Model mismatch

The difference between the expected demands d used in the model and the realization of an actual demand d causes a model mismatch. The scheduler corrects this error after the observation of this demand in a reactive manner by the production decisions taken at the beginning of the next period. For example, the decisions taken in period i = 1 take the expected value of the demand di into account (di = 6) while the decisions taken in period i = 2 take the true value into account. When di = 0, the large storage causes a relatively low production in the next period [xiih) = 5) whereas in case of d = 12, a deficit results and the production of the next period is larger (x2(t2) = 12). The sequence of decisions is a function of the observed demands and thus the sequence varies over the scenarios. 

Example 2. Let us now test the robust linear regulator under model mismatch. For this application let us consider again the CSTR of Example 1. Here, we assumed that the CSTR model dynamics is given by... 

In Figure 7 we compare the performance of the robust regulator with that obtained from applying a nonrobust controller. Prom inspecting this figure, it is clear that the robust controller yields excellent response in the face of initial error conditions and model mismatch. In contrast, the nonrobust controller performance is influenced by the periodic reference signal and as a result, it does not settle down around a constant value (in fact, the off-set is periodic). 

Simulation studies carried out to compare the amount of desired product C obtained from on-line dynamic optimization strategy with that from off-line strategy, are cases where the perfect model (all parameters correctly specified) is used (nominal case), and where plant/model mismatch is introduced by changing parameters in actual plant i.e. pre-exponential rate constant (kf) decreased by 50% and activation energy (Ea) increased by 20% from their nominal values, as shown in Table 5. 

For the mismatch in the value of the desired product C = 7.8851 can be achieved at the end of batch for the on-line optimization strategy which is higher than that obtained from off-line strategy where the mismatch is not noticed (C = 7.6751). Similar results can be observed under the case of plant model mismatch in k as shown in Table 5. These results indicate clearly that the performance of batch reactor operation is improved via the proposed strategy. Due to similarity in their control responses, only the result for change in k is shown in Fig. 7. 

Figure 3.11 Closed-loop response of the product composition and reactor holdup for a 15% increase in production rate and 1.5% decrease of the product-purity setpoint, in the presence of plant-model mismatch, (a) Product stream composition and (b) reactor holdup and setpoint.

Sorensen and Skogestad (1994) developed control strategies for BREAD processes by repetitive simulation strategy using a simple model in SPEEDUP package. Wilson and Martinez (1997) developed EKF (Extended Kalman Filter) based composition estimator to control BREAD processes. The estimator was found to be quite robust and was able to estimate composition within acceptable accuracy, even in the face of process/model mismatches. Balasubramhanya and Doyle III... 

Modelling process-model mismatches and hybrid modelling in batch distillation (Mujtaba and Hussain, 1998 Greaves et al., 2001)... 

In Mujtaba and Hussain (1998), the detailed dynamic model was assumed to be the exact representation of the process while the difference in predictions of the process behaviour using a simple model and the detailed model was assumed to be the dynamic process-model mismatches. Theses dynamic mismatches were modelled using neural network techniques and were coupled with the simple model... 

In many chemical processes, especially inherently dynamic processes, it is not always possible to model the actual process. Therefore, the states predicted by using the model (Equation 12.1) will be different than that of the actual process and will result in process-model mismatches. The implementation of the optimal operating policies obtained using the model will not result in a true optimal operation. Regardless of the nature of the mismatches, a true process can be described (Agarwal, 1996) as ... 

The error ex(t) is in general time dependent and describes the entire deviation due to process-model mismatches. Structural incompleteness in the model,... 

At any time t, the true estimation of the state variables requires instantaneous values of the unknown mismatches eft). To find the optimal control policies in terms of any decision variables (say z) of a dynamic process using the model will require accurate estimation of ex(t) for each iteration on z during repetitive solution of the optimisation problem (see Chapter 5). Although estimation of process-model mismatches for a fixed operating condition (i.e. for one set of z variables) can be obtained easily, the prediction of mismatches over a wide range of the operating conditions can be very difficult. 

Greaves et al. (2001) modelled the actual process (Equation 12.2) by combining a simple dynamic model (of type Equation 12.1) and the process-model mismatches (eft)) model. 

Here, neural network techniques are used to model these process-model mismatches. The neural network is fed with various input data to predict the process-model mismatch (for each state variable) at the present discrete time. The general input-output map for the neural network training can be seen in Figure 12.2. The data are fed in a moving window scheme. In this scheme, all the data are moved forward at one discrete-time interval until all of them are fed into the network. The whole batch of data is fed into the network repeatedly until the required error criterion is achieved. 

Figure 12.4 illustrates a general optimisation framework (developed by Mujtaba and Hussain, 1998) to obtain optimal operation policies for dynamic processes with process-model mismatches. 

Dynamic sets of process-model mismatches data is generated for a wide range of the optimisation variables (z). These data are then used to train the neural network. The trained network predicts the process-model mismatches for any set of values of z at discrete-time intervals. During the solution of the dynamic optimisation problem, the model has to be integrated many times, each time using a different set of z. The estimated process-model mismatch profiles at discrete-time intervals are then added to the simple dynamic model during the optimisation process. To achieve this, the discrete process-model mismatches are converted to continuous function of time using linear interpolation technique so that they can easily be added to the model (to make the hybrid model) within the optimisation routine. One of the important features of the framework is that it allows the use of discrete process data in a continuous model to predict discrete and/or continuous mismatch profiles. 

Figure 12.4. General Optimisation Framework For Dynamic Processes with Process-model Mismatch...

In Greaves et al. (2001) and Greaves (2003), instead of using a rigorous model (as in the methodology described above), an actual pilot plant batch distillation column is used. The differences in predictions between the actual plant and the simple model (Type III and also in Mujtaba, 1997) are defined as the dynamic process-model mismatches. The mismatches are modelled using neural network techniques as described in earlier sections and are incorporated in the simple model to develop the hybrid model that represents the predictions of the actual column. 

B show the model and pilot plant predictions respectively. Figure 12.6 clearly shows that there are large process-model mismatches in the composition profiles although for a given batch time of tdiS = 220 min the amount of distillate achieved by the experiment was the same as that obtained by the simulation. These process-model mismatches can be attributed to factors such as use of constant Vmodei instead of a dynamic one constant relative volatility parameter used in the model and uncertainties associated with it actual efficiency of the plates. 

The four experiments done previously with Rnp (= 0.5, 1, 3, 4) were used to train the neural network and the experiment with / exp = 2 was used to validate the system. Dynamic models of process-model mismatches for three state variables (i.e. X) of the system are considered here. They are the instant distillate composition (xD), accumulated distillate composition (xa) and the amount of distillate (Ha). The inputs and outputs of the network are as in Figure 12.2. A multilayered feed forward network, which is trained with the back propagation method using a momentum term as well as an adaptive learning rate to speed up the rate of convergence, is used in this work. The error between the actual mismatch (obtained from simulation and experiments) and that predicted by the network is used as the error signal to train the network as described earlier. 

Figure 12.6. Experimental Simulation Results and Dynamic Process-model Mismatch Model (Rexp= 2).

Figure 12.6 also shows the instant distillate composition profile for / exp = 2 (which is used to validate the network) using the simple model coupled with the dynamic model for the process-model mismatches (curve C). The predicted profile (curve C) shows very good agreement with the experimental profile (curve B). Similar agreements have been obtained for the accumulated distillate amount and composition profiles (Greaves, 2003). 

If the output constraint, Eq. (4), were not present, then the choice m = 1 and a sufficiently large p n + m would stabilize the closed loop, in the absence of process/model mismatch. However, the presence of the output constraint destabilizes the closed loop. As p, the closed loop largest... 

There is uncertainty in the kinetics due to model mismatch, errors in fitting the model, and variations in mixing conditions, temperature, and particle size distribution. For this application, an overall uncertainty in the rate, dr /dt, of 40% was included to allow for these factors, with k, = 0.1, k2 = 7.5, and = 1.5. 

Keywords Measurement-based optimization Real-time optimization Plant-model mismatch Model adaptation Model parameterization. 

It is weU known that the interaction between the model-update and reoptimization steps must be considered carefully for the two-step approach to achieve optimal performance. In the absence of plant-model mismatch and when the parameters are structurally and practically identifiable, convergence to the plant optimum may be achieved in one iteration. However, in the presence of plant-model mismatch, whether the scheme converges, or to which operating point the scheme converges, becomes anybody s guess. This is due to the fact that the update objective might be unrelated to the cost or constraints in the optimization problem, and minimizing the mean-square error in y may not help in our quest for feasibility and optimality. To alleviate this difficulty, Srinivasan and Bonvin [23] presented an approach where the criterion in the update problem is modified to account for the subsequent optimization objective. 

Convergence under plant-model mismatch has been addressed by several authors [3,8] it has been shown that an optimal operating point is reached if model adaptation leads to a matching of the KKT conditions for the model and the plant. 

In order to overcome the modeling deficiencies and to handle plant-model mismatch, several variants of the two-step approach have been presented in the literature. Generically, they consist in modifying for the cost and constraints of the optimization problem for the KKT conditions of the model and the plant to match. The optimization problem with modifiers can be written as follows ... 

This last class of methods provides a way of avoiding the repeated optimization of a process model by transforming it into a feedback control problem that directly manipulates the input variables. This is motivated by the fact that practitioners like to use feedback control of selected variables as a way to cormteract plant-model mismatch and plant disturbances, due to its simphcity and reliability compared to on-line optimization. The challenge is to find functions of the measured variables which, when held constant by adjusting the input variables, enforce optimal plant performance [19,21]. Said differently, the goal of the control structure is to achieve a similar steady-state performance as would be realized by an (fictitious) on-line optimizing controller. 

Parameter identification is complicated by several factors (i) the complexity of the models and the nonconvexity of the parameter estimation problems, and (ii) the need for the model parameters to be identifiable from the available measurements. Moreover, in the presence of structural plant-model mismatch, parameter identification does not necessarily lead to model improvement. In order to avoid the task of identifying a model on-line, fixed-model methods have been proposed. The idea therein is to utilize both the available measurements and a (possibly inaccurate) steady-state model to drive the process towards a desirable operating point. In constraint-adaptation schemes (Forbes and Marlin, 1994 Chachuat et al., 2007), for instance, the measurements are used to correct the constraint functions in the RTO problem, whereas a process model is used to... 

Model predictive control node. MPC has several components. It has a model (usually an identified linear) of the world. It has a KD where past values of the manipulated variables (MVs) and controlled variables (CVs) are stored. In this KD other information is stored as MVs and CVs limitations, weighting factors, etc. The model uses the inputs to predict the future. This state is used in Behavior Generation module. In this module an optimization is performed to select the best action plan. This plan (a set of movements for the MVs along with CVs values) is set and sent to the regulatory level. Some preprocessing is implemented as well. The MPC module implements also a feedback loop to correct model errors (due to model mismatch with the actual plant). 

Schramm et al. (2001) have presented a model-based control approach for direct control of the product purities of SMB processes. Based on wave theory, relationships between the front movements and the flow rates of the equivalent TMB process were derived. Using these relationships, a simple control concept with two PI controllers was proposed. This concept is very easy to implement however, it does not address the issue of optimizing the operating regime in the presence of disturbances or model mismatch. 

The results may lead to a decrease in operating efficiency due to plant-model mismatch. 

RTO Fault Detection and Diagnosis. Halim proposed a method that uses gradient information from the plant, which is obtained via plant experiments, and the model to both quantify plant-model mismatch and monitor RTO system performance. In this work, the angle between the two profit gradients of the model-based RTO system and that obtained via plant experiments is used as an indicator of RTO... 

Halim, A. Detection and Diagnosis of Plant-Model Mismatch for Real-Time Optimization. M.Sc. thesis, University of Alberta, Edmonton, 2003. 

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