Flow and Temperature Control of a Tank System by Backstepping Method

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1 Flw nd eperture Cntrl f nk Syste by ckstepping Methd JinÐHu She ky Engineering University, Hchiji, ky, Jpn, she@cc.teu.c.jp Hirshi Odji Advnced Fil echnlgy Inc., Musshin, ky, Jpn, PXW@niftyserve.r.jp Hirshi Hshit ky Engineering University, Hchiji, ky, Jpn, hshit@cc.teu.c.jp Minru Higshiguchi ky Engineering University, Hchiji, ky, Jpn, hgsgh@cc.teu.c.jp ASAC he bckstepping ethd ws pplied t tnk syste t cntrl the flw nd teperture t the exit. A bckstepping cntrl lw nd rbust bckstepping cntrl lw were develped fr the plnt withut nd with uncertinties, respectively. he designed cntrl syste is syptticlly stble fr the plnt withut uncertinties nd glblly unifrly bunded fr the plnt with uncertinties.. INODUCION Over the pst few yers, cnsiderble nuber f studies hve been devted t new cntrl design ethdlgy: bckstepping ([]. Unlike feedbck lineriztin, bckstepping cn vid the cncelltin f useful nnlinerities. S, it ffers the prspect f re prcticble nnliner cntrl lw. his pper describes the pplictin f the bckstepping cntrl strtegy t tnk syste t cntrl the flw nd teperture t the exit. Mtheticl dels f the syste re first derived. hen, bckstepping cntrller is designed fr the ninl plnt. Hwever, there re usully se uncertinties in the theticl del f the plnt. chieve rbustness fr the cntrl syste, rbust bckstepping cntrller is designed by iprving the bckstepping cntrl lw t gurntee glbl unifr bundedness. Nenclture i te f inflw f the wter ( / s te f utflw f the wter ( / s i eperture f the inflw (K eperture f the utflw (K Air teperture (K h Height f the wter level ( A Crss-sectinl re f the tnk ( Φ i Heter supplied (W Euivlent het resistnce f the tnk (K/W 5. Dischrge cfficient f vlve ( / s ρ Density f wter ( kg / c p Specific het f wter ( J/ kg/ K. MAHEMAICAL MODEL OF HE ANK SYSEM he tnk syste studied here is shwn in Fig.. In this syste, cld wter is sent t the tnk fr wterwrks. he wter is heted in the tnk, nd then sent ut. he syste hs tw cntrl inputs: the rte f inflw nd the heter supply. he rte nd teperture f the inflw, the rte f the utflw, nd the teperture f the wter in the tnk re knwn. he rte f utflw is given by nd dh i =, A ( h ( > 0. ( Let Φ be the het tht the wter in the tnk pssesses, Φ be the het tht the utput wter tkes wy, Φ S be the het relesed t the ir nd Φ c be the het tht the inflw brings in, nd ssue tht the teperture f the wter in the tnk is unifr. hen we hve Φ =Φ Φs Φc Φi. (

2 i, i h, F i FIGUE : he tnk syste. = =, (0 where nd re the errrs between the rel nd the desired utputs. If we substitute (0 int (9 nd trnsfr the cntrl inputs int i = ( ( u Φ i = {( cpρ } ( cpρi uφ, ( nd A c d, (5 = c ρ, (6 = = h [, A] [ ], (7 = c ρ. (8 p Φ = ρ Φ p Φ s Φ c p i i Fr the bve reltinships, the del f the tnk cn be written s d = A A i d c pρ = ( cpρi i Φ i, where = /( Ac p ρ. (9 It is cler fr (9 tht the tnk syste is tw-input tw-utput nnliner syste. If we let nd be the desired utputs, then the cntrl bjective is t ke the utflw nd teperture trck the. In the design f such cntrl systes, the flw sub-syste nd the teperture sub-syste re usully cnsidered seprtely, nd the crreltin between the cntrlled utputs is ignred. Liner cntrl thery is inly used fr cntrl syste design. In prticulr, PID cntrllers re generlly designed fr ech linerized sub-syste ([]. In this study, we used the MIMO nnliner del directly s s t tke the nnlinerties f the plnt nd the crreltin between the cntrlled utputs int ccunt. Decpsing the utputs yields then the plnt del beces d = A u d c pρ = ( cpρi u u ( If we let Φ. ( x:= [ ], ( nd rewrite the del in the fr: then dx / = f ( x g( x u g( x uφ, ( 0 f( x = cpρ ( 0 g ( x = A c g ( x = pρ i (. (5 ( is the ninl del f the plnt. Since it is hrd t btin n exct del f rel plnt, it is useful t include uncertinties in the plnt del s fllws: with dx / = f ( x g( x u g( x uφ =, A ( (6

3 δ, δ. (7. DESIGN OF ACKSEPPING CONOL LAWS We first cnsider plnt withut uncertinties. If Lypunv functin is defined s Vx ( := x x=, (8 it is cler tht the fllwing cntrl lw u x k g x g x u Φ = α ( Φ x = : [ ( ( ] α ( k A k = k i k ( [ ( ] kes dv (9 = ( f gα gαφ negtive definite if k, k > 0 (0 re chsen. uφ = k{ vφ ( [ k i k ( ]} c ρ{ k ( p i k ( } v k c ρ ( p k { cpρi( v vφ} ( (- gurntees the glbl sypttic stbility f the syste ( t (, = (, fr ny k, k > 0. Prf: dv / = dv / ( v α ( u dα / ( v α ( u dα / Φ Φ Φ Φ = ( f gα gαφ gz gzφ α z[ u ( f gv gvφ ] αφ zφ[ uφ ( f gv gv Φ ]. S, if the cntrl lw is chsen t be Nw, bckstepping the plnt ( gives dx / = f ( x g ( x v g ( x v d v u. v = u Φ Φ Φ A new Lypunv functin is selected: ( α u =k( v α ( f gv gvφ x x g A k v k k v A cpρi = ( A kk A k k v cpρi = ( (, V : = V( x z z Φ, ( where z : = v α ( x. ( zφ : = vφ αφ( x hen we hve the fllwing le. Le : he cntrl lw A u = ( kk A ( k k v (- cpρi, αφ uφ =k( vφ αφ ( f gv gvφ x x g = k{ vφ ( [ k i k ( ]} c ρ{ k ( k ( } v p i kc pρ( k { cpρi( v v } ( then dv Φ = ( f gα g α k z k z < Φ Φ 0. ht gurntees the glbl sypttic stbility t (, = ( 00,, i.e. (, = (,. (QED,

4 sed n the bve bckstepping cntrl lw, rbust bckstepping cntrl lw is derived fr plnt with uncertinties (6: Le : he cntrl lw αφ α zφ{ uφ ( f gv gvφ} zφ Φ. Using Yung's Ineulity A u = ( kk A ( k k v cpρi δ k k sgn[ z ], uφ = k{ vφ ( [ k i k ( ]} cpρ{ ki( k( } v kcpρ( k { cpρi( v vφ} ( k sgn[ zφ ]{( cpρki k( δ kδ } (5 αβ λα λ β, where ( αβ, nd l is ny psitive nuber, yields = A ( δ δ ( 6A ( δ δ. 6A S, if the cntrl lw is chsen t be u α =k ( v α ( f g v g vφ g gurntees the glbl unifr bundedness f the syste: dx / = f ( x g( x v g( x vφ d v u v = (6 u Φ Φ t (, = (, fr ny k k, k > 0 nd k, k > 0. Prf: Defining k, > 0, α α α : = ( α sgn( y : sgn( y sgn( y fr y y y nd = [ ] δ = δ δ A ( yields = [ ] dv / = dv / ( v α ( u dα / ( v α ( u dα / Φ Φ Φ Φ = ( f gα gαφ gz gzφ α α z{ u ( f gv gvφ } z then k A = ( kk A ( k k v cpρi δ k k sgn[ z ], sgn[ v α ] α δ αφ uφ =k( vφ αφ ( f gv gvφ x g k v Φ sgn[ Φ αφ] α δ = k{ vφ ( [ k i k ( ]} c ρ[ k ( p i k ( ] v k { cpρ( cpρi( v vφ} c k k sgn[ z ]( ( ρ Φ ( k ( δ kδ, dv p i = ( f gα g αφ k z k z Φ

5 α α sgn[ z V k z δ z ] x sgn[ ] ( k αφ sgn[ z α k z δ z Φ ] Φ x sgn[ Φ ] Φ ( k k kz ( ( k kz δ 6A δ. ht iplies tht dv Φ / is negtive when δ δ X > 6A, where X:= [ z zφ ] ζ nd ζ : = in( k, k, k, k, i.e. the stte X is glblly unifrly bunded. (QED. NUMEICAL EXAMPLE Let the preters f the tnk syste be A= (. 5 = ( / s = 000 ( K/ W = ( K i = ( K (7 nd ( 0 = ( / s ( 0 = ( K (8 he desired utputs re = ( / s = 8. 5 ( K (9 he siultin results re shwn in Figs. nd. In Fig., the ninl plnt is cntrlled by the cntrl lw (. he preters f the cntrller re k = k = k = k = (0 It cn be seen tht the designed cntrl syste is stble nd the utputs rech the desired vlues fter 000 inutes. FIGUE : Siultin results fr the ninl plnt. In Fig., the plnt is ssued t cntin uncertinties, nd nd re white nise with the bunds 5 = sin( t = 0. 0 sin( t ( he preters f the cntrller (5 re chsen t be (0 nd

6 5. CONCLUSIONS In this pper, theticl del f tnk syste is first derived, nd bckstepping cntrl lw is develped fr the ninl plnt. hen, bsed n the cntrl lw, rbust bckstepping cntrl lw is develped fr plnt with uncertinties. he vlidity f these cntrl lws is denstrted by siultins. EFEENCES. M. Krstic, I. Knellkpuls nd P. Kktvic: Nnliner nd Adptive Cntrl Design, Jhn Wiley & Sns, Inc., 995. N. Sud, PID Cntrl, Askur Shten, 99 k FIGUE : Siultin results fr plnt with uncertinties. = k =. ( 0 he siultin results shw tht the designed cntrl syste is glblly unifrly bunded nd the utputs cnverge t the desired vlues fter 000 inutes.