DESIGN OF CONTROL PROCESSES IN DPS BLOCKSET FOR MATLAB & SIMULINK

Transcription

1 DESIGN OF CONTROL PROCESSES IN DPS BLOCKSET FOR MATLAB & SIMULINK C. Belavý, G. Hulkó, M. Mchalečko, V. Ivanov Department of Automaton, Informatcs and Instrumentaton, Faculty of Mechancal Engneerng, Slovak Unversty of Technology, Bratslava Abstract In the paper desgn of control processes for class of dstrbuted parameter systems s presented. Controlled system s nterpreted as lumped-nput/dstrbuted-output systems wth dynamcs modelled by fnte element method. Schemes of dstrbuted parameter control loops are arranged n software envronment Dstrbuted Parameter Systems Blockset for MATLAB & Smulnk, whch s thrd-party MathWorks product. In concluson, results of control of a meltng glass temperature feld are shown. 1 Introducton Nowadays s pad consderable attenton to the numercal dynamcal analyss of dverse processes, machnes, apparatuses, as dynamcal systems gven over complex shape of defnton domans. New software systems based on fnte element method (FEM) have been developed to treat these problems. In ths sphere of applcatons emerges a new dscplne computatonal scence and engneerng. The felds of quanttes to be studed are often montored by camera methods or approprate sensor felds. In accordance wth systems and control theory, all these real systems are dstrbuted parameter systems (DPS). Ther state or output varables are felds of varables, or spatal dstrbuted varables. In practce, controlled DPS frequently are found as lumped-nput/dstrbutedoutput systems (LDS). In the paper frst basc concept of LDS dynamcs and control wll be outlned. Further basc features of the DPS Blockset and procedure of control synthess desgn of glass meltng temperature feld wth dynamcs gven by FEM wll be descrbed. In concluson, results of control of a meltng glass temperature feld as DPS/LDS are shown. 2 Basc concept of DPS/LDS dynamcs and control In general, DPS are systems whose state or output varables, X(x,y,z,t)/Y(x,y,z,t) are dstrbuted varables or felds of varables, where (x,y,z) s a vector n 3D. In the mathematcal theory DPS are nterpreted as systems whose dynamcs s descrbed by partal dfferental equatons (PDE), Lons (1971). In the nput-output relaton, PDE defne dstrbuted-nput/dstrbuted-output systems (DDS) between dstrbuted nput U(x,y,z,t) and dstrbuted output varables Y(x,y,z,t), at ntal and boundary condtons gven, Fg. 1. U(x,y,z,t) Y(x,y,z,t) Fgure 1: Dstrbuted-nput/ dstrbuted-output system It s well-known, that formulaton and soluton of control tasks for real DPS based on DDS s too dffcult. Much better representaton of DPS s n the form of LDS, (Hulkó,1987, 1998) see Fg. 2. Y(x,y,z,t) Fgure 2: Lumped-nput/dstrbuted-output system: {U (t)} =1,n - lumped nput varables, Y(x,y,z,t) - dstrbuted output varable

2 Dynamcs of LDS s decomposed to tme and space components. In the tme dependency, there are for example dscrete transfer functons (1) between -th nput varable and correspondng partal dstrbuted output varable at pont x = (x,y,z) for =1,n. SH x, z (1) { ( )} = 1, n In the space dependency there are e. g. reduced transent step responses n steady-state: ΗHR x, { ( )} = 1, n Decomposton of dynamcs enables also to decompose the control synthess to tme and space control tasks n dstrbuted parameter control loop, see Fg. 3. (2) V Fgure 3: Dstrbuted parameter dscrete feedback control loop at unt samplng perod HLDS - LDS wth zero-order holds {H } on the nput, CS - control synthess, TS - control synthess n tme doman, SS - control synthess n space doman, K - tme/space samplng, Y ( x,t) - dstrbuted x, k x, t E x, k - control error controlled varable, W ( ) - control varable, V ( ) - dsturbance varable, ( ) Let the goal of control be to ensure the steady-state control error to be mnmal,.e.: mn E x, = mn W x, Y x, = W x, Y x, = E x, where. s a norm approprately chosen. ( ) ( ) ( ) ( ) ( ) ( ) (3) Frst, the approxmaton problem (4) n the block of the Space Synthess (SS) s solved on the H HR x,, set of reduced steady-state dstrbuted step responses { ( )} = 1, n ( ) ( ) ( ) = ( ) ( ) ( ) mn E x, k E k H HR x, E x, k E k H HR x, E n = 1 = 1 n (4) { } Further, the control errors vector E( k) = E ( k) vector components { E( k) } are fed through the nputs of controllers { ( )} control varables ( ) of the tme part of the controlled dstrbuted parameter system dynamcs { SH ( x, z) } = 1, enters the Tme Synthess block (TS), where, R z and the sequence of U k s generated. Tunng of controllers s done - accordng to sngle components n Durng the control process, for k, we get the followng relaton and thus, the control task (3) s accomplshed. n mn E x, E H HR x, = E x, (5) E ( ) ( ) ( ) ( ) = 1.

3 3 Modelng of temperature feld of glass meltng furnace Meltng glass process s typcal case of DPS. In the nput/output relaton t s possble to model t as LDS. Lumped nputs set flow rates of heatng medum (earth gas and ar) nto seres of burners located on both sdes of the glass furnace above molten glass. Temperature feld of the molten glass on the defnton doman Ω E2 (n cross-secton of the meltng space) s dstrbuted output varable, see Fg.4. Dynamcs of a meltng glass process as DPS s modelled n FEMLAB (COMSOL Multphyscs) software envronment, whch offers very effcent tool for modellng and smulaton scentfc and engneerng problems based on PDE. Soluton of formulated models s realsed by FEM. Is well know, that FEM s a method for solvng an equaton by approxmatng contnuous quanttes as a set of quanttes at dscrete ponts, often regularly spaced nto a so-called grd or mesh. Because fnte element methods can be adapted to problems of great complexty and unusual geometry, they are an extremely powerful tool n the soluton of mportant problems n heat transfer, flud mechancs, and mechancal systems. Furthermore, the avalablty of fast and nexpensve computers allows problems whch are ntractable. For the recuperatve gas glass tank furnace wth the cross flame was made two-dmensonal space model of the meltng glass dynamcs based on FEM. In the defnton doman s supposed glass meltng n steady-state and heatng by the mxture of gas and ar, whch enter nto four pars of burners. Dstrbuton of temperatures n molten glass n the defnton doman Ω E 2 s modelled by PDE of parabolc type Y d ( c Y) + ay = f t (6) wth constants d, c, a and Neumann type boundary condtons n c Y + qy = g (7) ( ) where n s the outward unt normal and q = 0, g = 0. Fgure 4: Scheme of recuperatve gas glass tank furnace Modelng process n Femlab s started va Model Navgator, whch enables to select Applcaton Modes. In ths case Heat Transfer and Space dmenson 2D was chosen and GUI was actvated. There through the toolbar menu both defnton doman Ω E2 and subdomans were drawn, see Fg. 5, further n Physcs menu parameters of equaton system were defned. For ths model dstrbuted transent responses from each nput actuated on subdomans were obtaned, see Fg. 6.

4 Fgure 5: Defnton doman Fgure 6: Dstrbuted transent response n steady-state from each lumped nput

5 On dstrbuted transent step responses n steady-state, pont wth maxmal value was determned. Partal dstrbuted transent response n ths pont was dentfed, see Fg.7 and dscrete transfer functon (1) was obtaned for =1, 4 nputs. These transfer functon are used for tme synthess. Reduced transent step responses n steady-state (2) for space synthess have been also expressed, see Fg. 8. Fgure 7: Identfcaton of partal dstrbuted transent responses from each lumped nput Fgure 8: Reduced dstrbuted transent responses n steady-state from each nput

6 4 Dstrbuted Parameter Systems Blockset for MATLAB & Smulnk The Dstrbuted Parameter Systems Blockset s a blockset for use wth MATLAB & Smulnk for dstrbuted parameter systems and ther applcatons n modelng, control and desgn of dynamcal systems gven on complex 3D domans of defnton, see Fg Fgure 9: DPS Blockset for MATLAB & Smulnk The blockset features: Engneerng methods for DPS modelng, control and desgn DPS models based on lumped-nput/dstrbuted-output systems, tme/space analyss, synthess and desgn tools Dstrbuted parameter PID, algebrac, state space and robust control schemes DPS Wzard demonstrates n step-by-step operaton dstrbuted parameter control loops arrangement and settng procedures Sute of blocks and schemes for DPS control practcally n any feld of techncal practce Interactve Control Servce for support DPS control solutons va the nternet The block HLDS models controlled dstrbuted parameter systems as lumped-nput/dstrbutedoutput systems wth zero-order hold unts. The DPS Control Synthess provdes feedback to dstrbuted parameter controlled systems n control loops wth blocks for PID, algebrac, state space and robust control. The block DPS Input generates dstrbuted quanttes whch can be used as dstrbuted control varables or dstrbuted dsturbances, etc. DPS Dsplay presents dstrbuted quanttes wth many optons ncludng export to AVI fles. The block DPS Space Synthess performs space synthess as an approxmaton problem. The block Tutoral presents methodologcal framework for formulaton and soluton of dstrbuted parameter systems of control. The block Show contans

7 motvaton examples: Control of temperature feld of 3D metal body, Control of 3D beam of smart structure, Adaptve control of glass furnace and Groundwater remedaton control. The block Demos contans examples orented to methodology of modelng and control synthess. The DPS Wzard n step-by-step operaton, by means of fve model examples on 1D-3D wth default parameters, gves a gude for arrangement and settng dstrbuted parameter control loops. 5 Smulaton of control process In the MATLAB & Smulnk envronment by means of the DPS Blockset dstrbuted parameter system of control s arranged, see Fg. 10. It s feedback control loop, where block Dstrbuted Parameter control Synthess ncludes both tme part and space part of control synthess. In ths case control system consst of four sngle parameter control loops, where dscrete PID controllers are used. For tunng of controllers, dscrete transfer functons (1) are at dsposal. Smulaton results of control process of temperature feld of meltng glass are on Fgures 11, 12. Control objectve s certan temperature requred by technology wth 10% dvergence at most. Glass meltng process s long-term process wth slow dynamcs. It s energy demandng process. Therefore, tme of steady state acqurement s not as crucal as fuel consumpton. Wth proper set up of controllers can be acheved consderable cost savng. Here controllers were tuned n order to assure aperodc runnng of quadratc norm of dstrbuted control error. Fgure 10: DPS control loop n DPS Blockset for MATLAB & Smulnk Fgure 11: Dstrbuted reference varable W(x,y,t) and controlled varable Y(x,y,t), t

8 Fgure 12: Control varables U (k) and quadratc norm of dstrbuted control error E(k) 6 Concluson In today s engneerng envronment huge amount of money are nvested nto development. Devces have to operate more precsely and react more robustly than ever before. FEM and FEA (Fnte Element Analyss) has shown us ts strong ponts. Moreover, together wth computer technologes headway t s takng vald poston n engneerng feld. Methodcal approach presented n the paper demonstrates smple possbltes, how to explot of dstrbuted dynamcal characterstcs, obtaned by numercal analyss of systems on complex defnton domans for control of dstrbuted parameter systems. The DPS Blockset for MATLAB & Smulnk provdes block-orented effcent software for ths knd of tasks. ACKNOWLEDGEMENT Ths work has been carred out under the fnancal support of the Slovak Scentfc Grant Agency VEGA to project Control methods of Dstrbuted Parameter Systems gven on Complex Defnton Domans wth Demonstratons n MATLAB & Smulnk (grant 1/2051/05) and the European Socal Fund - project No. JPD / References [1] Lons, J. L Optmal control of systems governed by partal dfferental equatons. Sprnger- Verlag, [2] Hulkó, G Control of Dstrbuted Parameter Systems by means of Mult-Input and Mult- Dstrbuted - Output Systems. In: Preprnts of 10-th World Congress of IFAC, Munch, [3] Hulkó, G. et al. Modelng, Control and Desgn of Dstrbuted Parameter Systems wth Demonstratons n MATLAB. Publshng House of STU, Bratslava, [4] Hulkó, G., Belavý, C. PID control of dstrbuted parameter systems. The IFAC Conference on CONTROL SYSTEMS DESIGN '03, Bratslava, [5] Hulkó, G. et al. (2003). Interactve Web-based Learnng Servce for Control of Dynamcal Systems Gven on Complex Defnton Domans. he 6-th IFAC Symposum on Advances n Control Educaton. Oulu, Fnland, [6] Hulkó, G. et al. Dstrbuted Parameter Systems Blockset for MATLAB & Smulnk thrd-party MathWorks product. Bratslava Natck, MA, USA, Contact nformaton: Doc. Ing. Cyrl Belavý, PhD. Department of Automaton, Informatcs and Instrumentaton Technology Faculty of Mechancal Engneerng, Slovak Unversty of Technology Bratslava Nám. Slobody 17, Bratslava, Slovak Republc Tel , fax e-mal: cyrl.belavy@stuba.sk