Predictivepid

ISA TRANSACTIONS

1

ISA Transactions 38 (1999) 11±23

Predictive PID R.M. Miller a,1, S.L. Shah a, R.K. Wood a,*, E.K. Kwok b a

Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, T6G 2G6, Canada b Department of Chemical Engineering, University of British Columbia, Vancouver V6T 1Z4, Canada

Abstract A new stochastic, predictive, proportional-integral-derivative (PID) control law is proposed which is mathematically equivalent to generalized predictive control (GPC) with a steady state weighting term. The main motivation of this paper is the extension of the classical PID algorithm on industrial computers to do advanced control without employing specialized software. The predictive PID constants and the internal model are chosen by equating the discrete PID control law with the linear form of GPC. The result is a long range predictive control law with a model based PID structure. Use of a ®rst order model yields a PI controller while a second order plant results in a PID structure. The process model order is restricted to a maximum of two although there is no restriction on the choice of GPC tuning parameters. Performance of the predictive PID scheme is shown, via simulation, to be identical to GPC. Results from the use of the predictive PID algorithm for the control of an industrial heat exchanger are also presented. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Predictive PID; Model-based control; Stochastic PID

1. Introduction The PID controller has a long history in the ®eld of chemical process control and therefore practitioners are well acquainted with its tuning and idiosyncrasies. However, many key processes in a chemical plant have dicult dynamics which are characterized by one or more of the following (a) long time delay, (b) inverse response, (c) frequent and severe disturbances, (d) signi®cant nonlinearities, (e) multivariable interactions, and (f) constraints. Moreover, recent advances in chemical process eciency including heat exchanger networks and reactive distillation are increasing the number of dicult control problems. In these * Corresponding author. 1 Currently with Honeywell Hi-Spec Solutions, 325 Rolling Oaks Drive, CA 91361-1266 USA.

dicult control situations, PID controllers can only be detuned to retain closed loop stability resulting in sluggish performance. Commercially available linear model predictive control (MPC) methods can address the above mentioned control diculties except for non-linearities. The widespread use and success of MPC applications described in the literature [1] attests to the improved performance of MPC compared to PID for control of dicult process dynamics. However, even with these outstanding results MPC still is not used in many chemical processes with dicult dynamics primarily because the licensing, hardware and commissioning costs of commercial MPC require signi®cant economic justi®cation which is dicult to quantify at the feasibility stage. Furthermore, long term maintenance of MPC requires a higher level of expertise and cost compared to PID maintenance.

0019-0578/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0019 -0 578(98)00041 -X

12

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

In this paper a compromise between MPC and PID is proposed (denoted as predictive PID). The main advantage of the predictive PID algorithm developed in this paper is that it is cost e€ective yet retains most of the advantages of MPC for single loop control. The algorithm involves the reformulation of generalized predictive control (GPC) [2] with steady state weighting, denoted here as gGPC [3,4], into a model based PID structure. Therefore, predictive PID preserves the same PID ideology that is familiar to operating personnel while the closed loop performance is that of MPC. In the following sections, the development, interpretations and industrial application of predictive PID will be described. 2. Generalized predictive control (GPC) Generalized predictive control (GPC) or GPC with steady state weighting is chosen as a basis for predictive PID for two major reasons. First, the receding horizon solution of GPC can be described easily in a linear polynomial representation. In comparison, the receding horizon form of DMC [5] is cumbersome because it is based on a convolution model. Second, GPC is a ``generalized'' strategy which includes all desirable properties of the industry proven MPC formulations. It is assumed that the plant is adequately represented by the ARIMAX model A…qÿ1 †y…t† ˆ B…qÿ1 †u…t ÿ 1† ‡

C…qÿ1 † …t†


…2:1†

^ ‡ j j t† ˆ G~ j
u…t ‡ j ÿ 1† ‡ fj y…t where the unforced term, fj , is fj ˆ

G j
u…t ÿ 1† Fj y…t† ‡ C C

…2:5†

The j ˆ 1; . . . ; N multistep predictor can conveniently be expressed in the vector form, y^ ˆ G~u ‡ f, where ^ ‡ 1 j t†; y…t ^ ‡ 2 j t† . . . y…t ^ ‡ N j t†ŠT y^ ˆ ‰y…t G ˆ ‰G1 ; G2 . . . GN ŠT u~ ˆ ‰
u…t†;
u…t ‡ 1† . . .
u…t ‡ N ÿ 1†ŠT f ˆ ‰f…t ‡ 1†; f…t ‡ 2† . . . f…t ‡ N†ŠT The GPC control objective is composed of a sum of squares prediction error term and a control action penalty term given by X N2 ^ ‡ j j t† ÿ w…t ‡ j†Š2 ‰y…t JˆE jˆN1

‡

Nu X

2



C ˆ Ej A
‡ qÿj Fj

…2:2†

Ej B ˆ CG~ j ‡ qÿj G j

…2:3†

Long range predictions of the plant output based on current and past data are given by

…2:6†

l…j†‰
u…t ‡ j ÿ 1†Š

jˆ1

where w…t ‡ j† is the setpoint, l…j† is the control weight, N1 is the minimum prediction horizon, N2 is the maximum prediction horizon and Nu is the control horizon. Minimization of the performance index (2.6) with respect to future
u yields the GPC control law u~ ˆ …GT G ‡ lI†ÿ1 GT …w ÿ f†

where A, B, and C are polynomials in the backward shift operator qÿ1 and y, u, and  are the predicted output, control input and a zero-mean white noise disturbance, respectively. The jth step ahead predictions of (2.1) require the Diophantine identities given by (the (qÿ1 ) notation is omitted for brevity):

…2:4†

…2:7†

3. GPC with steady state weighting (gGPC) A terminal matching condition, de®ned as the weighted square of the steady state error, is included in the GPC cost function (2.6) to derive GPC with g weighting (denoted herein as gGPC [4]). Computation of the steady state prediction by solving for the steady state value of (2.2) and (2.3) yields F s ˆ es A

…3:1†

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

G s
ˆ gs C ÿ es B where C…1† es ˆ A…1† gs ˆ

…3:2† …3:3†

B…1† A…1†

…3:4†

The orders of Fs and G s are nA and max…nB ÿ 1; nC ÿ 1†, respectively. The optimal steady state predictor is found by taking the limit of (2.4) as j approaches in®nity. ^ ‡ j j t† ˆ y…s ^ j t† lim y…t

j!1

ˆ gs

Nu X

t; t ‡ 1; . . . t ‡ Nu ÿ 1 based on the prediction horizon t ‡ N1 ; . . . t ‡ N2 . For the receding horizon implementation, only the ®rst element of u~ is implemented at the current interval and the remaining elements are discarded. This implementation can be expressed in a linear form by solving for the ®rst control move,
u…t†, in the control law. Because GPC is a subset of gGPC, the linear form of GPC will be a subset of the linear form of gGPC. For the ®rst control element of (3.6), it follows that
u…t† ˆ h…w ÿ f† ‡ hs …ws ÿ fs †


u…t ‡ j ÿ 1† ‡ G s
u …t ÿ 1† ‡ Fs y …t† f

…3:5† The gGPC cost function contains the ®rst two terms of the GPC cost function plus a steady state prediction error. Minimizing the objective function yields the gGPC control law u~ ˆ ‰GT ÿy G ‡  ‡ GTs ÿGs Šÿ1 ‰GT ÿy …w ÿ f† ‡ GTs ÿ…ws ÿ fs †Š

h ˆ first row of‰GT ÿ G ‡  ‡ GTs ÿGs Šÿ1 GT ÿy hs ˆ   the first row of‰GT ÿy G ‡  ‡ GTs ÿGs Šÿ1 GTs ÿ fs ˆ G s
uf …t ÿ 1† ‡ Fs yf …t† expanding (4.1) yields (

…3:6†

" C‡q

( " ˆ C

where ÿy ˆ y IN2 ÿN1 ‡1N2 ÿN1 ‡1  ˆ lINu Nu ÿ ˆ INu Nu 2 3 gs 0 . . . 0 6 . 7 .. 6 gs gs . .. 7 6 7 Gˆ6 . 7 .. 4 .. . 05 gs . . . . . . gs Nu Nu 1

…4:1†

where f

jˆ1

fs ˆ ‰ 1

13

. . . 1 ŠT ‰G s
u f …t ÿ 1† ‡ Fs y f …t†ŠNu 1 ws ˆ ‰w…t†; . . . ; w…t†ŠTNu 1

N2 X

#) G j hj ‡ G s hs

jˆN1 N2 X

#)

hj ‡ hs

( w…t† ÿ

jˆN1


u…t† N2 X

) Fj hj ‡ Fs hs y…t†

jˆN1

…4:2† which is in the linear form T
u…t† ˆ Rw…t† ÿ Sy…t† where

"

TˆC‡q

ÿ1

N2 X

"

N2 X

…4:3† #

G j hj ‡ G s hs

jˆN1

RˆC

#

hj ‡ hs

jˆN1

4. The linear form of GPC The gGPC control law is implemented in a receding horizon. At each interval, the control vector, u~ , is computed for the control horizon

ÿ1

Sˆ

N2 X

Fj hj ‡ Fs hs

…4:4†

jˆN1

with the order of the linear polynomials being:

14

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

nT ˆ max…nB; nC†


u…t† ˆ GCw w…t† ÿ GCy y…t†

nR ˆ nC and

where

nS ˆ max…nA; nC ÿ 1†

…4:5†

‡ …ÿKP ÿ 2KD †qÿ1 ‡ …KD †qÿ2

The incremental discrete PID control law is developed by starting with the non-interacting continuous algorithm given by … tC de…tC † e…tC †dtC ‡ KDC u…tC † ˆ KPC e…tC † ‡ KIC dtC 0 …5:1† where e…tC †, KPC , KIC and KDC are the error, noninteracting proportional, integral and derivative constants and the subscript c denotes continuous time, respectively. The non-interacting algorithm (5.1) was chosen as a basis because it is the simplest PID form and interacting forms can be easily derived from it. A ®rst order discretization of (5.1) results in the following discrete control law u…t† ˆ KP e…t† ‡ KI

t X

e…i† ‡ KD ‰e…t† ÿ e…t ÿ 1†Š

iˆ0

where t denotes sampled time and KP ˆ KPC KIˆ KIC TS K KD ˆ TDSC

…5:2†

…5:3†

‡ …ÿKP ÿ 2KD †qÿ1 ‡ …KD †qÿ2 Še…t†

…5:4†

A common industrial practice is to remove the setpoint signal from the proportional and derivative terms in (5.4) to avoid abrupt control actions following a setpoint change otherwise known as the proportional and derivative kick. The resulting setpoint on integral only (SPI) controller is described by

…5:6†

Observation of the GCw and GCy polynomials in the SP on I control law (5.5), shows that a PI controller is ®rst order in y…t†, a PID controller is second order in y…t† and PI/PID controllers are zero order with respect to w…t†. 6. The PI/PID form of gGPC It has been recognized by McIntosh [6] and Henningsen et al. [7] that a discrete PID control law such as (5.5) could be equivalent to standard GPC given some restrictions in the GPC formulation. The conditions under which gGPC is equivalent to PID are determined in the current work. For the deterministic (C ˆ 1) linear gGPC control law for a second order A polynomial and zero order B polynomial, a second order plant model without time delay, from (4.5), it follows that the linear polynomials T, R and S are of order 0, 0 and 2, respectively, so the control law (4.4) can be written as
u…t† ˆ r0 w…t† ÿ …s0 ‡ s1 qÿ1 ‡ s2 qÿ2 †y…t†

and TS is the sampling interval. The incremental control law is determined by applying the di€erencing operator to the control output.
u…t† ˆ ‰…KP ‡ KI ‡ KD †

GCw ˆ K1 GCy ˆ …KP ‡ K1 ‡ KD †

5. The discrete PID control law

…5:5†

…6:1†

where ri and si are the coecients of the R and S polynomials, respectively. Recall that the T polynomial has a leading 1, therefore, a zero order T polynomial is unity. Equating the SPI form of the PID control law (5.5) and the GPC controller (4.4) yields an exact match if GdCw ˆ R

…6:2†

GdCy ˆ S

…6:3†

The superscript d indicates a deterministic model representation. The PID tuning constants in (5.5)

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

can be expressed in terms of the linear gGPC coecients by equating (5.5) with (6.1) to yield KP ˆ ÿ…s1 ‡ 2s2 † ˆ s0 ÿ r0 ÿ s2 KI ˆ r0 K D ˆ s2

The GPC control law (cf. 4.2) can be expressed as (

" ÿ1

C‡q …6:4†

A ®rst order plant model results in a ®rst order S polynomial and an equivalent PI controller while a second order plant yields a PID controller from the relations in (5.5). A comparison of GPC and equivalent PID servo and regulatory response is shown in Fig. 1 for the unstable plant: ÿ0:015 GP ˆ 1 ÿ 1:95qÿ1 ‡ 0:935qÿ2 A step disturbance of 0.1 is applied/removed at the 100/150 and 250/300 sampling instants, respectively. The simulation shows an exact match between GPC and PID control while the control response is satisfactory for a dicult control problem. 7. The multistep long range predictor, GMP The PID controller proposed in the previous section does not o€er a practical solution to control problems with time delays or realizable second order plant models (i.e. with a ®rst order numerator). It was shown by Harris et al. [8] that a PI controller with a Smith predictor [9] is equivalent to a minimum variance controller (MVC) for models with time delay. However, a Smith predictor gives a d step ahead prediction only where d is the time delay. The practical extension of MVC is to a long range multistep prediction as in GPC. In addition, real processes are subject to random disturbances that are usually correlated. GPC compensates for correlated disturbances by including the disturbance model, C=A
, directly in the control law. A PID control law based on GPC with a non unity C polynomial will therefore not require ad-hoc detuning to reduce manipulated variable variance. This implies a stochastic weighted predictor must be developed so that the model based PID controller is equivalent to GPC.

15

( " ˆ C

N2 X

#) G j hj ‡ G s hs

jˆN1 N2 X

#)

hj ‡ hs

" w…t† ÿ

jˆN1


u…t† N2 X

# Fj hj ‡ Fs hs y…t†

jˆN1

…7:1† which can be rewritten as ( "

N2 X


u…t† ˆ C ( ÿ

#) hj ‡ hs

jˆN1 N2 X

w…t† )

Fj hj ‡ Fs hs y…t†

jˆN1

(" ÿ

N2 X

#

)

G j hj ‡ G s hs ‡ …C ÿ 1†
u…t ÿ 1†

jˆN1

…7:2† because there is always a leading 1 in the Cc polynomial. The ®rst term in the right hand side of (7.2) is of order nCc which does not ®t into the PID SP on I form unless Cc ˆ 1. In industry, most processes are operated in a regulatory fashion most of the time while changes in setpoint are typically infrequent. A zero order approximation of the servo term in (7.2) that accommodates the PID SP on I form is desired. The logical zero order is its steady state value. This requires that Cc is replaced by the sum of the elements of Cc . Such an approximation will always give a conservative or detuned servo response but have no e€ect on regulatory performance. The following expression is such an approximation (
u…t† ˆ ( ÿ

nC X

" cj

jˆN1

jˆ1 N2 X

ÿ

#) h j ‡ hs

w…t†

)

Fj hj ‡ Fs hs y…t†

jˆN1

("

N2 X

N2 X

# )   Gj hj ‡ Gs hs ‡ …C ÿ 1†
u…t ÿ 1†

jˆN1

…7:3†

16

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

Fig. 1. GPC and equivalent PID control response to a discrete second order plant.

In Section 6, it was shown that the linear gGPC polynomials R and S are related to the discrete PID terms GCw and GCy through (6.4), therefore, (7.3) can be expressed as follows
u…t† ˆ GCw w…t† ÿ GCy y…t† ÿ GMP GCy
u…t ÿ 1† …7:4† where

GMP ˆ

"

N2 P jˆN1

Gp ˆ

# G j hj ‡ G s hs ‡ …C ÿ 1† GCy

di€ers from the SPI form discussed previously. The setpoint ®lter in Fig. 3 has the same e€ect as removing the proportional and derivative action from setpoint changes. The control performance of GPC and the equivalent predictive PID algorithm is demonstrated in Fig. 4 for the second order plant:

…7:5†

The GPC law is an optimal multistep predictive control law, therefore (7.5) is an optimal multistep weighted predictor when used as an internal model for PID control. When the model order is restricted to a maximum of two the PID controller constants KP , KI and KD can be solved by (6.4). Fig. 2 shows the block diagram of the stochastic predictive PID control loop. The stochastic predictive PID controller can also be expressed in a model based PID form as shown in Fig. 3. This alternative representation uses all three control modes (P, I and D) on setpoint changes which

eÿ5s …3s ‡ 1†…5s ‡ 1†

A step disturbance of 0.05 is applied/removed at the 100/150 and 250/300 sampling instants, respectively. The simulation in Fig. 4 shows excellent performance for PID on a process with a signi®cant time delay. Fig. 5 shows the performance of the deterministic PID controller given by (7.9) for the plant model Gp ˆ

…ÿ2s ‡ 1† …6s ‡ 1†…3s ‡ 1†

where plant noise is correlated by Cp ˆ 1 ÿ 0:8qÿ1 with variance  2 ˆ 0:0005. The control response is satisfactory although the manipulated variable variance is excessive. Fig. 6 shows that the

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

Fig. 2. Block diagram of the stochastic predictive PID control loop.

Fig. 3. Alternate diagram of the stochastic predictive PID control loop.

Fig. 4. GPC and equivalent PID response to a second order plant.

17

18

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

Fig. 5. Response of deterministic PID to a stochastic process.

Fig. 6. Response of stochastic GPC and predictive PID to a stochastic process.

response of the stochastic PID controller is slightly more sluggish than GPC for servo response. However, the regulatory response is equivalent. The control variance in Fig. 6 is signi®cantly smaller than the control variance in Fig. 5. A stochastic control law compensates for

the correlation structure in the disturbance which results in less aggressive control action. The detuned PID servo response compared with GPC in Fig. 6 should not be a problem for most processes because regulation is the typical control objective.

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

8. Interpretations of the predictor, GMP Conceptually, GMP can be interpreted as a long range predictor in comparison with gGPC. Consider the special case where g is set to zero and C ˆ 1. The resulting model based PID controller can be interpreted by expanding GMP which follows since G j contains the nB ÿ 1 jth step ahead step response coecients for a deterministic plant model as can be illustrated by the following example. Let the ®rst order plus time delay plant be given by A ˆ 1 ÿ 0:9qÿ1 and B ˆ 0:1qÿ1 (note that the plant ZOH is removed from the model). The step response coecients are: 0.0, 0.100, 0.190, 0.271, 0.344, . . ., 1.000. For the ®rst step (j ˆ 1) G 1 ˆ 0:1 and for the second step (j ˆ 2) G 2 ˆ 0:19, therefore, G j contains the jth step ahead prediction and is of order nB ÿ 1. The predictor GMP can now be interpreted as an optimal weighted sum of j step ahead predictions from the minimum prediction horizon, N1 to the maximum prediction horizon, N2 . The interpretation of GMP is more dicult to see when the plant model is stochastic, i.e. Cc ˆ 6 1. Both Diophantine identities (2.2) and (2.3) contain the Cc polynomial. Therefore, the extension of the above interpretation to include stochastic disturbance

19

compensation is not as simple as inserting a 1=C ®lter in the feedback loop. All of the linear PID polynomials, GCw , GCy and GMP are strong functions of Cc . The Cc polynomial has the e€ect of detuning all of the linear PID polynomials. Fig. 7 shows the PID block diagram with expanded predictor GMP . 9. In®nite horizon GPC and PID In the practical application of GPC, it is recommended that a maximum prediction horizon, N2 , corresponding to 50 to 90% of the rise time of the process be used [4]. Furthermore, unstable control behavior may result from an N2 that is too short. Setting N2 to in®nity results in mean level control or steady state model inverse control [10,11]. The closed loop poles of a mean level controller are the same as the open loop poles [10], therefore, for an open loop stable process without model plant mismatch, mean level control response is guaranteed stable. Mean level control o€ers a conservative but robust approach to automatic control. Removing the ®nite horizon term from the gGPC control law (3.6) results in the mean level controller (denoted as 1GPC) expressed as

Fig. 7. Block diagram of the stochastic predictive PID control loop with expanded predictor, GMP .

20

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

   C Fs ÿ1 Gs C‡q
u…t† ˆ w…t† ÿ y…t† gs gs gs

…9:1†

A PID equivalent to 1GPC, denoted as 1PID, can be developed following the same procedure as in Sections 6 and 7. From (4.4), the linear polynomials for 1GPC are G s T ˆ C ‡ qÿ1 gs

Rˆ

C gs

Sˆ

Fs gs

C…1† gs

GCy ˆ

Fs gs

GMP ˆ

G s ‡ gs …C ÿ 1† Fs …9:3†

Substitution of the relations in Section 3 into (9.3) gives C…1†A…1† C…1† GCy ˆ A GCw ˆ B…1† B…1† GMP ˆ

B…1†C ÿ C…1†B ‡ B…1†
…C ÿ 1† C…1†A


KP ˆ

…a1 ‡ 2a2 †…ÿC…1†† B…1†

KI ˆ

C…1†A…1† B…1†

…9:2†

Equating the PID polynomials in (7.9) with (9.2) along the same lines of previous sections yields

GCw ˆ

The above polynomials now are independent of any GPC tuning parameters. Predictive PID controller constants, KP , KI and KD can be expressed in terms of the model parameters by comparing the coecients of (9.4) with the linear relations in (6.4) which results in

…9:4†

KD ˆ

C…1† a2 B…1†

…9:5†

Fig. 8 shows satisfactory control response using the 1PID controller for the same plant and disturbance as presented in Fig. 6. The 1PID controller de®ned by (9.5) no longer requires the solution of the Diophantine identities as was the case in the predictive PID controller developed in Section 7. This has several advantages. Implementation of this control law is very simple and easy to understand. The fact that the Diophantine identities are not required also means that the execution of the 1PID controller is very ecient in comparison to GPC or DMC. The adaptive implementation will also be very ecient

Fig. 8. Response of 1GPC and 1PID to a stochastic process.

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

because only simple multiplications are required to compute the control law. This adaptive implementation is discussed in Miller [12] and Miller et al. [13,14]. 10. Industrial application of predictive PID Predictive PID was implemented in a Honeywell TDC2000 control computer for control of a key unit in a fertilizer plant. Fig. 9 shows the simpli®ed process and instrument diagram of the process. Sub-cooled liquid ammonia is vaporized by E1 and superheated to about 300 F by the E2 con®guration. Superheated ammonia and compressed air enter the highly exothermic reactor, R3 which produces NO2 in a yield of less than 50%. The existing controllers for TC1, TC2 and TC3 are PID, PID and DMC, respectively. The undesired byproducts, NOx (other than NO2), are produced in signi®cant quantities if the catalyst temperature of R3 drops below 1495 F, while the catalyst degrades prematurely if the temperature exceeds 1505 F. Control of the catalyst temperature is further complicated by variance in the ammonia temperature and the

21

ambient air temperature. E1 and E2 use process steam which has a signi®cantly higher variance in pressure than utility steam. Furthermore, the temperature transmitter, TT2, is located several metres downstream of the E2 parallel con®guration which results in a signi®cant time delay for the TC2 control loop. TC2 was chosen to upgrade to predictive PID because the existing PID controller was known to cause increased variability in the catalyst temperature of R3. PID controller constants and the internal model, GMP , were computed using an open loop model of E2 and (6.4) and (7.12), respectively. Predictive PID was implemented by means of a conventional TDC2000 PID control loop with the addition of a simple user de®ned function for GMP and the removal of proportional and derivative terms from setpoint changes (a built in option). The total time required for this implementation was about one hour not including process identi®cation. A comparison of predictive PID and the existing PID during nominal operating conditions is shown in Fig. 10. The variance of the ammonia temperature is signi®cantly higher when E2 is controlled by the PID controller compared to the

Fig. 9. Simpli®ed schematic and instrument diagram for the NO2 process.

22

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23

Fig. 10. Comparison of PID and predictive PID for control of the NO2 process.

period that is controlled by the predictive PID controller. The FCOR method described by Huang et al. [15] was used to assess the performance of PID and predictive PID in Fig. 10 relative to minimum variance. An FCOR measure of 1.0 indicates minimum variance type control performance while measures approaching zero indicate poor performance. The PID controlled response resulted in an FCOR measure of 0.14 while predictive PID yielded a measure of 0.26 which represents an 86% improvement relative to minimum variance.

A mean level formulation of PID is determined by setting the ®nite horizon weight to zero in the predictive PID control law. The resulting 1PID controller can be expressed as a simple function of ARIMAX model parameters. Implementation of the predictive PID controller on a TDC2000 control computer for control of a key heat exchanger in a fertilizer plant required only one hour. The performance of the predictive PID scheme was demonstrated to be superior to the existing PID controller for a series of disturbances to the steam pressure and during nominal operation.

11. Conclusions

Acknowledgement

Long range predictive stochastic PI and PID control laws are determined by equating the linear polynomials in GPC with PID constants plus an internal model for ®rst and second order plants, respectively. The internal model GMP, which can be interpreted as a multistep weighted predictor, exists for models with time delay. There are no restrictions on the gGPC tuning parameters N1 , N2 , Nu , l, y and from which the predictive PID controller is based. Predictive PID is equivalent to GPC for all cases except for an approximation of stochastic servo control.

The ®rst author (R.M.M.) would like to thank NSERC (Natural Sciences and Engineering Research Council of Canada) for ®nancial support in the form of a postgraduate scholarship. References [1] S.L. Shah, Model-based predictive control: theory and implementation issues, Presented at ADCHEM '94, Kyoto Research Park, Kyoto, Japan, 1994. [2] D.W. Clarke, C. Mohtadi, P.S. Tu€s, Generalized predictive controlÐpart I. The basic algorithm, Automatica 23 (2) (1987) 137Ð148.

R.M. Miller et al. / ISA Transactions 38 (1999) 11±23 [3] K. Kwok, Long range adaptive predictive control, Ph.D. thesis, University of Alberta, 1992. [4] K. Kwok, S.L. Shah, Long range pedictive control with a terminal matching condition, Chemical Engineering Science 49 (9) (1994) 1287±1300. [5] C.R. Cutler, B.L. Ramaker, Dynamic matrix controlÐa computer control algorithm, Proceedings of the 1980 Joint Automatic Control Conf. Pt. 1, WP5-B, San Francisco, CA, 1980. [6] A.R. McIntosh, Performance and tuning of adaptive generalized predictive control, M.Sc. thesis, University of Alberta, 1988. [7] A. Henningsen, A. Christensen, O. Ravn, A PID autotuner utilizing GPC and constraint optimization, Proc. 29th IEEE Conf. on Decision and Control, Honolulu, HI, pp. 1475±1480, 1990. [8] T.J Harris, J.F. MacGregor, J.D. Wright, An overview of discrete stochastic controllers: generalized PID algorithms with dead-time compensation, Can. J. of Chem. Eng. 60 (1982) 425±432.

23

[9] O.J.M. Smith, Closer control of loops with dead time, Chemical Engineering Progress 53 (5) (1957) 217±219. [10] D.W. Clarke, C. Mohtadi, P.S. Tu€s, Generalized predictive control part II. Extensions and interpretations, Automatica 23 (2) (1987) 137±148. [11] A.R. McIntosh, S.L. Shah, D.G. Fisher, Analysis and tuning of adaptive generalized predictive control, The Canadian J. of Chemical Engineering 69 (1991) 97±110. [12] R.M. Miller, Adaptive Predictive PID, M.Sc. thesis, University of Alberta, 1995. [13] R.M. Miller, S.L. Shah, R.K. Wood, Adaptive predictive PID, Proc. ISA/95, pp. 1±10 Toronto, 25±27 April 1995. [14] R.M. Miller, K.E. Kwok, S.L. Shah, R.K. Wood, Development of a stochastic predictive PID controller, Proc. 1995 American Control Conference, pp. 4204±4208, Seattle, WA, 1995. [15] B. Huang, S.L. Shah, K.E. Kwok, On-line control performance monitoring of MIMO processes, Proc. 1995 American Control Conference, pp. 1250±1254, Seattle, WA, 1995.