a control-relevant identification strategy for gpc

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 31, NO. I , JULY 1992 915

Special Issue Technical Notes A Control-Relevant Identification Strategy for GPC

David S. Shook, Coorous Mohtadi, and Sirish L. Shah

Absstruct-This note addresses the question of a suitable "control- relevant identification" strategy for a class of long-range predictive controllers. It is shown that under certain conditions the best process model for predictive control is that which is estimated using an identification objective function that is a dual of the control objective function. The resulting nonlinear least squares calculation is asymptotically equal to a standard recursive least squares with an appropriate (model and controller-dependent) FIR data prefilter. Experimental results demonstrate the validity and practicality of the proposed estimation law.

I. INTRODUCTION

Long-range predictive controllers have achieved wide-spread ac- ceptance and success in industry because of their ability to monitor and control the process over a finite-time horizon [l]. Unlike minimum-variance or generalized minimum-variance controllers, the long-range predictive control criterion is based on the minimization of a quadratic cost function over a nonmyopic or finite-time horizon as well as design features to handle hard constraints on inputs, outputs, and states of the process [ 2 ] . Self-tuning application of such long-range adaptive predictive controllers (EHAC [ 3 ] , EP- SAC [4], GPC [5 ] ) have centred upon input-output or equation- error models of the type

A ( q - ' ) y ( t ) = B ( q - ' ) u ( t - 1) + € ( t )

where

operator q ~ I ,

A ( q - ' ) and B ( q - ' ) are polynomials in the backward shift

y ( t ) is the process output at time instant t , u ( t ) is the control action at time instant t , E ( t ) is the stochastic realization of process or measurement noise

or disturbances of the type

where T ( q - ' ) and D ( q - ' ) are polynomials in q - ' . ( t ) is assumed to be a zero-mean uncorrelated random sequence

with a variance g'. For random-walk or Brownian motion type disturbances, T = 1 and D = A or 1 - q- ' . The knowledge of D is important for asymptotic disturbance rejection. (See [6] for a discussion of the internal model principle and [7] for a more complete discussion of the role of noise models in disturbance rejection.)

Manuscript received December 13, 1990; revised October 11, 1991. Paper recommended by Associate Editor at Large, M. P. Polis. This paper was presented in part at the 1990 American Control Conference, San Diego, CA.

D. S. Shook is with Novacor Chemicals Ltd., Red Deer, Alberta, Canada. C. Mohtadi is with Eurotherm Ltd, Durrington, Worthing, W. Sussex,

S. L. Shah is with the Department of Chemical Engineering, University of

IEEE Log Number 9200623.

England.

Alberta, Edmonton, Alberta, Canada T6G 2G6.

A . Overall Control Criterion

The objectives of process control are regulation and setpoint tracking. Many controllers measure the quality of control in terms of the variance of the control error ( y s p - y ( t ) ) (where y,, is the setpoint and y ( t ) is the controlled variable). This provides a reasonable performance function which penalizes large excursions from the setpoint and is easy to analyze.

The original self tuning regulator (STR) of Astrom and Witten- mark [8] was an early attempt at minimum variance control. The STR was not particularly successful because it could not withstand significant amounts of measurement noise or model-plant mismatch. This drawback was caused by the simplistic design of the STR, not the overall objective. GPC is more successful because it is much less sensitive to high frequency noise, whether actual noise or caused by model-plant mismatch. The high frequency sensitivity is reduced because the controller considers many predictions in the near future, an approach known as long-range predictive control (LRPC).

The GPC cost function at time t , in its simplest form, is given by

where ? ( t + j 1 f ) is the prediction, or estimate, of y ( t + j ) based on information available at time 1 . j ( t + j 1 t ) is therefore the best guess the controller has of the futuJe value of the controlled variable. NI and N2 are called the prediction horizons, since the controller does not look beyond N2 steps into the future and looks no closer than NI steps.

The GPC control action calculation is of course only one part of an adaptive controller. The prediction used by GPC must come from a model, in turn provided by a process identification scheme. Typically, some variant of the popular recursive least squares (RLS) algorithm is used. In the particular case of GPC, process identification was originally added as an afterthought, without considering the relevance of RLS to GPC; RLS was used almost by default. If standard RLS is used with GPC then a number of ad hoc modifica- tions must be made to RLS for robust adaptation and control in practice because of the inevitable structural model-plant mismatch. In other words, because the model cannot describe the process completely, RLS must be modified before it can be used safely. One common ad hoc fix is the use of a low-pass data pre-filter in the process identification in addition to the filtering resulting from the noise model (see, for example, [9]-[ll]).

In this note an overall adaptive control objective is proposed. In an attempt to find the most relevant identification method for adaptive GPC, an adaptive controller is designed to minimize a cost function that considers the actual control quality, as opposed to the predicted control quality of GPC (2). At time t , the overall control objective is given by

Obviously, y(t + j ) is unknown at time t. Nevertheless, this is the goal of the adaptive controller. The adaptive controller must therefore contain the process identification and prediction as part of

0018-9286/92$03.00 0 1992 IEEE

916 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 31, NO. 7, JULY 1992

itself, rather than having them added on in an ad hoc way. If the whole cost function is minimized with respect to the model parameter estimates and the control action, then we have a finite horizon dual controller [ 121. Unfortunately, dual control is impractical because of the nonlinear dependence of the control action on parameter estimates and their uncertainties. Instead, a certainty equivalent formulation is proposed. The overall control criterion of (3) can be rewritten as

This can be rearranged to give the following three terms:

N2

- 2 C ( ~ s p - p ( t + j I f ) ) ( ~ j ( f ) ) ( 5 ) j = N ,

where e j ( t ) = y ( t + j ) - j ( t + j I t ) is the j-step ahead prediction error.

Obviously, any approach that minimizes the three terms indepen- dently (e.g., a certainty equivalent controller) will be suboptimal if the control problem is not neutral or certainty equivalent [13]. The difficulty lies in calculating the control action when there is parameter uncertainty -addition of “probing” may help to reduce the parameter uncertainty and thus improve long term control quality even though it will increase the cost of the first term in (5). Nevertheless, it is instructive to design a certainty equivalent adaptive controller using this cost function.

The first term on the right-hand side of (5) corresponds exactly to the GPC cost function at time t , as in (2). GPC is one controller that minimizes the value of this term for a given model, and is one certainty-equivalent control algorithm for the specified adaptive control objective (3). A “cautious” controller could also be formu- lated to take parameter and measurement uncertainty into effect, but development of such a controller is beyond the scope of the present work.

The second term is the identification objective. Expanded, it is given by the following expression:

N2 N7

J ,D= 2 E , ( ‘ ) * = 2 [ y ( f + J ) - P ( t + j l t ) ] ’ . (6) j = N , j = N ,

So the optimal identification method for the overall control objective (3) must provide the model that predicts best, not just one step ahead, but also two steps ahead, three, and so on up to N2 steps ahead. This makes sense because GPC uses these long-range predictions. In contrast, the commonly-used least squares (LS) method is only concerned with one-step-ahead predictions. In the presence of structural model-plant mismatch, the least squares model will not give the best long-range predictions. The use of RLS thus reduces the effectiveness of adaptive GPC because the LS models are suboptimal. Better control can be arrived at by using an identification method satisfying the objective in (6) above.

The role of the identification scheme is then to minimize the second moment of the prediction errors over the horizon used by the controller. This is distinct from the least squares objective, which attempts merely to minimize the one-step-ahead prediction errors. Thus, a model or prediction obtained by minimizing any criteiron

other than (6) will not provide the best control. What is needed is a control-relevant identification method, i.e, one that provides the best predictions in the range required by the controller. The development and implementation of this method, known as long-range predictive identification, or LRPI, will be discussed in the next section.

The third term on the right-hand side of (5) is a cross term combining the effects of the identification and control. It is ignored in all certainty-equivalent control. The expected value of this term is zero, if the prediction errors for all j are zero mean and uncorrelated with the predicted control errors. Because of noise or model-plant mismatch, the prediction errors are normally highly correlated with the actual control errors. They are relatively uncorrelated with the predicted control errors, as long as the model is reasonably accurate. Lu and Fisher [14] have stated that this term may be considered to be zero on average subject to weak conditions if the model is identified using a regression method, since the prediction errors are orthogonal to the predictions. In the present study this cross term is assumed to be zero asymptotically as the number of observations tends to infinity.

II. DEVELOPMENT OF LRPI

The identification objective is to minimize the value of the second term in (5). This is sufficiently different from the LS objective to change adaptive control performance. The measurable cost minimized by LRPI is chosen to be the following regression:

where the number of predictions considered Np = N2 - NI + 1 . In words this corresponds to minimization of the mean sum of squares of j-step-ahead prediction errors. The prediction errors considered are those over the horizon used by the GPC controller, and the mean is with respect to all previously gathered data. This cost function is merely an expression of the standard regression objective. It is nonlinear because the long-range predictions ( j > 1) are nonlinear in the model parameters [ 151.

Implementation of LRPI using various techniques (including Newton-Raphson and Gauss-Newton) is discussed at length in

The cost function expressed in (7) is merely an extension to long-range prediction of the standard least squares regression objective, which is normally written thus

~151.

1 ‘ JLS = - [ Y ( k + 1 ) - j ( k + 1 I k ) I 2 . (8)

t k = l

Although (7) is simple, it is computationally difficult to solve for the parameters. The j-step-ahead predictions are nonlinear in the parameters for j > 1 , so the usual least squares closed-form solu- tion is not available.

Example 1: The following second-order discrete process was simulated to illustrate the models that result through minimization of objective functions expressed in (7) and (8)

~ ( t ) = -1 .4y( t - 1 ) + 0.5y( t - 2) + O . l ~ ( t - 1 )

+ 0.05u(t - 2) .

The process input U was white noise. A first-order model structure

y ( t ) = B,y( t - 1) + & u ( t - 1)

was chosen. The step responses of the two optimum models are shown in Fig. 1 . It is clear that the LRPI model (with NI = 1 and N, = 10) provides a better match to the step response over the first

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 37, NO. 7, JULY 1992 911

The LRPI cost function, JLRPrr may be written in terms of the 1.6

1.4- GPC predictor

1 t -Nz 1 N2

JLWI = - c--c t - N2 k = l Np j = ~ ,

. [ y ( k + j ) - ( ___ ";lr) + T A u ( k + j - l))]'. (10)

The j-step-ahead prediction error at time k (using filtered data) is contained within the large :quare brackets. To analyze the dependence of the model { A , B } on the design variables, it is first desirable to remove the process output y from (10).

The j-step-ahead Diophantine equation may be written thus Time (samples)

Fig. 1. Step responses of LS and LRPI models (for LRPI N , = 1 and N2 = 10).

10 steps than the LS model. In particular, the LS model fits the first point well, but later ones poorly. The step responses are shown for two reasons: first, they are simple, easily understood examples of long-range prediction, and second, the step response parameters are themselves important to GPC .

When the GPC control action is calculated, a vector of future errors is multiplied by the pseudoinverse of a matrix of step response coefficients from N I to N,. If these values are badly underestimated (as for the LS model), the resulting controller will have too high a gain. The closer estimates from the LRPI model will result in better control. n

A. Implementation of LRPI Through Filtering

A full derivation of LRPI including stochastic extensions and interpretations is given in [15]. The discussion here will be limited to the deterministic case with structural model-plant mismatch introduced by underestimating the order of the process transfer function. The identification cost function is analyzed in terms of the frequency-domain implications of long-range prediction, in a man- ner similar to that in [16], [17]. Using Parseval's theorem in the limit as the number of observations tends to infinity, the LRPI cost function can be transformed from time to frequency domain. Equa- tion (7) may then be expressed as the sum of N, individual cost functions, one for each value of j .

The process is assumed to be of the form

( 9 )

where A0(q- ' ) and BO(4-I) are polynomials in the backshift operator q - I .

The process model is given as [ 181

t ( f ) q 4 - y A ( q - ' ) y ( t ) = B ( q - ' ) u ( t - 1) + ___

A where A and B are in this case lower order polynomials than A' and Bo, respectively. The noise model

LI

is imposed by GPC. The j-step-ahead GPC predictor is [ 181

F. E . B j ( t + j J t ) = ' y ( t ) + I A u ( t + j - 1).

T T

F.( 4- l ) Ej( q - ' ) a( q - ' ) A 1 - q - j L = T ( q - ' ) T ( 6 )

and this may be applied to y ( k + j ) in the cost function to give

1 t -N2 1 N2

JLRPI = - c--c t - N2 k=l Np j = N ,

I' A A E . B y ( k + j ) - L A u ( k + j - 1 ) . ( 1 1 )

T

The true process description (9) may now be used to express y ( k + j ) in terms of u(k + j - 1)

E , A A B O E . B -- u ( k + j - 1) - L A u ( k + j - 1) [ T A' T

This can finally be rearranged to give the following equation:

1 f - N 2 1 N2

- [ (7) E, A A (; - $)U(. + j - I)]*. (13)

The contents of the large square brackets are still the j-step-ahead filtered prediction error at time k , as in (10).

We now have an expression that gives the mean square prediction error as an explicit function of the design variables NI and N, , the noise model T ( q - ' ) , the actual process, the process model, and the input sequence. The LS cost is the same, but with NI and N, both equal to 1

k= 1

The noise model is present in both of these expressions as the A / T term. If, in addition to the noise model, an additional filter L ( q - ' ) were used for identification, then the LS cost function would change to the following:

To compare (13) and (15) in a meaningful way, it is useful to examine the frequency domain properties of the different models. The frequency domain provides a useful framework for analyzing


L - - - - - - I

Predlctlve

different processes and models regardless of their orders. On the other hand, it is difficult to compare a second-order process and a first-order model through examining the parameters themselves. The use of Parseval's theorem applied to (15) yields

where @,,(a) is the power spectrum of the input u ( t ) .

different j-step-ahead cost functions The LRPI cost function may in turn be written as the sum of Np

The summation may be moved within the integral

What is interesting about this equation is that the only term that explicitly depends on the prediction horizons (required to be within the summation) is the magnitude of Ej. To make this clear (18) may be rewritten as

The only difference between the LS and LRPI costs is the L filter in the LS equation and the summation of Ej in the LRPI cost. If all other parts of the cost functions are the same (same input, same model, same actual process) then the LS cost is equal to the LRPI cost if the following condition holds:

for all W . Fortunately, if L ( q - ' ) is of order N2 then there is a unique L which satisfies this criterion. The L filter is found by the spectral factorization technique of Peterka [ 191. For on-line applications, one iteration of the method is carried out each time step, as convergence is quite rapid. The L filter is also a stronger function of the prediction horizons than of the process model [15], so the change in L ( q - ' ) is quite small when the model changes. A detailed discussion of the L filter is included in [15].

On-line implementation of adaptive GPC using LRPI is shown diagrammatically in Fig. 2. The LRPI filter L is calcualted using the previous process model and then applied to the current data. The filtered data are then used to update the process model, which is then used by GPC to calculate the new control action.

III. EXPERIMENTAL RESULTS Experimental studies were carried out on a pilot-scale stirred tank

heater in the Department of Chemical Engineering at the University of Alberta. A schematic diagram of the equipment is shown in Fig. 3. It consists of a double-walled glass tank 50 cm high with an inside diameter of 14.5 cm. Cold water enters the tank and is heated by a steam coil. The water leaving the tank passes through a long 1 inch copper pipe with four thermocouples placed along its length. The different thermocouples provide a choice of transport delays.

+ dlsturbances

II 1

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 37, NO. 7, JULY 1992 979

STIRRED TANK HEATER

cold water in

steam t Fig. 3 .

condensate

Schematic diagram of stirred tank heater equipment.

- $ 45 0 - '2 40

35

- 0 100 200 300 400 500 600 700 800

100

c 80

2 60

40 - 0 " 20

'0 100 200 300 400 500 600 700 800 time (samples)

Fig. 4. Process input and output for GPC with LS estimator.

503 45 c i

0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800

45 c 1

60 I I

lo: 100 200 300 400 500 €00 700 e60 lime (samples)

Fig. 6. Process input and output for GPC and LRPI.

-0.9

-0.95

2 -1

-1.05

-1.1 0 100 200 300 400 500 600 700 800

0.06

0.04

i; 0.02

'0 100 200 300 400 500 600 700 800 time (samples)

Fig. 7. Parameter trajectories for GPC with LS estimator.

-0.9 I

-1.1" . I 0 100 200 300 400 500 600 700 800

lo: 100 200 300 400 500 600 700 8bO lime (samples)

Process input and output for GPC with LS estimator and an ad hoc filter.

second-order filter of 5(1 - 0.8q- ' ) * (the factor of 5 was used to keep the steady-state gain of the second filter equal to unity), and LRPI. Naturally, the data filtering for LRPI included the effects of the A and T filter of the controller disturbance model.

The results of the first set of experimental trials are shown in Figs. 4-6. Control with a first-order filter applied to the estimation is obviously unacceptable. Use of a second-order filter results in significantly better control, very similar to the LRPI results. There are large excursions when the disturbance enters the process because of the long dead time and relatively detuned controller.

The large fluctuations in the parameter estimates and the control signal in the first case, shown in Fig. 7 , are caused by the emphasis placed on the high frequency part of the prediction error due to insufficient signal filtering for identification. The use of a variable

0.04 - U - - 9

0.02 -

'0' 100 200 300 400 500 600 700 800 o/ 100 200 300 400 500 600 700 8AO time (samples)

Fig. 8. Parameter trajectories for GPC with LS estimator and an ad hoc filter.

forgetting factor results in large parameter fluctuations, but is required if the "true" parameters are known to vary [21].

The model parameters for GPC control with the ad hoc filter and the LRPI filter, shown in Figs. 8 and 9, show no such behavior. It is remarkable how similar the process temperature and parameter trajectories are for these two cases. The only apparent difference is in the convergence rate. The LRPI parameters do not move quite as quickly as the LS parameter during the changes in process. The difference is likely insignificant, since the closed-loop control performance is almost identical.

II I ~


z . o i r ,~ ,’I -1.05

-1.1 0 100 200 300 400 500 600 700 800

-’--I I

I I I ‘0 100 200 300 400 500 600 i o 0 800

lime (samples)

Parameter trajectories for GPC and LRPI. Fig. 9.

l o o

i

10-2 10-1 100 10’ 10-2

10-3

Normalized frequency

Fig. 10. Evolution of LRPI filter ( 0 0 * . . 0 line shows the ad hoc T filter, whose FIR approximation (1/5(1 - 0.8q-’)) is also the initial value of the L filter).

Fig. 10 shows the evolution of the LRPI filter during the run. The initial value of L was a FIR approximation 1 / ( 5 - 4 q- ’). Over the course of eight samples the filter converged to near its final value. The ad hoc T filer is also shown in Fig. 10, and it can be seen that the ad hoc filtering is very similar to the actual “optimal” L filter, for this particular process and tuning. If the sampling frequency were changed then this ad hoc filter may not be suitable. LRPI has also been used in conjunction with constrained GPC as reported in WI.

IV. CONCLUSIONS The use of an overall control objective in the design of an

adaptive long range predictive controller has been shown to result in the use of a multistep identification and control strategy. The identification strategy is “control-relevant’’ and thus makes the overall adaptive control “optimal.” The use of a long-range predictive identification strategy is demonstrated to reduce the variance of long-range prediction errors and to improve the quality of control over GPC plus RLS. The usefulness and practicality of the sug- gested approach is demonstrated by successful simulation and experimental evaluations.

REFERENCES [l] C. E. Garcia, D. M. Prett, and M. Morari, “Model predictive

control: Theory and practice-A survey,” Automatica, vol. 25, pp. 335-348, 1989.

C. Cutler and B. L. Ramaker, “Dynamic matrix control-A com- puter control algorithm,” in Proc. JACC, San Francisco, CA, 1980. B. E. Ydstie, L. S. Kershenbaum, and R. W. H. Sargent, “Theory and application of an extended horizon self-tuning controller,” AIChE J . , vol. 31, no. 11, pp. 1771-1780, 1985. R. M. C. de Keyser and A. R. van Cauwenberghe, “Extended prediction adaptive control,” in Proc. IFAC Symp. Identification Syst. Parameter Estimation, York, UK, 1983. D. W. Clarke, C. Mohtadi, and P. S. Tuffs, “Generalised predictive control-Part I. The basic algorithm,” Automatica, vol. 23, no. 2,

B. A. Francis and W. M. Wonham, “The internal model principle of control theory,” Automatica, vol. 12, no. 5, pp. 457-467, 1976. A. R. McIntosh, S . L. Shah, and D. G. Fisher, “Experimental evaluation of adaptive control in the presence of disturbances and model-plant mismatch,” Adaptive Control Strategies for Industrial Use, (Springer-Verlag Lecture Notes in Control and Information Sciences, vol. 137), S. L. Shah and G. Dumont, Eds. New York: Springeg-Verlag, 1988, pp. 145- 174. K. J. Astrom and B. Wittenmark, “On self tuning regulators,” Automatica, vol. 9, pp. 185-199, 1973. A. R. McIntosh, S. L. Shah, and D. G. Fisher, ‘Analysis and tuning of adaptive GPC,” Canadian J . Chem. Eng., vol. 69, pp. 97-110, 1991. C. Mohtadi, “On the role of prefiltering in parameter estimation and control,” Adaptive Control Strategies for Industrial Use (Springer-Verlag Lecture Notes in Control and Information Sciences, Vol. 137), S. L. Shah and G. Dumont, Eds. New York: Springer- Verlag, 1988, pp. 121-144. E. P. Lambert, “Process control applications of long range prediction,” D.Phi1. dissertation, Univ. Oxford, 1987. A. A. Feldbaum, Optimal Control Theory. New York: Academic, 1965. J. W. Patchell and 0. L. R. Jacobs, “Separability, neutrality and certainty equivalence,” Int. J. Contr., vol. 13, no. 2, pp. 337-342, 1971. W. Lu and D. G. Fisher, “Nonminimal predictive control,” Chem. Eng. Sci., vol. 47, no. 4, pp. 809-820, 1992. D. S. Shook, C. Mohtadi, and S. L. Shah, “Identification for long range predictive control,” IEE Proc. D, vol. 138, no. 1, pp. 75-84, 1991. B. Wahlberg and L. Ljung, “Design variables for bias distribution in transfer function estimation,” IEEE Trans. Automat. Contr., vol. AC-31, no. 2, pp. 134-144, 1986. L. Ljung, System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987. D. W. Clarke, C. Mohtadi, and P. S. Tuffs, “Generalised predictive control-Part 11. Extensions and interpretations,” Automatica, vol. 23, no. 2, pp. 149-160, 1987. J. Bohm, A. Halouskova, M. Karny, and V. Peterka, “Simple LQ self-tuning regulators,” in Proc. 9th IFAC World Congress, Bu- dapest, Hungary, 1984. R. Kulhav):, “Restricted exponential forgetting in real-time identification,” Automatica, vol. 23, no. 5, pp. 589-600, 1987. R. Kulhav): and M. Karny, “Tracking of slowly varying parameters by directional forgetting,” in Proc. 9th IFAC World Congress, Budapest, Hungary, 1984. R. K. Mutha, “Constrained long range predictive control,” Master’s thesis, Univ. Alberta, Alberta, Canada, 1990.

pp. 137-148, 1987.

Identification for the Control of MIMO Industrial Processes

Ton C. P. M. B a c k and Ad. A. H. Damen

Abstract-A procedure for the identification of industrial processes with the intention of control system design is proposed, discussed, and

Manuscript received December 15, 1990; revised October 25, 1991. The authors are with the Measurement and Control Group, Faculty of

Electrical Engineering, University of Technology, Eindhoven, The Nether- lands.

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a control-relevant identification strategy for gpc

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