Multivariable Generalized Predictive Controller: Analysis, Tuning and Application

Copyright © IFAC 12th Tri ennial World Con)!rl'ss. Sydney. Australia. 1993

MUL TIV ARIABLE GENERALIZED PREDICTIVE CONTROLLER: ANALYSIS, TUNING AND APPLICATION A.A.R . Coelho and W.e. Amaral* LA CO.'lIDEEIUFPA. CP . 6025. Bc/cm . PaTli, Brazil *DCAIFEEIUNI CA M P. c.P. 6 10 1. Cllmpill
Abs~rac~.

Multivariable generalized predictive contro l ler haB received attention in recent years and different academic and industrial implementation. h.ve being reported in the literature. The succe •• of this control .trategy i. ba.ed on controlling plants with several kind. of complexitie •• non-minimum phase respon.e and nonlinearitieB. To obtain an adequate clo.ed-loop performance, it i. necessary to adjuBt the tuning parameter. and we i ghting matrices of the controller . This paper pre.ents analytic expre •• ions for tuning. with respect to special multivariable control performance goal. .uch a. the .tability cloBed-loop sy.tem. Simulation examples illustrate how the closed-loop proceBs response may be adju.ted to different control tuning parameterB. It ' s also included computational re.ult. with the application of the predictive control law in a high purity binary di.tillation column.

Key Words. Adaptive control, dynamic stability,

identification,

proce ••

control, clo.ed-loop .y.tem, predictive control

parameters for stabilizing unBtable and min imum phaBe procesB, will be developed the GPC.

1. INTRODUCTION Since the concept of predictive control introduced by Richalet (1978) and Cutler (1980), the generalized predictive controller (GPC) ha. been the .ubject great deal of academic and industrial re.earche •• GPC u.e. a receding-horizon prediction philo.ophy and i. ba.ed on two .teps. Fir.t, it u.e. a predictive model to foreca.t the proce.s output over a time horizon. The output is composed by one term dependent only of the pa.t inputoutput mea.urement., and another that i. a function of the future control input •• Second, it determine. future input. to meet control objectives in term. of a desired reference (Clarke,1989).

nonMIMO

The performance and tuning characteristics of multivariable predictive contro l strategy is al.o evaluated to control a high purity binary di.tillation column . This proce.. ha. .ome interesting featureB at least from a control point of view. strong interaction., nonlinearitie., dynamic. varying with the load and setpoint and disturbanceB. Although the literature is rich in describing algorithms a nd approaches, few paper. present reBults of the nonlinear process top and bottom compoBitions for the predictive control strategy . The paper is organized as followB. Next Bection pre.ent. the characteri.tic. of the multivariable proce.. model, k-Btep-ahead predictor and predictive control law. The cloBed-loop sy.tem analysis are then described in the section 3. Finally, .ections 4 and 5 pre.ent the computational results and conclusions.

Several author., for the ca.e of monovariable GPC, provide heuri.tic. and mathematic. condition. for adju.ting the clo.ed-loop .y.tem pole. in a u.er.pecified location (McInto.h,1990 Criea11e(1990). I n the literature, few paper. concentrate on the development of theoretical expre •• ion. which analy.i. the multivariable generalized predictive controller. Koivo (1980) extended the .elf-tuning control pre.ented by Clarke (1975) to MIMO linear .y.tem•• Thi. cont r ol etrategy need. to know exactly the interact or matrix of the .y.tem (the time delay to the multivariable ca.e) and cannot handle .ystems that are both un.table and non-minumum pha.e. Dugard (1984) generalized the extended horizon approach propo.ed by Yd.tie (1984) in order to control MIMO .y.tem., and .howed that this .trategy need to know the interactor matrix. Shah (1987) extended the GPC to multivariable determini.tic ca.e. Finally, Kinnaert (1989) derived the MIMO GPC in a .tocha.tic framework and .howed the influence of the de.ign parameter. of the controller on the clo.ed-loop dynamic. in the .tate-.pace domain. In order to obtain an explicit form for the characteri.tic polynomial of the clo.ed-loop IYltem for the input/output de.cription (tran.fer function form) and the properties a •• ociated with the choice of the different horizon. and de.ign

2. MtJLTIVARIABLE GENERALIZED PREDICTIVE CONTROLLER 2.1. Process Representation The linear di.crete-time process is described by the following Contro l led Auto-Regressive Integrated Moving-Average (CARI MA) model •

where A(z-i), B(z-i), C(z-i) are polynomial matrices with ordera nA, nB, nC, reapectively. The variab l e. y, u, e are (p x 1) oU~fut, input'_i zero mean white-noise vector. and ~(z ) - (1 - z )I . p

2.2. Predictive Model For multivariable discrete .ystems i t ia known that the interactor matrix, ~ (z), represents a

93

time case case form

delay structure analogous to the monovariable (Dugard,1984 , Shah,1987). In the general the interact or matrix ha. a lower triangular and satisfiea the relationship I

lim z -1{ (Z)A- 1 (z-l)B(z-l) _ It

z

where

(2 )

T T EU(t) - ['7U (t) ... '7u (t+HU-l)]T

(17)

YFA - [;r: (t+d/t) ... ;r: (t+d+HY-l)]T

(18)

YR - [y:(t) ... y:(t+HY-l)]T

(19)

--+ 00

for any finite and non.ingular X (Wolovich,1976). For computing the k-step-ahead .ignal Yr (t) is defined as Yr(t) - t.(z

-1

)~

predictor,

the

The GPC algorithm i. implemented in a recedinghorizon approach. At each .ampling time, a new vector EU(t) i. computed u.ing the Eq . (16), but only it. fir.t p line. '7u(t) are applied to the proc••••

(3)

(z)y(t-d)

where d is a maximum advance in ~ (z) and t.(Z-l) i. an user-.pecified tuning matrix. The k-.tep-ahead predictor Yr (t+k/t) can be decompo.ed a.

3. CLOSED-LOOP SYSTEM ANALYSIS

(4)

In the generalized predictive control the tuning parameter., (HY,HU,t.,A,r), are user-specified. The overall closed-loop .y.tem behaviour ( •• rvo and regulatory) dep.nds upon the choic. of these parameter. (McIntosh,1989 Clark.,1989 I Crisalle,1990). Several works, showing that the clo.ed-loop system behave. in the desired faahion, can b. find in the literature for the monovariable predictive algorithm. The.e preliminary re.ult. were ba.ed on simulation and experimental implementations in the control of plants that exhibt diff.rent kind. of complexitiea. Table 1 present. two procedure. to adju.t the SISO GPC tuning parameter •.

where (5) (6)

(7)

(3(z

-1

t.(z

-1

-

) - ,\(z )~C

*

(z

-1

-1

) + z

-hd-1--1

(8)

H(z)

)-Fk(z,z

-1

*

)A (z

-1

)'7+z

d-k

Gk(z

-1

)

Theoretical result. for the multivariable generalized predictive controller in standard feedback control will be developed to characterize the clo.ed-loop proce.. .tability. U.ing the.e equations, it is po.sible to analyse the effect of varying the prediction horizons as well as the weighting matrices on the clo.ed-loop dynamic a for multivariable aystems. In the literature few paper. have b •• n pr.sented mathematical expre.sions for the clo.ed-loop proce.. which analy.i. the MIMO GPC in terms of the tuning paremeter •• Recentely, Kynnaert (1989) ahowed aome tuning re.ult. from the .pace-.tate model.

(9)

(10)

~]

H _ [ :: Ho·.... H

HY-l

(11)

..... H

HY-HU

and H., are the elements of the matrix ;(Z-1) dim H

z

with

(p.HY , p.HU).

2.3. Predictive Control Law

Table 1

The idea of the generalized predictive controller i. to predict the proce.. output for a .et of steps (larger than the time delay of the proce •• ). To reduce the magnitude of control actions and improve robustness to modelling error., the variance of the output can be minimized .ubject to a conatraint on the variance of the manipulated input. Con.ider the objective function

Tuning procedures for the SISO GPC

parameter author

Hl

Clarke

d::!:Hl::!:n

HY

HY~2n-l

McInto.h Hl-nB+l

HY~HU+

Hl-l

HU

l::!:HU::!:HY

A

A -

o

•

n

= ma.x(nA,

nB"t-d>

0

< A

0

< 1

A - A [B(l) lZ 0

HU-nA+l A

0

where

A [B(l) lZ

ml tr(HTH) HU

I

Hl- \.1"'I'i.t ia.t hori.zon

3 . 1. Predictive Control and Clo.ed-Loop System (13)

For the purpo.e of the clo.ed-loop process analy.i., the multivariable g.neralized predictive control law, Eq.(16), can be rearranged in the form I

(14)

e u (t+k) - '7u(t+k)

(15) and y e!R r

P

(20)

is the reference .ignal, HY and HU are

The objective of writting the predictive control law in a polynomial matrix form i. for analy.is purpose. the clo.ed-loop transfer function and the clo.ed-loop pole. can be determined. A•• uming a con.tant reference vector, the Eq.(20) i. equivalent to Eq.(16) with

-1

called the output and input horizons , A(z ) and r are the reference and control weighting matricea, reapectively. The minimization of the Eq.(12) EU(t) gives

with

re.pect

to

HY

R(z

(16)

-1

) -

[

+ z

I I'

94

-1 \'

L

k= cl

(21)

HY

k~d

T(Z-l)

6

T k

A(z

-1

4.2. High Purity Binary Distillation Column ){ (z)

(22) Di.tillation columna constitute a major part of mo.t ch.mical proce.sing plants. The purpose of a di.tillation column i. to split the feed into two or more product. with different compositions from that of the f •• d. The objective of the control .y.tern i. to ke.p theae product compositiona at th.ir d.aired level. The distillation column control problem can be a difficult task. This is mainly due to the poaaibility of having strong interactiona present at high purities. Nonlinearities come into play when there are changes in the process operating point.

HY

S(Z

-1

6

) • Jd

6

T •

[

T

and 6

[HTH +

6d

T k

C\(Z

6 Hy

•••

-1

(23)

)

(24)

]

i. the fir.t p-row

block

of

the

matrix

r ]-lHT.

Con.id.ring the output horizon HY horizon HU ~ HY - d + 1 and an r.f.r.nc. v.ctor (y (t) Y )/ r

~

r

0, the input u •• r-.p.citi.d the r •• ulting

The high purity binary distillation column under atudy i. described in detail by Coelho (1991). For .imulation purpo.es the configuration described by Fig.2 waa selected. Process variables, distillate (X ) and bottom (X ) compositions, are controlled D u by the reflux rate (R) and the energy delivered to the reboiler (~), respectively. Since

clo.ed-loop .y.tern pole. are Pct. (z

-1

•

).R (z

-1

)A(z

-1

)\7(z

-1

-1.

)+z

B (z

-1

-1

)S(z)

(25)

Con.id.r the .y.t.m d •• crib.d by the CARlHA mod.l, Iq.(l), the multivariabl. g.n.raliz.d pr.dictiv. control law, Iq.(20), and the clo •• d-loop pol •• .quation, Iq.(25), with

di.tillation pr.ssure is assumed to be constant the t.mperatur.. of stages 1 and 27 (closely r.lated to Xu and X ) are used as process outputs. D

r •

~

0 / HY • HU + d - 1 / 1

HO

~

HY - d + 1 (26) In ord.r to .how the robuatness and an adequata performance of the GPC in a difficult industrial proc •• a, different disturbances were applied to the column. Th. reaults ahown in Fig.3 were obtained for a atep change in feed composition (10000 to 10500), feed concentration (0.8 to 0.72) and a.tDoint in bottom comDosition (0.0006 to 0.0025). For both cases the design parameters used in the implementations were (HY,HU,Ll.,rb,rt,TO,A,P,

Th.n, the clo •• d-loop .y.t.m pol •• are giv.n by Pct.(z

-1

). z

-1.

B (z

-1

-1

)BoLl.(z

-1

(27)

){(z)

Proof. The proof i. giv.n in Co.lho (1991). From the .quation •

R (z

-1

)B(z

-1

*

) • B (z

-1

)R(z

-1

(28)

)

T)

Th.

it i. po •• ibl. to .how that the root. of B·(z-l) are the root. of B(Z-l) and th.r.for. the clo •• dloop pol.. are the op.n-loop z.ro. of the proc ••• (Wolovich,1976 / Co.lho,1991)~lIn particular ca.e wh.n {(z) i8 diagonal and Ll.(z ) . lp' the Iq. (28)

(forgetting factor), P (resetting matrix) and (filter factor) are aasociated with the algorithm with covariance resetting procedure improve the adaptation level of the estimator. thia aet of control parameters, the process indicat.d a good cloaed-loop behaviour diff.rent .ituationa.

.implifi.d to P t.(Z-l) • B·(Z-l)B~i

(29)

C

In ord.r to illu.trat. the influ.nc. of tuning paramet.r. on the p.rformanc. of the multivariabl. g.n.raliz.d pr.dictiv. controll.r two .imulation r •• ult. of diff.r.nt proc ••••• are pre.ent.d. 4.1. Clo •• d-Loop Pol •• Ar. Op.n-Loop Proc ••• Z.ro. Conaider a .tabl. and non-minimum pha.e plant r.pr ••• nt.d by the following di.cr.t. lin.ar mod.l with p • 2 •

5. CONCLUSION The multivariable generalized predictive controller for a MIMO system and described by a CARlHA model with lower triangular interactor matrix wa. developed for closed-loop system analy.i ••

+ AiZ-i]y(t)

wh.r. Al

-

0.4 0.0] [1.00.0] [-0.30.0 ] ,Bo • ,B1 [ 0.0-0.9 0.01.0 0.01.4

GPC can b. adjuated so that robust behaviour is achi.v.d in order to cope with complexities plant •• Rewritting the multivariable control law, it wa. po •• ible to analyse the atability and tuning param.ter. on the closed-loop output. In the literature the.e results were developed for SISO ca ••.

In ord.r to v.rify the clo •• d-loop .y.t.m b.haviour the following parameter. wa. con.id.r.d. y. . . 1 (p.riod • 40), yrz 2 (period 70), Ll. ( z

-i

) . 1\ ( z

-i

) •

T

RLS to For has for

For diatillation column, the most aevere di.turbance ia a change in feed rate. So, 10' chang. in f.ed flow rate wae used to analyae the column control performance with the objective of .imultaneous regulation of the composition of both product str.ams (Fig.4). The output and input horizon was changed to 10 and 3. As can be seen, the clo •• d-loop performance was excellent, the top and bottom compositions were maintained within the .pecifications.

4. IXPIRIMENTS AND COMPUTATIONAL RESULTS

[12

(5,2,1 ,0.0001,0.0001,4,0.9995,0.001,0.1). 2 param.tera TO (sampling time,minutes), A

12

and HY • 4. HU wa. .et

to

1 It wa. dernon.trat.d that, if the weighting matrix of the control aignal ia zero and output horizon i. equal to the input horizon, then the closedloop pol •• are open-loop process zeros.

and chang.d to 2,3 and 4. Th. l.ngth of .ach .imulation i. 100 .ampling p.riod. and the int.ractor matrix i. {(z) ZI • Th.r. i. a 2

clo.ed-loop pol. going to the po.ition -1.4 that i. an in.tabl. z.ro of the op.n-loop proc... and anoth.r going to 0.3. To modify thi. in.tabl. b.haviour k •• ping HY HU it i. n.c ••• ary to adju.t r ~ 0 l.ading to a .tabl. ay.t.m (Fig.1).

It - wa. alao ahowed that to decrease the computation .ffort of the multivariable GPC the input horizon muat be adjusted by HU· HY 1. Utilizing thi. choice a best tradeoff between

9S

computational .ffort and clo •• d-loop .yst.m behaviour is guarant •• d . Th. computational r •• ult. with the application of GPC in a high purity binary distillation column have demon.trat.d a ad.quat. clos.d-loop proc.ss b.haviour with small ov.rshoot, fa.t r.spon.. and th.r.for., show.d it •• lf a v.ry robust controll.r. The incr.as. in the pr.diction horizon improv.d .v.n mora the clo ••d-loop dynamic of the nonlin.ar proc •••• 6. REFERENCES Clark.,D.W. and P . J.Gawthrop (1975) . S.lf-tuning controll.r. Proc. IEE. Clarke,D.W. and C.Mohtadi (1989). Prop.rti.. of g.n.raliz.d pr.dictiv. control. Automatica,25, 859. Coelho,A.A.R. (1991). Adaptiv. control for multivariabl. proc... • th.or.tical analy.i. and .imulation . D. Phil. Th •• i., UNICAMP, Brazil. Cri.all.,O.D., D.E.S.borg and D.A.M.llichamp (1990). Th.or.tical analysis of long-rang. pr.dictiv. controll.r.. Am.rican Control Conferenc., San Di.go. Cutler,C.R. and B.C. Ramak.r (1980). Dynamic matrix control - a comput.r control algorithm. JACC, San Franci.co. Dugard,L. G.C.Goodwin and x.Xianya (1984) . Th. role of the int.ractor matrix in multivariabl. .tocha.tic adaptive control. Automatica, 20, 701. Kinnaert,M . (1989). Adaptiv. g.n.raliz.d pr.dictiv. controll.r for MIMO .y.tem •. Int. J. Control, 50. Koivo,H.N. (1980). A multivariabl • • • If-tuning controller. Automatica. McInto.h,A.R., D.G . Fish.r and S.L.Shah (1989). Se1.ction of tuning param.t.r. for adaptive generaliz.d pr.dictiv. control. Am.rican Cont. Conf., San Diego. Richal.t,J . A. , A. Rault, J.L.T.stud and J.Papon (1978). Mod. 1 pr.dictiv. h.uristic control. proc •••• applications to an industrial Automatica, 14, 413. Shah,S.L., C.Mohtadi, and D.W.Clarke (1987) • Multivariabl. adaptive control without a prior knowl.dg. of d.lay matrix. Syst.ms & Control L.tt.r., 9, 295. Wolovich,W . A. and P. L.Falb (1976). Invariants and canonical form. und.r dynamic comp.n.ation. SIAM J. Cont. and Opt., 14, 996. Yd.ti.,B.E. (1984). Ext.nd.d horizon adaptive control. 9 World Congr.ss of IFAC, Hungary .

Fig. 2. Configuration control of the column

:::l~:~ o

ooo~

o

0000

di.tillation

CB' Me al / "

'''d tlow chona. , •• 4 'Iow <:"ono. - ------- -- -- -- - - - - 12' L..J__________________

o (houri

(hourl

0 · 00100

130

0 . 00075 o . ooo~o

o. 0002~ , .. 4 concentration chonge t t •• d conc'ntrotlon change O. OOOO().L..-'---- -- -- - - -- - - -- - - - - - - - 1201----'-------------- -_______

2

(hourl

130

0 . 0 0 <4

o

'B~'2>

00 3

o 002 0 . 00 1

)

"'pol., c•••••

.. tpoi nt chanQe

0 .00 0

'20L-L-------------__

Fig. 3. Bottom output and input tor .tep chang••

'.O OOO~o,m he

' ' ' ' O t",M OI/ . 12.~O~

_2

~

__LL____

~

Sampllno I nt.r'lol

::::~: :::~

-,

____

.0

- . ~----~~------------~

' 00

'00

.0 Sampling i nt .,vol

o

2 (hourl

0

(hour)

Output and .. , po i nt (')

0.003

'B'"' he

0 . 0 02

0 . 00'

'0 Sampling i nt.r v ol

100 0.00 0

Sampling Inflrvol

' ' '[

aB' Mc., / '

~:: .~ o

' 0

(h ou rl

rig. 1. Proc ••• r •• pons. for HY • HU , r • 0 (top) and MY - HU , r ~ 0 (bottom)

rig . 4 . OUtput. and input. for a feed flow chang.

96