Extension of dynamic matrix control to multiple models

Computers and Chemical Engineering 27 (2003) 1079
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Extension of dynamic matrix control to multiple models Brian Aufderheide 1, B. Wayne Bequette * Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, NY 12180-3590, USA Received 6 February 2003; accepted 7 February 2003

Abstract The purpose of the paper is to extend dynamic matrix control (DMC) to handle different operating regimes and to reject parameter disturbances. This is done by two new multiple model predictive control (MMPC) schemes: one based on actual step response tests and the other on a minimal knowledge based first order plus dead time models (FOPDT). Both approaches do not require fundamental modeling. As a benchmark comparison, the two controllers are compared with a nonlinear model predictive controller (NL-MPC) using an extended Kalman filter (EKF) with no initial model/plant mismatch. The application example is the isothermal Van de Vusse reaction, which exhibits challenging input multiplicity. Simulations include disturbances in the feed concentration, kinetic parameters, and additive input and output noise. The two controllers have comparable performance to NLMPC and in the case of multiple disturbances can outperform NL-MPC. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: Multiple model predictive control; Dynamic matrix control; Extended Kalman filter; Nonlinear estimation

1. Introduction Model predictive control (MPC) has been the most successful advanced control technique applied in the process industries. The formulation naturally handles time-delays, multivariable interactions and constraints. Particularly in the petrochemical industry, MPC has often been tuned for robustness rather than a high level of dynamic performance. In addition to conservative tuning, performance has been limited by the use of linear models and the standard ‘additive output disturbance’ assumption to compensate for plant-model mismatch. Many chemical processes have nonlinear dynamics such as steady-state multiplicity or desired operating regions with distinctly different input
/output behavior. This behavior has sparked much recent research on nonlinear model-based control; our focus here is on nonlinear MPC. The basic idea is to:

* Corresponding author. Tel.: /1-518-276-6683; fax: /1-518-2764030. E-mail addresses: [email protected] (B. Aufderheide), [email protected] (B.W. Bequette). 1 Present address: Keck Graduate Institute, 535 Watson Dr., Claremont, CA 91711, USA.

i)

use a model to predict future deviations from a setpoint over a ‘prediction horizon’ forming an objective function to be minimized, ii) by adjusting a ‘control horizon’ of manipulated input moves, iii) implementing the first move and measuring the resulting output at the next sample time, iv) updating the model and returning to (i). Nothing in this procedure precludes the use of a nonlinear model for prediction. In practice, there are a number of complications that result from the use of a nonlinear model. Linear models, linear constraints and a quadratic objective function results in a convex optimization problem easily solved using quadratic programming. With nonlinear models the optimization problem is no longer convex and convergence to the global optimum is not guaranteed. Also, for large-scale systems the computation time may be a significant fraction of the sample interval. A review of early nonlinear MPC approaches is provided by Bequette (1991), while Henson (1998) provides a more current review. In the remainder of this section we review fundamental and multiple model predictive control (MMPC) studies. Section 2 presents the application example, followed by our MMPC (Section 3) and extended

0098-1354/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00038-3

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Nomenclature Aw Bw Ca Cain Cb Cw d ˜/ /d dave dmax e F i j J k K k1 k2 k3 kp l L m M N Nm p P Q R Rv Rw S Save t u Du uss u0 V W x xw yss /y ˆ/ /y ˆave/ /y ˆc/ y ybias y0 Greek symbols d o

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disturbance model dynamic matrix disturbance model input matrix concentration of reactant A in reactor feed concentration of reactant A in reactor concentration of reactant B in reactor disturbance model output matrix additive disturbance term pseudo-additive disturbance term which includes baseline information term(s) weighted average of the indvidual model’s additive disturbances maximum distance between extreme models predicted error between model and desired trajectory volumetric flow rate to reactor coefficient index for step response matrix, S model index number objective function to be minimized specific number of actual sample time Convergence factor for recursive Bayesian scheme rate constant for A 0/B rate constant for B 0/C rate constant for A/A 0/D model gain step index for objective function Kalman filter gain Bayesian scheme summing index number of control moves for optimization to minimize model horizon for step response models number of models in model bank probability of model being equivalent to plant prediction horizon for optimization output weighting matrix for objective function input weighting matrix for objective function output noise covariance matrix unmeasured disturbance covariance matrix step response coefficient matrix weighted average step response coefficient matrix time F manipulated input, the dilution rate ð Þ/ V change in manipulated input over one sample time steady state input of model prior to step response initial input of plant prior to control initiating volume of reactor weight of model state vector augmented disturbance state steady state output of model prior to step response predicted model output weighted average predicted model output corrected model output actual plant output baseline bias adjustment for model bank initial plant output prior to initiating control probability cutoff for Recursive Bayesian Scheme model’s residual (deviation from actual plant output)

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Kalman filter (EKF)-based NMPC (Section 4) formulations. Simulation results are presented in Section 5, followed by a discussion and conclusions in Section 6.

1.1. Fundamental model approaches Various early versions of nonlinear model predictive control were focused on die development of efficient techniques for solving the differential equations and the optimization problem. More recent efforts, similar to the work in linear model predictive control, have focused on formulations to guarantee stability (using end point constraints, etc.). A nice tutorial overview of NMPC is provided by Allgo¨wer, Badgwell, Qin, Rawlings and Wright (1999). Garcı´a (1984) proposed a strategy that was a natural extension of quadratic dynamic matrix control (QDMC). In this NL-QDMC approach, a nonlinear model was integrated once over the prediction horizon, then a linear perturbation model was used for the optimization iterations. This resulted in a convex optimization problem and enabled the use of proven QDMC technology. A major limitation was the standard ‘additive output disturbance’ assumption, which can only be applied to open-loop stable processes and can result in poor performance for disturbances that enter at the plant input. This NL-QDMC approach was extended by Gattu and Zafiriou (1992), who used a steady-state Kalman filter for state estimation; this allowed the application to open-loop unstable systems. A further improvement was the EKF-based nonlinear model predictive controller (NL-MPC) strategy of Lee and Ricker (1994). The EKF was applied to an augmented state vector, enabling the estimation of physical parameters or unmeasured states in a styrene polymerization reactor with multi-rate output measurements. Doyle and Wisnewski (2000) applied a similar strategy to a Continuous Kamyr Digester.

1.2. Nonlinear input
/output model approaches Unfortunately relatively few fundamental modelbased strategies have been applied in industry, primarily due to the effort and difficulty in developing a good nonlinear model. A fundamental model based on first principles often requires parameters that take a lot of time and effort to obtain experimentally. Nonlinear input
/output models, such as Hammerstein, Volterra, or artificial neural networks, do not require much in terms of basic fundamental knowledge about a system. However, they still need a great deal of process data/ training to be able to describe the system behavior over a wide range of operating conditions.

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1.3. Multiple linear model approaches Much of the early work in multiple linear modelbased approaches was inspired by challenges in the aerospace industry. Although aircraft dynamic behavior is known to be nonlinear, specifications are normally developed based on linear models at a number of equilibrium flight conditions. A multiple model adaptive control (MMAC) approach was developed by Athans et al. (1977), with a specific application to F-8C aircraft. Schott and Bequette (1997) review MMAC and apply the technique using multiple PI controllers in several simulation examples. 1.4. Multiple model predictive control Kothare, Mettler, Morari, Bendotti and Falinower (2000) develop a linear parameter varying (LPV) model of a nuclear steam generator. The continuous schedule of the parameters based on the steam load is obtained by interpolation of a fixed set of models. Azimzadeh, Palizban and Romagnoli (1998) fit local model parameters based on the responses of a fundamental model of a batch fermentation process. MPC is used to compute a sequence of optimal set points of pH and temperature, which are then supplied to PID controllers. To handle nonlinear problems without using fundamental models, we have developed a MMPC strategy that relies on the use of a bank of linear models to describe the dynamic behavior over a wide operating range. A recursive Bayesian scheme assigns weights to each model. The combined-weighted model is then used for the predictions in the optimal control move calculation. Our initial research efforts in this area were applied to the control of cardiac output and mean arterial pressure of canines (see Rao, Palerm, Aufderheide & Bequette, 2001; Rao, Aufderheide & Bequette, 2003). In this paper we will be extending dynamic matrix control (DMC), to handle different operating regions and input disturbances. This is done by using a multiple model framework of step response models. Two different model banks will be tested: one uses actual step responses for the different operating conditions and the other is a minimal knowledge based approach using only first order plus dead time models (FOPDT). The minimal knowledge bank requires only the range of gains, dominant time constants, and time delay approximations for minimum phase behavior of the system. Different operating regions and disturbances are handled by the overall model bank switching to a more appropriate model(s) using a Recursive Bayesian algorithm. These two controllers are compared with the fundamental nonlinear model predictive control EKFbased strategy of Lee and Ricker (1994) which will have no initial model/plant mismatch and will serve as a benchmark.

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2. Van de Vusse reaction system The Van de Vusse reaction consists of two decomposition reactions of A taking place in parallel. The desired product is the concentration of B, Cb (mol B l 1). k1

k2

A0B0C

k3

AA 0 D

(1)

The equations that govern the system are: dCa dt

k1 Ca k3 Ca2 (Cain Ca )u

dCb k1 Ca k2 Cb Cb u dt

for u

(2) F V

(3)

where Ca and Cain are concentrations of A (mol A l 1) in the reactor and in the feed, respectively. The manipulated input, u, is the dilution rate in min 1. The rate constants, k1, k2, and k3 have two sets of kinetic parameters used in this simulation study: 1st set is k1 /5/6 min 1, k2 /5/3 min 1 and k3 /1/6 mol A l 1 min 1 and 2nd set is k1 /5/4 min 1, k2 /3/5 min 1, and k3 /6/5 mol A l 1 min1. The first set is the nominal set used in all cases. The second set is used as a test of model uncertainty. The steady state plots for different feed concentrations are shown in Fig. 1. The control objective is to operate as near as possible to the optimum point to maximize the concentration of B. Operating points on the left side of the optimum are non-minimum phase. As the dilution rate is increased the right half plane zero

moves to the left half plane and the gain now becomes negative. The gain at the optimum point is zero. Control is very challenging since the desired operating point has (zero) gain and the dynamics on either side are very different. While the right hand side of the optimum can be controlled in almost a dead beat fashion, the left hand side has a significant inverse response that requires the controller to not only change gain signs but be detuned significantly as well. No fixed gain controller will suffice for this system as was demonstrated in the work of Schott and Bequette (1995) where an MMAC strategy using internal model based-proportional integral derivative control was implemented successfully. For more details on Van de Vusse reaction system see Sistu and Bequette (1995).

3. Multiple model predictive control Fig. 2 provides a schematic of MMPC. There is one single constrained optimizer which uses a weighted model bank as a prediction model. In essence, it is a multiple model adaptive estimator coupled with a single model predictive controller. The multiple model structure provides flexibility to handle a system with large variability and is required when designing a single nonlinear model is not practical or possible. The advantage of MMPC is that a large number of models can be used and constraints explicitly handled. The issues for MMPC are determining the number and type of models to encompass the plant behavior and the need

Fig. 1. Steady state curves for Cain /6 and 10 mol A l 1 for two different sets of kinetic parameters.

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Fig. 2. Schematic of the MMPC strategy (Rao et al., 2001).

to detune the controller so that it is sufficiently robust in the possible range of prediction models stemming from the model bank. The multiple model adaptive estimator was typically designed to do online identification of fault conditions or disturbances using a bank of models or Kalman filters running in parallel to the actual plant (Menke & Maybeck, 1993). The multiple model adaptive estimators used in this paper are banks of step response models and are essentially an extension of DMC to a multiple model framework. Two sets of model banks are tested */one is based on actual step responses on the plant and the other is based on minimal plant knowledge on the range of gains, dominant time constants, and time delays. Both sets do not require any fundamental knowledge of the actual plant. The use of step response models is by far the most prevalent model used in predictive control starting from initial work on DMC by Shell Oil in the 1960s and 1970s (see Cutler & Ramaker, 1980). Although limited to describing only one fixed linearization of an open-loop stable plant, it is intuitive, very simple to implement, and requires no fundamental modeling whatsoever. Our objective is to expand single model DMC to a bank of step response models to describe different operating conditions, disturbances, and desired set points. 3.1. Single model DMC optimization To better understand the implementation of the multiple model banks we will briefly give an overview of DMC optimization that will be referred to later to show exactly how existing DMC programs can be utilized for a multiple model bank of step response models. A general form of the optimization problem at time step k is:

min

kP X

Du(k)...D(kM1)

eT (l)Qe(l)

lk1

kM1 X

DuT (l)RDu(l)

lk

(4)

objective function to minimize w:r:t: Du The model predicted error is: e(l)r(l) yˆc (l)

(5)

The prediction and corrected prediction models are: y(k) ˆ

N X

S(i)Du(k i)S(N)u(k N 1)

i1

prediction model ˆ corrected prediction model yˆc (k) y(k)d(k)

(6) (7)

where d(k )/y (k )/yˆ/(k) is the constant additive disturbance term and S(i) is the ith step response coefficient matrix. N is the model horizon and typically is equivalent to the settling time. The input constraints to be met: umin 5u(l)5umax u(l 1)Dumax  u(l)u(l 1)Dumax u(l)u(k M l) for all l k M 1

(8)

where at step l, yˆc/(l) is a vector of corrected model predicted outputs, e(l) is a vector of model predicted errors, r(l) is the desired output trajectory, u (l ) is the vector of manipulated variables, and Q and R are the output and input weighting matrices. Absolute and velocity constraints on the manipulated variable are included. There is a prediction horizon of P steps with M control moves. The optimization is a Quadratic Programming problem which searches for values of Du over the control horizon that minimize the objective function subject to the absolute and velocity constraints specified. Although

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M control moves are calculated only the first move is implemented. At the next sampling time the weighted model bank is updated and the optimization is done again. 3.2. Actual step response models as a model bank The following common process control issues can be dealt with by using a multiple model bank of step response models:
/ difficulty in bringing a process from start-up to its final desired output,
/ changing from one set point to another operating region,
/ rejecting known input disturbances such as changes in feed stock or concentration fiom upstream processes. The concept is to identify step response models for different operating regions and/or for different parameter disturbances. Each step response model then is included in the model bank. For operating regions, this is fairly straightforward since each step response model can be taken at a different steady state for the process. The number of step responses will be based on the range of operability desired and the allowable maximum distance between adjacent models. To handle parameter disturbances requires that the new set of parameter(s) be relatively constant over the identification of the step response models. For example, problems with feed concentrations to a reactor due to poor control of upstream processes can be modeled by holding the feed concentration constant to the reactor and then doing a step response in the manipulated input such as the dilution rate. By doing this over different feed concentrations, a bank of step response models can be developed to handle that specific input disturbance. Even problems with different feed stocks, temperature profiles, or changes in catalyst which cause the process to behave sufficiently differently so as to affect process performance can be handled by having step response models for each of these conditions. It is clear that depending on the range of operability and the number of parameter disturbances to be rejected could require a large number of step response models and, therefore, a great deal of testing. But that level of testing may not be nearly as much as required for developing a fundamental model, fitting a nonlinear input
/output model, or training a neural network. Depending on how the step response model was developed, it is quite possible that past experimental tests on the process could be used for initial models in the bank since some averaging and other identification techniques may have been used to develop the nominal step response model for the process being operated by DMC. If the one time cost of testing

down time is deemed too expensive an approach then the minimal knowledge model bank discussed in the next section is a possibility. The question becomes how to best select, switch, and/ or blend the bank of models to provide a prediction model for the constrained optimization. This is an issue in any multiple model control framework. One approach is to have a supervisory level which determines when each model is used, such as traditional gain scheduling where the actual input or output determines when a model is used. The difficulty with this approach is how to utilize modeled input disturbances. Another approach is to develop a ‘global’ model that is typically done by neural nets or fuzzy models. The advantage here is that modeled disturbances and different operating regions can usually be incorporated. The problem for developing this ‘global’ model is that it may require such a large data set or such extensive training to be impractical in an actual plant The approach we use is a probabilistic one. Given this model bank and the past history of model predictions, what is the likelihood that this model is equivalent to the actual plant? In essence, what occurs is a form of scheduling by the models’ residuals, the error between the actual output and the predicted output. The resulting weights for each model are bounded between zero and one and the sum of all the weights equals one. The weighting scheme will be discussed in more detail in Section 3.4. The mathematical implementation provided next can also be used for a process with multiple inputs and outputs without any modification beyond the inputs and outputs becoming vectors. In general the steady state information for step response models are included as follows: y(k) ˆ

N X

S(i)Du(k i)S(N)(u(k N 1)uss )

i1

yss

(9)

where uss and yss are the actual steady state values prior to doing the step response. For the multiple model situation it will be easier to split up: y(k) ˆ

N X

S(i)Du(k i)S(N)(u(k N 1)u0 )

i1

ybias

(10)

where ybias S(N)(u0 uss )yss

(11)

and u0 is the initial steady state input of the plant prior to MMPC taking place. This calculates the predicted bias, ybias from a step response model that was taken at a different steady state (uss , yss ) than the process is at initially (u0 ). This is necessary since most if not all the

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models in the model bank will be taken for a different operating region than the process could be at initially. For a single step response model, the underlying assumption is that the process is always initiated at the exact same steady state conditions of the step response so no bias term is required. The weighted model bank with Nm step response models and individual weights wj: yˆave (k)

Nm X

wj (k)yˆj (k)

(12)

j1

yˆave (k)

Nm X

wj (k)

j1



X N

Sj (i)Du(k i)Sj (N)

i1

 (u(k N 1)u0 )ybiasj yˆave (k)

N X



(13)

Save (i)Du(k i)Save (N)

i1

 (u(k N 1)u0 )ybiasave

(14)

where Save 

Nm X

wj Sj

and

ybiasave 

j1

Nm X

wj ybiasj

(15)

j1

The actual additive disturbance term is: d(k) y(k) yˆave (k)

(16)

However, for this formulation it is easier to simply lump both the bias term and the actual additive disturbance term together to form a pseudo-additive disturbance term: ˜ d(k) y(k) yˆave (k)ybiasave

(17)

The corrected prediction model for the optimizer becomes the following: yˆc (k l)

N X

Save (i)Du(k l i)Save (N)

i1

˜  (u(k l N 1)u0 ) d(k)

(18)

for l /1, 2, . . .P where P is the prediction horizon. Now this can be applied to any DMC code using the step response coefficients, Save, and d˜/(k ) for the additive disturbance term. The one major difference though is that the step response coefficients for the weighted model bank as well as the average bias can change at each sample time. It is very tempting to look at this implementation and decide to remove the ybias terms since in the end this will end up in a pseudo-additive disturbance term. For the prediction model that is the case. However, it is critical to include these when

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actually calculating the residuals for each model since this is how the weights will be determined. By keeping the bias term, the steady state information is preserved and the models will have values that are correct for their specific region of state space. Likewise for a single model standard DMC approach it is typical to exclude the S (N )(u(k/N/1) term. This term maintains the step response model at steady state into the future when time has exceeded the model horizon of the step response model. What is often done is to let this ‘rollover’ effect go to the additive disturbance term. For standard DMC, this works fine and the ‘rollover’ just causes a jump in the additive disturbance term and nothing happens to the control. In the case of multiple models this can be disastrous. The reason is very simple */weights are determined by the residuals. A very large change in each models’ residual value can ruin at least temporarily the model identification taking place and summarily, the control performance. For the Van de Vusse reactor, the bank of models was chosen to span the region of input space with added emphasis near the optimum and less at the extreme values of the input range. The modeled input disturbances are for changes in feed concentration, Cain, and for two sets of kinetic parameters k1, k2, and k3. The two sets of kinetic parameters can be thought of as a simulation for different feed stock, change in catalyst, or change in temperature profile. The model bank of actual step responses then is based on combining the following: uss / [0:4 0:7 1:15 1:25 1:4 1:7 2:3 3 3:6]; steady state baseline inputs for each feed concentration, Cain /[10 8 6]; and the two sets of kinetic parameters. The total number of models is for nine inputs, three feed concentrations, and for two sets of kinetic parameters for a combination of 54 models. 3.3. First order plus dead time models as a model bank based on minimal knowledge The number of actual step responses required to span properly the system for one or more disturbances and for different operating regions can be quite large. The objective here is to provide a set of step response models without requiring extensive experiments to be run on the plant leading to considerable downtime and loss of revenue. In this section, we will describe a minimal knowledge based model bank. Much of this work has been shown previously in Aufderheide, Prasad and Bequette (2001b) and Aufderheide and Bequette (2001a). FOPDT are used since as a minimum they will provide global features such as the plant’s gain and dominant time constant. The dead time can approximate any non-minimum phase behavior of the plant. To develop the model bank what is required is a first order plus dead time approximation of the plant

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behavior extremes. The model with the shortest dead time, quickest time constant and largest gain is the upper extreme. The lower extreme is the model with the longest dead time, slowest time constant and smallest gain. So the information that is required are the ranges on the dominant time constant, gain, and time delay. The gain, time constant, and time delay ranges form a rectangular polytope of a box with three dimensions. The key then for being successful with implementing a multiple model bank approach using FOPDT models is that the process dynamics and gain lie within this box. Obtaining the ranges, is not difficult to do and can be estimated readily from plant data and/or operator experience eliminating the need to do a lot of costly step responses on the actual plant. The real challenge to designing the model bank is given the ranges, how to determine best to cover the volume of the box with as few models as possible. To eliminate the need to know all the steady states for each model we use an individual additive disturbance term based on each model’s residual from the previous sample time. At start up all models begin at the initial conditions of the plant. Further discussion on model bank design will be specifically for the Van de Vusse reaction system. A design for a Multiple Input Multiple Output system is given in Rao et al. (2003). The Van de Vusse reactor has input multiplicity so there will need to be two groups of models present in the model bank (see Fig. 1). For a dilution rate less than the optimal, the plant has positive gain and an inverse response. For a dilution rate greater than the optimal, the plant has negative gain and no inverse response. Therefore, for this system it is necessary to know the ranges for the time constants and time delays for both sets of models. For the gains, both sets of models will approach zero at the optimum. So it is our choice on how small of a positive and negative gain we have in our model bank. We chose a gain of 9/0.05 mol B l 1 min 1) as the closest gains for models nearest the optimum. For a system undergoing an input multiplicity there exists a correlation between the length of time of the inverse response and the magnitude of the time constant. So as the dilution rate increases and approaches the optimum from the left side both the time constant and the inverse response will be longer. This occurs as the right half plane zero is crossing over the jvaxis to become a left half plane zero. This is very useful information since it allows us to have a much smaller model bank since we do not need to use all possible combinations of time constants and time delays. Likewise we know that as the models approach the optimum all gains will go to zero. So the smallest positive gain model will have the longest time constant and delay and the largest gain will have the shortest time constant and delay. Therefore, for the Van de Vusse reactor we will design a model bank that has models as evenly spaced as

possible between the two extreme models (see Fig. 3). As a design metric for explicitly stating what the maximum distance between models should be, we used a gap metric (Galan, Romagnoli, Arkun & Palazoglu, 2000). First we obtain the maximum distance, dmax, between each set of extreme models. The number of models, Nm is determined by how small a gap metric, g, we choose by dividing the maximum distance as follows:   dmax 1 (19) Nm ceil g The gain, kp, and time delay, u, ranges are split up by Nm/1 intervals such that the (j/1)th model’s gain and dead time are:   kpmax  kpmin (20) kpj1  kpj  Nm  1   u  umin (21) uj1 uj  max Nm  1 The time constant, tp, for each model is calculated so the first order plus dead time model passes through the points demarcated by the maximum distance line between the extreme models at the intervals determined by the gap metric. The model bank step response curves for a gap metric of 1.0 mol B l 1 are shown in Fig. 3. The ranges for the gains, time constants, and time delays for both sets of models are: Left side: gain range is 0.05
/7 mol B l 1 min 1. Time constant is from 0.5 to 3 min. Time delay is from 0.2 to 2 min. Right side: gain range is from /0.05 to /0.40 mol B l 1 min 1. Time constant is from 0.05 to 0.40 min. There is no time delay. This model bank uses a gap metric of 0.05 mol B l 1 and the ranges specified which results in a total of 149 models. Note that the model bank here is different than the one used in both Aufderheide et al. (2001b) and Aufderheide and Bequette (2001a) which did not have to account for changes in kinetic parameters but only in feed concentrations and was subsequently much smaller having only 56 models. The prediction models and the weighted model bank for a bank of step response models where no steady state information is known and an individual additive disturbance is included to provide a moving baseline for the models will be outlined here: yˆj (k)

N X

Sj (i)Du(k i)Sj (N)(u(k N 1)u0 )

i1

dj (k 1)y0 where

(22)

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Fig. 3. Step response plots for a unit in min 1 of dilution rate. 10 models based on a gap metric of 1.0 mol B l 1. Also shown maximum distance line demarcated.

dj (k 1)y(k 1) yˆj (k 1)

(23)

is the individual model’s additive disturbance term based on past sample’s error in predicted output from the actual output (i.e. its past residual). By not including this information, there will be an initial bias added to each model’s residual, making discerning between models that much more difficult, especially when the bias is an order of magnitude greater than the residual. The weighted model bank is: yˆave (k)

N X

Save (i)Du(k i)Save (N)

i1

 (u(k N 1)u0 )dave (k 1)y0

(24)

˜ d(k)y(k) yˆave (k)dave (k 1)y0

(27)

The prediction model for the optimizer becomes the following: y(k ˆ 1)

N X

Save (i)Du(k l i)Save (N)

i1

˜  (u(k l N 1)u0 ) d(k)

(28)

for l/1, 2, ...P where P is the prediction horizon. Again this can be applied to any DMC code using step ˜ response coefficients such as Save and d(k) for the additive disturbance term. 3.4. The recursive Bayesian weighting scheme

where dave (k 1)

Nm X

wj (k)dj (k 1)

(25)

j1

So this is not necessarily equivalent to the additive disturbance at (k/1)th time step since these are the current weights which most likely have changed from the previous time sample. The actual additive disturbance term is: d(k) y(k) yˆave (k)

(26)

Again, we combine the baseline information and the actual additive disturbance term to form a pseudoadditive disturbance term.

The weighting scheme we use is a probabilistic one which assigns weights a value from 0 to 1 inclusively and where all the weights sum to 1. This ensures that no prediction model can be physically unrealizable since the weighted model bank is always properly bounded and cannot exceed the extreme models in the bank. It also allows an exact model if it is present in the bank, to become the sole prediction model. When no single model describes the plant, the scheme provides interpolation between models by assigning fractional weights to each model */there is no need for any additional supervisory level as exists in many other multiple model control strategies. The recursive Bayesian weighting scheme is a conditional probability of the jth model in

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the bank being the true model of the plant given this population of models and this past history of residuals and probabilities. The probabilities are assumed to be stochastic in nature and Gaussian. The recursive Bayes theorem for the kth step and jth model is   1 exp  o Tj;k Ko j;k Pj;k1 2 pj;k  N (29)   X 1 T exp  o m;k Ko m;k Pm;k1 2 m1 where o j;k yk  yˆj;k is the jth model’s current residual. The algorithm is computationally inexpensive and rejects poor models exponentially fast so that large number of models can not only be handled computationally but also does not result in any serious degradation of controller performance. Increasing the value of the convergence matrix, K, is useful in the case of a bank of actual step response models where convergence to a single model is more desired. For the minimal knowledge based bank of FOPDT, more blending is desired and the convergence matrix, K, is set at lower values. The steady state probabilities for the recursive Bayesian scheme are 0 or 1. So when blending is desired as is the case with the FOPDT bank, the convergence matrix, K, is detuned such that the controller converges to the desired trajectory faster than the recursive Bayesian scheme converges to a single model. Note that when the controller reaches and maintains the desired set point further convergence of the model bank ceases. Since it is very likely that a model that initially does poorly at predicting the plant output, may be needed later due to a disturbance or change in operating conditions it is critical that all models be available for future selection. Therefore, to keep models alive in the bank an artificial small probability cutoff, d, is assigned to each of the models when p B/d so their probabilities never go to zero. These models are then excluded from being weighted such that 8 Pj;k > > for Pj;k  d > > > m1 : 0 for Pj;k d

input disturbance estimation */an EKF based model predictive controller. The Kalman filter is an optimal observer for linear systems, and the EKF extends its use to nonlinear systems and parameter estimation. A disturbance model is appended to the original state system to create an augmented state model. In this case, the augmented state was the inlet feed concentration, Cain. The only limitation to the EKF is that the number of augmented states cannot exceed the number of measured outputs to ensure bias-free estimates (Kozub & MacGregor, 1992). There is only one measured output so only one augmented state can be updated. Neither model-structure mismatch nor initial condition mismatch exists. Again the simulation results were done as favorably as possible to the EKF-based model predictive controller so as to provide the best benchmark possible to compare our MMPC strategy. For Cases 1 and 2, there is a disturbance in feed concentration so this disturbance model fits exactly those scenarios. Case 3, where a sudden change in kinetic parameters occurs, is the only simulation where the disturbance model does not match the actual disturbance. However, estimating a change in feed for this disturbance is still perhaps the best way to estimate this disturbance as well since there is again the limitation of estimating only one disturbance parameter for this SISO system. For a nonlinear plant whose discretized form can be described by the equations xk Ftx (xk1 ; uk1 ; dk1 ) yk  h(xk ; dk )

with a stochastic augmented disturbance model of the form xwk Aw xwk1 Bw wk1 dk  C w xwt

(32)

the Kalman filter takes the following form: Model prediction:



 xk½k1 Ftx (xk1½k1 ; uk1 ; C w xwk1½k1 )  w w A xk1½k1 xwk½k1 Sk½k1  Fk1 Sk1½k1 FTk1 Gw Rw (Gw )T

4. Extended Kalman filter

(31)

Measurement correction: 

 xk½k xk½k1  w Lk (yk  yˆk ) xwk½k xk½k1

(33)

(34)

where Ideally as control engineers, we would always prefer to have a good detailed working model for the system to be controlled. With this model, plant states and/or parameters can be updated on-line. The purpose of this paper is to compare our control strategy to a fundamental model-based, state space approach, with

Lk Sk½k1 JTk (Jk Sk½k1 JTk Rv )1 Sk½k (I Lk Jk )Sk½k1

(35)

Here, Lk is the Kalman gain and Skjk is the error covariance estimate. F, Gw and J are the linearized

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discrete state transition, input and output matrices of the augmented system, respectively. The tuning parameters are Rv, the output noise covariance, Rw, the unmeasured disturbance covariance and S0, the initial covariance of the state estimate error. For more details see Lee and Ricker (1994).

5. Results All simulations used a sample time of 0.1 min with constraints on the dilution rate of 0
/4 min 1. Velocity constraints on dilution rate were 9/1 min1. All cases start with a nominal feed concentration of 10 mol A l 1 and the 1st set of kinetic parameters with k1 /5/6 min 1, k2 /5/3 min 1 and k3 /1/6 mol A l 1 min 1. This has a steady state optimum of Cb /1.26 mol B l 1 at u/1.22 min 1. All controllers were tuned for Case 2 first. Case 2 was chosen since it involves not only a set point change but also a feed concentration disturbance along with input and output additive noises. Cases 1 and 3 use the same tuning as in Case 2. Tuning was done based on both the controlled response of the plant and the ability to estimate the plant accurately. The closedloop behavior desired was a combination of the quickest response and settling times but with as little chattering in the manipulated variable as possible. The recursive Bayesian scheme and EKF were tuned specifically to reduce the variability in the predicted outputs due to the additive input and output noises and their ability to match the actual outputs. In almost all cases shown here a reduction in input weighting would lead to a great deal of oscillations in the input */this is true for both EKF based MPC and MMPC simulations. The tuning for the EKF-based MPC was a prediction horizon, P /40, with one control move, M /1, and weights on the output, Q /17, and the input, R /7. The estimator tuning was for the measurement noise covariance, Rv /0.30, the unmeasured disturbance covariance, Rw /20, and the initial covariance of the state estimate error, S0 /1. The tuning for the minimal knowledge based multiple model predictive controller had a model horizon, N / 180, with a prediction horizon, P /60, with one control move, M /l, and weights on the output, Q /30, and the input, R /l. The tuning for the recursive Bayesian scheme was a convergence factor, K /1000, and a probability cut-off, d/0.0066. The tuning for the multiple model predictive controller with actual step responses had a model horizon, N /120, with a prediction horizon, P /50, with one control move, M /l, and weights on the output, Q /5, and the input, R /3. The tuning for the recursive Bayesian scheme was a convergence factor, K/180, and a probability cut-off, d /0.0184.

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5.1. Case 1: Disturbance in Cain from 10 to 7 (mol A l 1) The control objective is to try to maintain the output at 1.25 mol B l 1 during a disturbance in the feed concentration of A from 10 to 7 mol A l 1 from t/2 to 17 min (Fig. 4). During the disturbance the optimum Cb is 0.95 mol B l 1 at u /1.26 min 1. The new optimum input is very close to the optimum value for Cain /10 mol A l 1 at u /1.22 min 1. The EKF allows the prediction model to get much closer to the actual optimum than the multiple model approaches with banks of step response models. The recursive Bayesian scheme assigns weights by the variance of the models’ residuals and does provide some interpolation between models. However, there is no guarantee that the weighted model bank will have a prediction model that is very close to the optimum unless one of the models in the bank is actually close to the optimum. The large oscillations seen in the actual step response model bank are due to the fact that no single model approaches the optimum model for a feed concentration of 7 mol A l 1. These oscillations could be reduced greatly by including more models at more input values and at feed concentrations beyond 10, 8, and 6 mol A l 1, thus decreasing the gradation between step response models in the bank. An interesting feature of the Van de Vusse system is that although the previous steady state concentration can not be maintained in equilibrium it can be reached and surpassed in a transitory manner. The EKF-based MPC also tries to approach the set point having rather large input changes that hit velocity constraints at each sample time. But the disturbance is estimated very well and the controlled output becomes so close to the steady state optimum so there is little change in the actual output since the EKF is simply switching from models that are just to the left and right of the optimum. Recall at the optimum the system has zero gain so large input changes right near the optimum will cause only small changes in the plant output. The FOPDT model bank has models with gains as small as 9/0.05 mol B l 1 min1 and has oscillations much smaller than the actual step response model bank. Decreasing both the gap metric to less than 0.05 mol B l 1 in the model bank design and the minimum gain models could help reduce the oscillations further. All three controllers do a great job of returning to the original set point after the feed concentration disturbance ceases. Since the system has input multiplicity, some controllers approach the steady state input value to the right of the optimum and some to the left. A slight change in operating conditions or tuning could change the final steady state output for all the controllers to the other operating side. Basically this is due to which model is chosen right after the disturbance is removed be it the one to the left or right of the optimum.

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Fig. 4. Case 1. Cain disturbance from 10 to 7 from t/2 to 17 min. No noise.

Fig. 5 shows each controller’s ability to predict the output one-step ahead. All three methodologies do a good job of estimating the plant output one step into the future. The EKF, not surprisingly, does the best out of the three methodologies. What is impressive is how well the minimal knowledge bank predicts the plant output. Now, this must be taken with some reserve since all that is shown is one step prediction. Unfortunately, we do not know of any way to show the multiple model bank approach prediction further into the future since it is changing at each sample time. But never the less, the one step prediction is very good for the minimal knowledge bank and is almost as good as the actual step response bank. 5.2. Case 2: Setpoint change with feed concentration disturbance along with additive input and output noise The control objective is to increase the output from a steady state value of 1 to a set point of 1.25 mol B l 1 (see Fig. 6). From 25 to 40 min, the feed concentration goes from the nominal value of 10 to 9 mol A l 1. Throughout the simulation, 2% of additive input and output noise is included. The kinetic parameters are from the first set. This is the case where tuning for all three controllers was determined since we felt it was the most challenging. The EKF-based MPC provides a straight forward framework to handle noise, especially output noise. For the recursive Bayesian scheme which is a system to handle ‘jump-parameters’, noise is more of

an issue. This is especially true for the actual step response model which required considerable detuning in both the convergence matrix, K to 180, and in the relative weighting for the output and input of Q /5 and R /3. For systems with large amounts of noise, filtering of the input and output prior to using the recursive Bayesian scheme may also be required. For the minimal knowledge bank this is also an issue but not nearly as much as the actual step response bank. In fact, since each model has its own additive disturbance term extra excitation is necessary to help discern between models more readily. The result is that the tuning for the minimal knowledge bank has K /1000, Q /30, and R /1. These tuning values are still fairly detuned from values for the same simulation without noise (results not shown here). However, they are much more aggressive than for the actual step response model bank that includes steady state information and models are very easy to discern from each other. All three controllers do well at returning back to the desired set point when feed concentration returns back to its nominal value (Fig. 6). Again the magnitude of the oscillations is due to how closely the controller drives the output to the new steady state optimal value and how good a prediction model is present at that time. The actual step response bank does not have any models for a feed concentration of 9 mol A l 1, so the gradation is fairly large between models causing larger oscillations to be present. Fig. 7 shows the actual plant values versus the predicted outputs. The EKF once again does a slightly better job than the two

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Fig. 5. Case 1. Actual plant output vs. predicted output for each controller.

multiple model banks. This is not surprising since the EKF has no model mismatch, the disturbance model is for the feed concentration, and there is no initial

condition mismatch. What is surprising is that also once again both multiple model approaches estimate the plant output well too.

Fig. 6. Case 2. Setpoint change of 1 to 1.25 Cain disturbance from 10 to 9 from t /25 to 40 min. 2% additive noise to input and output.

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5.3. Case 3: Start up problem with a sudden change in kinetic parameters In this case, the plant is starting up at nominal conditions with a desired set point of 1.25 mol B l 1 (see Fig. 8). At 15 min (arrow on Fig. 8), the kinetic parameters are now the 2nd set of k1 /5/4 min 1, k2 / 3/5 min 1, and k3 /6/5 mol A l1 min 1 for the rest of the simulation. Feed concentration remains constant at 10 mol A l 1. All three controllers are able to reject the disturbance and return to the desired set point. The minimal knowledge bank of FOPDT step responses takes by far the longest time of over 17 min. The fact that the EKF-based MPC with a feed concentration disturbance model still was able to reject this disturbance is impressive. Looking at the actual plant output versus the estimated values (Fig. 9), the EKF is also impressive with only a small deviation right after the disturbance occurs. Time needed to return to the set point is almost 4 min. But by far the most impressive is how quickly the actual step response model bank rejects this disturbance. The reason being that models based on the 2nd set of parameters are immediately selected by the recursive Bayesian scheme after the disturbance. It takes less than 1 min to return to the set point. This is less than a quarter of the time as compared with the EKF-based MPC. In addition, it takes less than 2 min to reach the set point from start up, outperforming the EKF-based MPC in this aspect as well. Note that for

inputs greater than the optimum input, no inverse response is present and control can approach dead beat control. The actual step response model bank can be even quicker than that shown here but the controller is detuned to have the best performance for Case 2. The estimates for all three are extremely good and track the output very well (Fig. 9).

6. Discussion and conclusions The Van de Vusse reactor with its input multiplicity is a challenging SISO system. The optimum point that has the greatest productivity also has no gain, making it very difficult to maintain. The dynamics on each side of this optimum are very different with one side having significant inverse responses which get longer as the input approaches the optimum point, and the other side having very quick dynamics with no inverse response present whatsoever. The possible control performance for each side is very different too, since inputs greater than the optimum can approach almost dead beat control while the other side has significant non-minimum phase significantly limiting control performance. Using a quadratic objective function with a linear prediction model, results in a standard quadratic programming problem with assured convexity and guaranteed solutions. However, for this system it leads to oscillations in the manipulated variable when trying

Fig. 7. Case 2. Actual plant output vs. predicted output for each controller.

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Fig. 8. Case 3. Start up problem with sudden change in kinetic parameters at 15 min.

Fig. 9. Case 3. Actual plant output vs. predicted output for each controller.

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to keep the output at or near the optimal value. To help limit the oscillations, a heavy weighting is needed for the input cost. If a nonlinear prediction model for MPC was used, the optimum could be recognized explicitly, but even in this scenario some plant/model mismatch would be expected and lead to some oscillations. Out of the three controllers investigated in this paper, the EKFbased MPC caused the output to oscillate the least, even with the input oscillating at each sample time hitting velocity constraints. The multiple model banks did not have an exact fundamental model of this system and oscillated in the output much more than the EKF-based MPC. The minimal FOPDT model bank had models such that each adjacent model was within the design gap metric of 0.05 mol B and we chose the smallest magnitude gain in the bank to be 9/0.05 mol B l 1 min 1. The actual step response model bank was not designed with a maximum gap metric, and therefore, its spacing between models was much greater resulting in much larger oscillations than the FOPDT bank. Since the system has input multiplicity, the multiple model banks required must have both a positive and a negative gain set of models. This leads to the possibility that a weighted model bank could have a gain of zero if the recursive Bayesian scheme cannot adequately discern between models. For the actual step response model this is not a serious issue since it is so easy to discern between models especially when the steady state information is included. However, for the minimal knowledge bank this issue is not trivial and can occur. Discerning between models which have time delays and individual additive disturbance terms to provide moving baselines for each model can be difficult if not tuned properly. The purpose of having dead times approximate the time of an inverse response is to make it easier to design the model bank. Also when all models have the same sign gain the recursive Bayesian scheme will select the model which has a dead time that best approximates the time of the inverse response. However, with a model bank that has both positive and negative gains, some model misidentification will occur as can be seen at the beginning of Cases 2 and 3. The inverse responses initially are identified with some or all of the non-zero weighted models having negative gains to the detriment of control performance. The minimal knowledge bank also requires more excitation, resulting in much larger convergence factors (180 for actual step response model bank as compared with 1000 for FOPDT bank) and much more aggressive controller tuning (Q /5 and R / 3 for actual step response model bank as compared with Q /30 and R /1 for FOPDT bank). One possible help for this would be to use filtered individual additive disturbances for each model. The other serious draw back to the minimal knowledge bank is once the extreme gains, time constants and delays are known how does one carve up the space

between the extreme models? In the Van de Vusse reactor, a relationship between the gains, time constant, and delays was utilized greatly reducing the number of models in the bank. But what to do when no correlation is known a priori is still an open research question. In this paper a gap metric was used as a design parameter to determine the number of models in the bank. The concept here is to split the space up as evenly as possible and make each model more readily discernible for the recursive Bayesian scheme. This does not utilize any knowledge from Robust Control theory to help determine closed-loop performance as the criteria for model selection. How to meld the tools of robust control theory with the ability to discern and switch between models is another open area of research. The last drawback for the minimal knowledge bank is that the emphasis is on blending these FOPDT models together to form higher order prediction models. The recursive Bayesian scheme is not suited to blending models in this manner. Other weighting schemes could be considered to replace the detuned convergence factor, K, approach to achieve some blending. The recursive Bayesian scheme is ideal, however, for the actual step response model bank where one or two accurate models need to be selected out of a large bank of models. It is impressive how quickly the poor models are rejected and the weighted prediction model becomes an average of only one or two models. Actual step response models do not appear to have much difficulty in dealing with the positive and negative gains being present in one model bank since it is so easy to discern the few models that are able to predict the current plant output from those that cannot. The drawback to using an actual step response model bank is the number of experiments on the actual plant required, leading to possible down time and lost revenue. However, it is clear from results presented here that substantial financial savings from improved control could not only offset these one time expenditures but also show much greater long term profit. The limitation of an EKF for state and disturbance estimation is that the number of augmented states can be no greater than the number of measured outputs. Multiple disturbances or disturbance model mismatch are real issues for an EKF. Case 3 illustrated this limitation. The EKF-based MPC still gets the job done even when the kinetic parameters suddenly change and the disturbance model is for the feed concentration. However, the actual step response model bank reaches the set point from start up in almost half the time and rejects the disturbance in less than one fourth the time of the EKF-based MPC performance. The actual step response bank provides multiple modeled disturbance rejection and is not limited by the number of measured outputs. Sudden changes in parameters are handled rapidly by the recursive Bayesian scheme where EKF

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convergence tends to be slower. What the multiple model bank of step responses does not provide explicitly is the smoothness and gradation that an EKF does naturally. Nor does it handle noise as explicitly and as optimally as an EKF. The EKF performance could also be improved greatly if an additional measurement for the concentration of A was available. Not only would this allow another disturbance parameter to be estimated but there is no input multiplicity present in terms of the concentration of A. Another solution is to use a multiple bank of EKF’s, to try and reject more disturbances (Kesavan & Lee, 1997). What was not shown in this paper was how a poorly designed model bank that does not encompass the process dynamics appropriately will affect controller performance. The extreme model(s) would be locked in by the recursive Bayesian scheme and the only model correction would be the additive disturbance term. No further adaptation would be occurring. For a properly designed model bank, the extreme models should rarely be chosen since the process dynamics would be bounded accordingly. At each sample time, the recursive Bayesian scheme will select those model(s) that have shown the ability to predict the measured output well and a new weighted model will be calculated for the optimizer. When properly designed the weighted model bank’s additive disturbance term will be small and is present only to provide integral action for the model predictive controller to remove any offset from the setpoint. The multiple model bank of step responses is a relatively simple and straightforward extension of DMC. It can handle different operating regions and multiple parameter disturbances. Actual step response models can provide excellent performance that rivals and in the case of multiple disturbances even exceeds the performance of a fundamentally modeled EKF-based MFC. The advantage of using actual step responses is that no fundamental model is required. Its one drawback to implementation on an actual plant is the amount of down time to get the step responses needed to develop the model bank over the input range and for each disturbance. This down time, however, could still be much less than other ‘minimal knowledge’ controllers that require extensive data sets or training. The minimal knowledge based FOPDT model bank is useful as an initial controller where constraints must be met and no fundamental model is known and access to doing multiple step responses is not available. Obtaining the range of the plant’s dominant time constants, delays, and gains, is information that can be found quite readily from operator experience or from limited step response tests. Although its performance did not match with that of the multiple model bank of actual step responses or the EKF-based MFC, it was still very good and demonstrated that it can handle a wide variety of disturbances and operating conditions.

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In conclusion, we feel a multiple model predictive controller has great potential for handling systems with high variability where first principles models are difficult to obtain or when multiple disturbances are an issue.

Acknowledgements This work was supported by grants from the Whitaker Foundation, National Science Foundation (BES 9522639), and Merck.

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