- Email: [email protected]
Modifier Adaptation with Model Adaptation in Iterative Real-Time Optimization
Mario R. Eden, Marianthi Ierapetritou and Gavin P. Towler (Editors) Proceedings of the 13th International Symposium on Process Systems Engineering – PSE 2018 July 1-5, 2018, San Diego, California, USA © 2018 Elsevier B.V. All rights reserved. https://doi.org/10.1016/B978-0-444-64241-7.50110-5
Modifier Adaptation with Model Adaptation in Iterative Real-Time Optimization Afaq Ahmad*, Weihua Gao, Sebastian Engell Process Dynamics and Operations Group, Department of Biochemical and Chemical Engineering, TU Dortmund, Emil-Figge Strasse 70, 44227 Dortmund, Germany. [email protected]
Abstract This paper deals with the iterative real-time optimization (RTO) of chemical processes under plant-model mismatch. Certain RTO methodologies, called modifier adaptation add bias and gradient correction terms to the underlying model based on measurements of the true values of the objective function and of the constrained function. These affine corrections lead to meeting the first-order necessary-conditions of optimality of the plant in spite of plant-model mismatch. However, accurately computing the gradient corrections is a limiting element here, and to improve the model parameters were possible will speed up convergence. Also, the additional terms do not guarantee satisfaction of the second-order optimality conditions. So modifier adaptation should be combined with model adaptation such that first- and second- order optimality conditions are met at the plant optimum. In this paper, we compare different iterative RTO schemes that perform modifier adaptation and model adaptation. The performance of the approaches is evaluated by means of simulation results for the Williams-Otto reactor benchmark problem. Keywords: Real-Time Optimization; Modifier Adaptation; Effective Model Adaptation
1. Introduction Real-time optimization (RTO) can make a significant contribution to profitable and resource efficient operation of processing plants. However, the feasibility and optimality of the proposed operating points relies on the accuracy of the plant model. Due to plantmodel mismatch, the optimum of the real plant may differ significantly from the optimum that is computed from the available model and constraints may be violated by the theoretically optimal set-point. Therefore it is necessary to adapt the model or the optimization problem based on the available measurements. In recent years, several RTO methodologies have been developed which differ in how they utilize the information which is provided by the measurements. In the two-step approach (Chen and Joseph, 1987), the model parameters are estimated at the current operating conditions in order to match the model prediction with the plant outputs. The optimization step is then performed based on the adapted model to update the operating point. This iterative two-step scheme may potentially converge to the process optimum if the plant-model mismatch is mainly of parametric nature, and there is sufficient excitation to estimate the model parameters accurately. However, if there is a structural mismatch, the two-step scheme may not converge to the true plant optimum. Therefore Roberts (1979) proposed a modified two-step scheme, known as Integrated Acknowledgement: The research leading to these results has received funding from the Deutsche Forschungsgemeinschaft (DFG) under grant agreement number EN 152/41-1.
692
A. Ahmad et al.
System Optimization and Parameter Estimation (ISOPE) in order to handle structural plant-model mismatch. The ISOPE approach converges iteratively to a Karush-KuhnTucker (KKT) (also known as first-order necessary conditions) point of the plant by adding gradient correction terms to the cost function of the optimization problem. Tatjewski (2002) realized that the step of the estimation of the model parameters in ISOPE is not required to satisfy the optimality condition of the plant. Based upon this work, Gao and Engell (2005) extended the approach to handle process dependent constraints and applied it to the optimization of a chromatographic separation process. Later, this approach was labelled as modifier-adaptation (MA) in Marchetti et al. (2009). The MA scheme adds bias and gradient modifiers to the cost and to the constraint functions of the model-based optimization so that the first-order necessary conditions of optimality of the adapted model match those of the plant. However, the approach does not ensure convergence if the model adequacy condition is not met as shown by Marchetti et al. (2009) and Gao et al. (2015). A modification of the two-step approach is to estimate the model parameters such that the updated model predicts the gradient of the plant correctly at the current operating condition as proposed by Mandur and Budman (2015). However, the estimation of model parameters in order to fit the available plant measurements does not necessarily ensure that the model is adequate for optimization purposes (Forbes and Marlin, 1996). Model adequacy can be assured by using modifier adaptation with quadratic approximation (MAWQA) (Gao et al. 2016). However, the rate of the convergence depends on the collected data that is used to construct the quadratic approximation and is of course better when the model quality is high. Recently, Ahmad et al. (2017) proposed an effective model adaptation (EMA) scheme where an adequate process model is identified and used in the optimization problem. The key idea is to combine MA with EMA in the sense that the model that is used in the optimization problem is adapted only when the model with the updated parameter values is adequate, meaning that the first- and second-order necessary conditions of optimality are satisfied upon convergence. In this paper, we investigate the reliability and efficiency of different iterative RTO schemes in the presence of plant-model mismatch. A simulation study is performed using the benchmark Williams-Otto reactor as a case study. The RTO schemes are compared based on their ability to converge to the plant optimum and the number of plant evaluations that are required to converge.
2. Iterative RTO schemes The general model-based optimization problem at the (k)th iteration can be formulated as
(1)
where is the vector of outputs which is a function of the input variables and the set of model parameters . is the objective function that is minimized and represents the vector of the constraint functions. In the two-step scheme, the vector of model parameters is obtained as
Modifier Adaptation with Model Adaptation in Iterative Real-Time Optimization
693
(2) where represents the measured variables at the current operating conditions . Since the conventional model adaptation via least-squares value fitting of the plant measurements does not guarantee the matching of the gradients, Mandur and Budman (2015) introduced an algorithm (called simultaneous model identification and optimization (SMIO)) in which the model is adapted in two steps in order to handle structural plant-model mismatch. In first step the values of the selected model parameters are computed via Eq.(2). The second step is based on gradient fitting, where the model gradients are matched to those of the plant by estimating the change of the model parameters as
(3)
where
and
are the weighting factors for the objective and constraint functions. and are the plant objective and
constraint gradients at . A correction term is introduced to compensate the deviation of the predicted steady-state values of the outputs due to the adjustment of the parameters in the second step, where is the Jacobian matrix of the model output with respect to the model parameters. The accuracy of the correction term can be quantified by using the relative truncation error (4) In the MA scheme, the model-based optimization problem in Eq.(1) is adapted by the bias and first-order correction terms instead of model parameter adaptation. The updated optimization problem of the MA scheme can be written as follows:
(5)
where and are the adapted objective and constraint functions, represents the fixed model parameter vector, is the process objective value and is the vector of constraint functions. The operating condition at the (k)th iteration is updated as: (6) where
is the solution of Eq.(5) and
is a diagonal matrix of damping factors with
A. Ahmad et al.
694
. The process gradients are estimated by finite difference approximation via perturbing the system in each iteration as proposed in Gao and Engell (2005).
3. Model adequacy A criterion of model adequacy has been first discussed in Forbes and Marlin (1996). It states that a process model must produce a fixed point which is a local minimum for an RTO problem at the true process optimum . The model adequacy criterion for MA requires that must meet the first- and secondorder necessary conditions of optimality for the adapted optimization problem Eq.(5). As the MA scheme relies on the bias- and gradient corrections to ensure satisfaction of the first order KKT conditions of the plant, in addition the second-order necessary conditions of optimality need to be satisfied at . This leads to the criterion that the matrix of the reduced Hessian of the adapted optimization problem Eq.(5) should be positive definite. The Lagrangian function of the adapted optimization problem can be written as: (7) where
is the vector of Lagrange multipliers.
4. Modifier adaptation with model adaptation A possible way to handle the issue of model adequacy is to combine the idea of MA with model adaptation. Ahmad et al. (2017) introduced the effective model adaptation (EMA) scheme to identify an adequate process model for the adapted optimization problem Eq.(5) and proposed the modifier-adaptation with EMA (in short MAWEMA) algorithm. EMA ensures and speeds up the rate of convergence to the optimum . Let be the estimated model parameter vector at , then the updated parameters are accepted by EMA if and only if the following two conditions are satisfied: x x
is positive definite .
The first condition ensures the model adequacy criterion of the Lagrangian function . It is sufficient that the Hessian matrix of the adapted objective function is positive definite which is the same as that reduced Hessian is positive definite. To motivate the second condition, we perform a Taylor series expansion of around , omitting terms of order higher than 2: (8) The prediction error of the adapted objective function at , that is, , can be used to infer the difference of the Hessian matrices at . Hence, reducing the difference between and ensures that the adapted model parameters accelerate the rate of convergence to the process optimum.
Modifier Adaptation with Model Adaptation in Iterative Real-Time Optimization
695
5. Simulation study Different RTO schemes were tested on the Williams-Otto reactor benchmark problem which has been used to evaluate the performance of RTO schemes in many papers, e.g. Marchetti et al. (2010). The plant consists of an ideal CSTR in which three irreversible reactions take place whereas the presence of only two reactions is assumed in the model. The optimization objective is to maximize the profit which is defined as: (9) where . The manipulated variables are the feed flow rate and the reactor temperature . The model equations and the description of the variables can be found in Marchetti et al. (2010). The meaning and the values of the model parameters are: and are the activation energies and and are the pre-exponential factors of the reactions. The initial model parameters and are calculated at the initial conditions of the manipulated variables, and . Figure 1 demonstrates the performance of the four RTO approaches presented above. Due to the presence of parametric and structural plant-model mismatch, the two-step approach does not converge to the true plant optimum. The standard MA approach where model parameters are not adapted moves near to the process optimum in the first iteration but then starts to oscillate since the available model is not adequate as it does not meet the second-order necessary condition of optimality. By updating the model at each iteration, the SMIO scheme performs better than the pure MA scheme. However, the SMIO scheme converges to a sub-optimal point after the eighteenth iteration, which is shown in the insert in Figure 1. This is due to the fact that parameter values do not exist which Figure 1: Comparison of the performance of the minimize the objective gradient error iterative RTO schemes between the plant and model . The best performance was obtained by the modifier adaptation with effective model adaptation (MAWEMA). After the initial iteration, EMA identifies an adequate process model for the modified optimization problem and ensures the convergence to the process optimum. Since the updated model at the first iteration also reduces the difference between the Hessians of the plant and the model, faster convergence to the process optimum is observed as compared to SMIO. To illustrate the benefit of EMA, a simulation study was performed for different values of the initial feed flow rate and the reactor temperature. Figure 2 shows the comparison of the evolutions of the optimization objective over the iterations between SMIO and MAWEMA. Figure 2b shows that all trajectories obtained from the MAWEMA approach converge smoothly to the process optimum and satisfy the first- and second-
696
A. Ahmad et al.
order optimality conditions whereas large oscillations can be observed in SMIO for some of the initial operating conditions. The SMIO scheme is not always able to reach the process optimum as the updated parameters cannot adapt the model gradients to the process gradient.
Figure 2: Optimization runs from arbitrary initial operating conditions. (a) SMIO, (b) MAWEMA
6. Conclusions This paper highlights the importance of model adequacy in model adaptation which further enhances the performance of modifier adaptation. The key feature of MAWEMA is that a modified optimization problem is formulated to meet the first order KKT conditions and the second-order optimality conditions are satisfied via EMA.
References Ahmad, A., Gao, W., Engell, S., 2017. Effective Model Adaptation in Iterative RTO. In: Computer Aided Chemical Engineering. Vol. 40. pp. 1717-1722. Chen, C. Y., Joseph, B., 1987. On-line optimization using a two-phase approach: An application study. Ind. Eng. Chem. Res. 26(9), 1924-1930. Forbes, J. F., Marlin, T. E., 1996. Design cost: A systematic approach to technology selection for model-based real-time optimization systems. Comput. Chem. Eng. 20 (6-7), 717–734. Gao, W., Engell, S., 2005. Iterative set-point optimization of batch chromatography. Comput. Chem. Eng. 29 (6), 1401–1409. Gao, W., Wenzel, S., Engell, S., 2015. Integration of gradient adaptation and quadratic approximation in real-time optimization. In: 34th Control Conference (CCC). pp. 2780–2785. Gao, W., Wenzel, S., and Engell, S. 2016. A reliable modier-adaptation strategy for real-time optimization. Comput. Chem. Eng. 91, 318–328. Mandur, J. S., Budman, H. M., 2015. Simultaneous model identification and optimization in presence of model-plant mismatch. Chem. Eng. Sci. 129, 106–115. Marchetti, A. G., Chachuat, B., Bonvin, D., 2009. Modifier-adaptation methodology for real-time optimization. Ind. Eng. Chem. Res. 48 (13), 6022–6033. Marchetti, A. G., Chachuat, B., Bonvin, D., 2010. A dual modifier-adaptation approach for realtime optimization. Journal of Process Control 20 (9), 1027–1037. Roberts, P. D., 1979. An algorithm for steady-state system optimization and parameter estimation. Int. J. Syst. Sci. 10 (7), 719–734. Tatjewski, P., 2002. Iterative optimizing set-point control - the basic principle redesigned. Proceedings of the 2002 IFAC World Congress 15 (1), 49–54.
Modifier Adaptation with Model Adaptation in Iterative Real-Time Optimization Afaq Ahmad*, Weihua Gao, Sebastian Engell Process Dynamics and Operations Group, Department of Biochemical and Chemical Engineering, TU Dortmund, Emil-Figge Strasse 70, 44227 Dortmund, Germany. [email protected]
Abstract This paper deals with the iterative real-time optimization (RTO) of chemical processes under plant-model mismatch. Certain RTO methodologies, called modifier adaptation add bias and gradient correction terms to the underlying model based on measurements of the true values of the objective function and of the constrained function. These affine corrections lead to meeting the first-order necessary-conditions of optimality of the plant in spite of plant-model mismatch. However, accurately computing the gradient corrections is a limiting element here, and to improve the model parameters were possible will speed up convergence. Also, the additional terms do not guarantee satisfaction of the second-order optimality conditions. So modifier adaptation should be combined with model adaptation such that first- and second- order optimality conditions are met at the plant optimum. In this paper, we compare different iterative RTO schemes that perform modifier adaptation and model adaptation. The performance of the approaches is evaluated by means of simulation results for the Williams-Otto reactor benchmark problem. Keywords: Real-Time Optimization; Modifier Adaptation; Effective Model Adaptation
1. Introduction Real-time optimization (RTO) can make a significant contribution to profitable and resource efficient operation of processing plants. However, the feasibility and optimality of the proposed operating points relies on the accuracy of the plant model. Due to plantmodel mismatch, the optimum of the real plant may differ significantly from the optimum that is computed from the available model and constraints may be violated by the theoretically optimal set-point. Therefore it is necessary to adapt the model or the optimization problem based on the available measurements. In recent years, several RTO methodologies have been developed which differ in how they utilize the information which is provided by the measurements. In the two-step approach (Chen and Joseph, 1987), the model parameters are estimated at the current operating conditions in order to match the model prediction with the plant outputs. The optimization step is then performed based on the adapted model to update the operating point. This iterative two-step scheme may potentially converge to the process optimum if the plant-model mismatch is mainly of parametric nature, and there is sufficient excitation to estimate the model parameters accurately. However, if there is a structural mismatch, the two-step scheme may not converge to the true plant optimum. Therefore Roberts (1979) proposed a modified two-step scheme, known as Integrated Acknowledgement: The research leading to these results has received funding from the Deutsche Forschungsgemeinschaft (DFG) under grant agreement number EN 152/41-1.
692
A. Ahmad et al.
System Optimization and Parameter Estimation (ISOPE) in order to handle structural plant-model mismatch. The ISOPE approach converges iteratively to a Karush-KuhnTucker (KKT) (also known as first-order necessary conditions) point of the plant by adding gradient correction terms to the cost function of the optimization problem. Tatjewski (2002) realized that the step of the estimation of the model parameters in ISOPE is not required to satisfy the optimality condition of the plant. Based upon this work, Gao and Engell (2005) extended the approach to handle process dependent constraints and applied it to the optimization of a chromatographic separation process. Later, this approach was labelled as modifier-adaptation (MA) in Marchetti et al. (2009). The MA scheme adds bias and gradient modifiers to the cost and to the constraint functions of the model-based optimization so that the first-order necessary conditions of optimality of the adapted model match those of the plant. However, the approach does not ensure convergence if the model adequacy condition is not met as shown by Marchetti et al. (2009) and Gao et al. (2015). A modification of the two-step approach is to estimate the model parameters such that the updated model predicts the gradient of the plant correctly at the current operating condition as proposed by Mandur and Budman (2015). However, the estimation of model parameters in order to fit the available plant measurements does not necessarily ensure that the model is adequate for optimization purposes (Forbes and Marlin, 1996). Model adequacy can be assured by using modifier adaptation with quadratic approximation (MAWQA) (Gao et al. 2016). However, the rate of the convergence depends on the collected data that is used to construct the quadratic approximation and is of course better when the model quality is high. Recently, Ahmad et al. (2017) proposed an effective model adaptation (EMA) scheme where an adequate process model is identified and used in the optimization problem. The key idea is to combine MA with EMA in the sense that the model that is used in the optimization problem is adapted only when the model with the updated parameter values is adequate, meaning that the first- and second-order necessary conditions of optimality are satisfied upon convergence. In this paper, we investigate the reliability and efficiency of different iterative RTO schemes in the presence of plant-model mismatch. A simulation study is performed using the benchmark Williams-Otto reactor as a case study. The RTO schemes are compared based on their ability to converge to the plant optimum and the number of plant evaluations that are required to converge.
2. Iterative RTO schemes The general model-based optimization problem at the (k)th iteration can be formulated as
(1)
where is the vector of outputs which is a function of the input variables and the set of model parameters . is the objective function that is minimized and represents the vector of the constraint functions. In the two-step scheme, the vector of model parameters is obtained as
Modifier Adaptation with Model Adaptation in Iterative Real-Time Optimization
693
(2) where represents the measured variables at the current operating conditions . Since the conventional model adaptation via least-squares value fitting of the plant measurements does not guarantee the matching of the gradients, Mandur and Budman (2015) introduced an algorithm (called simultaneous model identification and optimization (SMIO)) in which the model is adapted in two steps in order to handle structural plant-model mismatch. In first step the values of the selected model parameters are computed via Eq.(2). The second step is based on gradient fitting, where the model gradients are matched to those of the plant by estimating the change of the model parameters as
(3)
where
and
are the weighting factors for the objective and constraint functions. and are the plant objective and
constraint gradients at . A correction term is introduced to compensate the deviation of the predicted steady-state values of the outputs due to the adjustment of the parameters in the second step, where is the Jacobian matrix of the model output with respect to the model parameters. The accuracy of the correction term can be quantified by using the relative truncation error (4) In the MA scheme, the model-based optimization problem in Eq.(1) is adapted by the bias and first-order correction terms instead of model parameter adaptation. The updated optimization problem of the MA scheme can be written as follows:
(5)
where and are the adapted objective and constraint functions, represents the fixed model parameter vector, is the process objective value and is the vector of constraint functions. The operating condition at the (k)th iteration is updated as: (6) where
is the solution of Eq.(5) and
is a diagonal matrix of damping factors with
A. Ahmad et al.
694
. The process gradients are estimated by finite difference approximation via perturbing the system in each iteration as proposed in Gao and Engell (2005).
3. Model adequacy A criterion of model adequacy has been first discussed in Forbes and Marlin (1996). It states that a process model must produce a fixed point which is a local minimum for an RTO problem at the true process optimum . The model adequacy criterion for MA requires that must meet the first- and secondorder necessary conditions of optimality for the adapted optimization problem Eq.(5). As the MA scheme relies on the bias- and gradient corrections to ensure satisfaction of the first order KKT conditions of the plant, in addition the second-order necessary conditions of optimality need to be satisfied at . This leads to the criterion that the matrix of the reduced Hessian of the adapted optimization problem Eq.(5) should be positive definite. The Lagrangian function of the adapted optimization problem can be written as: (7) where
is the vector of Lagrange multipliers.
4. Modifier adaptation with model adaptation A possible way to handle the issue of model adequacy is to combine the idea of MA with model adaptation. Ahmad et al. (2017) introduced the effective model adaptation (EMA) scheme to identify an adequate process model for the adapted optimization problem Eq.(5) and proposed the modifier-adaptation with EMA (in short MAWEMA) algorithm. EMA ensures and speeds up the rate of convergence to the optimum . Let be the estimated model parameter vector at , then the updated parameters are accepted by EMA if and only if the following two conditions are satisfied: x x
is positive definite .
The first condition ensures the model adequacy criterion of the Lagrangian function . It is sufficient that the Hessian matrix of the adapted objective function is positive definite which is the same as that reduced Hessian is positive definite. To motivate the second condition, we perform a Taylor series expansion of around , omitting terms of order higher than 2: (8) The prediction error of the adapted objective function at , that is, , can be used to infer the difference of the Hessian matrices at . Hence, reducing the difference between and ensures that the adapted model parameters accelerate the rate of convergence to the process optimum.
Modifier Adaptation with Model Adaptation in Iterative Real-Time Optimization
695
5. Simulation study Different RTO schemes were tested on the Williams-Otto reactor benchmark problem which has been used to evaluate the performance of RTO schemes in many papers, e.g. Marchetti et al. (2010). The plant consists of an ideal CSTR in which three irreversible reactions take place whereas the presence of only two reactions is assumed in the model. The optimization objective is to maximize the profit which is defined as: (9) where . The manipulated variables are the feed flow rate and the reactor temperature . The model equations and the description of the variables can be found in Marchetti et al. (2010). The meaning and the values of the model parameters are: and are the activation energies and and are the pre-exponential factors of the reactions. The initial model parameters and are calculated at the initial conditions of the manipulated variables, and . Figure 1 demonstrates the performance of the four RTO approaches presented above. Due to the presence of parametric and structural plant-model mismatch, the two-step approach does not converge to the true plant optimum. The standard MA approach where model parameters are not adapted moves near to the process optimum in the first iteration but then starts to oscillate since the available model is not adequate as it does not meet the second-order necessary condition of optimality. By updating the model at each iteration, the SMIO scheme performs better than the pure MA scheme. However, the SMIO scheme converges to a sub-optimal point after the eighteenth iteration, which is shown in the insert in Figure 1. This is due to the fact that parameter values do not exist which Figure 1: Comparison of the performance of the minimize the objective gradient error iterative RTO schemes between the plant and model . The best performance was obtained by the modifier adaptation with effective model adaptation (MAWEMA). After the initial iteration, EMA identifies an adequate process model for the modified optimization problem and ensures the convergence to the process optimum. Since the updated model at the first iteration also reduces the difference between the Hessians of the plant and the model, faster convergence to the process optimum is observed as compared to SMIO. To illustrate the benefit of EMA, a simulation study was performed for different values of the initial feed flow rate and the reactor temperature. Figure 2 shows the comparison of the evolutions of the optimization objective over the iterations between SMIO and MAWEMA. Figure 2b shows that all trajectories obtained from the MAWEMA approach converge smoothly to the process optimum and satisfy the first- and second-
696
A. Ahmad et al.
order optimality conditions whereas large oscillations can be observed in SMIO for some of the initial operating conditions. The SMIO scheme is not always able to reach the process optimum as the updated parameters cannot adapt the model gradients to the process gradient.
Figure 2: Optimization runs from arbitrary initial operating conditions. (a) SMIO, (b) MAWEMA
6. Conclusions This paper highlights the importance of model adequacy in model adaptation which further enhances the performance of modifier adaptation. The key feature of MAWEMA is that a modified optimization problem is formulated to meet the first order KKT conditions and the second-order optimality conditions are satisfied via EMA.
References Ahmad, A., Gao, W., Engell, S., 2017. Effective Model Adaptation in Iterative RTO. In: Computer Aided Chemical Engineering. Vol. 40. pp. 1717-1722. Chen, C. Y., Joseph, B., 1987. On-line optimization using a two-phase approach: An application study. Ind. Eng. Chem. Res. 26(9), 1924-1930. Forbes, J. F., Marlin, T. E., 1996. Design cost: A systematic approach to technology selection for model-based real-time optimization systems. Comput. Chem. Eng. 20 (6-7), 717–734. Gao, W., Engell, S., 2005. Iterative set-point optimization of batch chromatography. Comput. Chem. Eng. 29 (6), 1401–1409. Gao, W., Wenzel, S., Engell, S., 2015. Integration of gradient adaptation and quadratic approximation in real-time optimization. In: 34th Control Conference (CCC). pp. 2780–2785. Gao, W., Wenzel, S., and Engell, S. 2016. A reliable modier-adaptation strategy for real-time optimization. Comput. Chem. Eng. 91, 318–328. Mandur, J. S., Budman, H. M., 2015. Simultaneous model identification and optimization in presence of model-plant mismatch. Chem. Eng. Sci. 129, 106–115. Marchetti, A. G., Chachuat, B., Bonvin, D., 2009. Modifier-adaptation methodology for real-time optimization. Ind. Eng. Chem. Res. 48 (13), 6022–6033. Marchetti, A. G., Chachuat, B., Bonvin, D., 2010. A dual modifier-adaptation approach for realtime optimization. Journal of Process Control 20 (9), 1027–1037. Roberts, P. D., 1979. An algorithm for steady-state system optimization and parameter estimation. Int. J. Syst. Sci. 10 (7), 719–734. Tatjewski, P., 2002. Iterative optimizing set-point control - the basic principle redesigned. Proceedings of the 2002 IFAC World Congress 15 (1), 49–54.