Model based numerical state feedback control of jet impingement cooling of a steel plate by pole placement technique

Size: px
Start display at page:

Download "Model based numerical state feedback control of jet impingement cooling of a steel plate by pole placement technique"

Transcription

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. IEEE Trans.Autom. Control 2000, 45, [4]. C. Kravars, P. Daoutds, R.A. Wrght. Output feedback control of no mnmum-phase nonlnear processes. Chem. Engg. Sc., 1994, 49 (13), [5]. A. Isdor. Nonlnear control systems; Sprnger- Verlag: New York, [6]. N.H. El-Farra, P.D. Chrstofdes. Integratng Robustness, Optmalty and constrants n control of Non lnear processes. Chem. Engg. Sc. 2001, 56, [7]. N.H. El-Farra, P.D. Chrstofdes. Bounded Robust control of constraned nonlnear processes. Chem. Engg. Sc. 2003, 58, [8]. S. Dubljevc, N. Kazantzs. A New Lyapunov desgn approach for nonlnear systems based on Zubov s method. Automatca 2002, 38, [9]. C. Kravars, P. Daotds. Nonlnear State feedback control of second order non-mnmum phase nonlnear systems. omput. Chem. Eng. 1990, 40 (4/5), 439. [10]. M. Nemec, C. Kravars. Nonlnear model-state feedback control for non-mnmum phase processes. Automatca 2003, 39, [11]. J.M. Kanter, M. Soroush, W.D. Seder. 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

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:

More information

COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN

COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN Int. J. Chem. Sc.: (4), 04, 645654 ISSN 097768X www.sadgurupublcatons.com COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN R. GOVINDARASU a, R. PARTHIBAN a and P. K. BHABA b* a Department

More information

Time-Varying Systems and Computations Lecture 6

Time-Varying Systems and Computations Lecture 6 Tme-Varyng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy

More information

NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS

NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc

More information

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng

More information

The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

The equation of motion of a dynamical system is given by a set of differential equations. That is (1) Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence

More information

DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM

DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI

More information

FUZZY GOAL PROGRAMMING VS ORDINARY FUZZY PROGRAMMING APPROACH FOR MULTI OBJECTIVE PROGRAMMING PROBLEM

FUZZY GOAL PROGRAMMING VS ORDINARY FUZZY PROGRAMMING APPROACH FOR MULTI OBJECTIVE PROGRAMMING PROBLEM Internatonal Conference on Ceramcs, Bkaner, Inda Internatonal Journal of Modern Physcs: Conference Seres Vol. 22 (2013) 757 761 World Scentfc Publshng Company DOI: 10.1142/S2010194513010982 FUZZY GOAL

More information

Chapter 11: Simple Linear Regression and Correlation

Chapter 11: Simple Linear Regression and Correlation Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests

More information

A PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.

A PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY. Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

DUE: WEDS FEB 21ST 2018

DUE: WEDS FEB 21ST 2018 HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant

More information

A Hybrid Variational Iteration Method for Blasius Equation

A Hybrid Variational Iteration Method for Blasius Equation Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method

More information

Lectures on Multivariable Feedback Control

Lectures on Multivariable Feedback Control Lectures on Multvarable Feedback Control Al Karmpour Department of Electrcal Engneerng, Faculty of Engneerng, Ferdows Unversty of Mashhad June 200) Chapter 9: Quanttatve feedback theory Lecture Notes of

More information

On the Multicriteria Integer Network Flow Problem

On the Multicriteria Integer Network Flow Problem BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of

More information

Irregular vibrations in multi-mass discrete-continuous systems torsionally deformed

Irregular vibrations in multi-mass discrete-continuous systems torsionally deformed (2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected

More information

REAL TIME OPTIMIZATION OF a FCC REACTOR USING LSM DYNAMIC IDENTIFIED MODELS IN LLT PREDICTIVE CONTROL ALGORITHM

REAL TIME OPTIMIZATION OF a FCC REACTOR USING LSM DYNAMIC IDENTIFIED MODELS IN LLT PREDICTIVE CONTROL ALGORITHM REAL TIME OTIMIZATION OF a FCC REACTOR USING LSM DYNAMIC IDENTIFIED MODELS IN LLT REDICTIVE CONTROL ALGORITHM Durask, R. G.; Fernandes,. R. B.; Trerweler, J. O. Secch; A. R. federal unversty of Ro Grande

More information

COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

More information

EEL 6266 Power System Operation and Control. Chapter 3 Economic Dispatch Using Dynamic Programming

EEL 6266 Power System Operation and Control. Chapter 3 Economic Dispatch Using Dynamic Programming EEL 6266 Power System Operaton and Control Chapter 3 Economc Dspatch Usng Dynamc Programmng Pecewse Lnear Cost Functons Common practce many utltes prefer to represent ther generator cost functons as sngle-

More information

Note 10. Modeling and Simulation of Dynamic Systems

Note 10. Modeling and Simulation of Dynamic Systems Lecture Notes of ME 475: Introducton to Mechatroncs Note 0 Modelng and Smulaton of Dynamc Systems Department of Mechancal Engneerng, Unversty Of Saskatchewan, 57 Campus Drve, Saskatoon, SK S7N 5A9, Canada

More information

Some modelling aspects for the Matlab implementation of MMA

Some modelling aspects for the Matlab implementation of MMA Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton

More information

Introduction. - The Second Lyapunov Method. - The First Lyapunov Method

Introduction. - The Second Lyapunov Method. - The First Lyapunov Method Stablty Analyss A. Khak Sedgh Control Systems Group Faculty of Electrcal and Computer Engneerng K. N. Toos Unversty of Technology February 2009 1 Introducton Stablty s the most promnent characterstc of

More information

Robust observed-state feedback design. for discrete-time systems rational in the uncertainties

Robust observed-state feedback design. for discrete-time systems rational in the uncertainties Robust observed-state feedback desgn for dscrete-tme systems ratonal n the uncertantes Dmtr Peaucelle Yosho Ebhara & Yohe Hosoe Semnar at Kolloquum Technsche Kybernetk, May 10, 016 Unversty of Stuttgart

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t

Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t 8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes

More information

Indeterminate pin-jointed frames (trusses)

Indeterminate pin-jointed frames (trusses) Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all

More information

Appendix B. The Finite Difference Scheme

Appendix B. The Finite Difference Scheme 140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton

More information

Constitutive Modelling of Superplastic AA-5083

Constitutive Modelling of Superplastic AA-5083 TECHNISCHE MECHANIK, 3, -5, (01, 1-6 submtted: September 19, 011 Consttutve Modellng of Superplastc AA-5083 G. Gulano In ths study a fast procedure for determnng the constants of superplastc 5083 Al alloy

More information

MMA and GCMMA two methods for nonlinear optimization

MMA and GCMMA two methods for nonlinear optimization MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons

More information

EEE 241: Linear Systems

EEE 241: Linear Systems EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they

More information

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4) I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes

More information

Hongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)

Hongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k) ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of

More information

An identification algorithm of model kinetic parameters of the interfacial layer growth in fiber composites

An identification algorithm of model kinetic parameters of the interfacial layer growth in fiber composites IOP Conference Seres: Materals Scence and Engneerng PAPER OPE ACCESS An dentfcaton algorthm of model knetc parameters of the nterfacal layer growth n fber compostes o cte ths artcle: V Zubov et al 216

More information

Study on Active Micro-vibration Isolation System with Linear Motor Actuator. Gong-yu PAN, Wen-yan GU and Dong LI

Study on Active Micro-vibration Isolation System with Linear Motor Actuator. Gong-yu PAN, Wen-yan GU and Dong LI 2017 2nd Internatonal Conference on Electrcal and Electroncs: echnques and Applcatons (EEA 2017) ISBN: 978-1-60595-416-5 Study on Actve Mcro-vbraton Isolaton System wth Lnear Motor Actuator Gong-yu PAN,

More information

Second Order Analysis

Second Order Analysis Second Order Analyss In the prevous classes we looked at a method that determnes the load correspondng to a state of bfurcaton equlbrum of a perfect frame by egenvalye analyss The system was assumed to

More information

Speeding up Computation of Scalar Multiplication in Elliptic Curve Cryptosystem

Speeding up Computation of Scalar Multiplication in Elliptic Curve Cryptosystem H.K. Pathak et. al. / (IJCSE) Internatonal Journal on Computer Scence and Engneerng Speedng up Computaton of Scalar Multplcaton n Ellptc Curve Cryptosystem H. K. Pathak Manju Sangh S.o.S n Computer scence

More information

THE GUARANTEED COST CONTROL FOR UNCERTAIN LARGE SCALE INTERCONNECTED SYSTEMS

THE GUARANTEED COST CONTROL FOR UNCERTAIN LARGE SCALE INTERCONNECTED SYSTEMS Copyrght 22 IFAC 5th rennal World Congress, Barcelona, Span HE GUARANEED COS CONROL FOR UNCERAIN LARGE SCALE INERCONNECED SYSEMS Hroak Mukadan Yasuyuk akato Yoshyuk anaka Koch Mzukam Faculty of Informaton

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

On a direct solver for linear least squares problems

On a direct solver for linear least squares problems ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear

More information

Global Sensitivity. Tuesday 20 th February, 2018

Global Sensitivity. Tuesday 20 th February, 2018 Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values

More information

Solving Nonlinear Differential Equations by a Neural Network Method

Solving Nonlinear Differential Equations by a Neural Network Method Solvng Nonlnear Dfferental Equatons by a Neural Network Method Luce P. Aarts and Peter Van der Veer Delft Unversty of Technology, Faculty of Cvlengneerng and Geoscences, Secton of Cvlengneerng Informatcs,

More information

One-sided finite-difference approximations suitable for use with Richardson extrapolation

One-sided finite-difference approximations suitable for use with Richardson extrapolation Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,

More information

Adaptive sliding mode reliable excitation control design for power systems

Adaptive sliding mode reliable excitation control design for power systems Acta Technca 6, No. 3B/17, 593 6 c 17 Insttute of Thermomechancs CAS, v.v.. Adaptve sldng mode relable exctaton control desgn for power systems Xuetng Lu 1, 3, Yanchao Yan Abstract. In ths paper, the problem

More information

Simultaneous Optimization of Berth Allocation, Quay Crane Assignment and Quay Crane Scheduling Problems in Container Terminals

Simultaneous Optimization of Berth Allocation, Quay Crane Assignment and Quay Crane Scheduling Problems in Container Terminals Smultaneous Optmzaton of Berth Allocaton, Quay Crane Assgnment and Quay Crane Schedulng Problems n Contaner Termnals Necat Aras, Yavuz Türkoğulları, Z. Caner Taşkın, Kuban Altınel Abstract In ths work,

More information

A Bayes Algorithm for the Multitask Pattern Recognition Problem Direct Approach

A Bayes Algorithm for the Multitask Pattern Recognition Problem Direct Approach A Bayes Algorthm for the Multtask Pattern Recognton Problem Drect Approach Edward Puchala Wroclaw Unversty of Technology, Char of Systems and Computer etworks, Wybrzeze Wyspanskego 7, 50-370 Wroclaw, Poland

More information

Lab 2e Thermal System Response and Effective Heat Transfer Coefficient

Lab 2e Thermal System Response and Effective Heat Transfer Coefficient 58:080 Expermental Engneerng 1 OBJECTIVE Lab 2e Thermal System Response and Effectve Heat Transfer Coeffcent Warnng: though the experment has educatonal objectves (to learn about bolng heat transfer, etc.),

More information

Lecture 14: Forces and Stresses

Lecture 14: Forces and Stresses The Nuts and Bolts of Frst-Prncples Smulaton Lecture 14: Forces and Stresses Durham, 6th-13th December 2001 CASTEP Developers Group wth support from the ESF ψ k Network Overvew of Lecture Why bother? Theoretcal

More information

SIMULATION OF ROBUST CONTROL OF TEMPERATURE FIELDS IN DPS BLOCKSET FOR MATLAB & SIMULINK

SIMULATION OF ROBUST CONTROL OF TEMPERATURE FIELDS IN DPS BLOCKSET FOR MATLAB & SIMULINK SIMULATION OF ROBUST CONTROL OF TEMPERATURE FIELDS IN DPS BLOCKSET FOR MATLAB & SIMULINK C. Belavý, M. Mchalečko, V. Ivanov, S. Leľo Insttute of automaton, measurement and appled nformatcs, Faculty of

More information

The Minimum Universal Cost Flow in an Infeasible Flow Network

The Minimum Universal Cost Flow in an Infeasible Flow Network Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran

More information

Army Ants Tunneling for Classical Simulations

Army Ants Tunneling for Classical Simulations Electronc Supplementary Materal (ESI) for Chemcal Scence. Ths journal s The Royal Socety of Chemstry 2014 electronc supplementary nformaton (ESI) for Chemcal Scence Army Ants Tunnelng for Classcal Smulatons

More information

Real-Time Systems. Multiprocessor scheduling. Multiprocessor scheduling. Multiprocessor scheduling

Real-Time Systems. Multiprocessor scheduling. Multiprocessor scheduling. Multiprocessor scheduling Real-Tme Systems Multprocessor schedulng Specfcaton Implementaton Verfcaton Multprocessor schedulng -- -- Global schedulng How are tasks assgned to processors? Statc assgnment The processor(s) used for

More information

4DVAR, according to the name, is a four-dimensional variational method.

4DVAR, according to the name, is a four-dimensional variational method. 4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The

More information

PREDICTIVE CONTROL BY DISTRIBUTED PARAMETER SYSTEMS BLOCKSET FOR MATLAB & SIMULINK

PREDICTIVE CONTROL BY DISTRIBUTED PARAMETER SYSTEMS BLOCKSET FOR MATLAB & SIMULINK PREDICTIVE CONTROL BY DISTRIBUTED PARAMETER SYSTEMS BLOCKSET FOR MATLAB & SIMULINK G. Hulkó, C. Belavý, P. Buček, P. Noga Insttute of automaton, measurement and appled nformatcs, Faculty of Mechancal Engneerng,

More information

Feature Selection: Part 1

Feature Selection: Part 1 CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

A Simple Inventory System

A Simple Inventory System A Smple Inventory System Lawrence M. Leems and Stephen K. Park, Dscrete-Event Smulaton: A Frst Course, Prentce Hall, 2006 Hu Chen Computer Scence Vrgna State Unversty Petersburg, Vrgna February 8, 2017

More information

Lecture 10 Support Vector Machines. Oct

Lecture 10 Support Vector Machines. Oct Lecture 10 Support Vector Machnes Oct - 20-2008 Lnear Separators Whch of the lnear separators s optmal? Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron

More information

Lecture 16 Statistical Analysis in Biomaterials Research (Part II)

Lecture 16 Statistical Analysis in Biomaterials Research (Part II) 3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan

More information

Report on Image warping

Report on Image warping Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased

More information

Linear Approximation with Regularization and Moving Least Squares

Linear Approximation with Regularization and Moving Least Squares Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...

More information

Chapter - 2. Distribution System Power Flow Analysis

Chapter - 2. Distribution System Power Flow Analysis Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load

More information

COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD

COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,

More information

risk and uncertainty assessment

risk and uncertainty assessment Optmal forecastng of atmospherc qualty n ndustral regons: rsk and uncertanty assessment Vladmr Penenko Insttute of Computatonal Mathematcs and Mathematcal Geophyscs SD RAS Goal Development of theoretcal

More information

Digital Signal Processing

Digital Signal Processing Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over

More information

Robust Model Predictive Control

Robust Model Predictive Control Robust Model Predctve Control Formulatons of robust control [1] he robust control problem concerns to the control of plants that are only approxmately nown. Usually, t s assumed that the plant les n a

More information

A new Approach for Solving Linear Ordinary Differential Equations

A new Approach for Solving Linear Ordinary Differential Equations , ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of

More information

Advanced Quantum Mechanics

Advanced Quantum Mechanics Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton

More information

The Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites

The Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites 7 Asa-Pacfc Engneerng Technology Conference (APETC 7) ISBN: 978--6595-443- The Two-scale Fnte Element Errors Analyss for One Class of Thermoelastc Problem n Perodc Compostes Xaoun Deng Mngxang Deng ABSTRACT

More information

COS 521: Advanced Algorithms Game Theory and Linear Programming

COS 521: Advanced Algorithms Game Theory and Linear Programming COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton

More information

Lossy Compression. Compromise accuracy of reconstruction for increased compression.

Lossy Compression. Compromise accuracy of reconstruction for increased compression. Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost

More information

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1 P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the

More information

Finite Element Modelling of truss/cable structures

Finite Element Modelling of truss/cable structures Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures

More information

The Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method

The Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method Journal of Electromagnetc Analyss and Applcatons, 04, 6, 0-08 Publshed Onlne September 04 n ScRes. http://www.scrp.org/journal/jemaa http://dx.do.org/0.46/jemaa.04.6000 The Exact Formulaton of the Inverse

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

Chapter 13: Multiple Regression

Chapter 13: Multiple Regression Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to

More information

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS) Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve

More information

princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg

princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there

More information

Zeros and Zero Dynamics for Linear, Time-delay System

Zeros and Zero Dynamics for Linear, Time-delay System UNIVERSITA POLITECNICA DELLE MARCHE - FACOLTA DI INGEGNERIA Dpartmento d Ingegnerua Informatca, Gestonale e dell Automazone LabMACS Laboratory of Modelng, Analyss and Control of Dynamcal System Zeros and

More information

LOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin

LOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, M-H Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence

More information

Polynomial Regression Models

Polynomial Regression Models LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance

More information

The Feynman path integral

The Feynman path integral The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space

More information

Application of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems

Application of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &

More information

Fuzzy Boundaries of Sample Selection Model

Fuzzy Boundaries of Sample Selection Model Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) Fuzzy Boundares of Sample Selecton Model L. MUHMD SFIIH, NTON BDULBSH KMIL, M. T. BU OSMN

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

The Quadratic Trigonometric Bézier Curve with Single Shape Parameter

The Quadratic Trigonometric Bézier Curve with Single Shape Parameter J. Basc. Appl. Sc. Res., (3541-546, 01 01, TextRoad Publcaton ISSN 090-4304 Journal of Basc and Appled Scentfc Research www.textroad.com The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Heuristic Algorithm for Finding Sensitivity Analysis in Interval Solid Transportation Problems

Heuristic Algorithm for Finding Sensitivity Analysis in Interval Solid Transportation Problems Internatonal Journal of Innovatve Research n Advanced Engneerng (IJIRAE) ISSN: 349-63 Volume Issue 6 (July 04) http://rae.com Heurstc Algorm for Fndng Senstvty Analyss n Interval Sold Transportaton Problems

More information

Turbulence classification of load data by the frequency and severity of wind gusts. Oscar Moñux, DEWI GmbH Kevin Bleibler, DEWI GmbH

Turbulence classification of load data by the frequency and severity of wind gusts. Oscar Moñux, DEWI GmbH Kevin Bleibler, DEWI GmbH Turbulence classfcaton of load data by the frequency and severty of wnd gusts Introducton Oscar Moñux, DEWI GmbH Kevn Blebler, DEWI GmbH Durng the wnd turbne developng process, one of the most mportant

More information

A Local Variational Problem of Second Order for a Class of Optimal Control Problems with Nonsmooth Objective Function

A Local Variational Problem of Second Order for a Class of Optimal Control Problems with Nonsmooth Objective Function A Local Varatonal Problem of Second Order for a Class of Optmal Control Problems wth Nonsmooth Objectve Functon Alexander P. Afanasev Insttute for Informaton Transmsson Problems, Russan Academy of Scences,

More information

Module 3: Element Properties Lecture 1: Natural Coordinates

Module 3: Element Properties Lecture 1: Natural Coordinates Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

Statistical Energy Analysis for High Frequency Acoustic Analysis with LS-DYNA

Statistical Energy Analysis for High Frequency Acoustic Analysis with LS-DYNA 14 th Internatonal Users Conference Sesson: ALE-FSI Statstcal Energy Analyss for Hgh Frequency Acoustc Analyss wth Zhe Cu 1, Yun Huang 1, Mhamed Soul 2, Tayeb Zeguar 3 1 Lvermore Software Technology Corporaton

More information

Psychology 282 Lecture #24 Outline Regression Diagnostics: Outliers

Psychology 282 Lecture #24 Outline Regression Diagnostics: Outliers Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.

More information

Supporting Information

Supporting Information Supportng Informaton The neural network f n Eq. 1 s gven by: f x l = ReLU W atom x l + b atom, 2 where ReLU s the element-wse rectfed lnear unt, 21.e., ReLUx = max0, x, W atom R d d s the weght matrx to

More information