Full Order Observer Controller Design for Two Interacting Tank System Based on State Space Approach

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1 Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: Volum 6, Issu 7, July 07 ISSN Full Ordr Obsrvr Controllr Dsign for Two Intracting Tank Systm Basd on Stat Spac Approach Amitava Biswas, Garg Chakraborty Dpartmnt of Applid Physics, Calcutta Univrsity, 9, A. P.C oad, olkata , INDIA Dpartmnt of Applid Physics, Calcutta Univrsity, 9, A. P.C oad, olkata , INDIA ABSTACT Th control of liquid lvl in tank systm and flow btwn tanks is main problms in procss industris. Thus, th control of liquid lvl in tank systm and flow btwn tanks which is must b controlld. Th control of lvl of tank in th intracting systm is major task. Th main objctiv of this papr is to rmin th mathmatical modl of a coupld tank systm which will b usful for dsigning and to implmnt full-ordr obsrvr using th softwar packags for computr aidd control systm dsign in MATLAB. A stat fdback gain matrix is dsignd for th intracting tank systms with th hlp of polplacmnt tchniqu. Obsrvr stimation rrors ar prsntd by choosing th obsrvr(s) initial conditions. Th proposd study of this obsrvrs and obsrvr basd controllrs may find application in numrous nginring and scintific industrial problms. ywords: Liquid lvl systm, stat spac, stat obsrvr, pol-placmnt. INTODUCTION Liquid lvl control [] is ndd in various industrial applications,.g., in food procssing, watr purification systms, filtration, pharmacutical industris, tc. is wll known that th stat coupld two-tank liquid lvl systm. Th stat spac approach is a gnralizd tim domain mthod for modling [], analyzing and dsigning a wid rang of control systms and is particularly wll suitd to digital computational tchniqu. Th stat obsrvrs [] ar usd not only for th purpos of fdback control, but also in thir own right to obsrv stat variabls of a dynamic systm [3], which can b an xprimnt in progrss whos stat has to b monitord at all tims. In this papr w shall prsnt dsign mthodology of closd loop control in stat spac domain using stat fdback gain matrix [4]. This concpt of tchniqu is calld pol placmnt tchniqu. It will b shown that if systm is compltly stats controllabl thn pols of closd loop systm may b placd at dsird location with th hlp fdback gain matrix. In pol placmnt tchniqu, an assumption is mad that all stat variabls ar availabl for fdback to dsign th control systm. In practic, all stats ar not availabl during masurmnt. So it may b rquird to stimat unavailabl variabls. So th procss of stimation of unmasurd stats is calld obsrvation [5]. To build an obsrvr gain matrix is rquird. So th stat obsrvr gain matrix can b dsignd if th systm is compltly sat obsrvabl [6, 7]. Aftr dsigning th obsrvr it will b shown that th rspons of masurd and unmasurd sat of variabls. A control algorithm is implmntd in MATLAB softwar.. MATHEMATICAL MODELING OF TWO TAN INTEACTING LEVEL POCESS Th procss consisting of two intracting liquid tanks shows in Fig.. Th hight of th liquid lvl is h (cm) in tank- and h (cm) is tank-. Volumtric flow into tank- is q in (cm 3 /sc), th volumtric flow rat from q (cm 3 /sc), and th volumtric flow rat from tank- is q o (cm 3 /sc). Cross sctional ara of tank- is A (cm ) and ara of tank- is A (cm ). and ar th rsistanc paramtr (valv) in flow lin shown in Figur. Th diffrntial quations rlatd to two intracting tank systm ar givn blow: Volum 6, Issu 7, July 07 Pag 34

2 Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: Volum 6, Issu 7, July 07 ISSN (i) For tank-: A q q in Assum linar rsistanc to flow, Figur Two intracting tank systm h h q A qin h h h h q in () A A A Tim constant for tank- is A (ii) For tank-: A q q o h h h A h h A A Tim constant for tank- is A Aftr rarranging th quations objctiv transfr is H H A A s ( A A A A ) s () (3) Tabl : Spcification of two tank intracting procss and xprimntal rsult takn from ral tim systm Condition Final stat aftr stp chang Flow in LPH Hight of tank- (mm) Hight of tank- (mm) Ara of tank- = ara of tank- Tim constant of tank- Tim constant of tank cm = 0.0m 9 s 3.6 s Initial stat cm = 0.0m 9 s 3.6 s Now, Equations () and () bcoms (00 50) mm (00 00) LPH (50 30) mm (300 00) LPH 900 s / m 360 s / m Volum 6, Issu 7, July 07 Pag 35

3 Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: Volum 6, Issu 7, July 07 ISSN TWO TAN INTEACTING LEVEL POCESS USING STATE SPACE ANALYSIS arranging th quations () and (), w rprsnt th diffrntial quations in to stat spac quation. W choos th stat variabl x h and x h y (4) Hnc, stat modl of two lvl intracting tanks is drivd from quations (), () and (4), th stat quation is Th output quation is dx A A x A in dx x q 0 A A y 0 x x x (6) (5) 4. STATE FEEDBAC GAIN MATIX DESIGN FO TWO INTEACTING TAN POCESS Considr a control systm whr x = stat vctor (n-vctor) u = control signal (scalar) A = n n constant matrix B = n constant matrix W shall choos th control signal to b dx Ax Bu (7) u x (8) This mans that th control signal is rmind by an instantanous stat. Th ( n) matrix is calld th stat fdback gain matrix. Th stps involvs with finding stat matrix ar discussd blow: Stp-: Chck controllability condition for th givn systm: A ; B and C 0 Hr controllability matrix 00 Qc B AB W find that Qc 0 and rank of th stat controllabl tst. Qc 0 so th systm is compltly controllabl and it is stabl. k k and quating SI A B with th Stp-: By mntioning th dsird stat fdback gain matrix as dsird charactristic quation, w obtain SI A B 0 s k k 0 0 s s 0.00k 0.00k 0.7 s s k s k k 0 ( ) ( ) 0 (9) Volum 6, Issu 7, July 07 Pag 36

Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: www.ijaim.org Email: ditor@ijaim.

4 Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: Volum 6, Issu 7, July 07 ISSN Stp-3: Suppos th dsird location of closd loop pols ar at s = -0 and s = -00. Hnc, dsird charactristics quation is ( s 0)( s 000) 0 s 0s (0) Stp-4: Comparing quations (9) and (0), w gt k.095 and k Th fdback gain matrix is Ackrman s formula is usd to th stat gain matrix to writ th MATLAB program and find th stat fdback gain matrix as Th simulink diagram of th closd-loop control of two intracting tanks systm with u x is shown Figur and th rspons of th systm with initial condition is shown in Figur 3. Figur Closd-loop control of two intracting tank systm with u x. Figur 3 spons curv of closd-loop control systm with initial condition [0 ]. 5. DESIGN OF FULL ODE STATE OBSEVE GAIN MATIX FO TWO INTEACTING TAN POCESS In th pol placmnt tchniqu to th dsign of control systms, w assumd that all stat variabls ar availabl for fdback. In practic, all stat variabls ar not availabl for fdback masurmnt [9]. Mthods ar availabl to stimat immasurabl stat variabls without a diffrntiation procss. Estimation of immasurabl stat variabls is commonly calld obsrvation. A dvic (or a computr program) that stimats or obsrvs th stat variabls is calld a stat obsrvr [0], or simply an obsrvr. If th stat obsrvr obsrvs all stat variabls of th systm, rgardlss of whthr som stat variabls ar availabl for dirct masurmnt, it is calld a full-ordr stat obsrvr [, ]. Assum that th stat x is to b approximatd by th stat x of th dynamic modl of obsrvr to b x [ A C] x Bu y () whr x is th stimatd stat and Cx is th stimatd output. Th matrix obsrvr rror quation is dfind by x x [ A C]( x x) [ A C] () Volum 6, Issu 7, July 07 Pag 37 is calld obsrvr gain matrix. Hnc

5 Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: Volum 6, Issu 7, July 07 ISSN whr x x is rror vctor and th dynamic bhavior of this vctor dpnds upon ign valus of A C. Th stps involvs with finding stat matrix ar discussd blow: Stp (a): Chck obsrvability condition for th givn systm. Th obsrvability matrix T T T Q [ C A C ] W find that Q 0 and thrfor, rank of obsrvability matrix Q. So th systm is compltly stat obsrvabl. Stp (b): By dfining th dsird stat fdback gain matrix dsird charactristic quation, w obtain as SI A C 0 0 s s s s s s 0 and quating (0.49 ) ( ) 0 (3) SI A C with th Stp (c): Dsign a full-ordr stat obsrvr, assuming that th systm configuration is idntical to that shown in Figur 4. Assum that th dsird Eign valus of th obsrvr matrix ar s j and s j. Th dsign of th stat obsrvr rducs to th rmination of an appropriat obsrvr gain matrix. Figur 4 Closd-loop control of two intracting tank systm with full ordr obsrvr. Hnc dsird charactristics quation is ( s j)( s j) 0 s s 5 0 (4) Stp (d): Comparing quation (3) and (4), w gt and.5 Ackrman s formula is usd to th stat gain matrix to writ th MATLAB program. Th stat fdback gain matrix is Th simulink diagram of th closd-loop control of two intracting tanks systm with full ordr obsrvr is shown Figur 4 and th rspons of th systm and rror with initial condition [0 ] ar shown in Figur 5(a) and (b). Volum 6, Issu 7, July 07 Pag 38

6 Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: Volum 6, Issu 7, July 07 ISSN Figur 5(a) spons curv of actual output and stimatd output of systm with full ordr obsrvr to initial condition [0 ]. Figur 5(b) spons curv of rror of actual output and stimatd output of systm with full ordr obsrvr with initial condition [0 ]. 6. OBSEVE BASED CONTOLLE DESIGN FO TWO INTEACTING TAN POCESS Th transfr function of obsrvr basd controllr is givn by U ( s) [ SI A C B] Y ( s) This obsrvr basd controllr rprsntd by th block diagram as shown in Fig.6. Figur 6 Block diagram of th obsrvr basd controllr. Th MATLAB program [8] is usd to find th transfr function of obsrvr controllr for diffrnt valu of and. Th program is givn blow: Cas-A: W taking closd loop pol location and obsrvr pol location ar J [ 0 00] and L [ j j] which givs th valu of and Hnc, th Transfr function of obsrvr controllr is 73. s s^ +. s + 3 This controllr transfr function has pols and zros location all ar lft hand sid of S plan. So this controllr is stabl controllr. Cas-B: W taking closd loop pol location and obsrvr pol location ar J [ j j] and L [ 40 00] which givs th valu of and Transfr function: s s^ +.99 s This controllr Transfr function has pol location P, j.8960; j That mans thy situatd at imaginary axis of s-plan. So this controllr is unstabl controllr. This typ control systm is not accptd. Volum 6, Issu 7, July 07 Pag 39

Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: www.ijaim.org Email: ditor@ijaim.org Volum 6, Issu 7, July 07 ISSN 39-4847 6.

Aftr gtting th transfr function of obsrvr basd controllr, transfr function of controllr is implmntd in simulink block as shown in Figur 7.

7 Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: Volum 6, Issu 7, July 07 ISSN spons curv of Obsrvr Basd Controllr with Two Intracting Tank Procss In this sction ffct of stabl and unstabl obsrvr basd controllr on systm will b discussd. Aftr gtting th transfr function of obsrvr basd controllr, transfr function of controllr is implmntd in simulink block as shown in Figur 7. Figur 7 Closd loop control systm with obsrvr basd controllr By using simulink a closd loop systm is dsignd. Whr procss is connctd with controllr and usd stp as input. Hr w just compar th systm rspons with rspct to diffrnt transfr function of obsrvr basd controllr as shown in Figur 8(a) and (b). Figur 8(a) spons of Closd loop control systm with stabl obsrvr basd controllr Figur 8(b) spons of Closd loop control systm with unstabl obsrvr basd controllr 6.Comparison of spons curv to Initial condition with Dsignd and valus Cas-A: W taking of and shown in Figur 9. Cas-B: W taking and Figur 0.. Th rspons curv of th systm is. Th rspons curv of th systm is shown in Figur 9 : spons curv for cas-a. Figur 0 : spons curv for cas-b. Volum 6, Issu 7, July 07 Pag 40

Th stat fdback gain matrix () and th obsrvr gain matrix ( ) has bn rmind by dirct substation mthod and vrifid using MATLAB programming [8].

8 Intrnational Journal of Application or Innovation in Enginring & Managmnt (IJAIEM) Wb Sit: Volum 6, Issu 7, July 07 ISSN CONCLUSION Stat modl of intracting two-tank liquid lvl systm has bn dvlopd. Th stat fdback gain matrix () and th obsrvr gain matrix ( ) has bn rmind by dirct substation mthod and vrifid using MATLAB programming [8]. Aftr dsigning a obsrvr and gtting th rspons from obsrvr it was that stimatd variabls [7] taks quit mor tim to rach at its stady valu than actual variabls. Comparison study has mad on obsrvr basd controllr has dsign [0] by using diffrnt valu of and. Two obsrvr basd controllr has dsign for intracting two tanks systm. Cas-A typ obsrvr is mor accptabl bcaus it is stabl in opn loop as wll as closd loop systm. frncs []. Ogata, Modrn Control Enginring, 3rd dition, Prntic Hall, Nw Jrsy, 997. []. Dorf and. Bishop, Modrn Control Systms, Parson Education, Uppr Saddl ivr, Nw Jrsy, 005. [3] M. Darouach, Existnc and dsign of functional obsrvrs for linar systms. IEEE Transactions on Automatic Control, vol. 45(5), pp , 000. [4] C. Miral, A. umar, Dsigning a Controllr for Two Tank Intracting Systm, Intrnational Journal of Scinc and sarch, vol. 4, Issu 5, May 05. [5] B. Fridland, Control Systms Dsign: An Introduction to Stat Spac Mthods, Dovr Publications, 005. [6] Li Qi, Fang Yanjun, Song Jizhong,Wang Ji, Th Application of Fuzzy Control in Liquid Lvl Systm, Intrnational Confrnc on Masuring Tchnology and Mchatronics Automation, IEEE, ISSN : , 00. [7] D. Lunbrgr, Obsrving th stat of a linar systm, IEEE Transactions on Military Elctronics, Vol. 8, 74-80, 964. [8] D.G. Lunbrgr, Obsrvrs for multivariabl systms IEEE Transactions on Automatic Control, vol. (), pp , 966. [9] D.. Chaturvdi, Modling and simulation of systm using MATLAB and Simulink, CC Prss Taylor and Francis group Boca aton London Nw York 00. [0] M. Darouach, M. Zasadzinski, & S.J. Xu, Full-ordr obsrvrs for linar systms with unknown inputs IEEE Transactions on Automatic Control, 39(3), , 994. [] S. P. Bhattacharyya, Obsrvr dsign for linar systms with unknown inputs IEEE Trans. Automatic Control, Vol. AC-3, No. 3, pp [] S. Hui, S.H. Zak, (b). Obsrvr dsign for systms with unknown inputs, Intrnational Journal of Applid Mathmatics and Computr Scinc, 5(4), , 005. AUTHOS Amitava Biswas rcivd th B.S. dgr in Elctrical Enginring from Maulana Azad Collg of Tchnology (EC, Bhopal) in 999 and M.S. dgr in Elctrical Enginring from Univrsity of Calcutta in 00, rspctivly. H rcivd Ph.D.(Tch) dgr from Univrsity of Calcutta in 05. H startd his carr as a Lcturr (July 00) in th JIS Collg of Enginring. Dr. Biswas is prsntly attachd with th Dpartmnt of Applid Physics, Univrsity of Calcutta as Assistant Profssor sinc 06. Apart from taching, h is ngagd in rsarch in th fild of control nginring and application of picwis constant basis functions in systm and control. H is spcially working on picwis linar orthogonal functions and thir suitability in analyzing and synthsizing control systms and solving rlatd diffrntial quations. Till now, h has publishd tn (0) rsarch paprs in national and intrnational journals. Ms. Garg Chakraborty rcivd th B.S. dgr in Instrumntation and Control Enginring from Acadmy of Tchnology in 00 and M.S. dgr in th Dpartmnt of Applid Physics from Univrsity of Calcutta in 0, rspctivly. During 0-06, sh workd as an Assistant profssor in th dpartmnt of Instrumntation Enginring of Suprm nowldg Foundation. Prsntly sh workd as gust lcturr of Univrsity of Calcutta and Univrsity of alyani. Sh has publishd two rsarch paprs in intrnational confrnc. Volum 6, Issu 7, July 07 Pag 4

Rotor Stationary Control Analysis Based on Coupling KdV Equation Finite Steady Analysis Liu Dalong1,a, Xu Lijuan2,a

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