PREDICTIVE CONTROL BY DISTRIBUTED PARAMETER SYSTEMS BLOCKSET FOR MATLAB & SIMULINK

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1 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, Slovak Unversty of Technology, Bratslava Abstract Smulaton of model predctve control of castng mold temperature feld s presented. Temperature felds of the mould have been modeled and studed usng a fnte element method based software package, ProCAST, and numercal models n the form of lumped-nput/dstrbuted-output systems have been obtaned. For model predctve control synthess the Model Predctve Control Toolbox was used. Smulaton have been realzed usng the Dstrbuted Parameter Systems Blockset for MATLAB & Smulnk, whch s a thrd-party MathWorks product. 1 Introducton Most of the dynamcal systems analyzed n engneerng practce are dstrbuted parameter systems (DPS) gven by numercal structures on complex-shape 3D defnton domans. Dynamcal characterstcs obtaned by numercal methods offer wde possbltes for control of systems as DPS. In ths paper an enhancement of DPS Blockset, the Predctve Control Synthess Block s presented. Predctve control synthess s decomposed nto tme and space components. Space control problems are beng solved by approxmaton methods, problems concernng control n tme are based on predctve control synthess methods for lumpng systems. For software mplementaton, functons and methods from the Model Predctve Control Toolbox (MathWorks) are used. In ths paper, as a demonstraton of acheved results, predctve control of preheatng of a complex-shape castng de s presented. Temperature control n the castng de could have consderable mplcaton for the elmnaton of defects that may arse n the process. For the analyss of castng process dynamcs as DPS, benchmark of the castng processes was desgned. It has to be able to embrace typcal effects and crtcal moments of castng. Temperature felds of the castng mould have been modeled and studed usng ProCAST, whch s a FEM based software package. For control synthess purpose, numercal models n the form of lumped-nput/dstrbuted-output systems are made up by means of fnte element analyss. 2 DPS/LDS dynamcs and control Dstrbuted parameter systems, whose state or output varables X(x,y,z,t)/Y(x,y,z,t) are dstrbuted varables, or felds of varables, are n practce very frequently found n the form of lumped-nput/dstrbuted-output systems (LDS), (Hulkó, 1987, 1998) see Fg. 1. Fgure 1: DPS n the form of LDS: {U (t)} =1,n - lumped nput varables, Y(x,y,z,t) - dstrbuted output varable For ths DPS/LDS representaton t s possble to obtan dscrete transfer functons (1) between -th nput varable and the correspondng partal dstrbuted output varable at pont x = (x,y,z ). In space dependence there are reduced steady-state transent step responses (2): SH x, z (1) { ( )} = 1, HR ( x, ) n { } = 1, H (2) n

2 Decomposton of dynamcs outlned above enables decomposton of the control synthess to tme and space control problems n DPS feedback control loop, see Fg. 2. Fgure 2: DPS feedback control loop: 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, Y x,t dstrbuted controlled varable, W ( x, k) control varable, x, t E x, k control error K tme/space samplng, ( ) V ( ) dsturbance varable, ( ) In the SS block, approxmaton of dstrbuted control error E ( x, k ), on the set of reduced steady-state dstrbuted step responses { H HR ( x, ) }, s solved. n n mn E ( x, k ) E ( k ) HHR ( x, ) = E ( x, k ) E ( k ) H HR ( x, ) (3) E = 1 = 1 { } Next, the control errors vector E ( k ) = E ( k ) components { E ( k )} are fed through the nputs of controllers { ( )} varables ( ) enters nto the block TS, where the vector R z and the sequence of control U k s generated. Tunng of controllers s sutable to do accordng dynamcs gven by transfer functons (1). 3 Basc concept of Model Predctve Control Representaton of DPS n the form of LDS enables to use varous methods of control synthess well-known n lumpng systems control, ncludng the Model Predctve Control (MPC) as well. The MPC concept for sngle-nput sngle-output (SISO) systems s usually supposed acordng to Fg. 3. Fgure 3: Block Dagram of a SISO Model Predctve Control The man objectve s to hold a sngle output y, at a reference value (or setpont) r by adjustng a sngle manpulated varable (or actuator) u. Model Predctve Controller s desgned to acheve the control objectve. It requres a plant model, whch defnes the mathematcal relatonshp between the plant nputs and outputs. The controller uses t to predct plant behavor.

3 A Model Predctve Controller automates a target system (the plant) by combnng a predcton strategy and a control strategy. An approxmate lnear plant model provdes the predcton. The control strategy compares predcted plant sgnals to a set of objectves, then adjusts avalable actuators to acheve the objectves whle respectng the plant constrants. Such constrants can nclude the physcal lmts of the actuator, boundares of safe operaton, and lower lmts for product qualty. 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 defnton domans, see Fg. 4, or web ste The HLDS block models controlled dstrbuted parameter systems as lumped-nput/dstrbutedoutput systems wth zero-order hold unts. 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 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 gves a gude for arrangement and settng dstrbuted parameter control loops n step-by-step operaton, by means of fve model examples on 1D-3D wth default parameters,. Fgure 4: Lbrary of DPS Blockset for MATLAB & Smulnk

4 5 Modelng of temperature felds of the mould n castng process Temperature feld of the mould n the castng process s a typcal case of DPS, where the nput/output relaton s useful to be modeled as LDS. The benchmark castng plant s bult of the steel mold wth water-cooled copper nserts (chlls), electrc heatng elements (lumped nputs) and thermocouples, see Fg. 5. Fg. 5. Steel mold of the benchmark castng plant Temperature dstrbuton n the mold n the defnton doman Ω E3 s modeled by PDE of parabolc type n the form: T ( x, t) 2 a T ( x, t) = u ( x, t) (4) t where a = λ ρc s temperature conductvty (m 2.s -1 ). For lumped nput varables { } 1,5 { Ω } = 1,5 all necessary lumped and dstrbuted responses were computed by FEM n the software envronment ProCAST. u = n the form of step functons, whch affect on sub-domans 6 Smulaton of Model Predctve Control of temperature feld MPC controller desgn procedure outlned above s now appled for the control of temperature feld of the mold n the castng process. In the MATLAB & Smulnk envronment by means of both, the DPS Blockset and Model Predctve Control Toolbox, dstrbuted parameter system of MPC s set up, see Fg. 6. It s the feedback control loop, where the block MPC_DPS ncludes both, tme and space control synthess, see Fg 7. In ths case, the control system conssts of fve sngle parameter control loops, where MPC controllers are used, see Fg.8. MPC synthess controllers were adjusted n order to assure aperodc runnng of the quadratc norm of dstrbuted control error. Weghts of manpulated varables, manpulated varables rate and output varables were assumng 0.0, 0.1 and 1.0. Results of the MPC control process are on Fgures 10, 11.

5 Fgure 6: MPC DPS loop n DPS Blockset for MATLAB & Smulnk Fgure 7: Internal structure of MPC_DPS block Fgure 8: Block of MPC Controllers

6 Fgure 9: DPS Space synthess Fgure 10: Dstrbuted reference varable W(x,y,t) and controlled varable Y(x,y,t), t Fgure 11: Control varables U (k) and quadratc norm of dstrbuted control error E(k) 7 Concluson Methodcal approach presented n the paper demonstrates smple possbltes, how to explot dstrbuted dynamcal characterstcs, obtaned by numercal FEM analyss of systems on complex defnton domans for MPC controller synthess of DPS. The DPS Blockset for MATLAB & Smulnk and Model Predctve Control Toolbox provde effcent software for ths knd of problems.

7 ACKNOWLEDGEMENT The work has been fnancally supported by the ESF project JPD /NP1-047 PhD students for Modern Industral Automaton n SR, code No and by the Slovak Scentfc Grant Agency VEGA project Control methods of Dstrbuted Parameter Systems gven on Complex Defnton Domans wth Demonstratons n MATLAB & Smulnk (grant 1/2051/05). References [1] 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, [2] Hulkó, G. et al. Modelng, Control and Desgn of Dstrbuted Parameter Systems wth Demonstratons n MATLAB. Publshng House of STU, Bratslava, [3] Hulkó, G., Belavý, C. PID control of dstrbuted parameter systems. The IFAC Conference on CONTROL SYSTEMS DESIGN '03, Bratslava, [4] Hulkó, G. et al. 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, [5] Hulkó, G. et al. Dstrbuted Parameter Systems Blockset for MATLAB & Smulnk thrd-party MathWorks product. Bratslava Natck, MA, USA, [6] Hulkó, G. et al. (2005). Engneerng methods of control desgn for dstrbuted parameter systems. Preprnts of the 16th IFAC World Congress, Prague, Contact nformaton: prof Ing. Gabrel Hulkó, DrSc. Insttute of automaton, measurement and appled nformatcs Faculty of Mechancal Engneerng, Slovak Unversty of Technology Bratslava Nám. slobody 17, Bratslava, Slovak Republc Tel , fax e-mal: gabrel.hulko@stuba.sk