved. Stable temperature track wth respect to the reference temperature track was obtaned by usng the state feedback control algorthm. To solve ths prescrbed state feedback problem, an effcent pole placement technque was mplemented n two ways.e., ntally, the fxed poles and then varable poles n the complex plane. The control models were smulated n the MATLAB and SIMULINK envronments. The state feedback pole placement technque was found effcent n controllng the parameters n the gven applcaton. It was observed that the response of the non-lnear system s senstve to lnearzaton tme nterval. Better control s mplemented by ncreasng the frequency of adjustment of the closed-loop poles. KEYWORDS: Feedback Control, Jet Impngement Coolng, Pole Placement Technque, State Space Method, Sngle Input Multple Output Problem, Fxed Poles, Varable Poles, Temperature Control Nomenclature: K: The feedback gan matrx t: Tme (s) A: The coeffcent matrx n the state space equaton A : The coeffcent matrx n the state space representaton for th model A : The mean coeffcent matrx B: The coeffcent matrx of control nput T 1 : Temperature at the top surface of the steel plate (K) B : The coeffcent matrx of control nput or th model matrx of mean state varables X: The matrx of state varables X : The X : The matrx of state rates a e : The perturbaton n stran rates X X :The egen values X : The error matrx T 1 : The demand (K/s) P (S): The th number of transformable models a: The control nput (Stran rate) 1 ( s ) a : The mean stran rate (mean control output) A-BK: The feedback matrx T: The transformaton matrx I : The dentty matrx Z: The assumed state varable matrx r: The number of operatng condtons I. INTRODUCTION Heat treatment nvolves the method of controlled heatng, soakng and coolng temperature cycles that are appled to a materal or substrate, to mpart superor metallurgcal propertes to t. Heat treatment has wde applcaton and s most abundantly used n the feld of glass, ceramc, ron and steel producton ndustres. Issn August 2013 Page 37
2 The dynamcs of heat transfer systems are hghly nonlnear. Irrespectve of nonlnearty n of heat transfer systems, ar jet mpngement coolng systems also have model uncertantes whch ncludes large changes n the ar flow rate and varaton of parameters, the external dsturbances, leakages, and frcton. To overcome ths problem n order to gan profts, process desgners are creatng desgns and modellng n complex non-lnearty regons where process controllers can face stff the challenges. In model predctve control, the controller acton s the soluton to a constraned optmzaton problem that s solved numercally on-lne [1, 2, 3]. In contrast, dfferental geometrc control s a drect synthess approach n whch the controller s derved by requestng a desred closed loop response n the absence of nput constrants [4, 5]. In Lyapunov-based control, closed loop stablty plays a central basc role n a controller desgn. [6, 7, 8]. Kravars and Daoutds [9] presented a non-lnear state feedback controller for second order non-lnear systems. Nemec and Kravars [10] proposed a systematc procedure for the constructon of statcally equvalent outputs wth prescrbed transmsson zeros. Kanter et al. [11] developed nonlnear control laws for nput constraned, multple-nput, multple-output, stable processes. They addressed the nonlnear control of the processes by explotng the connectons between model- predctve control and nput-output lnearzaton. Kravars et al. [12] presented a systematc method arbtrarly assgnng the zero dynamcs of a nonlnear system by constructng the requste synthetc output maps. The method requres solvng a system of frst-order, nonlnear, sngular PDEs. Mckle et al. [13] developed a trackng controller for unstable, nonlnear processes by usng trajectory lnearzaton. Tomln and Sastry [14] derved trackng control laws for non-lnear systems wth both fast and slow, possbly unstable, zero dynamcs. Vander Scaft [15] developed a nonlnear state feedback H optmal controller. Devasa et al. [16] and Devasa [17] ntroduced an nverson procedure for nonlnear systems that constructs a bounded nput trajectory n the pre-mage of a desred output trajectory. Hunt and Meyer [18] showed that under approprate assumptons the bounded soluton of the partal dfferental equaton of Isdor and Byrnes [19] for each trajectory of an ecosystem must be gven by an ntegral representaton formula of Devasa et al. [17] Chen and Paden [20] studed the stable nverson of nonlnear systems. The present work uses the same contnuous-tme, model predctve control framework. Conceptually as the amount of nonlnearty s very less so the model does not requre any lnearzaton of the process equatons. Here the nonlnear terms are consdered as the stran rate n the convectve heat transfer correlaton. The controller ntroduced here s obtaned by requestng desred responses for the state varables. The nonlnear state feedback s derved by mnmzng a functon norm of the devatons of the controlled outputs from lnear reference trajectores wth orders equal to the (output) relatve orders. In the part of the work descrbed n ths chapter, we come up through a systematc evaluaton wth a process model that provdes the startng pont of evolvng a model-based control. At the same tme, we analyze the performance of a model-based nonlnear control wth the applcatons of an effcent pole-placement technque. The focal pont of the present analyss s a detaled modelng of the heat transfer and applcatons of effcent technques (pole-placement) to control the state varables nfluencng the process. Moreover ths chapter presents a nonlnear control method that s applcable to stable and unstable processes. The closed loop stablty s ensured by forcng all the process state varables to follow ther correspondng reference trajectores. The proposed control system ncludes a nonlnear state feedback and pole placement technque n both the forms of fxed poles and varable poles Pole placement technques Pole placement s a conventonal desgn methodology for lnear tme-nvarant control systems. The method s based on the fact that several performance specfcatons can be met by usng dynamc output feedback to adequately placed closed loop poles n the complex plane.. An extenson of the classcal pole placement problem s the regonal pole-placement problem, n whch the objectve s to place closed-loop poles n a sutable regon of the complex plane. In many real cases, the model uncertanty reflects on the parameters of the plant, whch has motvated extensve research efforts n parametrc robust theory [21, 22, 23]. Pole placement methods can be dvded manly nto two categores such as output feedback and state feedback methods. Usually domnant pole desgn technques are employed as t s not always possble to assgn all the poles of a system and thus one has to relax the desgn requrements and consder a partal pole placement that wll gve satsfactory overall performance. The pole placement problem s much easer to handle when the desgn s done n the state space doman and many technques exst for ths partcular case. In ths work, the plant s represented by proper state-space equaton. Robust pole placement problem s formulated as the problem of robustly assgned poles n a regon n two dfferent ways.e., by assumng some approprate fxed poles and varable poles n terms of the Egen values of the ( A BK ) matrx. Where K s the gan. Smlar condtons hold n [24] and [25]. Issn August 2013 Page 38
3 The man focus of ths work was to present a more effcent way of approachng Ackerman s formula to fnd the gan matrx so as to acheve the Egen poles. Apart from ths, the technque presented here was utlzed towards sgnfcant progress for the case of nonlnear Sngle-nput and multple-output (SIMO) systems.. As only the temperatures of the plate are controlled, typcally the plate may reman nsgnfcantly cool or the desred coolng temperature trajectory s followed backwards. That s why we focus on state trackng nstead of output trackng. To facltate the desgn of controller, we frst nvestgate the pole placement problem for SIMO systems usng dynamc gans, when all the state varables are avalable for feedback. Both the local and global pole placement control has been accomplshed n the MATLAB and SIMULINK envronments. In both the cases the results are shown n the comparatve mode between the demand and controlled states. II. THE DESIGN PROBLEM In ths work, we consder the modelng to evolve a dynamc model-based control of the ar jet mpngement coolng of steel plate. Here we propose the desgn procedure based on state space pole placement technques for the state feedback control desgn of jet mpngement coolng system. The desgn of State-Space models s not dfferent from that of transfer functons n that the dfferental equatons descrbng the system dynamcs are wrtten frst. For state-space models, models, nstead, the equatons are arranged nto a set of frst order dfferental equatons n terms of selected state varables, and the outputs are expressed these same state varables, Because the elmnaton of varables between equatons s not an nherent part of ths process, state models can be easer to obtan. The State models are drectly derved from the orgnal system equatons. The standard form of the State-Space model equaton s: X t AX ˆ t Bˆ a t (State Equaton) Here x s the State Vector, the vector of state varables, a s the control varable and A s the system coeffcent matrx. The matrces  and Bˆ are controllable. A fxed dynamc controller s desgned.e., a B ˆ 1 [AX ˆ X] The pole placement desgn of B n the form of full state feedback becomes equvalent to desgn K where, a e = KX e (Controller or Control Law) (1) For stablty all the Egen values of A must be such that the real part s negatve. Let us assume, Z TX (2) The equaton becomes, TX TAX TBa Z 1 TAT Z TBa 1 X T Z Z AZ Ba ; (3) Where A 1 TAT and B TB Let us consder the Egen values of equaton (3) be, Then, A I 0 A λi 0 X Plant a a - X - + A and B - ae K X e Fgure 1: State Feedback Confguraton Issn August 2013 Page 39
4 Snce,, the transformaton T does not alter the system stablty. The above dagram represents the correspondng control structure that evolves n the form of a feedback structure. In contrast to tradtonal feedback structure, K vares wth tme. Then feedng ths nput a t back nto the system, we obtaned, X t A BK X t (4) III. FULL STATE FEEDBACK DESIGN There are several motvatons for formulatng the desgn procedure n state space. One motvaton s a clear exposton of the controller structure. Another s that the controller structure ponts possbltes of reducng the order of the compensators usng state space technque [26]. A thrd motvaton s that the state space formulaton may accommodate desgn procedures for systems n whch the number of nputs or outputs s less than the number of operatng condtons. Let us consder the system P has r > 1 operatng condtons wth models P 1 (S), P 2 (S), P r (S). The state space representatons of the models P (S), = 1, 2 r, are: X AX Ba (5) n m Where the state X R and the nputa R. The pars A, B, = 1,2 r are controllable. Let us desgn a fxed statc controller a KX such that the poles of the closed loop system are λ A BK =, = 1 r, where are the desred Egen values for the th operatng condtons. For full state to have a soluton, t s necessary that the pars A, B, = 1 r, be controllable. Closely related to full state problem s the desgn of a fxed gan matrx K to stablze all the operatng condtons. The soluton to problem exsts a K such that: det si A B K p s, 1,..., r. (6) The nonlnear pole-placement equaton (6) can be consdered as Bass-Gura pole-placement formula [27], extended to multple operatng condtons. IV. NUMERICAL METHODS FOR THE SINGLE POLE PLACEMENT PROBLEM There are several methods for the pole placement problem some of the well known theoretcal formulae, such as the Ackermann s formula, Bass-Gura formula etc.. The prmary reason s that, these methods are based on transformaton of the controllable par A, B to the controller companon form. The computatonally vable methods for pole placement are based on transformaton of the par A, B to the controller Hesenberg form or the matrx A to the real Schur form (RSF), whch can be acheved usng orthogonal transformatons. The pole placement desgn n the present study s accomplshed n two ways, that s ) Assgnng the fxed poles arbtrarly n the complex plane. ) Assgnng the varable poles n the complex plane. Fgure 2: Block Dagram Showng Pole-Placement Control Model for Jet Impngement Coolng of Steel Plate (Temperature trackng Control). Issn August 2013 Page 40
5 V. STATE FEEDBACK CONTROL BY FIXED AND VARIABLE POLE PLACEMENT DESIGN As a frst attempt, a non-lnear model-referenced state-feedback trackng control by fxed (local) pole placement for a jet mpngement heat transfer system was mplemented. The closed-loop poles were adjusted arbtrarly at each of the tme ntervals based on the open-loop poles of the system. The arbtrary pole assgnment was performed n the SIMULINK envronment. The poles were selected on the bass of tral and error method, for dfferent smulaton tmes. For each tme segment the controlled output response and the reference output track were dentfed and compared. The results obtaned n the SIMULINK envronment for dfferent tme segments and tme steps adoptng the fxed pole placement control of the temperature track are dscussed n detal n the later sectons. After pckng the feedback gan matrx K to get the dynamcs to have some nce propertes (.e., Stablze A) λ A λ A BK, the selected poles were placed n the complex plane. Recallng that the characterstc polynomal for ths closed-loop system s the determnant of si A BK. Snce the matrces A and B*K are both 3 by 3 matrces, there were 3 poles for the system. By usng full state feedback we placed the poles at dfferent places arbtrarly. The control matrx K, whch gave the desred poles, was found by usng the MATLAB functon PLACE. For SIMO systems, PLACE use the extra degrees of freedom to fnd a robust soluton for K.e., mnmzes senstvty of closed loop poles to perturbatons n A and B. Then, the pole placement desgn procedures have been executed for the state feedback desgn of systems wth sngle operatng condton. To facltate the desgn, poles varyng wth tme were ntroduced. The same a KX controller usng strctly the state space technque was used for smulaton.e.,. Ths approach provded a clear exposton of the controller structures. The closed-loop poles were adjusted at each of the tme ntervals based on the open-loop poles of the system. The feedback gan was developed based on the plant models and the correspondng set of closed-loop poles. The response of the non-lnear plant to ths controller was presented for the full state feedback. In order to target a steady state of the system the Egen values of (A- BK) should be such that ther real parts are negatve. They were chosen n the vcnty of the open-loop poles of the system. If any of the open-loop poles have postve real part, the real part of the correspondng closed-loop poles were chosen to be zero. The tme varant Egen values of the matrx (A-BK) were computed then computed. VI. RESULTS AND DISCUSSIONS The man focus of ths work was to evolve a model based closed loop control analyss wth the applcaton of an effcent pole placement technque. On the way to establsh the necessary control model, we captured an mportant jet mpngement coolng of a steel plate. The smulatons were carred out n the MATLAB and SIMULINK envronment. Dfferent sets of results were generated for dfferent set of poles. The results dscussed below are both for fxed closed loop pole placement and varable pole placement problems. In the closed loop control phenomena, the mportant task was evaluaton of elements of the feedback gan matrx (K). 6.1 Closed- loop control results for fxed pole placement technque s observed that wth the applcaton of fxed poles such as two complex conjugate poles and one real pole, where all the real poles are negatve.e., poles are consdered more n the left half of the complex plane, the coolng rate controls upto -1 K /s and the nature of controlled temperature track s almost concdes over the reference temperature track as shown n the fgures 3 and 4. The poles selected ntally are (Complex Conjugate) and -100 (Real). Several trals were carred out by changng the pole locatons and plots were generated for the temperature track on the top surface of the steel plate to realze the controlled coolng rate. Issn August 2013 Page 41
6 Temperature, T1 (K) Temperature, T1 (K) Fgure 3: Effect of lnearzaton tme nterval on trackng control of Plate temperature by assgnng a fxed set of Tme Segments = 1 s Tme Steps = 100 D = -1 K/s Demand and Response Super-mposed Fxed Poles: For 0-1 Sec /- 1 (Complex Conjugate) -100 (Negatve Real) For Sec /- 1 (Complex Conjugate) -10 (Negatve Real) les Tme (S) Tme (S) Fgure 4: Effect of lnearzaton tme nterval on trackng control of Plate temperature by assgnng a fxed set of poles at Tme Steps of 100 and Tme segment of 1 second. Issn August 2013 Page 42
7 Temperature (K) Temperature (K) 6.2 Closed- Loop control results for varable pole placement technque Ths technque apples the Egen values of the matrx (A) where, the matrx A s a 3 by 3 matrx, so there wll be 3 poles for the system. Usng the MATLAB functon eg, the Egen values of the (A) matrx are evaluated to evolve the feedback gan matrx (K). The problem was solved as a closed loop feedback control model usng tme dependent poles. For the postve values of the poles, the stablty of the system was found margnal. Thus, at each lnearzed tme segment the approprate poles were assgned and the correspondng gan matrx was computed. The total computaton has been performed by the help of SIMULINK tools. The effects of lnearzaton tme segment on trackng control of Plate temperature by assgnng varable poles at dfferent tme locatons are presented n the fgures 5, 6, and 7 respectvely. The only pecularty found n ths case was that, the response was oscllatory n nature. These results clearly predct that, one of the domnant poles tend to more negatve sde, whle other two become less effectve. Fgure 5: Effect of Lnearzaton Tme Interval on Trackng Control of Plate Temperature by Assgnng a Set of Varable Poles at Dfferent Tme Locatons at Tme Steps of 100 and Tme Segment of 0.1 second. 800 Demand = -1 K/s Demand 740 Response Varable Poles = Egen values of (A-BK) Tme Segment = 1 s Tme Steps = Tme (S) Fgure 6: Effect of Lnearzaton Tme Interval on Trackng Control of Plate Temperature by Assgnng a Set of Varable Poles at Dfferent Tme Locatons at Tme Steps of 1000 and Tme Segment of 1 second Demand = -2 K/s Demand 780 Response Varable Poles: Egen Values of matrx (A-BK) Tme Segment = 1s Tme Steps = Tme (S) Fgure 7: Effect of Lnearzaton Tme Interval on Trackng Control of Plate Temperature by Assgnng a Set of Varable Poles at Dfferent Tme Locatons at Tme Steps of 100 and Tme Segment of 1 second. Issn August 2013 Page 43
8 VII. CONCLUSION A model based smulaton and control of the temperature track n jet mpngement coolng process of a steel plate has been developed that ncorporates the mplementaton of an effcent pole placement technque n two ways.e., one by the local poles and other by the global poles. The nonlnear control model developed here can be used to evaluate the performance n terms of physcal propertes and metallurgcal aspects. The conclusons beng obtaned from the work are: 1. Pole placement desgn procedures have been proposed for the state feedback desgn of systems wth sngle nput condton. To facltate the desgn, the concepts o coeffcent matrces (A and B) and local poles are ntroduced. 2. It s seen that the response of the non-lnear system s senstve to lnearzaton tme nterval. Better control s mplemented by ncreasng the frequency of adjustment of the closed-loop poles. 3. In the present algorthm, the mean soluton s utlzed at dscrete ntervals (Pece-wse lnearzed tme ntervals) of tme for evaluaton of the coeffcents. It s observed from both the fxed pole placement and varable pole placement that, n case of fxed pole placement, the smulaton tme s lmted upto1.4 seconds. Wthn ths smulaton tme, the demand temperature track and the response are concdng to each other wth very neglgble error. Whle, n case of varable pole placement, the smulaton tme s large for the same lnearzed tme ntervals and percentage of error s sgnfcant compared to the frst case. REFERENCES [1]. D.Q. Mayne, J.B. Rawlngs, C.V. Rao, P.O.M. Scokaert. constraned model predctve control: Stablty and Optmalty. Automatca 2000, 36 (6), 789. [2]. F. Allgower, A. Zheng. Nonlnear Model Predctve Control; Progress n Systems and Control Theory Seres; Brkhauser Verlag; Basel 2000; Vol. 26. [3]. S.L. de Olvera, M.V. Kothare, M. Morar. Contractve model predctve control for constraned nonlnear systems. 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Nonlnear controller desgn for nput-constraned, multvarable processes. Ind. Eng. Chem. Res. 2002, 41, [12]. C. Kravars, M. Nemec, N. Kazantzs. Sngular PDEs and the assgnment of zero dynamcs n nonlnear systems. Syst. Control Lett. 2004, 51, 67. [13]. M.C. Mckle, R. Huang, J.J. Zhu. Unstable, non-mnmum phase, nonlnear trackng by trajectory lnearzaton control. Proceedngs of the IEEE Internatonal Conference on Control applcatons, Tape, Tawan, 2004; p812. [14]. C.T. Tomln, S.S. Sastry. Bounded trackng for non-mnmum phase nonlnear systems wth fast zero dynamcs. Int. J. Control 1997, 68, 819. [15]. A.J. Van der Schaft. L 2- Gan analyss of nonlnear systems and nonlnear state feedback H control. IEEE Trans. Autom. Control 1992, 37, 770. [16]. S. Devasa, D. Chen, B. Paden. Nonlnear nverson-based output trackng. IEEE Trans. Autom. Control 1996, 41, 930. [17]. S. Devasa. Approxmated stable nverson for nonlnear systems wth non-hyperbolc nternal dynamcs. IEEE Trans. Autom. Control 1999, 44, [18]. L.R. Hunt, G. Meyer. Stable nverson for nonlnear systems. Automatca 1997, 33, [19]. A. Isdor, C.I. Byrnes. Output regulaton of nonlnear systems. IEEE Trans. Autom. Control 1990, 35, 131. [20]. D.G. Chen, B. Paden. Stable nverson of nonlnear non-mnmum phase systems. Int. J.Control 1996, 64, 81. [21]. J. Ackermann. (1993). Robust Control: Systems wth Uncertan physcal parameters. Sprnger-Verlag, New York, NY. [22]. B.R. Barmsh. (1994). New tools for robustness of lnear systems. Macmllan Publshng Co., New York, NY. [23]. S.P. Bhattacharya, H. Chapellat, L.H. Keel. (1995). Robust Control- The parametrc approach. Prentce Hall publshng Co, Upper Saddle Rver, NJ. [24]. Y.C. Soh, R.J. Evans, I. Petersen, R.E. Betz. Robust Pole assgnment. Automatca, 1987, 23, pages [25]. L.H. Keel, S.P. Bhattacharya. A lnear programmng approach to controller desgn. Proceedngs of the 36 th Conference on Decson and control. 1997, pages , San Dego, CA, USA. [26]. T. Kalath, Lnear systems, Anglewood Clffs, Prentce-Hall, [27]. R. W. Bass, I. Gura, Hgh order desgn va State space consderatons, Proc JACC, pp [28]. Chow, J.H. A pole placement desgn approach for systems wth multple operatng condtons, Proc. 27 th Conf. On Decson and Control, Austn, Texas. 1988, pages Issn August 2013 Page 44
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