Indian Journal of Chemical Technology Vol. 6, July 1999, pp. 225-236

) A state space model based multistep adaptive predictive controller (MAPC) with

~.

disturbance modeling and Kalman filter prediction

N Rao Sripadaa• & D Grant Fisherb

"Department of Chemical Engineering, Indian Institute of Technology, Madras 600 036, India

hDepartment of Chemical Engineering, University of Alberta, Edmonton, Canada T6G 2G6

Received 21 November 1997; accepted 10 JUlie 1999

A multistep adapti ve predi ctive control strategy ba'sed on a state space model of the process has been developed. [t can be compared with the Generalized Predictive Control algorithm. The emphasis in the development of the proposed control scheme is on modeling and elimination of disturbances. In the proposed scheme any prior information regarding the disturbances can be incorporated (by specifying certain polynomials and/or the noise covariances). If no prior information is available then the unknown unmodeled effects (such as noise, unmeasured load-disturbances and model process mi smatch) can be represented by a residual model which can best be . identified in a two-stage setting. This approach leads to satisfactory modeling of disturbances and good regulation via predictive control. Some important features of the proposed algorithm are: (i) it uses a state space model which allows separate modeling of u-to-y process dynamics, process and measurement noi se; thi s is not possible in an ARMAX-type input/output model where process and measurement noise appear lumped in the noise po lynomial ; (ii) it uses a Kalman Filter (KF) to generate the predictions of the output; the KF can be eas il y tuned via noise covariances and is a simpler and better alternative to specifying or estimating a noise polynomial: (i ii ) there is no need to solve a Diophantine identity on-line; the result is reduced computation; and (iv) if residu al modeling is used it leads to simpler and improved way of handling disturbances. The proposed control algorithm is presented for the single-input, single-output case. Applying the algorithm to multi variable processes is straightforward . Simulation examples are included to illustrate the advantages and performance of the proposed control scheme.

A self-tuning adaptive controller, broadly speaking, is a combinati on of a parameter estimation scheme and a suitable control strategy. The progress self-tuning control has made over the years has been summarized in a number of excellent review articles and text booksu 8.9.tH Self-tuning control has evolved from single-step predictive methods such as self-tuning regulator2 to multistep predictive control methods such as GPC4

..'i. A large impetus for developing the latter class of algorithms has come from such industrially successful but non-adaptive, model-based multistep predictive. control techniques such as DMC6

, IDCOM t7 and MOCCA22. However a single

self-tuning control algorithm that can be applied to a major class of practical processes has not been developed yet, although algorithms such as GPC4

.5

perform well in idealized situations. The objective of the work reported here is to

develop a comprehensive self-tuning adaptive

*For correspondence The lirst aUllior is presently with Padmasri Dr B Y Raju Institute o f Techno logy. Narsapur. Medak Distri ct. AP, India

controller that can be applied to practical industrial processes . The result is the multistep adaptive predictive controller (MAPC) reported in this paper. The major issues to be contended with in the application of adaptive control to industrial processes are handling of: (i) process time delays, (ii) noise, (iii) lo"ad-disturbances, (iv) parameter estimation, and (v) achieving user specified control performance. Some of these issues are briefly discussed before presenting an overview of the MAPC reported in this paper.

Process time delays Time delays are common in process systems and

are associated with flows and vary significantly . Therefore they are difficult to specify in advance and cause control performance to deteriorate. One approach to control processes with time delays is to use predictive control. However, single-step pledictive control methods are very sensitive to errors in the a priori es timate of the time delay and hence control detuning is often required. An alternative is to

226 INDIAN 1. CHEM. TECHNOL., JULY 1999

use predictive control based on the minimization of multi-stage cost function.

Process disturbances Regulation (or disturbance rejection) is more

important in industry than setpoint tracking. Offset­free rejection of load-disturbances with good dynamic behaviour requires the use of a proper load­disturbance internal model (cf. internal mod~l ·

principle) . Often the load disturbances are assumed to change as random steps at random times and although this gives good results, it is not enough to handle more complicated disturbances such as random exponential rises or load-disturbances that affect the process via a load-disturbance transfer function. The minimum variance strategy handles noise optimally if correct noise structure is employed in prediction. However, the use of a fixed noise model is often unrealistic as the spectral characteristics of the noise vary significantly with time since the noise results from several disturbance sources. On-line identificatiorl of the noise model is usually done in self-tuning control but is associated with convergence difficulties and can give poor results. As an alternative a state space model and a Kalman Filter can be used to generate mjnimum variance estimates of the process output and has the advantage that it is easy to tune on-line by changing the noise covariances. How the disturbances are modeled can influence both model parameter estimation as well as subsequent controller synthesis.

Model identification and parameter estimation Model identification is an important step in

adaptive control and is crucial to its success. Recursive prediction error methods are often used in on-line parameter estimation and one such algorithm which is widely used is the recursive least squares (RLS) or its variant, the recursive extended least squares (RELS). A number' of adhoc as well as theoretical modifications are added to the standard parameter estimation methods to enhance their performance in the self-tuning context. Shah and Cluett20 review a number of modified least squares estimation schemes. The identification scheme requires : (i) selection of an appropriate model structure, and (ii) estimation of selected model parameters by a suitable (ie. robust) estimation method.

Process constraints Constraints are important in process control and

when present they may cause problems in the

identification part of an adaptive controller. The proposed control algorithm can easil:}' incorporate constraints, but the contol calculations then are iterative. In the proposed scheme constraints are handled by the supervisory system, which adjusts the reference input to satisfy constraints. The supervisory system can also be used to generate other types of reference trajectories (eg. a filtered setpoint).

Using the supervisory system to implement the constraints has the advantage of dividing the control problem into two consecutive tasks: first the calculations on the reference trajectory ; and second the calculation of the control action so that the output tracks the computed reference trajectory. One advantage of this separation is that the supervisory control calculations can be done at a different (longer) sampling interval and can be stopped during periods of steady state operation .

In addition to the problems of time delays and disturbances, a practical adaptive control scheme must be able to: (i) overcome the problems caused by Non Minimum Phase (NMP) or unstable zeros, which arise in discrete time control due to fast sampling or which may be present inherently in the process; and (ii) handle unstable and poorly damped processes; although most processes are characterized by stable, well-damped, dead-time dominated dynamics, unstable and/or complex poles are not uncommon (eg. reactor control) and can present significant control difficulties.

In the development of the proposed control algorithm all the above issues have been addressed. Consideration of th~se issues have led to the adoption of following ideas in the present work: (i) use of multistep or longrange predictive control; (ii) state space model-based representation of process dynamics and other unmodeled effects (such as noise, load-disturbances and, model process mismatch ; (iii) use of robust estimation; (iv) use of prior information regarding for example, noise and load disturbances affecting the process; and (v) in the absence of any prior information use of residual modeling and two­stage identification. Although some of these ideas have been in use for some time, the major contribution of the present work is better disturbance modeling and treatment and integration of the various ideas into one single algorithm, so that it can be

SRIPADA & FISHER: MULTISTEP ADAPTIVE PREDICTIVE CONTROLLER 227

"

Conltnlntl

Proceu

','<-- -------' ",

Model TO

S)'1tem model .~meten

Fig. I- A high- level block diagram o/" the proposed MAPC.

applied to a majority of industria l processes. A number of researchers have considered state space modeling in model-based predict ive contro l area l

:1- 15

but in se lf-tuning control sta te space modeling has not been used much (except, for example, in LQG se lf­tuners), espec ially in the predictive control context.

Although the proposed control algorithm IS

inherently a multi va riabl e control algo rithm, it is presented for the si ngle- input. single-output (SISO) case for ease of under:--tandi ng and presenta ti on. The design of MAPC involves the follow ing ste ps: (i) process representa t ion (or model se lec ti on for es timati on and co ntroller synthes is), (ii ) model identification and parameter es timati on, (i ii ) predict ion of the process ou tput. (iv) control ca lculations, (v) the superv isory system ca lculati ons, and (vi) the executi ve subsystem. A hi gh-leve l sc hematic block diagram of MAPC is shown in Fig . I and the structure of the rest of paper cl ose ly foll ows thi s figure. The nex t six. sec ti ons of the paper discuss the six. main s t e p ~ of the proposed sc heme. Thi s is foll owed by a discussion of the convergence and stability issues of MAPC: and finally so me SISO simulated examp les to demonstrate some of its feature s.

Process representation The process IS assumed to be adequate ly

represe nted by the loca ll y I inear model:

. . . ( I )

where A( :::· ' )= I+{/ ,:::-'+ ... +0".::-" and B(:::-')=h ,.::-'+ .. . +17 ,,:::-". It may be not ed th at A and B can always be writt en thi s way by settin g appropriate coeffi cients to ze ro if nccessary. k is process pure time del ay in nUlllber of sampling interva ls. r(l) is a residu al that in c ludes all the unmodeled effec ts such as process and measurement nOise, unmeasured load­di slllrhances and model process mi smatch (MPM ).

Depend i ng on t he assumpti ons made about the di :-. turban ces affecting the process, r (l) can assu me seve ral forms kg. MA, 1M A, MA or IMA wi th a bias etc.). All thesc mode ls to represe nt di sturbances have ad vantages ;lIld di sad vantages. In the approac h foll owed in thi" paper it is dec ideu to incorporate all ava ilable pri or informati on regarding disturbances. The emphasis i:-- on a formulation which requ ires millimum or no prior informati on regardin g the dis turbances Foll owin g M'Saad e l 01 .16 it is decided ( 0 mudel r( I) hy :

rr(l )=Cc( 1)/1) ... (2)

F, C and D are polynomials in .::-1 (note that the argument .:: -1 is omitted for simpli city). F is a load­di sturbance intern al model. poss ibl y with -zeros on the boundary of the unit c ircle, ie. F=O when Z-I = I. It can n10del anything from a simple dc bias, random steps at random times or more general di sturbances including peri odic di sturbances. C and D are stable

228 INDIAN J. Cl-IEM. TECHNOL. , JULY 1999

polynomi als used to mode l co loured process noise. It may also be noted that if F is se t to Ll, then D is used to model load-di sturbances that have dynamics . {e(t) } is a zero mean, random, white no ise sequence (when

C=D= I and F=Ll= 1-z.' I, ret) models Brownian motion) or a random seq uence fo llowi ng Poisson distribution whic h means it is non-zero only at iso lated instants of time (used to mode l, for exampl e, random steps at random times) . Note th at r(! ) is pure ly a de te rmini sti c process when Fr(t)=O and pure ly a stoc hastic process whenF= I.

Thi s type of mode l is better su ited fo r industrial process cont ro l appli ca ti ons than a simpl e ARIMAX mode l used, for example, in GPc. Note that the above model is a ge nerali zation of the A RIMAX mode l. Alt hough one can model an arbitrary di sturbance using the ARlMAX mode l, the C polynomial may have to be of very hi gh order (o r equivalently A and B have to be over-parameterized to absorb D) because of the lack of denominator po lyn omial D. In what

fo llows, F is taken to be Ll, for simplicity. One must know or estimate F, C and D

polynomia ls for use in the mode l and hence the control scheme. One of the contributions of the paper is to provide a method (via a state space model, noise covari ances and KF) to avoid es timation and/or to pro vide a way to obta in the required information e ffi ciently (eg. residual modeling and two-stage identification). If one has prior information regarding the C and D (and F) polynomials this information can be direc tly used in the model. Alternative ly most of the times a t least the noise co vari ances can be specified . Then a stochastic state space model and a KF can be used for prediction. If no prior information is available at a ll, one can absorb C and D in F and est imate an AR model based on ret). Thi s is best done in a two-stage selling . ret) can be viewed as a measured s ignal and the identificat ion of the residual ll)odel is likely to be more successful than estimating the noise polynomia l in a single stage using Pseudo Linear Regression methods (eg. RELS) where estimates of the noise are used . The advantages and performance of residual modeli ng and two-stage

ide ntification have been reported by Sripada and Fi sher2J . In thi s paper, the following three cases are considered .

Case I

Pri or in fo rmati on in the form of C, D (and F=Ll) is ava il abl e . One can then absorb F and D in A and B

and wr ite down the fo ll owi ng state space model in observable canoni ca l form (Am=AFD and Bm=BFD where both Am and Bm are adj usted to be of degree n ):

x(t+ I )=<DX(t)+!\LlIl(l )+r e(t) vU)=Hx(t)+e(t )

where

() 0 o o

o

o l

o

r= and H = [00· .. 0 I]

o

(1

.. . (3)

.. . (4)

The dimensions of <D, A. , r and H are respectively

(n+k)x( ll+k), (n+k)x I, (n+k)x I and I x(n+k) . This state space mode l is already in innovations form and it is stra ightforward to see that the KF is g iven by,

x(t+ I /1 )=(l>x(l/t-1 )+A.Llu(t)+r[y(t)-Hx(tlt- l)] ... (5)

y(t)=Hx(tll-1 )+e(l) .. . (6)

Case 2

No prior information regarding C is available. D mayor lilay not be known . If D is known then it is absorbed in A and B as shown in casel . If D is

unknown then either it may simply be set to J or estimated along with A and B by over-parameterizing them. One can then use the following stochastic state space model (C is mode led by separate process and measurement noises) ,

x(t+ I )=cI>X(t)+A.LlU(t)+IW(t) .. . (7)

•

SR IPADA & FISHER: MULTISTEP ADAPTIVE PR EDICTIVE CONTROLLER 229

y(f)=Hx(f)+ V(f) . . . (8)

The state space model matrices (<1> and 1\) are d~fi ned as earlier based on the know ledge of A, B, D and F polynomilas . Here, for simplicity, r is set to I and w(t) is taken as [w I (f) ... wn(t)]', ie. w(t) is an n­dimens ional noise vector. For this case the full KF equations have to be used as di scussed in the fo ll ow ing sections .

Case 3 Absolutely no prior information is avai lable. Then

one can absorb C and D in F and wri te ~F,r(f)=£(t). Then the input/output model becomes,

A F,~ \'(t)=B F,!1I1(t-k)+£(t)

or Antv(t)=B" ,!1u(t-k)+£(t) and once aga1l1 one gets a . ' tate space model in innovations form as in case I. The state space model matrices must be written properly as before from the knowledge of the po lynomi als in the in pu t/ou tput model.

Rell1arks:(i ) If there is an auxiliary measured disturbance input. it can eas il y be included in the above modeh.

(ii) As discussed by Sripada and Fisher) -1 th is ki nd of mode ling disturbances is very att ractive when there is no prior informati on regardin g the disturbances . A and B are identified in a fi rst stage with proper precautions to obta in good estimates of the u-to-y model. The res idual model, F, , is best ident ifi ed in a second stage. Residua l model ing and two-stage identification has been found 10 give improved results compared with estimating a noise polynomial in a si ngle stage.

In what follows the proposed algo rithm IS presented fo r case 2. With slight modification (by noting that the sta te space model is already in innovations form) it can be adapted to the other two cases.

One-step-ahead Kalil/WI Filler prediclion A Kalman Filter can be used to etimate the states

of the stochas tic state space mode l gi ven by Eqs (7)

and (8) . The one-s tep-ahead pred icti on of the state has also been give n Il.lh.

x(t+ I 11)=</)x( /// - 1 )+t\~I1( I)+K ( l )e(t)

y(r)= Hx( I/I -1 )+e(t)

. .. (9)

... ( 10)

whi ch is opti mal ill the se nse of minImum van ance

estimation when the ; ~ all11an galll , K (f) , IS obtained from.

... ( II )

where the state covariance matri x. P(r/t-I), is updated accord ing to.

P(r+ I /1) =</)P(I/ I - 1 )¢ '- K(I )H P(r/t-I )¢'+rRw r' ... ( 12)

Here P{O/-I) is the initial covariance matrix and is chosen such that P«(}/- I )~O. The sequence {e(1):=y(l)­

H.r(l /t- I ») i:-, the inno vations sequence.

/(cl/I(/rk : If the mat rices, ¢, t\ , rand Hare consta nt then the stead y qate Kalman Filter can be med. whi ch gives the same assymptotic error covariance of the est imated state vector as the time varying. optimal KF. The steady state Kalman gain . K. is ca lculatcd using

.. . ( IJ )

whcrc " i ~ the ~ t l'ady state value or the covariance m;lt rix an d is compu ted by so lving the algebraic RiCGl tti cquation:

.. . (14)

It may he noted that K and P can be computed (/ pri(Jri and thereby reducc computation.

Model identification and parameter estimati on The ovcr;dl process model is composed of a mode l

or the u-t o-) process dynamics and a model or the disturhancc:-, . For ad~lp ti ve contro l, the parameters of these mode ls arc csti mated on-line from measured input/output data. {\ '(I)) and {u(t)). Such a mode l ide ntifi cation step forms an important part or an adapt ive contr(l l :-,c hemc and in the proposed scheme is show n a:-, the mode l ident ifica ti on block in Fig . I. The improved leas t squares (lLS ) algorithm23 is used as opposed to qandard RLS in thi s study ..

When the stochasti c state space model given in case 2 is uscd (where the C polynomial is modeled via the proces~ and mcas urement noise terms w(r) and 1'(1) res pecti vc ly) thc u-to-y dynam ic model given by

A and B pol ynomi als has to be es timated. If D is known then the data are filt ered by D before pass ing to the ILS al gorithm . If D is not know n, then either D can be assumed to be I or ca n be esti mated simultaneous ly wit h !\ and B by over-parameteri zing these polynomials.

230 INDIAN J. CHEM. TECHNOL., JULY 1999

i

j i:

, 1 -

o~ ~ f \ 1 l -1

0 50 100 150 200 250 300 t

Fig. 2-Selpoinl Iracking ability of MAPC; system res ponse: cO l1l i nuous. se l poi Ii l:dashed.

1. 5 .-----~-----------~---. ;

1 -

i: 0.5 I~ t~ !~~1 .! ~ ~ ~I"[i~

-05 L1 --~--~--~--~--~-~' o 50 100 150 200 250 300 t

, ::~~\~~r~Anl~~~IA~ j <., 'I rlll' 'WI '~i _O.~L----:~----,--:-:----~--~---:-:-:---~' o 50 100 150 200 250 300

t

Fig. 3-Performance of non-adapti ve MAPC in the presence of noise; system response:continuous, setpoil1l :clasheci .

Output prediction The MAPC uses an output predic ti on block as

shown in Fig . I . It is poss ible to obtain multistep­ahead prediction of the state, x(t+ilt) and hence y(t+ilt) for i> I using the KF from measurements available up to time t. The KF uses the one-step­ahead estimate of the sta te, x(t+llt) (di scussed earl ier), to give minimum variance es timates of the

future states and is given b/:

x(t+i1t)=$ix(tlt-1 )+$i-I K(t)e(t)+LjE [I , i) <l>i-j Aflu(t+j-I ) . ,. (15)

and y(t+i1 t)=Hx(t+ilt ) .. . ( 16)

for i2 1. With a little a lgebra one can write down the foll owing prediction equation for time t+N I to t+N2 in vec tor matri x notal ion

y=Gu+J ... ( 17)

where y=(y(t+N llt ), ... , y( t+N21t) )' , U= [flll (t) , ... ,

flu (t+Nu- I )], and J=[/( f+N I), ... , j( t+N2)], where j(t+i)

is the free response o f the system at t+i, based on past

inputs and is g iven by j( t+i )=H[$ 'x( rlr-1 )+<p,.1 K(r)e(r) ] . (Gu) IS d ue to the

contro l inc re ments to be implemented. G is a lower

triangul ar matri x w ith otherwi se. G is g ive n by

gN I

gN I + 1 G=

o gN I

gN2 gN2- 1

gij=H$i·jA

o o

for j 5, i

o o

and

gN 2 -Nu + 1

g'j=O

w he re {g i } are the step response coeffi c ients o f the sys tem (based on u-to-y dynamic mode l).

Remarks : (i) The proposed contro l sc heme uses a contro l h()riZ()Il. Nu, beyond whi ch the cont rol

inc rements are set to zero, ie. flu(t+i- l )=0 fo r i>Nu. (i i) Note that the output predic ti on does not req uire

so lvin g a s ingle or a bank of Dioph antine identities,

ego as in GPC ; in stead the pred icti on requires

ca lcul ati on of the Kalman F ilte r ga in, K(t), and $ i,

iE [I ,N2] whi c h a re the maj or computational steps.

However, ac tua l computation of $ i is simplified by noting that it is spa rse wih a spec ia l (canonical ) s truc ture. The worst case computation for obta ining the output predictio n g ive n by Eq . ( 17), for 11=3, N2= 10 and Nu=2, required for the proposed a lgorithm

(without ex pl o iting the spec ial structure of $ and exc ludin g KF cal culati o ns) are approx imate ly 8 10 flo ps whe re as G PC without usin g recursion of Diophant ine Identity requires about 1345 fl ops.

The multistep predictive controller The mult is tep predi cti ve contro lle r shown in F ig.

generates a trajec tory o f present and future contro l ac ti o ns such that the predic ted output trajec tory produced by the Kalman F ilte r is as close to [he re fe rence trajec to ry as poss ible ( in the sense that a user spec ified cost functi on is minimi zed). The supervisory sys te m (a lso shown in F ig . I) supplies the

refe rence traj ec tory vrd=[Yret{ t+N I), ... , )'ref (t+N2) ]' at time t . The cont ro l strategy is based on the minimi za tion of the fo ll ow ing multis tep. weighted, qu adrati c cost func ti on which is s imil ar to the one used in GPC~5 , DMC", MOCCA 22 etc.:

J(N I .N2.NU,r y.r Jl )=( 1/2)[ [vrcry ]TylVrcrY]+ll TuuJ . . . ( 18)

•

SRIPADA & FISHER: MULTISTEP ADAPTIVE PREDICTIVE CONTROLLER 23 1

1 .51--~--~--~ __ ~ _ __ -_

1 -

i: 0.5

o

O ··r-;'-~--~----;--~--~--~--

0.2

~ 0

~.2

Fig. 4-Performance of GPC in the presence of noise; system response:contlnuous. setpoint :dashed.

Here N22N I ?I; Ny:=N2-N I + I is a finite output

horizon; Nu is a finit e control horizon with Nu::;Ny ;

r y and r u are poss ibly time varyi ng weighting matrices on the track ing errors and control act ions respectively. Substituting for y by Eq . ( 17) and setting dJ/du=O, one obtains the followin g control-law: u=C'[Yrerf) ... ( 19)

where

C* :=[CT yc +rurICTy ... (20)

The control-law given by Eq. (19) involves very little computati on when the process model parameters are fix ed, si nce C' can then be computed a priori. In the self-tuning case, since the parameters change on­I' ne, C ' needs to be recalculated at every instant. It may be noted that the calculation of C' involves a matrix inversion which is a computationally significant step. The dimension of C is NyxNu but the dimension of the matrix to be inverted is only NuxNu . Because of the moving-window formulation it is generally possible to choose Ny as low as 3 or 5 and

Nu as low as 1 or 2. Thus C is usually a small matrix of the order of 5x2 and the matrix to be inverted is only of the order 2x2. The control-law is usually implemented in a receding-horizon sense, ie. only the

first control increment, Ll.u(t), is implemented and all the calculations are repeated at the next sampling interval. This means that only the product of the first row of C* and [Y,erf] needs to be computed .

Remarks: (i) The receding-horizon approach makes the control algori thm time invariant, ie . the same

cont ro l-l aw (c f. Eq. ( 19» is used at each instant, unlike in a fixed-hori zon LQG po licy.

(ii ) When the comp lete refe rence trajec tory IS

known ahead of time and the process mode l IS

relativel y time invariant with good parameters it is some times poss ible to impl ement the whole control vector. Il , ove r the contro l horizon of Nu and repeat the calcu lations every Nil intervals . Thi s saves' some computation .

Choice oj The controller parameters- The para­

meters NI , N2 , Nu and the weighting matrices r y and

r" allow one to shape the dynamic response of the controlled process as wel l as to stabilize NMP processes or processes with open-loop un stable or poorly damped dynamics . If the output hori zon and the contro l horizon are equal, the control problem is essenti ally s() lvi ng a set of linear algebraic equations for the contro l vector. By choosing the contro l hori zon less than the output hori zon in the problem formulation one then has to so lve an over-determined set of linear algebraic equations for the control vector. The proposed method solves this problem using the weighted least squares approach . Thus by assuming that the contro l increments beyond the control hori zon to be zero instead of allowing them to be free, the method derives a lot of fl exibility as well as reduces computati on . In particular this method

allows one to stabili ze NMP processes. Choosing Ll. u(t+i-l)=O for i>Nu can also be interpreted as using 00

-weighting on Ll.u(t+i-I) , iE [Nu+ I , N2] in the cost function .

Output horizon (N I, N2)-It has already been pointed out that using prediction over a range of points in the controller cost function rather than a single point makes the controller robust to incorrect specification of the process dead-time. If the time delay, k, of the process is exactly known then one can take NI =(k+ I ), since in that case the first future output that is directly affected by the current control action is at t+k+ I. Thus using y(t+ I It) to y(t+klt) in the cost function is redundant and hence by omitting these some computat ion can be saved . If the time delay is expected to vary or is not known wi th certa inity then one can set N I = I . In that case, even though C is singul ar a soluti on can be forced to exist by a proper choice of the weighting matrices (eg. c hoose rll=I). This is possible because of the over­determined set of equations for the solution of the control vec tor. Thus the proposed controller can perform wel l e ven if the knowledge of the process

232 INDIAN 1. CHEM. TECHNOL.. JULY 19')9

1.5

1 - V i r~\ i i 0.5

, , ~, ~\to/J'~ 0 ,........-

I -0 .5

0 50 100 150 200 250 300 t

O .~

0.2

; 0 ·\~W+··I -0.2 ~ I ~(\H , I -0 ~O -...!

50 100 150 200 250 300 I

Fig. 5- Pe rformance o f MAPC in the presence of no ise: >ys te rn response:co ntinuou s, setpoin t :das hed.

time dela: i" poor. Single-step cost functi ons do not offer thi " fl exibilit y and contro llers based on these may fail or r erform poorl y ' if the ti me delay is not ex actl y kn own , For N!VIP processes. NI can be set beyond the in itial wrong-way respon se in the rroces~ "tep response so that onl y the poi nts that approxh the "e tpoillt in the right directi on are included in the controll er cos t fun cti on.

N2 can be cho"e n "0 that N, ~5 to 10. A Itemati ve Iy th e output hor izo n. N) . C III be chosen e'-Ju al to the ri se time of the process step respon se and N2 is set approproate ly. Thi" i" adequate in most cases . [f the proces" i" ope n-loop un"t :lble or ha" co mple x poles. then ob\ ioLt"l y 1l1ore puint s muq be included in the llut Jut horizo n. One ap proach for se lection of N, is through a "i1l1ul :tt ion stud y m ing a rough llIodel of the proce"s. A .~ a ddaLt It N I can be "e l to I and N2 ca n be se t to

10. Mo"t pmce""e" gi\'l' "table response and good performance wi th th e"e "e!lings. If more conse rvative perforlllance IS de"irL'd. N2 ca n be incre:!sed ((l cover the entire () pen-I oup step response.

Crll/lm! !/(I,.i:: (ln (N{()- Usuall y there i" no ad\'an tage in takin g N{(>n where n is the order of the process . [ I' th.: re is no weig htin g th en for N{( ?n the controller gi\'l' " output dead-beat cont ro l. [n most ca"e" i\'{( can be se t to I or 2.

H'c igllfin g 1I/(//,.icl'S 0 -) . f u)- The we ighting 1l1a tri ce" (f ) :l1ll1 r ll ) arc es peciall y use ful in the mLtl ti \'ari ahk Cd\e becau"e they ca n be used to specify lh al an ) partic ul ar out put (i nput ) is more or less import ant th an th e uther outputs (input s) (ie.

re lali w we ighting among llu tput s and/or input s). The weighting 1l1atrices G ill also be u.'ed to pe nalize the control ;lcti on or to ensure that soft co nstraint s on the process var i ; l hle~ arc not violated. In numerical terms. the wc i!,! htin!,! matri ces arL' very helpf I to fo rce a stahle slliutilln to lhe co ntrol problem. For example. when (; i ~ pomly condit ioncd or al mos t sin gular then in the absence of any we ighting Eg . (1 9) can give poor solu tions. This 11Iay result 111 excess ive oscill ati ons or ri nging in the cont ro l in put or even closed-loop instab ilil y. By choosing t e weight ing 11Iatrices appropriate ly the matrix in version in Eq . (20) can be done more robu'stly (i e. th conditi on number of [CT)G+rllJ can be s ign ifica ntly reduced ). This can be done from a kn owledge of the process time scale (ie. sa mpling interval ). a rough step response of the proces.· and the desi red cl osed-loop performance. In the SISO case it is s impler to choose f )=! and rll=/~/ . I~> {) . so that

C'=[C'C+I,Jr IC'

in Eq . (20 ). There i ~ only one tunin g parameter. A. to be adjust ed on-line . If Ic=O th en there is no weighti ng on the control acti on. If A is large then the control action is heav il y pena li zed.

The superviso ry sys tem The MAPC assumes th aI the reference trajectory.

\ ·,,·I=[Yrc ,{r+N I) . .. .. .l'rd (I+N2» ), is available at time r. The supervisory sys tem whi ch is shown in Fig. I ge llerate s th ese values. The refence traj ectory may be const ant and eq u;t1 to a se tpo int ow r the output hori zon or lIlay \'ar). r or exa mple , the superv isory sy"tem can be used to generat e a sllloothed setpoint traj ec tory as in IDCOM 17 wh ere it is cons id ered that a smoothed approach to th e current se tpo inl. \ '51" is requ ired fro m the current output. In thi.\ case the ~ uperv l so ry system ca n ca lculate the reference trajectory from the fo ll ow in g first order lag model: .

.. . (2 1)

\',d{r+i)=o.\' rd(t+i-I )+( 1-0.)\\1' (1 ) . i= [. 2, .... N2 ... (22 )

where o.E [0, 1 j. MAPC is thu s capable of both time varying and time invar iant refe rence in put s. Another case is that tile refe rence trajec tory may be speci fi ed {/ priori by the process operat or. This :Jbility to foll ow prespecified reference traj ec tor ies IS useful 111

prac ti ca l applicati oll s and there are a number of real examples where such an abilit y is needed. Some examples are robot motion and batc h n.:actor control

•

SR IPADA & FISHER: MULTISTEP ADAPTIVE PREDICTIVE CONTROLLER 233

I

0.5 \1 ~ 0 l -0.5

- I 0 50 100 150 200 250

Fig. 6-Performance of non-adapti ve MAPC in the presence of 1 0ad-c1i sturbancc~ (No prior information ).

where the temperature of the batch may have to follow a spec ified trajec tory.

The reference inputs can also be adj usted on-li ne to satisfy process constrai nts. Constra int s on process variab les (eg. u and y ) are ve ry important in practical app lications and are usuall y ignored in the adapti ve control literature. Ampl itude constrain ts on the

contro l signal. II , a re espec iall y important. Satu rati on of the contro l s ignal may appear in an adapti ve sc he me_ eg .. ( i) during start-up or after parameters have converged; ( ii ) due to bad in itia lizati on; and (iii ) due to unatt ain ab le des ired performance (ie . Yref). When saturation of the input s ignal occurs the process ceases to be linear and its effec t is particularly de lete ri ous on the parameter estimation, as the esti ma tor then tends to identi fy a zero-gain process. The process can be fo rced to remain in the linear region by impos ing smoother reference changes or ca lcul ating the reference signa l such that the limits on u are never vio lated. The supervisory system can be used to ac hieve thi s task. The discuss ion of the procedure to do this is beyond the scope of the present papt'[o

It may b noted that the constraints ~an be handled by the predictive contro ller direc tl y. The advantage of usi ng the supervisory sys tem is that it admits more fl ex ibility.

The AI executive One of the requi rements of a practical adaptive

contro ller is that it should be easy to commjssion and operate. It is common in adaptive control practice to use heuri stic fixes or safe ty j ackets (eg. in parameter estimation) so that the adapti ve controller will work smoothly over a broader range of conditions . To do thi s it is usuall y sugges ted " " ,'9 that another block be added to di rect the overall operation of the adaptive contro LLer as shown in Fig. 1. T he functions of such a block can be mani fo ld , eg., ( i) to start-up or commission the adapti ve cont roller; (i i) to perform

mode-switch ing; and (ii i) to supervIse model

identification etc . T he knowledge based systems (AI) provide a

suitab le framework to imple ment these fun ctions. The deve lopme nt of the AI executi ve is not the obj ective of the work reported here. The AI executive is only highli ghted he re fo r the sake of completeness.

Conve rgence and stabili ty properties of MAPC The stabil ity and performance of MAPC depend in

a co mpl icated way on the convergence of the Kalman Filter and the stab il ity of the predic tive contro lle r pl us the paramete r estimates . If the KF con verges and g ives asymptotica ll y offset-free pred icti ons of the out put then the predic ti ve controller ensures that there is no contro l offset. T he dynamic performance of the contro ll er depe nds on its parameters (eg. N I, N2, Nu and the weighting matrices). The dynamic behaviour of the KF depends on the in iti al condi ti ons (eg.

P(O/- I» and the no ise covariances. No concrete results regarding the stability of the compos ite system are ava il able at thi s stage. However some remarks that are he lpful in understand ing the overall stability may be made as follows. The convergence propert ies of the KF are di scussed by Goodwin and Si n 'o and Kumar and varaiya t2

. The needed result is su mma­ri zed in the fo ll owing theorem.

Theoreml 2-Let Q be a square-root of Rw (ie .

QQ'=Rw) . Suppose (<l>,Q) is reachable, (ctJ ,H) is observable and Rv>O. Then lim, __ P(t+IIt)=P and P is the unique, non-negati ve definite solution of the steady sta te or algebra ic Riccatti equati on :

P=ct>[P-PH'(HPH'+Rvr ' H P]ct>'+Rw

In particular P is independe nt of the ini tial covariance matrix P(O/- l ).

Thus if the conditions of the theorem hold, which is not hard to verify, then the KF converges to a time invariant filter with steady state covariance given by P and the steady state gain K given by Eq. (1 3) ,

In the time varying or self-tuning case it is obvious that in order to gi ve proper estimates of the states (or predic tion .; of the output), the estimates of the process parameters should converge to those of the actual process in some sense (eg. frequency domain terms) . In the self-tuning case the KF gain matrix is computed on the basis of latest parameter estimates and the covari ance PC) available from the previous step . If the paramete r estimates converge, the KF gain converges to some steady state va lue as l~oo .

234 INDI AN J. CHEM. TECHNOL. , JULY 1999

lr------.------~------~~-----~--~~

0.5

i or-----~,L---_.,----------JL-~r----~L---

-0.5

Fig. 7-Performance of non-adapt ive MAPC in the presence of IOdd-di st urbances (Exact prior in fo rmati on).

Si nce the adapti ve contro ller is ex plic it at leas t a local stability resu lt can be roughly stated as fo ll ows: If the KF converges and the basic control ler is stable with in iti al es ti mates of the process parameters, 80 ,

and the converged (or steady state) KF, then it is reasonable to expect that the ti me varyi ng adapti ve cont ro ller wou ld be stable provided that 8(t) lies in a stab le region around 80 . The stabil ity of the controll er depends only on the process u-to-y model, ie. 8(t) . The parameter projecti on feature of ILS can be used to ensure that 8(t) stays in a stabl e region around 80 or some ot her nominal mode l for wh ich the sche me is f0und stab le, for all time. For simple processes (eg. first or second order) it is relative ly easy to define such regions.

Single-input, single-output simulation examples Si mulat ion examples are d iscussed in this section

which ill ust ra te .T lected fea tures of the proposed control sc heme. Where appropriate, comparisons with GPC arc included. The simu lati ons were carried out in MATLAB .

SCI{Joinl Im c/':ing--The setpoint tracking ability of the proposed con troller depends on the u-to-y model of the process. A second order underdamped (osc illatory) process. given by Ay(l)=BII(t) where A= 1_1.5::-1+.7::-2 and B=::I +.5::-2, was controlled wit h N I=1. N2=5 . Nu=2 and ",=0. 2A and 2B parameters we re es timated with 8(0)=[0;0; I ;0]' and P(O)=T. Fig. 2 shows the performance (.'I VS. t ) of the resulting adap ti ve controlle r. The response has settled down to normal one aft er £1 peri od of initi al adaptati on where the contro l was bumpy . Note how easy it is to obtain excell ent response with almost default settings of the controller parameters .

Handlin g - nois£'--Process measurements are typ icall y noisy . If the measurement noise is not taken into account in any prac ti cal control scheme the cont ro l signal may ex hi bit excess ive input/output variat ion whi ch is not des irable. The KF used in

MA PC optimall y rejects noise and passes the min imulll var ian ce estimates of the output to the contro lle r. The process used in thi s examp le is given hy AI'(I)=HII (I)+Ce(t)//:1 where A and B are as in above and C=I-::I+. 2::-2 . {cu) ) is a ze ro-mea·n wh ite noise sequence wi th variance 0.0 I. Since only noise is present and there are no load-dis turbances F and D are se t to /:1 and I res pective ly _ In MAPC the C polynomial is mode led by process and measurement noise ill the state space model. The process was controlkd with N I=I , N2=5 , Nu=1 and ",=0 and with i?w=O.O I and R,=O.O I. Fig. 3 shows the resul ting performance (\' vs_ 1 and II VS. I ) . Thi s can be compared with the performance obtained using GPC with C estimated_ Same contro ll er parameters were used. The re" ults arc sho 'vvn in Fig. 4 . A comparison (o f Fi gs ] and 4) "hows that MAPC performs better than GPC (both OLltput \ ' and input /I are better in case o r MAPC). The sum or abso lute erro rs (S AE) for ou tput has hee n calc ulated ror both cases from 50lh

sa mp l i ng ins tant to 200,h s<lmp l i ng instant. The SAE 1'0 1' MAPC is approximatel y 29.4 and th at for GPC is 40.15 whic h confirms the visua l comparison. The average ab~o l ute error over thi s peri od ex hi bited by MAPC is 0. 146 and by GPC is 0.20. It may be noted th at in both the cases the u-to-y model used corresponds to the actual proce ss_ With almost no effort and wi thout hav ing to use too much prior info rmati on (only noise covariances were used) improved performance was obta ined . The perfor­mance of GPC is the best that can be obta ined under the assumpt ion of no prior informat ion where as the performance of MAPC can be further improved by tuning the noise covariances.

The adaptive case (MAPC) is illustrated by Fig. 5. 2A and 2B parameters were estimated with 8 (0)=[0;0; 1 ;0]' and P(O)=1. It Illay al so be noted th at the noise polynomial C need not be es timated as well as no prior knowledge regarding it is requi red. The controller parameters were the same as before. Hand!in~ load-d isturhances--If an exact internal

model of the load-disturbances is used , then exact cancell ati on of the di sturbance occurs (depend ing on the controll er parameters)_ GPC models d is turbances ai; random :,leps at random times and hence if the disturbances have structure (ie_ va ry more slowl y than can be modeled by a sequence of steps) then it can give relatively poor perfo rmance_

In thi s example, the process is given by Ay(t)=Bu(t)+d(t) where A= I -.7z- 1 and B=Z-I- d(t) is

~

SRIPADA & FISHER: MULTISTEP ADAPTIVE PREDICTIVE CONTROLLER 235

1

o.~

-4.5

-1 ·0 110

'-----

100 150 200

Fig. 8-Performance of MAPC in the presence of load­disturbances (Residual modeling and two-stage identification).

t.6

~ 0

-4.5

-1 - ~

-1.50 50 100 150 200 250 300

t

Fig. 9-Performance of MAPC in the presence of MPM; system response:conti nuous. setpoint:dashed.

given by d(t)=e(t)/( 1-1.9i ' +.9i\ {e(t)} was non-zero at t=50, 90, 160, 180 and 225. The performance (y vs. t) of non-adaptive MAPC with A and B corresponding

to actual process and with F=11 and D=I (almost no prior information is used) is shown in Fig. 6. The performance of non-adaptive MAPC with F=/l and D=I-.9i' and C=I (corresponding to an exact knowledge of the disturbances) is shown in Fig. 7. To illustrate residual modeling and two-stage identification it is assumed that no information regarding the disturbances is available. An AR residual model with initial parameters set to zero and P(O)=I was identified and used. 10 AR parameters were estimated. A and B were taken to be the actual process polynomials (for fairness of comparison) . The results are shown in Fig. 8. The results show that residual modeling and two-stage identification can give excellent performance (in this case the performance is the same as the one obtained with exact knowledge of the disturbances) .

In all the runs the controller parameters were

NI =N2=Nu= I and A=O. Note that if required the process model parameters can also be identified.

Handling MPM-The proposed method works well even when there is MPM due to, for example, reduced order modeling (under-parameterization). To

illustrate this , the process given by 1/[(s+I)2(3s+1)2] was controlled using MAPC. The sampling period wa · taken as T=I time unit. 2A and 2B parameters were estimated with no filtering of data. The

parameters were first initialized to zero and identified in open-loop with random input excitation for first 50 sampling intervals before closing the loop. P(O)=I. The controller parameters were NI = I, N2=20, Nu=2 and A= I. Rw=O.1 and Rv=O.I . The results are shown in Fig. 9. The performance is satisfactory (in view of the highly conservative (detuning) controller parameters used) and illustrates the robustness of the controller.

In summary the simulation examples presented here show that MAPC performs as expected and can give excellent results.

Conclusions A multistep adaptive predictive control scheme has

been developed . It can be applied to practical industrial processes . The proposed algorithm is based on the integration of the following key ideas: (i) use of multistep or long-range predictive control; (ii) state space model based representation of process dynamics and other un modeled effects; (iii) use of Kalman filter for prediction of process output; (iv) use of robu st estimation methods; (v) use of prior information ; and (vi) use of mUltiple models (process dynamic model and residual model) and two-stage identification in the absence of any prior information. The proposed al gorithm compares favorably with GPc. Mode ling and elimination of disturbances has been gi ven special attention . The proposed scheme uses a more general model of the disturbances than the simple ARIMAX model used in GPc. The control algorithm can be adapted to any prior information .

In most self-tuning algorithms (including GPC) a distu~bance model is estimated on-line along with process model parameters . Such an approach is not likely to be successful for various reasons. The proposed scheme provides a way to avoid thi s difficulty via residual modeling and two-stage identification .

There is no need to solve a Diophantine Identity on-line in the proposed scheme. The result is reduced computation.

A set of preliminary SISO simu lated examples indicate excel lent performance and justifies further evaluation including experimental studies.

Acknowledgement The first author gratefully acknowledges the

financial support from the Pool-Scheme of CSIR, India.

236

Nomenclature A,B,C,D,F,F"AnvB", Ci. a",jj. bmi

e

I c.c' gi

H k K, K(t) NI N2 Ny Nu P, P(t)

y y

· 1 Z

Greek letters (l

6 A <l>.r.A ry.ru 9

Superscripts

Subscripts ref sp

Abbreviations AI AKFP ARMAX

ARIMAX

DMC GPC

IDCOM ILS

INDIAN 1. CHEM. TECHNOL., JULY 1999

polynomials in Z· I

coefficients of C,Am,Bm respectively noise tenn free response of the system matrices step response coefficients of the process state space model matrix process pure time delay Kalman filter gains minimum output horizon maximum output horizon output horizon control horizon covariance matrices of state estimation error square root of Rw noise covariances residual laplace transfonn operator process (control) input vector of control increments measurement noise process noise state vector process output vector of output predictions backshift operator

reference trajectory parameter differencing o perator, ( l-l l) weighting on control action state space model matrices weighting matrices

parameter vector

transpose

reference input setpoint

artificial intelligence adaptive Kalman filter prediction auto regressi ve moving average with exogenous input

= auto regressive integrated moving average with exogenous input dynamic matrix control generalized predictive control

identification and command improved lellst squares

IMA KF LQG MA MAPC

MOCCA

MPM NMP RLS RELS SAE SISO STR

References

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