Control ofa Nonlinear Coupled Three Tank System using Feedback Linearization

Transcription

1 Cntrl fa Nnlinear Cupled Three Tank System using Feedback Linearizatin Fatima Tahir Naeem Iqbal and Ghulam Mustafa Pakistan Institute f Engineering and Applied Sciences (PlEAS) Nilre Islamabad PAKISTAN fatimatahir.3m@gmail.cm naeem@pieas.edu.pk and gm@pieas.edu.pk Abstract - This paper presents the mdeling and cntrl f a nnlinear cupled three tank system (CTTS). In this wrk nnlinear state space dynamics f CTTS are derived. The bjective is t cntrl the level and tern perature f water in each tank. The cntrl is achieved using classical nnlinear feedback linearizatin technique. Input and utput scaling is dne accrding t the interface with a standard PLC. Design steps f nnlinear cntrl and input/utput respnses are shwn. Keywrds - Multi-Input-Multi-Output (MIMO) system Cupled Three-Tank System (CTTS) Feedback Linearizatin (FBL) Prgrammable Lgic Cntrl (PLC) I. INTRODUCTION Feedback linearizatin (FBL) is the mst widely used technique fr the cntrl f nnlinear systems. It is based n sme early wrk f Krener [1] and Brcket [2] accrding t which a large class f nnlinear systems can be exactly linearized by a cmbinatin f a nnlinear transfrmatin f state crdinates and a nnlinear state feedback cntrl law. A gd survey fthery can be fund in wrk dne by Isidri [3] Nijmeijer and Van Der Schaft [4] Sltine and Lie [5]. The central idea f this technique is the algebraic transfrmatin fthe nnlinear system dynamics int a fully r partly linear system s that cnventinal linear cntrl techniques can be applied. The main difference between feedback linearizatin and cnventinal linearizatin is that feedback linearizatin is achieved by exact state transfrmatins and feedback rather than by linear apprximatins fthe dynamics [5]. It results in cancellatin f the nnlinearities present in a nnlinear system in such a way that the clsed-lp dynamics fthe system is linear. This paper presents the cntrl f a nnlinear multi-inputmulti-utput (MIMO) system using the technique f feedback linearizatin by which the riginal nnlinear system can be transfrmed int a linear and cntrllable system by nnlinear state space change f crdinates and a state feedback cntrl law. The MIMO system under cnsideratin is a cupled three tank system (CTTS). II. COUPLED THREE TANK SYSTEM AND ITS MODELING The sketch f CTTS is shwn in Fig 1. It cnsists f ne reservir three tanks three pumps P; ~ ~ seven n-ff valves ~ V 2 ~ v: ~ v: ~ seven level sensrs L I L 2 L 3 L 4 L 5 L 6 L 7 fur temperature sensrs and five flw- meters F;. F 2 F; F 4 F The system has tw feedback lps f the flw f water; ne between tank-1 and tank-2 and the secnd between tank-2 and tank-3. Inputs t this system are vltages applied t the pumps and heat inputs. Outputs f the system are level and temperature f water in each tank. Levels are measured in centimeters and temperatures are measured in Kelvin. T mdel the system it is assumed that there are n thermal lsses thermal delays pump delays in the system mixing is perfect and hydraulic and frictin lsses are negligible. The system has been mdeled using Bernulli's law principle f cnservatin f fundamental quantities mass and energy. Accrding t Bernulli's law the flw' Fa' ut f an utlet having area f crss-sectin 'a' f a tank having height fliquid 'h' is given by: F: ==ax~2xgxh (1) In (1) 'g' is the gravitatinal acceleratin whse value is equal t 9.8 m / S2. Let 'V' be an n/ff valve having binary lgic r 1 accrding t it's OFF r ON state respectively then the effect f this valve can be intrduced as fllws: F: == V x a x ~2 x g x h (2) V2 P1 Q1 V3 ~ Tank-1 Tank-2 Reservir Fig. 1. Sketch fcupled three tank system Tank-3 V8 V /9/$ IEEE Authrized licensed use limited t: UNIVERSITY OF ALBERTA. Dwnladed n May 1421 at 19:3:5 UTC frm IEEE Xplre. Restrictins apply.

2 Equatin (2) shws that when valve is pen (i.e. V == 1) the utput-flw f the pipe is given by (1) and when valve is clsed (i.e. V == ) the utput-flw fthe pipe is equal t zer. Cnsider a pipe f height i.e. 'H' attached t the utlet. Then utflw is given by: Fa =Vxax~2xgx(h+H) (3) If 'v' is the vltage applied t the pump then utflw f the pump' Fp ut f a pump is given by: 1 x-- ~~ (11) Let 'T " "I'' and 'T j be the temperatures f water in the tank water cming int the tank frm t h inlet and water leaving the tank via r utlet respectively. Let 'R' be the density f water. Accrding t the law f cnservatin f fundamental quantity energy: d(fahc T) n m ---P-=fC 'LFT-fC 'LF.T dx p i=l lip j=l } Simplifying (6) applying prduct rule f differentiatin and then using (5) we get; Ah dt = ±F; (1; - T) dx F p = Kxv (4) ' K ' is the cnstant fprprtinality. The rate f change f vlume in a tank is equal t the difference f the inflw and utflw f the tank. Let 'n' be the ttal number f flws int a single tank and 'm' be the ttal number f flws ut f the same tank. Let 'A' be the area f the tank and 'h' be the level f the water in the tank. As area fthe tank is cnstant s change in vlume fwater in the tank is dependent nly n the level fwater in the tank. Let F;' be the input flw thrugh r inlet fthe tank and 'F j be the utput flw f the r utlet f the tank then accrding t the law f cnservatin f fundamental quantity mass: dh n m A x - = L ~ - L F j (5) dt i=l j=l i=l Using (1) t (7) the state equatins btained fr the three tanks are as fllws: (6) (7) (8) Reference [6] can be seen fr mre details. III. INPUT AND OUTPUT SCALING AND OPERATING POINT CONSTANTS Scaling is very imprtant in practical applicatins because it makes mdel analysis and cntrller design (weight selectin) much simpler and easier [7]. The specificatins f the system shuld be cnsidered very thrughly t have a better scaling. Inputs and utputs shuld be scaled prperly t get a maximum variatin f utput fr a maximum variatin f input. Table 1 shws the cnstants used with descriptin and numerical value. Fr details [6] can be seen. TABLE 1 CONSTANTS WITH DESCRIPTION AND VALUES Cnstant Descriptin Value with Units V 2 V 4 ~ ~ Valves ~~ Valves 3 5 Crss sectin Area f Tanks ern' 23 a Crss sectin f pipes.2419 crrr' AI' ~~ K I K 2 K 3 Gain Cnstants fpumps H I I Height fpipe thrugh ~ 135 em H I2 Height fpipe thrugh v: 3.48 em H 2 1 Height fpipe thrugh ~ em H 22 Height fpipe thrugh ~ 3.48 em H 31 Height fpipe thrugh ~ 3.48 em t; Input water Temperature 33 K 1\ 1\ 1\ Operating Vltages fpumps 12 V B. State Equatins/r Tank-2 dh2 = [-~a~2g(h21 + h2) - ~a~2g(h22 + h2)] ~ dt +~a~2g(h12 +~)+K3V3 -K2v2 (9) (1) VI' V 2 ' V 3 1\ 1\ 1\ Operating Pwers f Heaters 1 Watt QI' Q2' Q3 R Density f Water 1 Kg/ern' C p Specific Heat fwater 4185 J/Kg.K Authrized licensed use limited t: UNIVERSITY OF ALBERTA. Dwnladed n May 1421 at 19:3:5 UTC frm IEEE Xplre. Restrictins apply.

3 IV. CONTROLLER DESIGN USING FEEDBACK LINEARIZATION Cnsider a MIMO system fthe frm f(~) = ~ = [~l ~2 ~3 C!) sw ~5 ~6 ] (22) (x s-x6 ) X (x 4-xs ) '~4 =.19 '~6 = ~ = f(~)+ t gj(~)uj - j=l- YI = ~(x) Y2 =hz(x) (14) (15) (16) Ym = hm(x) (17).[ and ~j 's are smth vectr fields and h j (x) 's are smth functins defmed n an pen set f R". Then fr CTTS Xi = hi' u i =Vi 1 ~ i ~ 3 (18) x j = ~_3Uj = Qj_34 s ] ~ 6 (19) ~ =!:!.(~) = [~ C!) hz W ~ (~:) h4 hs h6r (2) = [ X 2 X 3 X 4 X s X 6 ]T -.845~ (~3.48+XI ~~3.48+X2).845(~3.48+X2 ~~3.48+X3).11 ~l =.11(x 4-33) '~3 = A. Step 1: Equilibrium Pint.19 (21) T find the equilibrium pint f a system we need t slve the fllwing: If m L1j=n i=l (26) Then nnlinear system can be exactly feedback linearized [3] which is true fr CTTS. There are tw cnditins relevant t vectr relative degree which must be satisfied fr the MIMO system. a. Cnditin N.1 Fr each Yi fr at least ne ~j Fr utput Y I = ~ (~) = LgjL11j-Ihi(~) 7= 1 ~ i.] ~ m Ii =1~ L~~ = ~ = L LO i; = 8(L~~) (X) = a(xl ) (X) gl 1"1 a gl - a gl_ X - X - =[1 ] ~l (~) =.11 7= L L O h = 8(L~~) g (x) = 8(x]) g (x) =.11 *- g2 1 ax _2 - ax _2_ L g3 L~~ = L g4 L~~ = L gs L~~ = L g6 L~~ = Fr utput Y2 = hz (~) = X2 x=o T which equilibrium pint a system will cnverge depends n the initial cnditins. Optimizatin tlbx f MATLAB is used t slve fr equilibrium pint. xinilial is the vectr representing initial values fthe states. xinitial =[ ]T (23) The technique used t fmd the slutin uses large scale algrithm t slve fr the prblems with bund cnstraints. Different ptins can be set accrding t the requirements. Fr CTTS the upper bund fr the levels is 25 em as the lwer bund is ern. Fr temperature fwater in each tank the upper bund is 373 K and lwer bund is 273 K. The equilibrium pint thus btained is: [ J ~ = X2 X 3 X 4 Xs X 6 (24) = [ ]T This equilibrium pint is needed in the verificatin f the tw cnditins fr a MIMO system t have a particular vectr relative degree. B. Step 2: Vectr Relative Degree Let 1j be the relative degree fy i then 1j is equal t the number f times utput Yi needs t be differentiated until any fthe inputs appears explicitly in the expressin. ~ 1j =11 ~ i ~ 6 (25) r 2 =1~ L~hz = h2 = x 2 L g2 L~h2 = = (27) (28) (29) (3) (31) (32) (33) Authrized licensed use limited t: UNIVERSITY OF ALBERTA. Dwnladed n May 1421 at 19:3:5 UTC frm IEEE Xplre. Restrictins apply.

4 L g3 L~h2 =.11 :;t: (34) Lg L~h2 = Lg4L~h2 = LgsL~h2 = Lg6L~~ = (35) Fr utput Y3 = ~ = x 3 r 3 = 1 ==> L~~ = ~ = x 3 (36) Lg L~~ = Lg2L~~ = Lg4L~~ = LgsL~~ = Lg6L~~ = (37) Fr utput Y 4 = h = x 4 Fr utputys = h S = Xs Lg3L~~ = -.11:;t: (38) r4 = 1 ==> L~h4 = b = x4 h O.Oll(x~ - 33) LgL f 4= :;t: O.OII(x~-x~) L g2lfh4 = :;t: L LO h =.19 g4 f 4 X:;t: 1 Lg3L~h4 = LgsL~h4 = Lg6L~h4 = r S = 1 ==> L~hs = hs = Xs Lg L~hs = Lg2L~hs = Lg4L~hs = Lg6L~hs = Oh =_O.OII(x~-x~) Lg3Lf 5 O:;t: X 2 (39) (4) (41) (42) (43) (44) (45) (46) C. Step 3: Feedback Law The cntrl law which cnverts the nnlinear mdel fthe CTTS int an exact linear representatin is given by: s. = :1-1 (!!: - Q)= [U l U 2 U 3 U 4 Us U 6 ]T (52) and L hex) = a(hi(~» fl- Lf'i ~ (~) L f r2 h 2 L r3 f ~ b(x) - r - - L 4 f h 4 Lfrsh s L f r6 h 6 Lf~ L fh 2 Lf~ L fh 4 Lfhs L fh 6 I(x) ax ~ (~3.48+-~r X-2 ).845(~3.48+X2 -~3.48+X3) (53) (54) (55) (56) (47) Fr utput Y6 = h6 = X6 r6 = 1 ==> L~ h6 = h6 = x6 LgiL~h6 =1 ~ i ~ 5 L LO h =.19 g6 f 6 X:;t: 3 (48) (49) (5) Hence first cnditin fr defining relative degree 1j at the pint.!o is satisfied. b. Cnditin N.2 The matrix ==>~=[Ul U 2 U 3 U 4 Us U 6 ]T (57) U 1 = 9.991w ~ x (x 4-33)w 4 U2 =9.991w l ~762+25xl w ~ x (x s - 33)w 4 U3 = 9.991w l ~762+25xl w ~ x (x s - 33)w (x 6 -x 5 ) ( w ~762+25xl x 3 (x -xx) U 4 = xl W 4 L Lfr6-1~(x) g6-6x6 shuld be a nn-singular matrix at.! =.!O. This cnditin is als satisfied. Since bth cnditins are satisfied s exact linearizatin fthe system is pssible. (51) U 5 = x 2 ( W ~762+25xl (x -xx) U 6 = x 3 ( W ~ x 2 (x - xx) Using the nnlinear feedback cntrl law given by (57) the linear system btained is: (58) Authrized licensed use limited t: UNIVERSITY OF ALBERTA. Dwnladed n May 1421 at 19:3:5 UTC frm IEEE Xplre. Restrictins apply.

5 This system bservable. cmpletely IS state cntrllable and Step 4: Cntrller Design using Ple Placement Since the system linearized using feedback linearizatin is cmpletely state cntrllable s it can be stabilized using ple placement techn ique [8] accrding t which W = -KTz (6) D. Ples can be placed arbitraril y fr a cmpletel y state cntrllable system. Let the arbitrarily selected lcatin fr ples be: P = [ ] (61) S the cntrller btained using ple placem ent technique is as fllws: I 2 3 K= 4 V. 5 6 (62) S using the feedback law given by (57) the resultant verall clsed-lp system has linear dynamics given by (59) and (63) and it is cmpletel y state-cntrllable. Let Cj Cz and C3 be the cntrllers used t cntrl states f tank-i tank-2 and tank-3 respectively. Then cntrller CI includes cntrl input s ul Uz u5 ' u4 Wz WI ' w4 ' cntrll er Cz includes cntrl inputs w5 and cntrll er C3 includes cntrl inputs u3 ' u6 ' w3 ' w6 Fig 3 shws the flw chart having the three tanks being cntrlled using Cj Cz and C3 which take states f tank-i tank-2 and tank-3 respectively as inputs and gives cntrl signa ls t cntrl the states. Als there is cmmunicatin betwe en the cntrllers and interactins are als present. Cntrllers Cz and C3 depend n cntrller CI and states f tank-i. As there is cmmunicatin and interactin between the cntrllers the breaking f a lp can result in the instability. Results and Cnclusin The clsed lp transfer functin matrix f the verall system is shwn in (63) which is same as it shuld be accrding t the frm given in [3]. Step respnse f the clsed lp system is as shwn in Fig 2 which shws that all states are stabilized. This is the respnse t unit step input s all utput s can be stabilized t any desired value by applying gain. I s+1 s +2 H(s) = s +3 s +4 s +5 T ank2 ( X2 x) (63) s +6 Step Respnse Fig. 3. Flw chart shwing cntrllers with interactins 8 REFERENC ES [I) [2) [3) [4) [5).1!:---!:---!:----!:--- -~----:'--- _! time (s e c) [6) A. J. Krener "On the equivalence f cntrl systems and linearizatin f nnlinear systems" SIAM J. cntrl and pt. II : R. W. Brckett " Feedback invariants fr nnlinear systems" Prc. VII IFAC Cngress Heisinki pp Albert Isidri "Nnlinear Cntrl Systems" 3rd editin SpringerVerlag Berlin Heidelberg New Yrk H. Nij meijer and A. Van Dar Schaft "Nnlinear dynamical cntr l systems" Springer-Ve rlag New Yrk 199. J. J. E. Sltine and E. W. Li "Applied Nnlinear Cntrl" Prentice Hall Englewd Cliffs N.J Fatima Tahir "Stability Analysis f Sampled-Data Cntrl f Nnlinear Systems" PlEAS Library Nilre Islamabad 27. Fig. 2. Step respnse fthe clsed lp system Authrized licensed use limited t: UNIVERSITY OF ALBERTA. Dwnladed n May 1421 at 19:3:5 UTC frm IEEE Xplre. Restrictins apply.

6 [7] S. Skgestad and I. Pstlethwaite "Multivariable Feedback Cntrl: Analysis and Design" Jhn Wiley & Sns Ltd England 25. [8] Katshuhik Ogata "Mdem Cntrl Engineering" 4 th editin Prentice Hall f India New Dehli 23. Authrized licensed use limited t: UNIVERSITY OF ALBERTA. Dwnladed n May 1421 at 19:3:5 UTC frm IEEE Xplre. Restrictins apply.