generic model control (gmc)

Contput. &e-m. Engng, Vol. 12, No. 6, pP. 573580, 1988 0098~1354/88 $3.00 + 0.00 Printed in Crest Britain. All rights reserved Copyright 8 1988 Pergamon Press plc

GENERIC MODEL CONTROL (GMC)

P. L. LEE’ and G. R. SULLIVAN~~

Department of Chemical Engineering, University of Queensland, St Lucia, Queensland.4067, i4ustralia and sDepartment of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada NZL 3Gl

(Received 28 January 1987; final revision received 2 September 1987; received for publication 5 October 1987)

A~ct-This paper presents a general framework for process controllers that rely upon a timdel to approximate plant hehaviour. By careful selection of a performance index and an approximate plant model, it is shown that single-loop PI control, feedforward and decoupling control, multivariable regulators, time horizon matrix controllers, internal model control and process model based control can all be derived. Furthermore, nonlinear process models can be imbedded directly into the controller without resorting to linearization. This unifying framework is illustrated with a number of examples to highlight the utility of such an approach.

INTRODUCTION

The need for improved process control has become more evident in recent years. The continued impact of energy conservation measures, the tighter integration of process designs and the continued high cost of capital have all lead to a greater demand for improved control performance. In addition, the means to implement improved control strategies has also increased with the availability of cheaper and more powerful computer hardware.

Since the beginning of this decade, there has been increased interest in multivariable control techniques. This has been illustrated by the work on dynamic matrix control (DMC) (Cutler, 1983), model algo- rithmic control (MAC) (Richalet et al., 1979) and internal model control (IMC) (Garcia and Morari, 1982). These techniques are all similar in the sense that they rely on models to predict the behaviour of the process over some future time interval, and the control calculations are based on these model predictions. The models used for these predictions have usually been linear approximations of the process of experimentally obtained step response data. Un- fortunately, real chemical processes generally behave in a quite nonlinear manner, and some chemical and biochemical processes change behaviour over a period of time. For these processes, one would like to imbed nonlinear process models in the controller itself.

There is a strong desire to develop a generic control structure that directly imbeds nonlinear process models such that many of the current “modem” control algorithms are subsets of this generic control structure. Furthermore, it is desirable that such a control structure would lead to an easy online imple- mentation. The work described in this paper illus- trates that by suitable problem formulation and a reasonable choice of a performance index, a number

TAuthor to whom all correspondence should be addressed.

of different control strategies can be derived. This paper first presents the underlying structure of this approach and then presents a number of examples to highlight the utility of the method.

CONTROL STRUCTURR

Consider a process described by equation:

f=f(x,u,d,t),

Y = r&x),

the following

(1)

(2)

where x is the state vector of dimension n, u is the vector of manipulated variables of dimension m, d is

the vector of disturbance variables dimension 1 and y is the output vector of dimension p. In general, f and g are some nonlinear functions. It follows from equations (1) and (2) that:

where

jr = G,f(x, a, d, t ),

In classical optimal control, the trajectory of Y is usually compared against some nominal trajectory, (y)*(t), as a measure of system performance. As an alternative, consider the performance of the system to be such that:

(jr)‘(t) = r*(Y), (4)

where r* represents some arbitrary function to be specified. Performance specifications using time de- rivatives of the output variables have been previously considered in robotic manipulators (Guo and Sardis, 1985).

Let us examine one reasonable choice of this specification function r*. When the process is away from its desired steady state y* we would like the rate

573

574 P. L. LQ and 0. R. SULLIVAN

of change of y. 9, to be such that the process is returning towards steady state, i.e.

jr= K,(~)(Y*-Y), (5)

where K,(t) is some diagonal matrix. In additon, we would like the process to have zero

offset, i.e.

f = KAt 1 (Y* - Y) dc (6)

where K,(r ) is some diagonal matrix. In the following we will consider that K,(t) and K,(t) are constant with respect to time.

Good control performance will be given by some combination of these objectives, i.e.

W* = K, (Y* - Y) + K, (Y' - Y) dt. (7)

Suitable choices of K, and K2 can be made to achieve a variety of responses in y(t) as detailed later in the Examples section.

We would like to choose u(t) such that the system follows (f)*(t) as closely as possible. This transforms into the optimal control problem, OCl:

given x = f(x, II, d, t),

Y = t3(x), choose u(t) such that 1 u I< a, to minimize

s ‘/ Mx, u, d, r )= wh(x, u, d, 111 dt, 0

where:

(8)

h(x, u, d, r )

= G,f(x, II, d, t) - K,(y+ - y) - K, s (Y* - y)dr (9)

and W is an appropriate positive-definite weighting matrix. If the control is feasible with respect to the constraints, the dimensions of m and p are the same and at least one element of II appears in each of the m equations represented by equation (9). then the solution of CKX is given by the solution of m equations in m unknowns:

G,f(x, u, d, t ) - K, (Y* - Y)

-K, (y*-y)dt =O. (10) s

Otherwise the solution reduces to the choice of u to minimize the instantaneous value of h(x, u, d, t ) at every point in time. This will involve performing a single time step online integration.

In general, the exact process model is rarely known, and an approximate model is introduced such that equation (10) becomes:

c;,il(x, u, d, r ) - K, (Y’ - Y)

-K2 s

(Y’-y)dr =O. (11)

where T and t? = at/ax represents the approximation to the true model.

The control manipulations that satisfy equation (11) must result in good control if the process model f and & are reasonable and (y)*(r ) is reasonable such that we operate within the control action constraints. Any degradation in control performance due to interactions of nonlinearities can be explained through the choice of P and c, to be used within the control law of equation (11).

In circumstances where the dimensions of m and p are equal and the “difference order” of the system (Freund, 1975) is equal to 1, explicit solutions of equation (11) are possible. This is similar to the approach used in “nonlinear decoupling” for systems with the difference order not equal to 1, as outlined by Freund (1973, 1982).

The control law has four desirable characteristics:

1. The control law as represented by equation (11) has within its structure an approximate process model.

2. Any inaccuracies introduced by this approximation will result in y not tracking (y)*(r) but this will be compensated by the integral term in the control algorithm. This integral term in the control law ensures that the controller is robust despite modelling errors.

3. It is a single-step law that has time horizon characteristics as defined by equation (11). The solution from the last control step is a good approximation to the solution of the current control step.

4. There is no need to perform an online integration of the process model providing that all states are observable.

In the following, for the ease of presentation, we shall assume that the output variables y and the state variables x are identical. If this condition is not met, then online integration of the state equations will be required to estimate the output variables. Let US

consider some examples of this control algorithm.

Case l-single-loop PI control

Let

g(x, II, d, I ) = Iu, (12)

i.e. no model of the process. Therefore, from equations (11) and (12):

u=K,(x*-x)+K, (13)

If K, and K2 are chosen to be constant diagonal matrices, then single-loop PI controllers result. The pairing between states and manipulated variables can also be dictated by alternate structures of K1 and K2.

Case Z-linear multivariable regulator

IA

9(x, II, d, t ) = Ax + Bn + Dd (14)

where A, 6, D are constant matrices. We assume that B is invertible in this and subsequent examples. Then solution of equations (11) and (14) yields:

u=B-‘K,(x*-x)+B-‘K, (x*-x)dt s

Generic model control 575

Substituting equation (18) into equation (16) and rearranging yields the closed loop response as:

ir=(A-B8-‘&x+(D-Blk’b)d+Bk’

[ K,(r*-x)+K+x*-x)dt]. (19)

- B-‘Dd - Bm’Ax. (15) The first two terms of equation (19) quantify the plant-model mismatch whereas the last two terms represent the control to compensate for the mismatch. The GMC formulation given by equation (11) has generalized this approach for the nonlinear case.

This form of the control law is similar to that derived for a linear multivariable regulator which includes pseudo-integral states (Newell and Fisher, 1972). The second last term provides feedforward, while the last term provides a decoupling of state interactions.

Case 3-time horizon matrix controllers

Dynamic matrix controllers (DMC) (Cutler, 1983) and universal dynamic matrix control (UDMC) (Morshedi, 1986) are examples of a general class of time horizon matrix controllers. Basically these controllers solve the optimal control problem, OCl. DMC builds a linear input-output model of the process and uses a performance index to force x(t ) to x*(t). UDMC essentially solves OCl using a nonlinear process model and a quadratic performance index involving x(t). Both of these approaches require solution for a number of control intervals in the future resulting in a large optimization problem. The advantage of the formulation given by equations (7x11) is that the optimal solution for the current control interval can be determined without calcu- lation of future control moves, thereby significantly reducing the size of the optimization problem. The time horizon characteristics of the solution are em- bedded in the PI terms of equation (11). However, it should be observed that time horizon matrix controllers do allow consideration for constraints on future control moves which may impact the current control calculations (Cutler, 1983; Morshedi, 1986).

Case I-internal model control (IMC)

IMC may be viewed as an integral only controller on the plant-model mismatch, plus a model pre- diction of the nonminimum phase elements. As a comparison, consider the linear regulator problem where:

f(x,u,d,t)=Ax+Bu+Dd

and choose

(16)

?(x, u, d, t ) = Ax + lh + bd

which gives the control law:

(17)

u=B-‘K,(x*-x)+8-‘K2 (x*-x)dt I

Example l-response specification

Consider a multivariable system such that:

ir=f(x,u,r) (20)

and that the process model used is perfect, i.e.

?=f. (21)

From equation (1 l), the control law is:

- b-‘bd - b-‘Ax. (18) f(x,u,t)-K,(x*-xX)-KKz (22)

d

Case 5-model reference adaptive control

Model reference adaptive control (MRAC) has been successfully demonstrated to control a pilot plant evaporator (Oliver et al., 1973a,b). MRAC requires the design engineer to specify the form of the control law and the desired system response, and then adapts the controller parameters to make the plant behaviour match that desired. Thus, both GMC and MRAC allow the user to specify a desired response of the plant. However, GMC does not require the explicit specification of the control law and is not limited to linear process models.

Case G-process model based control

The use of a nonlinear process model in a control algorithm derived from fundamental deterministic considerations has been demonstrated for distillation and particulate drying systems (Cott et al., 1986; Forbes et al., 1984). It can be shown that this is equivalent to using equation (11) directly as the control law with appropriate choices of K, and K,. The advantage of such an approach is that the nonliner behaviour of the process is more accurately modelled. In addition, any approximations made in the model derivation will be compensated for in the integral term in the control law. This makes such an algorithm robust in the face of modelling errors.

EXAMPLES

The purpose of this section is to illustrate how the control law specified by equation (11) can be used in practice. A number of examples illustrating specific points have been chosen to meet this objective.

576

and

P. L. LEE and G. R. SLXLIVAN

1.2

r

R=K,(x*-x)+K* (23)

It can be seen that by different choices of K, and K, the performance specification [equation (7)] can be altered for each variable separately. The single vati- able case is illustrated in Figs 1 and 2 for two different choices of these parameters. Thus, one can use these values to select any “reasonable” desired response for the system. “Reasonable” implies that the parameters are chosen in relation to the system’s natural dynamic response. How well the system matches this performance index will be governed by how closely the chosen model matches the plant bebaviour. This will be illustrated further in the following examples.

The results of these two examples can be further generalized by examining the structure of equation (23). If Laplace transforms of equation (23) are taken, the resulting transfer function becomes:

x 2zrs + 1

>= rY+2rCs + 1’

where

(24)

This system does not yield the same response as a classic second-order system (Stephanopoulos, 1984). However, similar plots to the classic second-order response showing the normalized response of the system x/x * vs normalized time f/r with 5 as a parameter can be produced and is shown in Fig. 3. The design procedure can be specified as follows:

1. Choose c from Fig. 3 to give desired shape of response.

2. Choose r from Fig. 3 to give “appropriate” timing of response in relation to known or estimated plant speed of response.

0 10 26 30 40 50

Time (a)

Fig. 1. GMC profile specification for set-point change (Example 1); k, = 0.25, k, = 0.25.

0.01 I I I I I

0 10 20 30 40 M

Time ($1

Fig. 2. GMC profile specification for set-point change (Example 1); k, = 0.1, k2 = 0.001.

3. Calculate k, and k2 using the following equations:

k=Zr- I 5’ (25)

Example Z-parametric errors

Consider a single-variable system such that:

R=bu (27)

and that the process model used is chosen as:

p=6U. (28)

From equation (1 1), the control law is:

u -6-l k,(x* -x)+k+x*-x)dt] (29)

and

i=bb-fk,(x*-x)+k+x*-x)dt]. (30)

1.4r

Fig. 3. Generalized GMC profile specification (Exardple 1).

Generic model control

Thus it is seen that if b and d are the same, the system performance will perfectly follow the desired performance index as dictated by the choice of k, and k,. If 6 does not equal b, then some degraded performance will occur. This is illustrated in Fig. 4 for b = 1, 6 = 1.8 and k, = k2 = 0.25.

Example 3-parameter update

It is clear from the previous example that parametric errors in the model degrade the system performance. A means of updating the steady state parameters in the chosen process model can also be easily derived from equation (11). Consider a single- variable system such that:

i=ax+bu

and that the process model is chosen as:

p=cix +6u.

From equation (ll), the control law is:

(31)

(32)

u=6-‘dx+d-’ k,(X+-XX) [

and

+k+x*-x)dt] (33)

k=ux-b6-‘cix+b6-’ k,(x*-x) [

++x*-x)dt]. (34)

At steady state, if d # a and d # b, fwill not equal zero. We would like to update our model parameters such that f does equal zero, i.e.

3=o=cix*+6u*, (35)

1.6 r

1.6 ‘:

1.4 I : : :

1.2

I:‘.;-

: ;

: :, :“,

* 1.0 ; ; ,I ’ .v \ I : I :

\I

xc16; ’ ii I,“,’

0 ,o 20 x) 40 SO 60 70 60 60 100

Time (s)

Fig. 4. Impact of parameter error on GMC controller (Example 2); k, = 0.25, k, = 0.25; b = 1.0; 6 = 1.8.

*r

- f - QX + bu lsoecitlcation)

---F&i +%,

-lJ- , , , , , , , , , , 0 20 40 60 60 100 120 140 160 160 200

Time (s)

Fig. 5. Model parameter update for GMC controller (Exam- ple 3); k, =0.25, k2 = 0.25; a = 1.0, b = 1.0, d = 1.0,

6 = 1.1.

This is then a strategy for updating the steady state or gain parameters in our process model. This strategy is illustrated in Figs 5 and 6. Figures 5 and 6 show the response to both a step increase and decrease in the process setpoint. During the step increase the control law ‘in equation (34) is used with initial parameter estimates. When steady state is obtained, the parameter update strategy of equations (35) and, (36) is employed. Thus the response to the step decrease includes better estimates of the model parameters.

2

1

% . x

C

-1

- f . ax + bu (specification)

---1-:x + Ab”

0 20 40 60 80 loo 120 140 160 180 200

Time (5)

Fig. 6. Model parameter update for GMC controller (Exam- ple 3); k, = 0.25, k, =0.25; n = 1.0, b = 1.0, h =0.8,

6 = 0.6.

578 P. L. Lse and G. R. SULLIVAN

Figure 5 shows the result for the case when the parameter describing the dynamic response is correct,

2-

,q , ’ i.e. when B = a. As can be seen, the parameter update strategy has returned the system performance to exactly that desired. In the case where the parameter describing the dynamic performance is not known X exactly, improved performance still results as shown in Fig. 6. Updating of ci will require dynamic parameter estimation and this can be shown to be an extension of the steady state procedure.

: :- > c 8 \ I’

Example 4-constraint action 01 I I

Most processes also include constraints. These \_I

\~,~_~~~~_~~~~L._

constraints can occur on both the manipulated variables and the desired values of the controlled vari- - ~u~cm(specificaflonl

ables. The effect of these constraints is often to limit --- JUl~O.2 the controlled performance. -1- , , , , , , , , , ,

Consider again the process such that: 0 10, 20 30 40 Jo 60 70 80 so 100

Ti = bu (37) Time (5)

and that the process model is chosen perfectly, i.e. Fig. 7. GMC with constraint on control variable (Example

4); k, = 0.25, k, = 0.25; b = 6 = 1.0.

f=bu (38)

and also impose a constraint such that: and that the process model is chosen as:

luI<a. (39) T=Ax+Bu (46)

This is the same process and model as bonsidered From equation (1 l), the control law is: in Example 2. The problem as stated in this example more closely resembles the general structure as out-

u = B-‘Ax

lined in OCl. With W set at 1, the control law is given by the choice of u that minimizes:

+B-‘[K,(x*-x)+K,J(x*-x)dr] 47)

[ bu -~,(x*-x)--L2~(x*-x)df]i. (40) an~zAx_BB_,k

The solution to this problem is:

u=b-’ k,(x*-x)+k+x*-x)dr] [

(41)

if lu(<a or:

u=a (42)

if (u ) > a. The effect of adding this constraint for a = 0.2 is shown in Fig. 7. As can be seen the inclusion of the manipulated variable constraint has limited the performance of the control system such that the desired trajectory can no longer be followed.

Example 5-modelling errors

+Bk!-’ K,(x*-x)+K2 s 1 (x*-*)dt ., (48)

Let us consider the case of structural errors in our process model. Firstly let us assume ‘that A = 0 and that B = B. The results are shown in Fig. S.,It can be seen from this figure that reasonable control performance is obtained.

Secondly, let us consider the case where:

A=A

and

(49)

8= 3.0 0.0 I 1 0.0 0.5 . (50)

Consider a system such that:

Z=AxfBu

with

A= -0.02 -0.05

-0.05 -0.2 1

This represents 50% parameter errors in the diag-

(43) onal terms and no estimates of the off-diagonal terms. The results are shown in Fig. 9, and it can be seen that the control performance has been degraded.

Thirdly, consider the case where: (44) rl\=A X51)

and

B=[;:; -;:;]

and

(45) 8=

2.0 1.0

[ 1 1.0 1.0 ’ (52)

Generic model control

1.2

r ,x, ,xA; f-Ax + 6u hpdflcation)

0.01 I I I J

0 100 200 300 .400

Time (8)

Fig. 8. Impact of modelling errors on multivariable GMC controller (Example 5); k, = 0.1, k2 = 0.001, (both vari-

ables); A = 0, ES = B.

which represents the correct maganitude but a wrong sign in an off-diagonal term. Here the control performance is reasonable, as can be seen from Fig. 10. Note that the performance specification for each of the state variables can be different as shown in Fig. 10. The performance specification for x, has been altered from that used in Figs 8 and 9.

Example 6-noniinearities

As the final example, consider a process such that:

x=--ax+bU2. (53)

However, use a linear process model:

f= --ax + bu, (54)

x,,xe,f.Ax + Bu (spKifiCOtl0n~

0.0 ;t

I I I I

100 200 300 400

Time (~1

Fig. 9. Impact of modelling errors on multivariable GMC controller (Example 5);P, =_O.!, k2 = 0.001 (both variables);

-‘-O0_

Time W

Fig. 10. Impact of modellihg errors on,multivariable GMC controller (Example 5); k, =O.l, k2 = 0.001 (for x,);

k, = 0.05, k, = 0.001 (for x2); A = A, 0 # 0.

i.e. the parameter values are known, but the non- linearity is not modelled. Using equations (11) and (53), perfect control would be given by the nonlinear control law:

(55)

This is anlogous to the results obtained from nonlinear decoupling (Freund, 1973, 1982). However, our process model will yield:

u=b-‘[ar+k,(x*-x)+k,J(x*-x)df]. (56)

1.2

0.6

* I 0.6

‘= -i

i

- f m-x + u* (specifieotlon)

--- s.-l+u

L I I I I I I I I I

10 20 30 40 60 60 70 60 so 100

Time (~1

Fig. 11. Comparison of linear and nonlinear GMC control- . I_ . _. . -. . - --- A=A; B#B. lers (example 6); k, = 0.1, Ic, = 0.001,

580 P. L. Lne and G. R. SULLIVAN

The results of using this strategy are shown in Fig. i 1. It again can be seen that modelling errors do degrade the performance of the controller.

CONCLUSIONS

This paper has presented a framework to unify many ideas that have been prevalent in process control for some time. By considering performance in terms of the time derivative of the output variables, a generic control law can be derived that embeds an approximate process model. This formulation leads to a single-step solution of an optimal control problem. Many different classes of controllers are shown to be a subset of this generic controller through suitable choices of the model approximation.

Future work will examine the stabilty characteristics of this control algorithm, the application to processes with dead-time (Doerr et al., 1987), the application of nonlinear steady state models for industrial control problems (Cott et al., 1988) and performance comparisons with traditional and mod- ern control algorithms on complex chemical engineering examples (Lee et al., 1987).

Acknowledgernears-The authors would like to thank IBM Canada Ltd for supplying computer facilities used in this work and financial support for PLL and the Natural Sciences and Engineering Research Council of Canada for their research support. The authors also acknowledge the useful comments from W. M. Grimm at the University of Duisburg (F.R.G.), Professor W. Wilson and B. Howie at the University of Waterloo.

REFERENCES

Cott B. J., P. M. Reilly and G. R. Sullivan. Selection techniques for process mode1 based controllers. Proc. Annl AZChE Mg, Miami (1986).

Cott B. J., R. G. Durham, P. L. Lee and G. R. Sullivan. Process model-based engineering: Parts l-3. Comput. Chem. Engng (1988) (in press).

Cutler C. R. Dynamic matrix control: an optimal multivariable algorithm with constraints. Ph.D. Thesis, Uni- versity of Houston, Texas (1983).

Doerr D., B. A. Howie, P. L. Lee, G. R. Sullivan and M. G. Whaley. Inherent models within PI controllers. Znd Engng Chem., Process Des. De0 (1987).

Forbes J. F., B. A. Jacobson, E. Rhodes and G. R. Sullivan. Model based control strategies for commercial grain drying systems. Can. J. Chem. Engng 62, 773 (1984).

Freund E. Decoupling and pole assignment in nonlinear systems. Electronic L&t. 9, 373 (1973).

Freund E. The structure of nonlinear systems. Znt. J. Control 21, 443 (1975).

Freund E. Fast nonliner control concepts for nonlinear computer controlled manipulators. Znt. J. ,Robot. Res. 1, 65 (1982).

Garcia C. E. and M. Morari. Intemal~model control 1. A unifying review and some new results. Znd. Engg Chem. Process Des. Dee. 21, 308 (19821.

Guo G. L. and G. N. Sardis. Gptimal/PID formulation for control of robotic manipulators. IEEE Znt. Cdnf. Robot. Autom. March 25-28, St Louis (1985).

Lee P. L., R. B. Newell and G. R. Sullivan. Generic model control-a case study. Can. J. Chem. Engng (1987).

Morshedi A. M. Universal dynamic matrix control. Proc. Chemical Process Cdntrol 3, Asilomar, California. pp. 547-577, Elsevier, New York (1986).

Newell R. B. and D. G. Fisher. Exnerimental evaluation of optimal, multivariable regulatory controllers with mode1 following capabilities. Automatica 8, 247 (1972).

Oliver W. K., D. E. Seborg and D. G. Fisher. Mode1 reference adaotive control based on Lianunov’s direct method. Part i: theory and control system*design. Chem. Engng Commun. 1. 125 (1973a).

Oliver W. K., D. E. Seborg and D. G. Fisher. Model reference adantive control based on Lianunov’s direct method. Part 2: hybrid computer simulation and experi- mental application. Chem. Engng Cornmutt. 1, 133 (1973b).

Richalet J.. A. Rault. J. L. Testud and J. Pawn. Model predictive heuristic’ control: applications to industrial processes. Automatica 14, 413 (1979).

Stephanopoulos G. Chemical Process Control: An Zntro- duction to Theory and Practice. Prentice-Hall, Englewood Cliffs, New Jersey (1984).

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