Application of generalised predictive control to the paper machine benchmark

Control Eng. Practice, Vol. 3, No. 10, pp. 1483-1486, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/95 $9.50 + 0.00

Pergamon 0967-0661(95)00155-7

APPLICATION OF GENERALISED PREDICTIVE CONTROL THE PAPER MACHINE BENCHMARK

TO

C.-M. Chow, A.G. Kuznetsov and D.W. Clarke

Departmentof EngineeringScience, Universityof Oxford, ParksRoad, Oxford, UK (Received March 1995; in final form June 1995) A b s t r a c t . This paper presents an application of Generalised Predictive Control (GPC) to the paper machine benchmaxk. The plant features a relatively large dead-time and severe retention level variation at the wire section. From simulations, results show that the controller has good performance when the the retention levels are at their nominal values, and that there is large output variation if the retention varies as specified. K e y Words. Multivariable control systems; predictive control; paper machine control.

1. I N T R O D U C T I O N

inal multivariable plant into a set of scalar subsystems using various decomposition techniques (Cloud and Kouvaritakis, 1988; Kouvaritakis el al., 1992). T h a t design proceeds by forming individual scalar GPC loops.

Since its inception in the late eighties, modelbased predictive control (MBPC) has attracted much attention both in the academic circle and industry. The popularity is mainly due to its relatively simple time-domain formulation and good performance. Within the generic framework of MBPC, a whole family of methods has been proposed. Generalised Predictive Control (GPC) (Clarke et al., 1987) is one of the most popular members of the family, which has quickly received widespread acceptance. The basic idea of G P C is to predict the process output over a longrange time horizon using a mathematical model. A quadratic cost function which includes errors between the predicted o u t p u t sequence and the set-point trajectory is then minimised to give an optimum control vector. The first element of this vector is actually applied to the plant and the procedure is repeated at every sample, leading to a so-called receding horizon strategy.

In this paper the formulation in (Mohtadi et al., 1987) is used mainly because it has inherited one of the most important attributes of scalar G P C simplicity - which has contributed to its success in industry. Also, when compared to the decomposition methods, it has a higher level of abstraction. For instance, the relative performance or importance of outputs can be adjusted simply by choosing different weightings in the cost function. The aim of this exercise is to investigate how effective a fixed GPC scheme is in controlling this paper machine.

2. MULTIVARIABLE GPC Central to GPC is a model of the plant in a controlled auto-regressive integrated moving-average (CARIMA) form,

G P C was formulated in a single-input-singleoutput context in the original papers by Clarke et ai. The extension to the multivariable case was first accomplished by Mohtadi et al. (1987). Subject to a suitable choice of controller parameters the requirement for a system interactor matrix (Wolovich and Falb, 1976), which had been the source of problems for single-step-ahead adaptive controllers, is removed. This makes the algorithm an attractive candidate for adaptive control applications. Another property is its invariance to input-output pairing. There is no need to compute the relative gain array of the system to determine pairing. Later ideas about putting G P C in a multivariable context include transforming the orig-

=

+

(1)

where Jl(q-1), B(q -1) and 7"(q -1) are polynomial matrices in the backward shift operator,

y(t) u(t)

= = =

[ul(t) 2(t) -..It, [ul(t) u2(t) ...It, 2(t) .-.]r,

are the output vector, input vector and white noise vector respectively and A = (1 - q . - 1 ) is 1483

C.-M. Chow et al.

1484

the difference operator. For a square system of dimension p, the cost function is defined as p

N

J(t) = Z

Z

Pi(j) [wi(t + jlt) - ,9,(t + jlt)] 2

i=l j = l

by solving Diophantine equations. However, this is not favoured in practice due to the added complexity, especially in an adaptive context. The output prediction vector can now be written in a compact matrix form

Nu

+ "#,~ Aui(t + jlt) 2

iJ = ~ ~, + :f .

j=0 The cost function becomes where N is the prediction horizon, Nv the control horizon, ~i(t + j]t) the future output predictions, wi(t + jlt) the future set-point sequence, Pi the constant weighting on control moves and #{(j) _ { Pi 0

V j 6 [Yli, gu{], otherwise ,

To derive the control law, consider the following quantities:

G=

GN

O

O

"" "

GI

0

...

".,

.°

GN- 1

6 ~pN ×pNu, .

......

= [ m u l ( t l L ) - . . m%(tl t)

mul(t +

lit). • •

Aup(t + N~ - lit)] r ~ R(p×NU), /t = [~l(t + l i t ) . . . y p ( t

__ ~ ( Q T ~ .~_ A) ~ n~"2 ( f - 1to)T ~

+

const.

Minimisation of the cost gives

the weightings on the tracking errors with ~{ being constants. The constants Nli and Nui are the lower and upper costing horizons respectively. The prediction horizon N is chosen according to N = max{Nu~, N22, ... Nu~}. The subscript i denotes the association with the ith channel of the system. Note that T(q -~) is a design polynomial matrix and is not estimated. It forms an important part in the design of GPC.

G1 G2

J(t) = (w - y)r (w - ~) + ~ A ~

+ lit) #x(t + 2It)-. •

9p(t +/~lt)] r e R (p×N), w=[w~(t + llt)...wp(t + lit) wx(t + 21t)... %,(t + Nit)] T 6 RO,×N), I = [f~(t + lit)...fp(t + lit) fa(t + 2It)... yp(t + Nit)] r e RO,×N), A = diag{~l • • •~p Pl"" "Pp""" P l " ' " P~,} ~pNu XpNu M = diag{tq (1)... pp(1) p1(2).., pp(2) • • • p l ( g ) } 6 R pNxpN where Gj is the j t h step response coefficient matrix of the transfer function matrix, 0 is a null matrix of dimension p and fi(t + jlt) are the free response predictions which are obtainable by iterating forward in time 7"- 1.mAy(t) = '7"- l ~ A u ( t ) ,

assuming that the future deviation of inputs is zero. Alternately the predictions can be acquired

= [i o o

...

o] ×

(QTMQ + A) -1 QTM (w -- I), where I is an identity matrix of dimension p. One can follow some simple guidelines to obtain appropriate controller paramenters for good performance in many cases. The upper control horizons N2i usually reflect the desired closed-loop settling times. Nu is usually chosen according to the complexity of the plant. The more unstable open-loop poles (or highly underdamped poles), the higher the Nu recommended. Control weighting is used to restrict control activities. The relative importance of the outputs can be adjusted using-the tracking error weightings. It has been found that the r61e of 7-(q -1) is of prime importance in practice (Lambert, 1987; M6htadi, 1988; Robinson and Clarke, 1991; Yoon and Clarke, 1994). In short, it enhances the robustness and performance of the closed loop in the face of unmodelled dynamics. Through T(q -1) disturbance-rejection properties can also be tailored. Note that only one control horizon for all inputs is assumed here; otherwise the input-output pairing has to be considered which immediately foregoes one advantage of multivariable GPC.

3. CONTROLLER DESIGN The design consists of the following: selection of sampling interval; specification of a model structure; obtaining a least-squares model from suitable data and determination of GPC parameters. No on-line estimation is used as the attempt is made to solve the problem by a fixed GPC. As mentioned before, the sensors give a reading every 30 seconds, so the logical choices of sampling interval are multiples of 30 seconds. In view of the rather fast dynamics of the plant and the desire to have a short closed-loop rise-time, the sampling interval is chosen as 30 seconds in this work.

Application of Generalised Predictive Control In the plant equation (1) there is no restriction on the form of the polynomial matrices. To simplify controller synthesis and implementation A(q-1), T(q -1) are assumed to be diagonal in this paper, i.e. "A(q-1) = [ all(q-i)0

0]

a22(q-1)

,

B(q -1) = [ bll(q-1) b21(q-1)

bl2(q-1) ] b22(q-1) ,

T(q-i) = [ tll(q-1)O

0 t22(q -1) ]"

This choice of T(q -1) also allows an easier interpretation, and hence selection, of its elements. Due to the complexity of the plant and the use of diagonal .A(q-1), it was decided to estimate 4 parameters in each of the aij(q -1) and 5 in the bij(q -1) polynomials. A dead-time of two samples is also assumed in every entry of the transfer matrix .A(q-i )-i B(q-1). Having decided the model structure, the polynomial matrices A(q - i ) and B(q - i ) can be estimated. Note that the retention variation is left out in the determination of this nominal model. Pseudo-random-binary-sequences (PRBS) with a rate one-tenth of the main sampling rate are applied to both inputs for 10,000 seconds. The levels of the PRBS are chosen to drive the plant around its operating region. The input-output data are then used with an estimation filter 0.1618/(1 /~q-1)5 where ~/ = 0.3 to obtain a least-squares model. The purpose of this filter is to shift the emphasis of the estimation towards a lower-frequency region. The filter coefficient ~ is chosen such that the resulting filtered data has roughly the expected speed of the closed-loop. Following the above procedure the nominal model is found to be bll(q -1) =

O.O044q-2+O.O573q-3-O.1290q -4 +0.0984q-5- 0.0282q -6

bl~(q -1) = 0.0171q- ~+0.5351q -3-1.0846q -4 +0.7354q-5-0.1844q -6

b21(q-1) = O.O068q-2+O.O683q-3-O.1544q-4 +0.1059q-S-0.0235q -6 b22(q-1)

= -O.O025q- 2-O.1224q-Z +O.2383q-4

-0.1349q-5+0.0189q -6 a11(q - i ) _-- 1 - 2.3147q -~ + 2.0467q -2 -0.8553q -3 + 0.1439q -4 a22(q -1) : 1 - 2.2766q- 1+ 1.8496q- z

-0.6532q-a +O.O973q-4 Now GPC parameters can be selected. T(q -1) is probably the most crucial parameter to tune as it determines the robustness bound and the

1485

Table 1 Summary of GPC parameters Parameters

Values

/Vii N~2 N2i N22 gu ~i

1 1 5 5 1 1

Pl P2

I0 -6 10-6 all(q-I)(1 - 0.3q - i ) ag.2(q-1)(1 - 0.3q -1)

tll(q -1) t22(q -1)

disturbance rejection property. Since it is chosen as diagonal, tll(q -1) and t22(q -1) correspond respectively to the dry-weight and ash-content loop. Here tll(q -1) = all(q-1)t*(q -1) and t22(q -1) = a2~(q-1)t*(q -1) where t*(q -1) = (1 - 0.3q-l). In choosing t* a compromise must be reached between disturbance rejection and smooth control. It was decided that t* = (1 - 0.3q -1) gives the right amount of filtering. For more information about the selection of this parameter see (Yoon and Clarke, 1995). In the literature a so-called default setting for GPC is N l i = 1, N~i = 10, Nu = 1, -~ = c where e is a vanishingly small positive number. This is recommended because of the fact that in practice many processes are well-damped and open-loop stable. This choice of GPC parameters gives rise to a mean-level-like control policy which places closed-loop poles near the open-loop positions. This is a low-gain type of control which is often referred to as robust. In this work, this default setting is chosen except that N2i is set to 5 to accelerate the closed-loop. The relative importance of both loops is assumed to be the same; therefore ~1 = 1 and P2 = 1. Table (1) summarises all key parameters. Since the plant is easy to stabilise, the real challenge of the problem being to cope with the large retention variation. If this variation is considered as an unmeasurable disturbance, the effect of it on t:he outputs can be reduced by choosing tll and t22 such that 1/tll and 1/t22 have a larger bandwidth or slower roll-off. The resulting control will, however, become more oscillatory due to the decrease in the robustness bound. A compromise must therefore be made during tuning. Notice that although the measured flow re~tes of filler and thickstock are available they aJ:e not used in the controller as the difference between the command and actual flow rates are considered insignificant.

1486

C.-M. Chow et al.

The simulation results are presented in the paper submitted by the organisers. It is worth pointing out that with t*(q -1) = (1 - 0.Tq-1) the resulting control for the tests with retention variation is much less oscillatory, and should be the preferred setting if active, oscillatory response is prohibited (Chow et al., 1994). This verifies the trade-off between disturbance rejection and smooth control.

4. CONCLUSIONS Multivariable Generalised Predictive Control (GPC) has been applied to a benchmark paper machine, which features a large deaxl-time and severe filler retention variation at the wire section. The basic multivariable GPC algorithm has been reviewed, and the choice of estimation and controller parameter has been discussed. A few simulation results are included which show the general performance of the controller. Simulation studies indicate that multivariable GPC performs well with the retention levels at their nominal values. However, large variation is observed at the outputs when the filler retention level varies. It is noted that in this case there is a trade-off between the amount of output deviation from the set-points and the smoothness of the control.

5. ACKNOWLEDGEMENTS The first author would like to express his gratitude to the Croucher Foundation of Hong Kong for financial support. The authors would also like to thank the UK SERC for its support of the work on predictive control.

6. R E F E R E N C E S Chow, C.-M., A. G. Kuznetsov and D. W. Clarke

(1994). Application of multivariable generalised predictive control to the Simulink model of a paper machine. In: Proceedings of the 3rd IEEE conference on Control Applications. Glasgow, UK. pp. 1675-1681. Clarke, D. W., C. Mohtadi and P. S. Tufts (1987). Generalised predictive control. Part I : The basic algorithm and Part II: Extensions and interpretations. Automatica 23(2), 137-160. Cloud, D. J. and B. Kouvaritakis (1988). Characteristic decomposition and the multivariable generalisation of predictive self-tuning control. IEE Proc. Pt. D 135(3), 165-181. Kouvaritakis, B., J. A. Rossiter and A. O. T. Chang (1992). A singular value decomposition approach to multivariable generalised predictive control. IEE Proc. Pt. D 139(4), 349-362. Lambert, E. P. (1987). Process control applications of long-range prediction. D.Phil thesis, Department of Engineering Science, Oxford University. Mohtadi, C. (1988). On the role of prefiltering in parameter estimation and control. In: Workshop on adaptive control strategies for industrial use. Lodge Kananaskis, Alberta, Canada. Mohtadi, C., S. L. Shah and D. W. Clarke (1987). Multivariable adaptive control without a prior knowledge of the delay matrix. Systems and control letters 9, 295-306. Robinson, B. D. and D. W. Clarke (1991). Robustness effects of a prefilter in generalised predictive control. IEE Proc. Pt. D 138(1), 2-8. Wolovich, W. A. and P. L. Falb (1976). Invariants and canonical forms under dynamic compensation. SIAM J. Cont. 8J Optim~sation 14(6), 996-1008. Yoon, T.-W. and D. W. Clarke (1994). Adaptive predictive control of the benchmark plant. Automatica 30(4), 621-628. Yoon, T.-W. and D. W. Clarke (1995). Observer design in receding-horizon predictive control. Int. J. Control 61(1), 171-191.