Nonlinear Model Predictive Control: A Self-Adaptive Approach

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Ind. Eng. Chem. Res. 2010, 49, 4782–4791

Nonlinear Model Predictive Control: A Self-Adaptive Approach Ivan Dones,† Flavio Manenti,*,‡ Heinz A. Preisig,† and Guido Buzzi-Ferraris‡ Chemical Engineering Department, NTNU; Sem Sælands Vei 4, N-7491 Trondheim, Norway, and CMIC Department “Giulio Natta”, Politecnico di Milano; Piazza Leonardo da Vinci 32, I-20133 Milano, Italy

Model predictive control (MPC) is an online application based on dynamic models. Its application faces two major obstacles: (i) computational constraints and (ii) the need to accurately simulate the process by a model that properly predicts how the plant will behave in the future. Implementation of MPC is not always possible in large-scale or industrial applications due to the computational complexity of MPC and to the dimensionality of the models. To facilitate MPC implementations, this paper proposes a self-adaptive approach based on simplified (or reduced-order) nonlinear models. The proposed methodology yields an MPC that adjusts the dimension of the model according to both the current process conditions and the control objectives. The self-adaptive approach is described and validated on an industrial case study, a C4-splitter. 1. Introduction Even though model predictive control (MPC) is still mainly perceived as an academic technique, it is nowadays an appealing methodology for process units that are characterized by long transients and high product requirements such as distillation columns and polymer reactors. The benefits of MPC have been confirmed in a series of academic and industrial contributions1–5 and, as reported in the recent work of Bauer and Craig,6 there are many potential industrial applications for MPC in the process industry. The MPC algorithm is not new, as it has recently celebrated its official 30th birthday,7–10 and the algorithm can be traced back to Propoi11 almost half a century ago. Developments and improvements of MPC have never stopped. The suggestions of new variants of the base algorithm indicate that many problems are still not resolved, such as (i) computation time exceeding the real-time requirements; (ii) insufficient accuracy or loss of stability; (iii) inability of reaching the performance specifications; and (iv) extension of the application domain of the classical MPC. For example, the extension of the application domain is considered by the work of Abel and Marquardt,12 who proposed a scenario-based approach for extending the MPC application to batch processes by validating it on the Luyben reactor.13 Pistikopoulos’s idea of a multiparametric MPC14–16 yields an explicit representation of MPC, which can be implemented on a microchip. Christofides and co-workers17–19 showed MPC applications where the ordinary differential equation (ODE) or the differential and algebraic equation (DAE) systems (usually constituting the dynamic model constraints) have been replaced by partial differential equation (PDE) and partial differential-algebraic equation (PDAE) systems. Another variant is proposed by Lima and co-workers,20 where the usual differential-algebraic constraints are replaced by a Takagi-Sugeno21 fuzzy model to characterize, in an easy way, those systems that show strong nonlinear behaviors such as polymer/copolymer reactors. Again, the key-point to the success of the method is the reduction of the computational complexity. * To whom correspondence should be addressed. Tel.: +39 02 2399 3273. Fax: +39 02 7063 8173. E-mail: [email protected]. † NTNU. ‡ Politecnico di Milano.

An interesting and promising approach is to combine a series of models and use each of them according to some predefined situations and operating conditions. For example, Dougherty and Cooper22 introduced a multimodel adaptive control strategy for multivariable MPC, and Guiamba and Mulholland23 developed and implemented an adaptive linear dynamic matrix control algorithm to control an integrated pump-tank system consisting of two input and two output variables. Cameron and co-workers proposed a multimodel approach24,25 to achieve robust control and global stability. Apart from all the proposed techniques available in the current literature, in the knowledge of the authors no papers propose to self-manage a set of (nonlinear) models to automatically adapt the model in the MPC algorithm to the current operating conditions of the plant. Specifically, the algorithm should adjust the ability of the model to predict the dynamic regime in which the plant is currently operating. Therefore, if the dynamics increase, the model complexity should be increased, while it can be decreased when the dynamics cease. This research activity is aimed at proposing, developing, and validating the feasibility of a self-adaptive (nonlinear) MPC approach which automatically selects the best level of detail of the mathematical model. An MPC using a simplified model is in many cases feasible while the MPC using the full model is not. Enabling the algorithm to adjust the model to the changes in the dynamics makes it possible to boost the performances of the MPC. A literature review on optimal control is presented in section 2. Essential aspects of MPC theory are reviewed in section 3. The self-adaptive MPC is introduced in section 4 where the underlying algorithm and the implementation of simplified nonlinear models are explained in details. The validation case study and numerical results are in sections 5 and 6, respectively. 2. Short Review of Optimal Control Literature Optimal control aims at optimizing the performance of a plant considering the dynamics in the order of minutes/hours. Optimal control includes MPC and real-time dynamic optimization (or set-point optimization), which are the lowest levels of the overall supply chain integration pyramid. These levels are also called continuous optimal control levels as no decisional variables are usually involved.26–28 The typical optimization problem is therefore an NLP, a nonlinear programming problem.

10.1021/ie901693w  2010 American Chemical Society Published on Web 04/20/2010

Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010

In contrast, the highest levels of the supply chain often depend on market laws and enterprise-wide optimization, where decisional variables (of Boolean and/or integer nature) are involved to describe scheduling and planning issues.29–34 In this case the optimization problem becomes a so-called mixed-integer linear/ nonlinear program (MILP/MINLP). 2.1. Model Predictive Control. Some authors35,36 reported in their works that only a small number of advanced industrial controllers are applied by the field, most of which use linear, fitted models. Today just a few applications are based on detailed first-principles models (i.e., on differential and algebraic equation systems), although the model is considered an essential step in the vertical hierarchy integration.37 There could be some issues that make difficult the implementation of detailed first-principles models and one of them is the effort in developing and solving detailed dynamic models of large-scale industrial plants in realtime. However, the well-known and consolidated technology behind model predictive control proves its feasibility and online application. In addition, the industrial application of MPC gives a relevant added value in most of the cases. For these reasons, large-scale industries are considering the implementation of these solutions. As reported in recent studies,1,2,38–40 the profit of the plant can increase. A fast payback time can also be guaranteed.5,41 2.2. Real-Time Dynamic Optimization. Real-time dynamic optimization is based on the same moving horizon methodology that is used in MPC; real-time dynamic optimization introduces a series of economic tasks in addition to the quadratic objectives of the MPC. Real-time dynamic optimization defines optimal production trajectories and paths, and allows the plant to operate optimally also when it undergoes persistent disturbances, such as load changes, grade change transitions, or process transients in general. As the MPC and the real-time dynamic optimization are quite close from a mathematical point of view, some authors solve in one shot both the MPC and the real-time dynamic optimization by adopting a vertical decomposition.42–47 Conversely, other authors solve the control and the optimization problems using the one-layer technique; some alternative formulations, such as single level control,37 closed-loop,5,48 one-layer,49 self-optimizing control,50,51 and single-layer46 to quote a few, have been proposed in the scientific literature. These methods either linearize the economical objective function or simplify the optimization problem.5,49,52,53 Recently, these academic efforts have evolved into the development of toolkits for the solution of dynamic optimization problems for DAE systems, for example DynoPC.28

Figure 1. Architecture of model predictive control.

tion sequence is to estimate at first the output predictions by using the model of the plant. Then, the errors between predicted and reference trajectories are calculated. The next step is to compute the sequence of the future control actions by minimizing an appropriate quadratic objective function. However, only the first control action is implemented. This procedure is iterated by means of moving the time horizon.54,55 In the last decades, MPC based on a nonlinear model (nonlinear MPC) has started to be studied. This has mostly been motivated by the following reasons: nonlinear MPC is able to handle nonlinearities both in process dynamics and in economical objective functions; it can be based on first-principles mathematical models and on nonlinear semiempirical models; it allows simultaneously solving the predictive control (quadratic optimal control problem) and the dynamic optimization (economical optimal control problem). The basic architecture of model predictive control is reported in Figure 1. Assuming an online implementation of this technique, the plant provides data to the model predictive controller at each sampling time. Specifically, the plant data are sent to the optimization algorithm, which includes an objective function, a dynamic model, and usually (according to the mathematical model type) a numerical integrator to solve differential equation systems. If the real-time dynamic optimization is to be solved as well, economical data and market scenarios must be provided to the MPC structure and one or more terms or economical objective functions must be defined. 3.1. Objective Function. Many optimal control problems such as data reconciliation, regressions, and rival models,56–60 etc. can be formulated as a minimization of a least-squares objective function subject to equality and/or inequality constraints. MPC enters this family and its generalized formulation is often the following one:

{∑

k+hp-1

k+hp

3. Optimal Control Problem Formulation The MPC algorithm was developed by Cutler at Shell Oil Company in 1979. At that time it was based on linear models, and it was called dynamic matrix control (DMC).8 Its basic idea is to use a time-domain step-response model (called the convolution model) of the process to predict the future responses and to obtain the optimal movements of the manipulated variables to optimize the process yield and efficiency. Using this approach, one obtains the future output responses matching the optimal trajectories in the HP (prediction horizon) by finding the best values of the manipulated variables in the HC (control horizon). This is exactly the concept of a least-squares problem of fitting HP data points with HC coefficients. The aim of this predictive control is to drive future outputs close to the reference trajectory while optimizing an objective function. The computa-

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min

u(k)....u(k+hc-1)

2 ωj(yj - yset j ) +

j)k+1

∑

2 ω1,l(ul - utar l ) +

l)k k+hp-1

∑

(i-1) 2 ω2,l(u(i) ) l - ul

l)k

{

}

(1)

subject to

ymin e yj e ymax j j umin e ul e umax l l (i) (i-1) ∆u(i),min e ∆u(i) e ∆u(i),max l l ) ul - u l l convolution model

where yj - yjset is the deviation between the jth controlled variable and its set-point; ul - ultar is the deviation between the lth manipulated variable and its steady-state target; ul(i) - ul(i-1) is the incremental variation between ith and (i -1)th time

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intervals of the lth manipulated variable; coefficients of the diagonal semipositive definite matrices ω are the weighting factors; HP is the prediction horizon; HC is the control horizon; min and max superscripts indicate lower and upper bounds, respectively, for both manipulated and controlled variables. Despite the fact that most real-world processes are approximately linear only within a limited operating range, linear models are used in a much larger operating range. In many cases the feedback in the controller makes the controlled system sufficiently robust to cope with these additional linearization errors. But if the specifications for the controlled system cannot be met anymore, one may have to use the nonlinear models, thus implementing nonlinear MPC, also abbreviated as NMPC.6,35 NMPC uses a nonlinear model directly in the control application; therefore the nonlinear model becomes a core component of the control application.61 The nonlinear model may be in the form of an empirical data fit (e.g., artificial neural networks) or a high-fidelity model based on conservation laws of mass and energy (first-principles model). As in linear MPC, NMPC requires the iterative solution of an optimal control problem on a finite prediction horizon. In contrast to the linear MPC where a linear optimization problem is solved in each step, NMPC solves in every step a nonlinear problem that is much more computationally expensive both in the computation of the optimization criterion as well as the solution of the model (especially for large models). Due to limitations in computation time, it is therefore interesting to investigate the use of simplified nonlinear models (instead of full nonlinear models) thereby at least reducing the costs associated with the computation of the model, without imposing any structural modification to the resulting nonlinear programming problem of NMPC. 4. Self-Adaptive Nonlinear Model Predictive Control In this section, we propose a self-adaptive NMPC (SANMPC) that uses a set of gradually simplified models. For a starter, we assume that there exists a master model, which describes the process sufficiently accurate such that the tuned NMPC based on this model meets the specs of the controlled process in any dynamic domain in which the plant may be operating. It is also assumed that there exists an index for measuring the relative fidelity of the models. In the case of reduced-order models, this index is usually but not always associated with the order, but may be refined with further simplification for each group of models with the same order. The algorithm hinges on the ability to detect the changes of the dynamic regime, which triggers an adaptation cycle and the ability to measure the gap between the responses of the different simplified models with respect to the full model. This gap measurement is used to decide which model is being used for the next step. The decision on changing the model is therefore based on this gap measure. If the algorithm detects that the currently used model is not sufficiently accurate, characterized by a too large gap, each next more complex model will be tried until the gap criterion is met. Conversely, if the system dynamics decrease, the current model is successively replaced by the next simpler model until the limit of the acceptable gap is reached. It is because of this adaptive nature that we call the algorithm “self-adaptive”. Analogously, if computations are so expensive as to undermine the online feasibility, the self-adaptive nature of the proposed algorithm should automatically lead to the selection of a reasonable reduced model. As this paper is mainly aimed at demonstrating the feasibility of the proposed algorithm of

SANMPC, the search for the best compromise between model prediction accuracy and time computing reduction is neglected for the time being. It is worth noting that the iterative procedure to identify the reduced model for the optimization routine has a computation need much lower than the optimization routine, since only a very small portion of a single simulation is needed to check the response of models. Thus, testing all models costs less than 1 s, while the optimization routine to calculate the manipulated variables requires minutes. 4.1. The Proposed Algorithm of SANMPC. In the algorithm we propose, the simplified models are classified based on the number of dynamic equations used in the system of equations. We assume that the higher is the number of dynamic equations, the closer is the simplified model to the full model and the better is the description of the plant dynamics. The set of simplified models ranges from a γlow-equations model to a γhigh-equations model (γlow e γhigh). The SANMPC algorithm selects the γhigh-equations model when the plant input observer notices a variation in the working conditions. The plant estimator can identify two types of variations in the working conditions: (1) Changes in set-point (e.g., step variation of the product requirements). This is what is called a servo problem. (2) Changes in some conditions due to disturbances (e.g., step variation in operating conditions). This is what is called a regulation problem. Set-point changes are known a priori, so the model for the SANMPC is set to the γhigh-equations model exactly when the servo problem occurs. On the other hand, the dynamics of the disturbances may not be immediately detected. We assume that the worst case is when the disturbance occurs at time t0, but it is detected at time t0 + ∆t (where ∆t the discretization time of the control action in the plant). In this case, the SANMPC algorithm will automatically select the γhigh-equations model with a delay equal to ∆t. After noticing the variation in the plant (either different requirements, or disturbances), the manipulated variables MVs are calculated using the γhigh-equations model for the integration in the optimization procedure (eq 1). To check whether the order of the model can be smaller in the next SANMPC call, first the γlow-equations model is tested for a reasonably small portion of a one-shot simulation from t0 to t0 + HP (therefore quite cheap in terms of computation time) with the same manipulated variables MVs as those calculated (the optimized input trajectory). Then, the procedure checks if the performances of the γlow-equations are acceptable by comparing the simplified model with the full model, simulated from t0 to t0 + HP with the optimized manipulated variables MVs (again, one-shot simulation, cheap in terms of computation time). If the γlow-equations model does not adequately fulfill the requirements (defined by some selection/rejection criteria), then the procedure compares the full model with a simplified model whose order is ∆γ higher than the previous one (so with γlow + ∆γ equations). This is iteratively repeated until the simplified model fulfils the selection/rejection criteria. The key of this iterative procedure is that the selected portion of oneshot simulations of the models is particularly small and hence fast to perform (seconds or fractions of seconds). Once the procedure has identified the most simplified model that does not differ too much from the full one (in agreement with the selection/rejection criterion), that simplified model is used for the next SANMPC call at time t0 + ∆t. The flowchart


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Figure 2. Flowchart of the SANMPC procedure. The panel with the darker filling shows the optimization procedure. This one is the time-consuming part; the part we aim to run with the most simplified model as possible. Panels with thicker borders show the iterative procedure to identify the proper level of model simplification.

in Figure 2 illustrates the algorithm of our proposed SANMPC. numeq is the variable for the number of equations used in the model. 4.2. The Model Bounds. The user has the option to define the bounds γlow and γhigh for the set of models used in the SANMPC algorithm. The most conservative approach is to set γlow equal to the smallest model size one can generate, and γhigh equal to the size of the full model. To ease the search algorithm in section 4.1, the user can narrow the acceptance window for the minimum and maximum simplified model. This may be done on the basis of previous experience or simply by visual inspection. Alternatively, a mathematical procedure can be proposed; the mathematical procedure seeks the current boundaries of the acceptance window.

In the example in section 6 we do narrow the acceptance window: we do closed-loop simulations with an NMPC based on the full nonlinear model and different simplified models in the presence of simple step input disturbances and/or with different grade changes. For a certain model with γ* equations, we obtain dynamics that are almost identical to the full model’s behavior; γhigh can therefore be set equal to γ*. The peculiarity of the proposed procedure is that a systematic mathematical procedure for the definition of the upper bound is not strictly mandatory; even qualitative consideration can be enough. As a matter of fact, the procedure allows the use of models that are more detailed than the upper bound, if this is required. As an extreme case, if it is found that even the γhighequations model is not appropriate, the algorithm proceeds further on by moving the upper bound toward the full model,

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thus selecting models with more equations than γ*. Therefore the proposed algorithm includes the possibility of automatically correcting the upper bound. If one already knows that some of the simplified models will most likely never be used (because they are not sufficiently accurate to properly reproduce the dynamics of the full model and of the plant), then one can increase the value of γlow. On the other hand, the “risk” one can meet is to select the lower bound too high, and to exclude a priori models that are instead applicable when operating close to steady conditions. If this is the case, it will only make the procedure less efficient in terms of computation time (in the optimization part) when the plant is close to steady conditions, but still applicable. 4.3. Selection/Rejection Criterion. In section 4.1 we mentioned the need for a criterion to establish whether or not a simplified model is acceptable. One can suggest several kinds of selection/rejection criteria. In this work, in order to accept a simplified model, the absolute value of the difference between the variables controlled by the SANMPC in the simplified model and in the full model should never exceed a certain tolerance εtol decided by the user. - yfull |yreduced k k | < εtol

(2)

The stricter the tolerances εtol are, the less the most simplified models will be used. In this work, we use the control horizon for the evaluation of the measure. Therefore, in eq 2, k ) 1,...,HC. Other selection/ rejection criteria are also possible; for example, one can evaluate the integral difference of the outputs between simplified and reduced model over a certain time span. 4.4. Other Remarks about the Proposed SANMPC Algorithm. For the SANMPC procedure, a set of simplified nonlinear models is required. In the most generic case, the model simplification may be any combination of reduced-order, local linearization, or meta-models of any part of the full model, like the thermodynamics. Several model simplification methods are available, and the choice of one method instead of another depends on the case study and on the confidence the user has with a specific methodology. The SANMPC algorithm here proposed is limited to all those model simplification techniques that reduce the number of the dynamic equations contained in the full model. For instance, if a plant is modeled using a network approach,62,63 then lumping64 the capacities (i.e., the control volumes) reduces the number of differential equations.65 If the model has multiple time-scale behavior, then singular perturbation can be used for the reduction of the dynamics of the model.66,67 For one-dimensional distributed systems, the aggregation model reduction method68,69 is another solution to reduce the model’s dynamics. In addition, we quote the balanced truncation method.70,71 This model reduction method reduces the dynamics of the model and maintains the same input-output behavior of the full model. Apart from the model simplification method, the algorithm in Figure 2 can be slightly modified and developed in an alternative way. One could establish a “dynamic regime detection” criterion and then select a proper simplified model for the specific dynamic regime the plant is passing through. The set of different simplified models must be ordered according to a “fidelity index” that maps the models with the dynamic regimes. In the SANMPC algorithm we propose, the dynamic regime detector is not explicitly implemented, since before every optimization the proposed algorithm compares the responses of the simplified model with the full model’s responses. If the

simplified model is sufficiently accurate, that model is implemented. The simplicity of the proposed algorithm avoids the creation of a mapping table between the dynamic regimes and the order of the model. 5. Validation Case: Distillation Column The validation case here considered is the control of distillation columns. As an example, we used a dynamic equilibrium model of a 92-trays C4 splitter. Since no plant data were available, they were substituted with data obtained by simulating the full model. The same model was previously described in other contributions,72,73 and only the fundamental explanations are reported hereinafter. 5.1. Dynamic Model. First principles distillation models yield a tridiagonal block structure, each block representing the properties of a single distillation stage. In this study, each stage of the column is considered as an equilibrium dynamic flash containment, where the liquid and gas phases are two uniform lumps in mechanical, thermal, and chemical equilibrium. The dimension of the state-space is 3NC + 4 states per column stage, where NC is the number of components. All thermodynamic properties are calculated using an external, portable, and easy-to-plug-in thermodynamic package.74 The package accepts temperatures, volumes, and masses, returning Helmholtz’s energy and its derivatives with respect to the inputs. Krishnamurthy and Taylor75 pointed out that many numerical solution methods (such as Newton’s method for the solution of algebraic system) require derivatives, and some of them (with respect to thermodynamic and transport properties) may be unavailable, urging the user to adopt approximations when required. Having available Løvfall’s74,76 stand-alone thermomodules makes any numerical approximation obsolete. Derivatives are computed from analytically derived expressions internal of those modules, thus providing a world of flexibility for their use in advanced algorithms. The pressure drop along the column was accounted by Krishnamurthy and Taylor: To express the column pressure drop as a function of tray (or packing) type, for each stage one should consider a hydraulic equation and make the pressure drop an unknown variable. In opposition to Krishnamurthy and Taylor, in our proposed distillation model the description of the pressure drop is not made explicit but is hidden in the resistance that the gas streams have to face to flow upward in the column. The law defining the gas streams is a valve-like equation derived from Bernoulli’s theorem;77 this equation contains the pressure drop (in form of dry and wet tray resistances) multiplied by the square root of the pressure difference between two consecutive stages. 5.2. Adopted Control Scheme. The process control scheme adopted is reported in Figure 3. Conventional loops are still used to control the top pressure by manipulating the cooling stream and the temperature in the stripping section, specifically the temperature of tray 74 (counting from the top to the bottom) by manipulating the reboiler duty. The multivariable NMPC is adopted to control the top and bottom product purity. Also, the NMPC manipulates the reflux stream and the set-point of the proportional-integral temperature controller at the tray 74. 5.3. Aggregation Method. The key of the SANMPC procedure is to generate a series of simplified models of the master full model. The model simplification technique adopted in the case study is Linhart’s and Skogestad’s aggregation method.69 The aggregation method is derived from the original version created by Le´vine and Rouchon.68 The method is based on partitioning the distillation column into “compartments”, where


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Figure 4. Grade change requirement to top composition at 100 s; application with NMPC (nonadaptive).

Figure 3. Case study and its control scheme. Table 1. Computation Speed Increment for Some of the Most Used Reduced Models size of reduced model with respect to the full model

approximate increment of computation speed

15% 10% 5%

2 times faster 3 times faster 7 times faster

each compartment comprises a number of consecutive trays. By assigning the total holdup of all trays of a compartment to a single “aggregation” tray within the compartment, a timescale separation caused by the difference between the large compartment holdup and the small tray holdups is made explicit in the model. This time-scale separation can be used for reducing the dynamic order of the model by applying quasi-steady-state assumptions to the trays within each compartment. As implemented by Linhart and Skogestad69 the aggregation method is a methodology to derive a simplified model. Among its advantages is the perfect steady-state agreement with the original model, a simple derivation, and a good control of the simplified model complexity. As described in Linhart’s and Skogestad’s recent work, the aggregation method sets to quasisteady-state all the dynamic equations of each compartment, namely the mass and energy balances of the trays. Recalling the definition of the proposed equilibrium full model, each time a tray is set to quasi-steady-state, the model’s dynamics are reduced by NC + 1 equations. The advantage of this method in computation time is that the algebraic equations generated by quasi-steady-state assumption can be solved off-line. The immediate consequence is that the burden faced by the numerical integrator is reduced. This is an old methodology, largely used in the past when computer power was much more limited than today. With Linhart’s and Skogestad’s aggregation method, the well-organized structure of the Jacobian is preserved and some very performing numerical integrators may relevantly benefit of it.78 6. Numerical Results 6.1. Simplified Dynamic Distillation Models. The dynamic models are implemented in Visual C++ and integrated by numerical solvers belonging to BzzMath library as already

Figure 5. Grade change requirement to top composition at 100 s; application with NMPC (nonadaptive).

discussed elsewhere.79,80 As previously reported, first of all we ran some closed-loop simulations where the NMPC is implemented without the adaptive part, i.e. with fixed simplified models. We have always considered reboiler and condenser being represented by dynamic equations, since their holdups are larger than the trays’ holdups. Concerning the dynamics of the trays, we considered various models, where the number of dynamic trays was always different. In particular, we did not consider specifically the location of the dynamic trays, just their number. Some significant models are the following: (i) full dynamic model, 92 dynamic trays; (ii) 15-dynamic-trays model, with the dynamic tray numbers being 1, 7, 15, 23, 31, 39, 45, 47, 55, 63, 71, 74, 79, 87 and 92; (iii) 5-dynamic-trays model, with the dynamic tray numbers being 1, 23, 45, 74 and 92. According to Linhart,69 the increment of computation speed is difficult to evaluate since it depends on the adopted model reduction method, and the same model reduction method can benefit differently on the computation speed of different applications. For the sake of completeness, solving off-line the algebraic equations, the aggregation method has the effects reported in Table 1 on the computation speed of a binary distillation column model. 6.2. Application of NMPC: Servomechanism Problem. In this section we show the qualitative test of the behavior of the plant-model in closed-loop for a grade change in the top composition (Figure 4). This test is used to determine the model bounds for the SANMPC. The model with 15 dynamic trays can be considered as the model to be used when disturbances or variations in the

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Figure 6. Grade change requirement to top composition at 100 s; comparison between NMPC and SANMPC.

operating conditions are noticed by the estimator. In fact, in nonadaptive NMPC, the 15-dynamic-trays model is able to predict the behavior of the plant similarly to the full model, from the beginning of the disturbance until the new steadystate. Adding more dynamic trays has no further evident benefits on the responses (even though not reported for sake of readability). The most reduced model we studied is the 5-dynamic-trays model, as suggested by Linhart and Skogestad69 for controlled distillation models. As shown in Figure 4 and Figure 5, the 5-dynamic-trays model can be used to calculate the optimal input trajectory only when the plant is approaching the new steady conditions. As reported in section 4.2, the user should be not excessively concerned about the finding of the bounds, since the SANMPC algorithm will automatically adjust the model size. 6.3. Application of SANMPC: Servomechanism Problem. As reported above, the aggregation method was applied traywise. For this reason we decided to redefine the general procedure reported in Figure 2, and to make it easier to communicate for this specific example. Instead of focusing on the number of equations to be reduced, we focus on the number of trays that are considered as dynamic or that are set to quasisteady-state (thus potentially solvable off-line). The result is the same, since the number of dynamic equations is proportional to the number of dynamic trays. However we found easier to communicate if we show the number of dynamic trays instead of the number of equations. In this context, the procedure is the same as that of Figure 2, where the variable numeq corresponds to the number of dynamic trays to be used in the simplified model. γlow is the number of dynamic trays in the most simplified model (i.e., γlow ) 5), while γhigh (15 in our case) is the number of dynamic trays to be used for the higher dynamic part. To use the smallest models in the largest possible extent, the increment ∆γ must be taken as 1; however, in the tests we noticed that even ∆γ ) 2 performs reasonably well. Figure 6 shows the dynamic behavior of the top composition, and Figure 7 confirms that our proposed SANMPC is able to calculate an optimal input trajectory much closer to that of the full model than that of a 5-dynamic-trays model. However, the latter one can be used, with beneficial effects for the computation speed, in the part of the simulation closer to the new steadystate (Figure 8). In few words, the philosophy of the proposed SANMPC is to use the smallest possible model (in terms of dynamic containment) in each part of the dynamic process: not smaller (to not compromise the quality of the optimal input trajectory), not bigger (to not use more computing time).

Figure 7. Grade change requirement to top composition at 100 s; comparison between NMPC and SANMPC.

Figure 8. SANMPC; number of dynamic trays used in the model for the control task.

Figure 9. Step disturbance to feed rate at 100 s; comparisons between NMPC and SANMPC.

6.4. Application of SANMPC: Regulatory Problem. Tests were also run to check whether the proposed SANMPC is robust in terms of disturbance rejections. From steady conditions, a step disturbance (+10% with respect to the nominal value) is injected in the feed rate. Even though the magnitude of the success is not so evident as in the reported servomechanism case, again the SANMPC emulates the optimal trajectories calculated by the full model better than an NMPC using a 5-dynamic-trays model. We also remind the reader that the scale in Figure 9 is smaller than the scale in Figure 6.


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we recommend the use of a technique that allows one to obtain a cascade of models from a full model by the use of an algebraic procedure. 8. Future Works

Figure 10. Step disturbance to feed rate at 100 s; comparisons between NMPC and SANMPC.

Future works can be addressed to define alternative gap measurements to select the proper model in the SANMPC algorithm. Other possible future activities regard the definition of the upper bound γhigh as a function of the disturbance/grade change. As already mentioned, if it was not possible do conduct this study, one could have always used a conservative approach by setting γhigh close enough to the full model size (or behavior). In addition, other model simplification techniques can also be tested and validated with SANMPC. In the algorithm proposed in this study, the model simplification simplifies the dynamic content of the model and gives the possibility of reducing the size of the DAE system. In the most generic case, the simplification may be any combination of reduced-order, local linearization, or meta-models of any part of the model, such as the thermodynamics. Acknowledgment This work is supported by the StatoilHydro Mongstad Pilot Project. Luisella Keller, Andreas Linhart, Johannes Ja¨schke, Ramprasad Yelchuru, and Prof. Sigurd Skogestad are acknowledged for valuable discussions. Literature Cited

Figure 11. SANMPC; number of dynamic trays used in the model for the control task.

7. Conclusions We propose a self-adaptive nonlinear MPC procedure. This algorithm uses simplified nonlinear models in the calculations of the optimal input trajectory. In addition, the level of the simplification is automatically adapted to the dynamics of the plant: when the used model is much too simplified (according to the selection/rejection criterion), then a less simplified model (i.e., closer in size to the full nonlinear model) is automatically chosen. Therefore the proposed procedure fulfils the calculation of the optimal input trajectory with the most simplified model that does not excessively deviate from the behavior of the full model. This novel approach could also easily be applied to realtime dynamic optimization problems. As the use of dynamic optimization is increasing quickly,44 this could be an interesting field of application for the new approach. Distillation is offered as a case study. A full model is used as plant estimator and a cascade of simplified distillation models is obtained with the aggregation method. The results are promising, and, as expected, they show that the most simplified models are automatically used closer to steady conditions. Vice versa, in higher dynamic ranges, more dynamic states must be used in the model, and once again the selection of the proper model is done automatically. The procedure is generic and can be easily applied to other cases and to other model reduction and simplification techniques. Concerning the model simplification (or reduction) techniques,

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ReceiVed for reView October 28, 2009 ReVised manuscript receiVed March 19, 2010 Accepted April 8, 2010 IE901693W

Nonlinear Model Predictive Control: A Self-Adaptive Approach

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