Passivity-based controller for chemical processes

Rev. Téc. Ing. Univ. Zulia . Vol. 25. Nº 1, 3 -11. 2002

Passivity-based controller for chemical processes 1

Osear Camacho 1 , Rubén Rojas , Alpha Pemía-Espinoza and Mercedes Pérez de la parte2

1

1Postgrado en Auiomatización e Instrumentación,

Grupo en Nuevas Estrategias de Control A vanzadas (GINECA) , Facultad de Ingeniería,

Universidad de Los A ndes. Mérida 5 101 , Venezuela. Te /.: 58-2 74 -2402891.

Fax: 58-274-2402890. E-mail: ocam [email protected].

2DepartamenLo de S istemas y A u tomática, Escuela Superior de Ingenieros ,

Un iversidad de SeviLla. Camino d [os Descubrimientos, s / n_ Sevilla 41092, España

Abstract This p ape r presents lh e developme nt of a Passivity -B ased ControUer (PBCr) from a Firsl-Order­ Plus- Dead -Tim e model of thc process. This approach res ults in a fixed structure controller that d epe nds on lh e ch aracterisUc para meters of the model. T his a llows a unique controller of adj u stablc parameters that can be u sed in s evera] processes. Compu ter simula lions on a n onlinea r che m icaJ proC'ess judge the conlroll er perfonnancc . Thc sim uJaUon r esu ll showed ("ffe tiveness and good p erfonnanc for lh stud­ ied case. Key words: PassiVit.y-based control. non linear chemical proces ses. dead -time .

Conirol basado en pasividad para procesos químicos Resumen E n esle trab Jo se pres nta e l d sarrollo de un con trolador basado en Pas ividad derivado a partir del modelo de Primer Orden Mas T iem po Mu erto (POMTM) de l proces o. El resultado de esta prop ues ta es un controlador de estru ctura fija qu e depend e de los parámetros caracleristicos del mod elo. Por tanto s e cuenta con un controla dor único de parámetros ajustables que puede ser empleado e n dife rentes proce­ sos. Para la eval u ación del controlador se realizaron simulaciones s obre u n proceso químico no lineal. Los resultados muestran la er: ctiVidad y bu en d esem peño del controlador en e l cas o estudiado.

Palabras clave: Control b asado en p asivida d , procesos qu ímicos no-lineales, líempo mu rto .

l. Introduction The theory aboul pa s ivity for non linear process control W G S [¡rs tly s ludied by Willems [11 who built a gen eral lh ory of dissipalive sys te ms. Popov [2 ] inlrodu ced the concep t of passivity as a fun dame ntal property of feedback syslems . A syste m is said lo be passive ir lhe increase in lh e s tored cn e rgy. in a given time interval. is lower or equallo the energy supplied to the ys­

tem, in th same interva l. This characteris lic made a pa ssive system sLable in Lhe sense of Lia pu n ov . The passivity stra tegy uses the prop­ erty of sta bility of lile passive systems to d el;ve a fe dback control la w . Also another con¡;ept. the d amping iryection, is uscd to ensure Lh e asymp­ lolically convergence of Lhe system to the desired equilibrium s latc . Th e passiVity meU10d has fo u nd m any p plications in the electrical and mechanical fields

Rev. Téc. Ing . Univ. Zulia. Vol. 25, No . 1, 2002

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Camacho y col.

wilh good resu lts [3. 4]. Recently. these concepts als o h a ve b een a p p lied to chem ical and b iological p rocesses [5. 6 ]. To develop a Passivity-Based Con lroller . PBCr. knowledgc of th e p rocess rnodel r ela ting the con1.rolled variable , X c(t). t o the manip ula ted variable, U(t). is n ecessary. Malhem atica l rnode1s of t he sys tem s olve m ost of lhe control problem . In the ind ustry. t.he re are rnany cornpIex pro­ cesses whose accurale rna lh ema tical mod els are nol available or difficult to obt.ain . As far as ch emical processes are con cem ed . the d evelop ­ ment of a complet e rnod el is d ifficu It d u e mainly to lh e complexi ty o[ Lhe process itself. and to the lack of knowledge o[ sorne p rocess parameters. Second. mosl p roces s models relating the con­ trolJed and the manipula lcd variables are of h igher-order. Generally, t he PBC procedure pro­ duces a compiex controller , which could contain fou r or more parameier s resu lting in a d ifficult l uning j obo Th erefore. lhe u se of lhe tradlUonal p ro e du res of PBC would present d isadvant a ges in lheir application lo chemical pr ocesses. Jt is a com m on proc dure Lo u s e app roxl­ mating syslem m odels fo r controIler designo JI Lhe

frequ ency responses of fue a pproxirnaUng mod el and the original sysiem coincide wilhin drawi ng a ccuracy. so do lhe elosed-Ioop res ponses of both s ysi ms [7]. An efficienl m odelin g rnelhod al ter­ n a tive for p roces s control ls lh e u se of empírica! rnodels. wh ich u se low or de r lin ar models wilh dead time . Many times, First-Ord r -Plu s Dead ­ ti m e (F'OPDT) models are a dequat for ch em ical proces s control analysis and d esign [8] . Som e times these reduced m odels pres en l uncertain­ ties arising frorn imperlecl rnod eling of lhe pro­ cess no nlineariLies Lhat contribute lo perform­ ance degradation of the controUer [9[. The aim of lhis pa per 1s lo design a PBCr b ased on an FOPOT m odel of the actual p rocess. The overall idea is 1.0 d evelop a gen eral PBCr . wh ic h can be u s ed for self-regulating ch emieal processes. It is expeet ed that due to lhe r obust­ ness s h own by PBCr . the propo sed eontroller wo u ld deal wiLh m odel u neertainlies and pertur­ bations . This arUcle is organized as follows. Sec­ Lio o 2 pres ents s orne hasic concepts of the P BC th ory. S clion:3 sh ows th e pr ocedu re to de s ign a PBCr u s ing the FO PDT model. Tu n ing equ tions

for the controller a r c also given in th is seclion. In Section 4 the simula tion of the PBCr for a nonlin ­ ear che mical reac tor is p r esen ted . Section 5 con ­ eludes the pap er.

2. Passivity Background Definitions Considering a S ISO system represen ted by

z = J(z) + g(z)u

( 1)

y = h(z)

where z E Z e R" is the state vector. u E U e R is the control inp ut a n d y E yc R is lhe outp u l func tion of the syslem . The vector fi eldsfiz) and g(z)are assumecllo b e s m oolh veelor fields 00 Z. Ass u m ing tha t an associa led energy s tor­ age fun c lion S:Z H m+ exists . Th e suppiy r ate is defined as a fu n ction w, U x Y H m.

Definition 1 Sys tem (1) is Said lo be passive with resp c l Lo t he s upply rate w = u y ifthc foll owing rela tion holds

s( z( t))

(2)

::: u(t)· y(t)

Thi relation is caBed th e pas s ivity inequ al­ ily [6].

Definition 2 Syslem (2) is said to b ecorne passive wiili respect to th e storage fu nction S if t h ere exisls a regular affine feedback law of lhe fonu u = a (z) + ,8(z )w

a (z ) E R; ,8(2) E ffi

(3)

wh ere ,8(Z) is a o on -zero scalar fu n ction in Z. a nd s u ch lha t thc losed -Ioop syslem (1 )- (3 ) becomes passive with re pect to a n ew scalar c ntrol in p ut w[6].

3. Passivity-Based Control from an FOPDT Model of tbe Process ThiS section pre s ents the d ev lopment of a paC r from an F OPDT rnodel of the process.

Rev. T' C . Ing. Univ. Zulia . Vol. 25. No. 1. 2002

Passivily -b ased con trolJer for chemical p rocesses

5

o and

1 b ut beyond the a pproximatio n brakes down . On oth er hand. the Taylor series approxi­ maU on improves as f.o increase . The fi rsL~ o rde r Taylor approxim a Lion or the Padé app roximaUon can be considered a s good approximaUons for Lhe dea dtime term for ch emical processes as has been shown previously [10] .

3 .1 Study of the process model: FOPDT The Fl rst Order plus Dead -Time (FOPDT) m odel of Lhe process is exp r essed by

(Ke-'' ' )

X; (s) U (s) rs + 1

(4)

If the firsL-order Taylor series app roxima­ tion is su bsLilu led jnlo equation (4) . it is obtained

wher e X I (s) is th e Laplace transform of the con­ trolled variable. lhe lran srrú Uer ou lput . and U(s) is lhe Laplace l ransform oflh c manipulated vari ­ a b l . the control1er outpu t. 8 0th X J (s) a n d U(s) ar deviaUon variables . The chara cteris tlc pa­ rameters of the process are: dead t ime. to . static gain. K. and tim con slant. r. Th e dead time can be approximaled in two different ways. A firs t-order Taylor series ap­ proximation lo Lhe dea d time tenn p roduces

X; (s) U (s )

Equ aUon (7) r epresents a minimum phase system and equalion (8) re presenLs a non ­ m ínimum p hase system. Up to n ow . Ule pass ivi­ ta tion procedu r e h a s been applied to minimu m phase syslems [4. 61. Sín ce . the Padé approxima ­ Uon p rod u ces a non~ mini mum phase syst m; lhe actu al p rocedure can not be drrectly usee! to syn­ thesize tll e passive conLroller for Lhis kind of sys­ tems . Therefo re, the Taylor series a ppr oxim a Lion is chosen to design the controJler.

Figure 1 sh ows a comparison amo ng the dead time term and Lhe firsl-order TayJor series and Padé Approximalions . The figu re shows that lh e Padé Approximation works very well between 12~

On Lhe othcr s ide. it was h own in [ 1 1] tha t the first.-ord er TayJor series a pproximation can

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~5

(8)

(rs

U (s)

(6 )

1+ 0.5tas

(7)

K (l- 0.5t as ) + 1)(1 + O.5tos )

...X¡ (s)

-- = -

The bove approxima tion can a lso be writ­ len as a first~ o rde r Padé Approximation

= 1- O·5tos

K

+ l )(tos + 1)

(r s

an do if it is r eplaced thc Padé pproxim Lion into equation (4). the follow"ing resuJt is obta ined

(5)

e~l o'

=

r

.,.

.. •..,

-:--- ---­

.J _ ___ .. _ ...... ___ _ _ ' _ __ .. _ .. _

,

,

J ~ , ~--------------~~--~~--~----~----~----~----~----~ e,i J[ O

x Figure 1. Compa rison among e

-x

(1). T,tylor (2 ) ancl Pa dé (3) approximations.

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Camac ho y col.

u

U~,---_(_1S_:_J_)-lI-X_2_(S_)-~'II"H

I

Using an auxiliary variable X 2 (s) the system m odel giv n by equalion (7) can be schematized as s hows Figure 2. Now , the model ofLhe process can be reWIit­ ten in staie variables a s follows: =( xz -x¡)D -A

x,

'flx.

= (x 2

-

x¡ )D - A -~

x2

2

= -T- - "' yD + W -X

(14)

2

y=x¡

This system has a passivity inequality ofthe form S(x) ~ x.W, S o , t ransforma tion (13) results in a passive system operator relaUng th e n ew in­ put W and tile output variable XI ' Now, the overall passive system can be re­ WIitten in the canon ical form [1 2]

x = - R(x )x -

fx. = - -x 2 + B + CU 2 r

(1 3 )

Sub Uluting this cxpr s s ion in lo lhe sys­ tem model (9) we obtain

be used lO h an d le U 1C dead ti m e ierm with out any p roblem for chemical processes.

l

~C (w.3.. - x¡D - B) x 2

Figure 2 . Bl ock d iagram realiza lion

of the syslem mod el given by equ tion (7).

x,

=

J(x)x + M (x)W

(1 5)

(9)

in lhis ca s e

y=x¡

-R(X)=(-~l

wh ere

(10)

-Yr)

-J(X\~ -~)

with M(x)

~I = (D+ ~ ) =

(xJ

(16)

where R(x) is a p ositive semi-defi nite matrix. and and

Wc a ssume that lhe va lue of the varia bles XI are always p ositive for all Um s (xtt) ~ O).

X2

Fol1owing fue passivi ta tion procedu re pro­ po ed by Sira and Angu lo [6 1: for chemical pro­ cesses , the following stora ge function can be u sed : 1 T S(x)= - xx 2

(1 1)

wh ere x is the s l t e vecto r. For the sys te m given by equation (9), the torage fu n clion d erivaltve is obtained

J{x) is an a n U-symmetric matrix .

3.2. Passivity Based feedback controller synthesis Th derivation of the control law was done based on a modified storage [uncUon Sct(x) a n d damping u-yection tbrough feedb a ck [6J , Th e modified storage funcHon, S d(X). is relaled to the Lya p unov slorage runcUon . 11s prope rties wer e u sed to der ivate an auxilia ry system frcm which lhe contr ol law was obta ined. The damping inJec­ tion was used to ensure lhe a ymptotically con­ vergence of the system to th e desíred equilíbriu m s tate.

2

.

2

S(x) = x¡x 2 D - X. D -

Ax. +

~

Bxz - -

r



+ Cxp (12) .

To ensure the passivily ineq uality fulfill­ m ent, equalion (21. ii is essen( la l lo define a sia le-dep n d n l inpul coordina te transfo rma­ líon as follows:

Then a passivity- based controller was pro­ posed for a system gíven in lhe equation (15) form o considering Ule following modifie d slorage fun cUon

Rev. Téc. Ing. Un ív. Zulia. Vol. 25. No. 1, 2002

(17)

Pas sivity-based controll r for chemical processes

7

where xd is an a uxiliary state vector to b e defined latero Along the solution of the system (l5). the function Sd(x,xd) have the followmg time deriva­ tive

(he controllaw via feedback. Then, x2d is s et to a desired eonstant equilib rium value x2 . Tnis is re ­ 1él.~~ d to the component valu~ in the equilibrium state ofthe origin al state vector (z). It has to be in the in lcrval 0< x2 < ;;, to be consistent with the coordinate transfonnation (lO), where the con­ stant value A was taken as positive.

Completing tenns in the right s lde and a dd­ ing a damp ing iryection term of the fonn - R di (x - xd). so that Rm == R + Rdi b a positive d e­ fined matrix. asymptotic stability is ensure. For sim plicity ~i is taken to be a constant m atrix.

Lctting. thus, X 2d = x2 , and after sorne mathematica l manipulations the dynamic feed­ back controller is obtained (24)

RUi

=

( ~O R¿O)

. ; Wlth

~.

R¿ > O

[19)

So the modified storage function derivative can be wIi tt n as follows: Sd(X , x d

)

== [x - xJ[(R+ Ru, )(x - x ct )

xd

-

J(x - x d

)­

-Rxd - Jxd + Ru,(x-xd )+ MW , (20)

Now, let the a uxiliary ,'ta le vector xd(t) sat­ isly lhe following state eq uaüon representation

Finally, replacing W in equation ( 13). \.he expression of the general passivity controller ba sed on fue FOPDT moc\el is obtain ed

fu = ~ [- D(x¡ - x

1~d

= - (D +

ld ) -

R)x2

-

x

2 )]

~ )X¡d+ Dx'l. + ~(-"i -

+ ti (25) -"id )

Thu s . a Passivity-Ba sed controllcr of [iXed structu re has b een d esigned with adjusting pa­ ramet rs:

(21)

C, D. Note that is a time varying linear syste m for the a uxiliary stalexd' Then, R¡ and R2 are ch osen so that fu e time d erivative of Sd(x.xd) satisfies (22)

lt follows that the

eelor x(t) exponentially asymptotieally converges towards the auxiliary vector trajectory x d(t). S ubs tituting R(x), J(x) and M(x) as defined. equation (1 6). in equ ation (21) the fo llowmg state equation is oblained

(23)

This system is s tab le in the Lya p u nov sens e wilh s torage function given by equ ation (17). Be­ ca use of linear charaeteristics, it is easy to obtain

~

and R,

e anc\ D d epending entirely of the charae­ len tic parameters of Che process : K. r and too ~ and R, are tuning parameters chosen to satisfy (18).

On th e other hand, e and D de termine the agressivHy ofthe controll r, while ~ and R, deter­ mine the d amping factor and the setLing time of the proces s. In sorne cases, the obtained PBCr , p res­ ents steady tate error differen t of zero . II cou ld be 'a u s ed by th c imperf c tion of the modeling. The FOP DT model is a redu ced orde r model that presents uncerlainties arising from imp erfe ct. knowledge of the system, and the process non ­ linear effects that. c ontribute to performance d cgradation ofthe controller. In order to improve th eontroll er closed loop performa nce an inLc­ gral tenn was a d d ed . Then eq uation (25) wa s m odified . a n d the fin al controller based on l h e FOPDT m odel is

Rev . Tée. lng. Univ. Zu lia. Vol. 25 , No. 1. 2 002

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Camacho y col.

j iD X 'rl -

:,1

~ (-DI: - "')- I~ -~, Id' - R,lx, - x,1] + ti + ~ )x,rl + D X2 + R,(x¡ - X'c1)

Ma nlpulaled Va riable u

Controlled Variable ."

x

Internal Variable

(26)

Figu re 3 shows Lh e m odified s(ructure of the controller . The _~ dynamic is represent ed by the second line in equ aLion (26).

4. Evaluation oí the PBCr designed ThiS s ecUon pres en t a simula tion test of the proposecl conlroller , previously c1esignecl, in the regulation of an exolhennic continuously s ti rred tank reactor (CSTR) system. The process m odel assumes that [8 ]: the re­ actor a n d Lhe jacket are perfecUy m ixed ; al] the volumes ancl phys ical properties remai n con ­ stant and lhe heat losses may be neglect d .

(27)

, INTEGRAL TERM

Figure 3. PBCr scheme. (m 3 ), Pe is t he de nsity of the coolant in Lhe j acket (kg/m 3 ). C{;c is l he specific heat of lhe coolanl P kg'l oC ). Pe i5 lhe coo)ant rate (m 3 /sl. T ó iS t h e coolanl inIel lem peratur (OC). ka is lhe Ar­ rhen ius frequ ency p aramctcr (m 3 S ' l kgmor l ). E is th a ctivation energy of the reaclion (J / kgmol). R is lbe ideal gas law constant (J kgmo r l {{- l) . FCmax is the m aximu m Oow Lhrough lhe control valve (m 3 / s) and a is fue rangeability parameter. Sim ula lions were performed using fue fol­ !owing valu es [8]:

Cp = 1.81 15 X 10 5 J kgmor 1 S , I; 3

where z¡ is the c oncentraU n of lhe reaclan t in the reaclor (kgmol/m 3 l. Z2 is the temperalure in llie reactor (OC) . 2 3 is the jacket temperatu re (oC) , u !s lhe conlroller output signal limited lo the in­ terval [0 ,1] and represenls t h e position fu nction of an equal percentage valve . F is the feed ra le (m 3 /sl. V is lhe r actor volu me (m 3 l. CAi is lh e concenlration of lhe r acla nt in lhe feed (kgmol/ m 3 ), k 1s the rcaction rale coefficient (m 3 kgmol' I s' 1). Ti is the temperatu re of the feed (OC), L\H R is (:h e heal ol" lhe re clion , assumed to be cons lan t (J kgmor11. pis the c1ensity oU he reac­ tor contents (kgmol/m 3 ), C p is the hea t capaci ty of the reactanls (J kgmor l S ·l). UT i the overall h cat lransfer coefficient (J s' 1m 2 ° . 1). AT is the heat transfer area (m 2 ), Ve is Lh e jacket volume

2

3

Ve =1.82 m ; AT = 5 .4 0 m ; Pe = 1000 kg/m ;

ka = 0. 0744 m 3 Cpc

S·I

= 41 84 J kg

kgmor l ;

1 De

l;

7 E= 1.182 x 10 J/ k gmol; F Cmax

F= 7 .5 X 10,3 m 3 /s; a

=0.020 ro 3 /s;

= 50;

H R = -9 .86 X 10 7 J kgmor l ;

UT

=3 550 J

S · I m,2

°c

l

;

CAí = 2 .88 kgmol/ m 3 ;

T ei = 27°C; Tí =66°C

and th equilibrium states of lhe process for a non -zero constanl u as:

Rev. Téc. Jng. Univ. Zulia. Vol. 25.

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Passivi ty-b ased conlroller for c h emical processes

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a

L - (OC)

' l

106.4

I

0 75

\/·----··---------·-··--··..-··---i (o)

106.2

--

b

Valv e Pos¡lion

106 6,--,.,...tl-----~------.-------.-~]

106

07

\Jv~~------------i '")

105_8 105.6 1054

1

1052 105

i

104 .8 104 _6

06 5

06

0 55 0 .2

, 5

0.5 T rrn e (5 )

0 .4

0 .6

0.8

\ 2 T,m e (s)

1.4

16

1.8

2 .2

x 10~

x 10 '

Figure 4. C losed -Ioop sys lem response and control signal whe n CA; changes by 10% . (a) wilho ut integral term o(b) with integral termo Z2 (OC )

Zc ('C )

106

106 K, K i 5 0C!~ , K -50%

10 5.5

10 5.5

105

105

104

104 5

104

104

o

0 .5

o

1.5 Ti m e (s)

X

02

0 .4

0_ 6

08

10