Convolution Model Based Predictive Controller for a Nonlinear

Dec 12, 1998 - The steady-state gain is derived from the first principle model of the system. The application of this special convolution model is as ...
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Ind. Eng. Chem. Res. 1999, 38, 154-161

Convolution Model Based Predictive Controller for a Nonlinear Process Arpad Bodizs,* Ferenc Szeifert, and Tibor Chovan Department of Chemical Engineering Cybernetics, University of Veszprem, P.O. Box 158, 8201 Veszprem, Hungary

The model predictive control (MPC) of a distributed parameter nonlinear laboratory heating system is studied. A nonlinear convolution model consisting of a linear dynamic and a nonlinear steady-state part is applied as the model of the process in the MPC algorithm. The dynamic part is represented by a relative impulse response model (IRM). The steady-state gain is derived from the first principle model of the system. The application of this special convolution model is as simple as the use of the transfer function model; however, it is valid on the whole operating range. MPC algorithms employing different models of the process are compared by simulation and physical tests. 1. Introduction In case of feedback systems the source of instability is mainly due to that the effect of the manipulated variable is delayed on the output. The slower the system is (or maybe it is an inverse response system), the more difficult the design of a suitable controller is. This problem is treated by model-based predictive controllers (MPC) which have had more and more industrial applications recently. Through prediction of the future error values these controllers are able to resolve an optimal control problem over a future time interval (prediction horizon). The several realizations differ in the type of applied models, in the form of the objective functions, in the considered limitations and in the methods of treating the model error.1 Despite the fact that the literature of MPC has just a 20 years old history, during these years its control concept became widely accepted in the academic area2-8 and in the industrial area9-11 as well. If the literature of model-based predictive control is studied, the most conspicuous feature is that the choice of an adequate (sometimes nonlinear) model has a distinguished role in the control design procedure of these controllers. The solutions range from the application of impulse response and step response models12,13 to the employment of neural network models14,15 and fuzzy models16,17 or recently to Hammerstein models.18 Other authors show a preference for state space models19,20 or ARMA and CARIMA models.4,5 The main improvements in the models, yielded with time, consist in their nonlinear feature. Using Hammerstein models (which are composed of a linear dynamic element in series with a nonlinear static element) in MPC is a new approach as well. Fruzzetti et al.18 showed that better controller performance can be obtained if Hammerstein models are employed instead of linear ones. It should be noted here that an analogous model design approach was applied in the modeling of the system presented below. Namely, a convolution model which consists of a steady-state nonlinear part in series with a dynamic part was put into practice. In fact, this approach belongs to the wellknown gain-scheduling concept for nonlinear control. * To whom correspondence should be addressed. Fax: +361-2063422. E-mail: [email protected].

Figure 1. Scheme of the physical system.

This paper applies transfer function, first principle, and nonlinear convolution models and aims to show the way toward a control-related model design and the implementation of predictive control in case of a nonlinear distributed parameter system with deadtime. To investigate the obtained models, impulse response models (IRMs) were computed and analyzed. By employment of different kinds of system models, the control performance of the controllers was tested by simulation and laboratory experiments too. 2. The Studied Physical System The studied physical system is a laboratory-sized heating system (Figure 1). The thermal agent (mains water) passes through a control valve (CV). The flowrate is measured with flowmeter Faq and the temperature with thermometer Tin. Then the water passes through a pair of metal pipes which contain a cartridge heater each. The heaters are linked parallel with each other, and each has a performance of 1 kW. The outlet water temperature is measured with thermometer Tout. The data acquisition and the control is carried out by a PC equipped with a process interface card. The control valve is operated by two digital signals (CVO, CVC). Physically the system consists of four analogue inputs (inlet temperature, outlet temperature, valve position, flow rate), two digital outputs (open and close of the valves), and one analogue output (heater control signal).

10.1021/ie980338q CCC: $18.00 © 1999 American Chemical Society Published on Web 12/12/1998

Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 155

The effective performance of the cartridge heater is given by

[

Q ) QM u -

]

sin(2π‚u) 2π

(1)

where QM is the maximal performance (in W) of the cartridge heater and u ∈ [0, 1] the relative input variable. Details on the computation of the performance of the cartridge heater are provided in the Supporting Information. This Q(u) relationship will be used in first principle modeling of the physical system. 3. Modeling for MPC Despite its physical simplicity, the system is characterized by some problematical properties. These are nonlinearity, distributed parameters, and delayed behavior. The guiding principle in model building was to obtain models that satisfy the model necessities of predictive control and not to design models with an accuracy over all limits. The fact that the model accuracy over a given limit does not serve to increase the performance of predictive controllers is mentioned by many authors.17,21 From the large number of system modeling methods, transfer function models and first principle models were studied. Further investigation was made on the basis of the IRMs generated from the designed models. Through this plan, employing IRMs, linear methods are used in modeling of a nonlinear system. 3.1. The First Principle Model. First principle modeling is a widespread process modeling approach in the field of chemical engineering. In the simulation of distributed parameter systems, a system of PDEs has to be solved. In our case the first principle model is the heat balance of the system. The physical system can be decomposed into three main elements. These are the cartridge-heater (CH), the moving thermal agent (w), and the wall of the pipe (W). A further element is the environment (env). Considering these elements, three heat balance equations were obtained:

∂TCH ) Q(u) - R1A1(TCH - Tw) (VFCp)CH ∂t ∂Tw ∂Tw (VFCp)w + (FFCp)w ) ∂t ∂z R1A1(TCH - Tw) - R2A2(Tw - TW) (2) ∂TW ) R2A2(Tw - TW) - RenvAenv(TW - Tenv) (VFCp)W ∂t Here, Q(u) is the sinusoidal performance of CH (eq 1) and z ∈ [0, 1] the dimensionless lengths. The values of TCH(0,z), Tw(0,z), TW(0,z), and Tw(t,0) ) Twin are known (as initial and boundary conditions). Through spatial discretization of the system of equations, a cascade of completely stirred tanks (CST) was obtained. To analyze the dynamics of the system, IRMs were generated from the first principle model. The IRMs were computed in the close neighborhood of a few working points in the operating range. Figure 2 shows the IRMs at a constant flow-rate value (F ) 70 L/h) and at different values of the manipulated variable (us1 ) 0.05, us2 ) 0.25, us3 ) 0.45, us4 ) 0.65, us5 ) 0.85). Figure 3 shows the IRMs at constant manipulated variable (us

Figure 2. Discrete IRMs generated from the first principle model (∆u: (s) 0.05-0.1; (- - -) 0.25-0.3; (O) 0.45-0.5; (/) 0.65-0.7; (+) 0.85-0.9. F ) 70 L/h).

Figure 3. Discrete IRMs generated from the first principle model (key: (s) 25 L/h; (- - -) 50 L/h; (O) 70 L/h; (/) 100 L/h; (+) 150 L/h. us ) 0.45).

) 0.45) and at different flow-rate values (F1 ) 25 L/h, F2 ) 50 L/h, F3 ) 100 L/h, F4 ) 150 L/h). The system was excited by using a single step of ∆u ) 0.05 in each case. The impulse responses were divided by the steadystate gains. In fact, these are relative impulse response models but for simplicity the attribute “relative” is omitted. The main feature of these IRMs is that their integral is equal to unity. Figure 2 (IRM vs u, while F ) constant) shows a notable result. The impulse response models are almost identical with each other. This suggests that at a given flow rate the system dynamics are the same over the whole operating interval; the differences derive only from the value of the steady-state gains. In Figure 3 (IRM vs F, while u ) constant) the impulse response models are different. Therefore, when the flow rate is changed at a given value of manipulated variable, the system shows different dynamic properties. 3.2. The Transfer Function Model. The inputoutput approach of modeling is common in process control engineering. The area of system identification has matured into an established collection of basic techniques that are well understood and known to perform successfully in practical application.22 From control aspect the physical system can be regarded as a SISO system, namely, the input (u) is the heating control signal and the output (y) is the outlet water temperature (while the flow rate is a load

156 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999

The process gain is calculated on the basis of the steady state heat balance of the system (∂TCH/∂t ) ∂Tw/ ∂t ) ∂TW/∂t ) 0), by integrating on the interval z ∈ [0, 1]:

(FFCp)w(Twout - Twin) )

∫01(TW - Tenv) dz

Q(u) - (RA)env

(6)

Expressing the heat loss with the driving force (Tw Tenv), and integrating the function Tw(z) by employing its linear approximation, we observe that the global steady-state heat balance takes the following form: Figure 4. Process gain vs process input at different flow rates (key: (s) 25 L/h; (- - -) 50 L/h; (O) 70 L/h; (/) 100 L/h; (+) 150 L/h).

disturbance). Thus the data collection procedure implies the input and output variables. The exciting signal was a pseudo random binary signal (PRBS) and was set approximately to the middle of the operating interval (u ) 0.4-0.5). The input-output pairs were collected with a sampling time T0 ) 2 s. Applying an ARX model

G(z) )

1 + a1z-1 + ... + anaz-na b1 + b2z

-1

-nb+1

+ ... + bnbz

z-nk

0.9350z-4 1.0000 - 1.5856z-1 + 0.6110z-2

2

)

- Tenv (7)

A2 ) Aenv ) A R)

1 δW 1 1 + + R2 Renv µW

(8)

The process gain (K) can be defined as follows:

K)

dTwout

(9)

du

At the same time the differentiation of eq 7 leads to

K)

∂Q RA ∂u (FFCp)w + 2 1

(10)

By differentiation of eq 1 and replacement of Q in eq 10, an easy-to-use equation results for K:

K)

QM (FFCp)w +

Equation 5 represents a convolution of the gi relative finite impulse response model and the uk-i (i ) 1, ..., N) past input values over the N model horizon. At each sampling the convolution is multiplied by a K steadystate gain which depends on the us steady-state input. In eq 5 the product-sum represents the dynamic part of the model, while the K(us) and ys represent the steady-state part of it. The first problem in the application of the model is the calculation of the steady-state gain (K(us)), which has to be valid in each point of the operating interval. At first sight the evaluation of the dynamic part of the model is not too difficult since the impulse response models (if F ) constant) are of the same shape (see Figure 2). So it is possible to fit a given function (polynomial, exponential, etc.) in these impulse response models.

Twin + Twout

Here

(4)

Since it is a linear model it can be employed in MPC only in the close neighborhood of the identification point. 3.3. The Convolution Model. As was shown in section 3.1., under special conditions (constant flow-rate) the physical system has the same dynamic behavior over the whole operating range. Consequently we should be able to express the model to be used in model based predictive control, as a combination of a steady-state and a dynamic part. The most obvious solution can be obtained by using a finite discrete convolution model, which can be expressed as follows:

(

Q(u) - RA

(3)

the following transfer function model (TF) was obtained:

G(z) )

(FFCp)w(Twout - Twin) )

[1 - cos(2πu)] RA 2

(11)

The obtained analytical relationship (plotted in Figure 4) is a function of two variables (system input and flow rate). Despite its simplicity the value of K computed with eq 11 fits properly the process gain computed with the first principle model and the measured data too. To express the value of K in eq 5, the (us, ys) steadystate input-output pairs should be calculated. If Q(u) from eq 1 is replaced in eq 7, the relationship between us and ys can be obtained (ys ) Twout ) Ts):

(

(FFCp)w(Ts - Twin) + RA

Twin + Ts 2

[

)

- Tenv )

QM µs -

]

sin(2πµs) (12) 2π

From this, Ts can be expressed as a function of us:

Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 157

Ts )

1 (FFCp)w +

{ [

RA 2

QM us -

[(FFC )

p w

-

]

sin(2πus) + ... + 2π

}

RA T + RATenv 2 win

]

(13)

Beside the relationship ys ) ys(us), us ) us(ys) is to be determined. This cannot be obtained explicitly from eq 12; consequently an iterative algorithm is to be used. The developed algorithm is based on the Newton method (for details see Supporting Information). Using eq 11, a set of steady-state characteristic-curves were calculated for the system (Figure 5). This shows the values of the steady-state input-output pairs on the whole operating range at given values of the flow rate. Thus the steady-state part of the model (with a nonlinear feature) is considered to be specified. The further investigations deal with the determination of the dynamic part of the model (determination of the relative finite impulse response model gi). If the shape of the curves in Figure 2 is examined, it can be established that they are formally analogous with the density function of residence time distribution of a cascade consisting of CSTs. This density function can be expressed as

t nc ncτ t exp -nc φ(t) ) τ (nc - 1)! τ

( )

( )

(15)

The parameter r is a function of the flow rate. The r(F) relationship is described by the following correlation:

F (150 )

r ) 0.064

0.11

Table 1. Results of the Fitting F (L/h)

nc

τ (s)

r

25 70 150

1.93 1.73 1.58

39.66 34.01 31.44

0.0504 0.0588 0.0636

(14)

where nc is the number of the elements of the cascade and τ is the residence time. The parameters nc and τ were determined from the equality of the corresponding first- and second-order momentum of the φ(t) function and the impulse response models generated from the first principle model. The results of fitting data at three different flow rates are contained in Table 1. According to these data nc ) 2 was chosen over the whole operating interval. The last column of Table 1 contains the value of the parameter r ) nc/τ. Thus, φ(t)swhich is in fact the impulse response modelscan be expressed as follows:

g(t) ) r2t exp(-rt)

Figure 5. Steady-state characteristic curves of the system at different flow rates (key: (s) 25 L/h; (- - -) 50 L/h; (O) 70 L/h; (/) 100 L/h; (+) 150 L/h).

(16)

The relative IRMs generated from the first principle model and the convolution model are shown in Figure 6. Thus the convolution model is determined through the following steps: (1) calculate the value of r with eq 16, using the value of the measured flow rate (F); (2) calculate the impulse response model with eq 15; (3) choose the value of the reference points us and ys, and compute the value of ys through eq 13 or the value of us considering the Newton algorithm; (4) calculate the value of the steady-state gain with eq 11. 3.4. Model Validation. The presented models can be compared by the integral of square errors criterion (ISE). The input signal is shown on the top of Figure 7 while the measured and calculated temperatures are shown on the bottom part of the figure. The outputs of

Figure 6. IRMs of the first principle and of the fitted model (key: (s) first principle model at 25 L/h; (- - -) first principle model at 150 L/h; (+) fitted model).

the convolution and first principle models are so uniform that their graphic representations cannot be distinguished. On the basis of the ISE criteria the first principle model is the most accurate one on the whole operating range. Table 2 shows numerically the difference between the models. Of course in a close neighborhood of the working point (us ) 0.45), the transfer function is an accurate model too. Since the transfer function model is valid only around a working point and the first principle model is difficult to use due to its complexity, the application of the convolution model is suggested in MPC algorithms. 4. MPC Control Implementation After the above presented models were developed, the testing and the tuning of different controllers can be carried out through simulation. Therefore the problem of real control becomes much simpler, since we have the opportunity to select proper models for the given control needs. The controllers simulated in this way can be employed in real control without more ado. On the other hand, because of model/plant mismatch and other model deficiencies (parameter uncertainty, the incapability of

158 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 L

ISE )

Figure 7. Comparison of models (key: (s) measured data; (- - -) transfer function model; (+) first principle and convolution model, 70 L/h). Table 2. Comparison of the Models by ISE Values reference model

measured

first principle

first principle transfer function convolution

1088 20066 1594

0 24661 216

reflecting noises and disturbances), the real control solutions will not fully agree with the simulation results. In our case the first principle model was chosen to perform the simulator role. At the same time the model role in the MPC algorithm can be satisfied by the transfer function model and by the convolution model. 4.1. The Studied MPC Controllers. Mathematically, the applied dynamic matrix controller is the solution of a conditional optimization problem and means the evaluation of p

min

q

ek+i2 + λ∑(uk+i - uk+i-1)2] ∑ i)1 i)1

[

uk,uk+1,...uk+q

(17)

where the mathematical model signifies the conditions. The analytical formulation of the controller can be expressed in the form

∆u ) (AT‚A + λ‚I)-1‚AT‚E

(18)

where A is the dynamic matrix, E is the vector of openloop prediction error over the prediction horizon (p), λ is the move suppression coefficient, and ∆u is the manipulated input profile computed for the next q sampling instants, also called the control horizon. In the paper the dynamic matrix controllers (DMC) employing the above presented system models (transfer function, convolution) are investigated. This means two types of dynamic matrix controllers: DMCtf; DMCconv. In the case of the DMCtf controller the applied impulse response model derives from a transfer function model. In the case of the DMCconv controller the step response model derived from the convolution model is used. As was shown in the design of the convolution model, the starting relationship (eq 5) provides the step response model in every point of the operating interval. To compare the studied controllers the integral of the square errors (ISE) criterion was used as performance index

L

∑ ek2 + χk)1 ∑ (uk - uk-1)2 k)1

(19)

where ek ) wk - yk is the control error and L is the performance time horizon. The criterion differs from that used for the comparison of models, since it employs a punishment term for the variation of the manipulated variable, weighted with χ. The controllers were first tested using the simulation technique, and then they were studied on the physical system. It should be noted that, in the performance evaluation of the controllers, the knowledge of the future set point values over the prediction horizon was employed. 4.2. Controller Tuning. There are several recommendations in the literature for tuning MPC controllers. For instance the model horizon has to be equal to the settling time, the prediction horizon should not be lower than the time corresponding to the maximal IRM coefficient, and the control horizon should be q ∈ [1, 6] in practice.23 The move suppression coefficient (λ), if q ) 1, is proposed to be zero. If a dynamic control action (high values of ∆u) is intended to be carried out, the value of λ should be low, and in the case of a less dynamic action (low values of ∆u), the value must be higher. The above tuning recommendations can be applied in case of both controllers (DMCtf and DMCconv) and are considered to be the starting points of the following discussions. The control quality analysis was accomplished by simulation, computing the performance-index for different values of p, q, and λ, over a given performance time horizon (L). The best control solution can be selected on the basis of the obtained control profiles and values of performance indexes. Because the obtained control profiles are roughly the same (except the case of q ) 1), their graphical representation was omitted, and the best controller parameters were selected on the base of the performance indexes. When the ISE values for DMCtf and DMCconv are compared, Table 3 reflects that the minimum value of the performance index can be located in the point p ) 10, q ) 2 (N ) 60, λ ) 5). Moreover, the choice of a DMC with a low value of the prediction horizon is confirmed by the fact that a high value of p could lead to large-sized matrixes and thereby to memory deficiencies. Since in real control high values of q can yield more dynamic control actions and stability problems can arise, the control horizon has to be chosen low too; therefore, the point N ) 60, p ) 10, and q ) 1 was chosen. 4.3. Control with DMCtf. Despite the linearity of the model employed in DMCtf the control action for servo disturbances is good (Figure 8). The controlled variable settles without overshoot and the control moves are fairly smooth too. The high performance (low value of the performance index) is attributed, first of all, to the attractive features of DMCs to control properly processes with difficult-to-handle properties (delay time, nonlinearity, distributed parameters). However it should be repeated that the future setpoint values are considered to be known. This possibility contributes significantly in increasing of the controller performance. Applying the parameters obtained through simulation (N ) 60, p ) 10, q ) 1) in the control of the physical system, a low increase of the ISE is observed (Figure 9). This is attributed to the presence of measurement

Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 159 Table 3. Performance-Index Values of DMCtf and DMCconv (N ) 60, L ) 100 s, χ ) 5) parameters

ISE

p

λ

q

DMCtf

DMCconv

10 10 10 10

0 5 5 5

1 2 3 6

17.4 17.5 18.9 18.5

16.6 13.0 13.3 13.1

20 20 20 20

0 5 5 5

1 2 3 6

26.5 21.2 18.7 18.5

26.5 14.3 14.0 13.1

30 30 30 30

0 5 5 5

1 2 3 6

31.5 21.5 18.4 18.6

32.3 15.0 14.3 13.6 Figure 10. Servo DMCconv control (simulation, N ) 60, p ) 10, q ) 1, χ ) 5, ISE ) 35.6).

Figure 8. Servo DMCtf control (simulation, N ) 60, p ) 10, q ) 1, χ ) 5, ISE ) 40.3). Figure 11. Servo DMCconv control (measured, N ) 60, p ) 10, q ) 1, χ ) 5, ISE ) 51.6).

Figure 9. Servo DMCtf control (measured, N ) 60, p ) 10, q ) 1, χ ) 5, ISE ) 48.8).

noises. After all, the dynamic matrix controllers employing transfer function models performs satisfactorily in the close neighborhood of the identification point. Further experiments showed that getting away from the identification point the performance of the controller becomes much worse. To extend their applicability, it would be necessary to execute identifications in a number of points of the operating interval. 4.4. Control with DMCconv. Further performance increase is achieved if the IRMs generated from the convolution model are employed (Figure 10). In real control the N ) 60, p ) 10, and q ) 1 controller

Figure 12. Servo DMCconv control (simulation, N ) 60, p ) 10, q ) 1, χ ) 5, ISE ) 42.0).

parameters give roughly the same performance as that of DMCtf (Figure 11). As expected DMCconv manages to solve the control problem at different steady states (Figures 12 and 13), and the obtained performance values do not change considerably. The servo performance was tested over a wide range (Figures 14 and 15), and a good control performance was achieved. The increase of the performance index is mainly due to the required larger control actions. The regulatory performance was tested through simulation by varying the flow rate (70-40-70 L/h) (Figure 16). The controller

160 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999

Figure 13. Servo DMCconv control (measured, N ) 60, p ) 10, q ) 1, χ ) 5, ISE ) 54.3).

Figure 16. Regulatory DMCconv control (simulation, N ) 60, p ) 10, q ) 1, χ ) 5). Table 4. Performance Indexes of Controllers (S ) Simulation, M ) Measured, / ) Unstable, L ) 1000 S, χ ) 5) ISE control DMCtf DMCconv

parameters

S

M

N ) 60, p ) 10, q ) 1, λ ) 0 N ) 60, p ) 10, q ) 2, λ ) 5 N ) 60, p ) 10, q ) 1, λ ) 0 N ) 60, p ) 10, q ) 2, λ ) 5

40.3 37.5 35.6 24.0

48.8 / 51.6 188.3

The differences between results of the simulation and the physical experiments are basically attributed to the model/plant mismatch and of course to some effects of real world measurements (e.g., noise). While in case of simulation the minimum of the performance index is by q g 2, the system is nearly unstable in the physical experiments when q > 1. Figure 14. Wide range DMCconv control (simulation, N ) 60, p ) 10, q ) 1, χ ) 5, ISE ) 244.0).

Figure 15. Wide range DMCconv control (measured, N ) 60, p ) 10, q ) 1, χ ) 5, ISE ) 449.5).

tuned for servo problems performs satisfactorily in regulatory control too. 4.5. Comparison of Controllers. It is important to note that beside the results presented so far, ISE values were also calculated without assuming the knowledge of the future setpoint values. A considerable performanceindex increase was observed. Consequently, the performance superiority of DMCs is attributed in a high proportion to the involvement of future setpoint values. To compare the studied controllers, the values of the performance indexes given in Table 4 are considered.

5. Summary Model predictive control of nonlinear systems with time delay has some difficulties. In this paper the application of different models in the MPC algorithm was investigated. The studied process is a laboratory heating system with the above difficult-to-handle properties. The investigations of the system gave almost identical relative IRMs on the whole operating interval at constant flow rate. This result suggested the possibility of separating the steady state and dynamic part of the system behavior. The idea was incorporated in a convolution model, consisting of a nonlinear steady state and a dynamic part. The generalization of this method needs further research. Transfer function and nonlinear convolution models based DMC algorithms were tested by simulation and physical experiments. For simulation the first principle model was used. The implementation of the nonlinear convolution model is almost as simple as that of the transfer function model; however, the first is valid over the whole operating interval without the need for further identification efforts. Both the simulation and the physical tests showed the good control performance of the suggested solution both in servo and in regulatory modes. Acknowledgment This work was supported by the Hungarian Academy of Sciences (Grant No. OTKA T023157) and by the Hungarian Ministry of Education (Grant No. PFP-3011/ 1997).

Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 161

Notation Lower Case ai: system identification parameters (denominator) bi: system identification parameters (numerator) e: control error k: discrete sampling time (index) n: number of sample nc: number of elements of a cascade of CSTs na: number of system identification parameters in denominator nb: number of system identification parameters in numerator nk: discrete deadtime p: prediction horizon q: control horizon r: parameter of the convolution model s: steady-state condition (index) t: time (s) u: manipulated variable (u ∈ [0, 1]) w: setpoint (°C) y: controlled variable (°C) z: dimensionless length (z ∈ [0, 1]) Capital Letters A: area (m2) A: dynamic matrix Cp: heat capacity (J/kg K) E: vector of the open-loop prediction error F: flow rate (m3/s) IRM: impulse response model ISE: integral of the square errors I: unit matrix K: process gain L: performance time horizon (s) N: model horizon Q: effective performance of the cartridge heater (W) QM: maximal performance of the cartridge heater (W) T: temperature (K) T0: sampling time (s) V: volume (m3) Controllers MPC: Model Predictive Controller DMC: Dynamic Matrix Controller DMCtf: DMC employing transfer function models DMCconv: DMC employing an convolution model Indexes 1: interface cartridge heater-thermal agent 2: interface thermal agent-wall CH: cartridge heater env: environment w: thermal agent (water) W: wall Greek Letters R: δ: φ: µ: λ: F: τ: χ: :

heat transfer coefficient (W/m2K) wall thickness at heat transfer (m) density function heat conduction coefficient [W/mK] move suppression coefficient density [kg/m3] residence time (s) weighting factor model error

Supporting Information Available: Details on the computation of the performance of the cartridge heater

and the applied Newton algorithm (2 pages). Ordering information is given on any current masthead page. Literature Cited (1) Byun, D. G.; Kwon, W. H. Predictive Control: A Review and Some New Stability Results; IFAC Model Based Process Control: GA, 1988. (2) Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. Algorithmic Control of Industrial Processes. 4th IFAC Symposium on Identification and System Parameter Estimation; Tbilisi, USSR, 1976. (3) Rouhani, R.; Mehra, R. K. Model Algorithmic Control (MAC): Basic Theoretical Properties. Automatica 1982, 18, 401. (4) Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Generalized Predictive Control - Part 1. Basic Algorithm. Automatica 1987, 23, 137. (5) Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Generalized Predictive Control - Part 2. Extensions and Interpretations. Automatica 1987, 23, 149. (6) Garcia, C. E.; Prett, D. M.; Morari, M. Model Predictive Control: Theory and Practice - A Survey. Automatica 1989, 25, 335. (7) Muske, K. R.; Rawlings, J. B. Model Predictive Control with Linear Models. AIChE J. 1993, 39, 262. (8) Katende, E.; Jutan, A. Nonlinear Predictive Control of Complex Processes. Ind. Eng. Chem. Res. 1996, 35, 3539. (9) Park, S. The Application of An Optimised DMC Multivariable Controller to the PCC Catalytic Cracking Unit. Proceedings of the Advanced Process Supervision and Real-Time Knowledge Based Control Symposium; University of Newcastle-upon-Tyne: Newcastle-upon-Tyne, U.K., 1998. (10) Wright, G. T.; Edgar, T. F. Nonlinear Model Predictive Control of a Fixed-Bed Water-Gas Shift Reactor: An Experimental Study. Comput. Chem. Eng. 1994, 18, 83. (11) Ricker, N. L.; Lee, J. H. Nonlinear Model Predictive Control of the Tennessee Eastman Challenge Process. Comput. Chem. Eng. 1995, 19, 961. (12) Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. Model Predictive Heuristic Control: Application to Industrial Processes. Automatica 1978, 14, 413. (13) Cutler, C. R.; Ramaker, B. L. Dynamic Matrix Control-A Computer Control Algorithm. Proc. JACC 1980, WP-5-B. (14) Montague, G.; Willis, M. J. Artificial Neural Network Model based Control. Proceedings of Advances in Model-Based Predictive Control Conference; Oxford, England 1993; p 65. (15) Tan, Y.; de Keyser, R. Neural Network-Based Adaptive Predictive Control. Proceedings of Advances in Model-Based Predictive Control Conference; Oxford, England, 1993. (16) Skrjanc, I.; Matko, D. Fuzzy Predictive Controller with Adaptive Gain. Proceedings of Advances in Model-Based Predictive Control Conference; Oxford, England, 1993; p 65. (17) Linkens, D. A.; Kandiah, S. Long-Range Predictive Control Using Fuzzy Process Models. Trans. IChemE, A 1996, 74, Jan. (18) Fruzzetti, K. P.; Palazoglu, A.; McDonald, K. A. Nonlinear Model Predictive Control Using Hammerstein Models. J. Proc. Cont. 1997, 7, 31. (19) Reid, J. G.; Mahmood S. A State Space Analysis of the Stability and Robustness Properties of Model Predictive Control (MPC). Proc. ACC, 1986, WP1, 335. (20) Lee, J. H.; Gelormino, M. S.; Morari, M. Model Predictive Control of Multi-Rate Sampled-Data Systems: A State-Space Approach. Int. J. Control 1992, 55, 153. (21) Subawalla, H.; Paruchuri, V. P.; Gupta, A.; Pandit, H. G. Comparison of Model Based and Conventional Control: A Summary of Experimental Results. Ind. Eng. Chem. Res. 1996, 35, 3547. (22) Ljung, L. System Identification-Theory for the User; Prentice Hall: Englewood Cliffs, NJ, 1987. (23) Shridar, R.; Cooper, D. J. A Tuning Strategy for Unconstrained SISO Model Predictive Control. Ind. Eng. Chem. Res. 1997, 36, 729.

Received for review June 1, 1998 Revised manuscript received September 30, 1998 Accepted October 9, 1998 IE980338Q