Extension of modifier adaptation for controlled plants using static open-loop models
Introduction
Process optimization consists in determining the values of input variables that maximize a given performance criterion (such as economic profit or product quality), while meeting all the safety, environmental and operational constraints. Although generally bounded, the values of these manipulated variables are typically not fixed at the design stage. The problem can be formulated mathematically as the following nonlinear program (NLP):
where u is the nu-dimensional vector of inputs, ϕp is the cost function, and gp is the ng-dimensional vector of process constraints. Here, the subscript (·)p indicates a quantity related to the plant, and this problem will be referred to as the plant optimization problem.
If a plant model is available, numerical optimization techniques can be used to compute a local (Biegler, 2010) or even the global (Floudas, 1999) solution. The problem to be solved then reads:
where ϕ is the modeled cost function, g is the ng-dimensional vector of modeled plant constraints, and θ is an nθ-dimensional vector of model parameters. Clearly, if the model matches the plant perfectly, solving Problem (1.2) provides a solution to Problem (1.1). Unfortunately, this is rarely the case, since the structure of the model functions ϕ and g as well as the parameter values θ are likely to be incorrect. This structural and parametric mismatch implies that the model-based optimal inputs u*(θ) will probably not correspond to .
When an accurate model is not available, one typically relies on plant measurements to help the optimization process, which is the field of real-time optimization (RTO). Various RTO techniques are available in the literature to solve Problem (1.1). These techniques can be classified in two broad families depending on whether a process model is used or not.
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If no model is available or if the model is too detailed to be used for numerical optimization in real time, evolutionary techniques can be utilized. With these techniques, the plant inputs are changed repeatedly based on observing the plant response, similarly to the way an operator would do it. Although the early works in this field date back to the 1940s and 1950s (Hotelling, 1941, Box and Wilson, 1951, Brooks, 1959, Brooks and Mickey, 1961, Spendley et al., 1962, Box and Draper, 1969), the resulting methods (such as the simplex algorithm, the steepest descent, and evolutionary operation) are still quite popular, mainly due to their simplicity. They basically follow the same successive stages: (i) initialize the inputs, (ii) apply the inputs to the plant, (iii) measure or estimate the plant cost and constraints ϕp and gp, (iv) compute an educated modification of the inputs, and (v) go back to Step (ii) and repeat until convergence.
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On the other hand, model-based RTO methods apply a numerical optimization algorithm to a model of the plant to calculate a solution. The difference with offline numerical optimization is the fact that real-time measurements are incorporated in the optimization framework to compensate the effect of modeling errors and disturbances on both feasibility and performance. Hence, the model functions and are modified, and the model-based optimization Problem (1.2) is solved iteratively. Model-based RTO can be seen as a combination of evolutionary operation and numerical optimization, as the advantage of using process measurements (which are representative of the actual behavior of the plant) is combined with numerical optimization and its ability to handle large, nonlinear and constrained systems. Measurements can be incorporated in the optimization framework in three distinct ways (François and Bonvin, 2013): (i) adapt the process parameters and use the updated model for optimization (e.g. the two-step or two-stage approach, Jang et al., 1987), (ii) add correction terms to the cost and constraint functions and repeat the optimization, and (iii) directly adapt the manipulated inputs through an appropriate feedback strategy. This paper focuses on modifier adaptation (MA), a technique of Class (ii).
MA has received growing attention recently among the methods that do not adapt the model parameters, but modify the cost and constraint functions using measurements (Tatjewski, 2002, Gao and Engell, 2005, Marchetti et al., 2009). Typically, measurements are used to implement zeroth- and first-order corrections to the cost and constraint functions, while the model parameters are kept unchanged. The key feature of MA is to modify the necessary conditions of optimality (NCO) predicted by the model via input-affine corrections to the cost and constraint functions. As a result, the adequacy conditions (Marchetti, 2009) are much easier to meet than the corresponding conditions for the two-step approach (Forbes et al., 1994), especially in the case of structural plant-model mismatch. This is a very valuable property since structural mismatch is almost invariably present in complex plants (i.e., there are always simplifying assumptions made during the modeling stage). However, experimental gradients must be estimated for the plant, an onerous task that has received much attention in the literature in recent years (Marchetti et al., 2010, Bunin et al., 2013).
Although MA has been designed to deal with plant-model mismatch, the model must still satisfy the following conditions:
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Condition 1: it is adequate for real-time optimization (Forbes et al., 1994),
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Condition 2: it encompasses the same degrees of freedom as the plant that is used to generate the measurements.
The main contribution of this paper is to propose several extensions to the standard MA formulation, which can be applied when Condition 2 does not hold. Some of these extensions have already been mentioned in two conference articles (Costello et al., 2013, Costello et al., 2014), but they are detailed and analyzed hereafter.
The paper is organized as follows. After a short review of the standard MA scheme, a motivating example highlighting the implications of Condition 2 is discussed in Section 2. Section 3 presents three extensions that can deal with the case where the plant and the model have different sets of decision variables. These methods are tested in simulation on a controlled continuous stirred-tank reactor in Section 4. It is shown that all methods are capable of converging to the plant optimum despite parametric uncertainty, plant-model mismatch and the use of an open-loop model. After brief concluding remarks in Section 5, the way one can construct a convex model approximation for the closed-loop system using the open-loop model is described in the Appendix.
Section snippets
Problem formulation
Modifier adaptation (MA) collects plant information to correct for differences between the plant and model optimization problems. This is done by successively applying different values of u to the plant, each time waiting for the plant to settle to steady state and observing its performance. The measured cost and constraint values corresponding to the input uk applied at the kth iteration are:
where and are realizations of a zero-mean random
Modifier-adaptation extensions
We show next how the standard MA scheme can be altered to optimize a closed-loop plant that does not have the same degrees of freedom (the setpoints r) as the available open-loop model (the inputs u). Three algorithms will be presented. With all three, the actual degrees of freedom for RTO are the setpoints r, with the estimated experimental gradients being ∇rΦE, and ∇rGE,j, j = 1, …, ng. The algorithms will differ in (i) the choice of the decision variables (either u or r), and (ii) the way the
Simulated example: controlled Williams–Otto reactor
Methods UR, UU and RR are illustrated on the Williams–Otto reactor (Williams and Otto, 1960). We will use the model from Roberts (1979), which has become a standard test problem for real-time optimization techniques (Marchetti et al., 2010). Although the original problem is an open-loop reactor, the aim is here to optimize the reactor in closed-loop operation. The open-loop plant is an ideal continuous stirred-tank reactor with the following reactions:
Conclusions
Modifier adaptation is a very appealing RTO method, whose principal strength lies in its capacity to converge to the plant optimum despite disturbances and plant-model mismatch. Although incorporating measurements in the optimization framework, MA still relies on a plant model, and it is typically assumed that the plant and the model have the same degrees of freedom. However, this assumption may not hold in practice, in particular in the context of controlled processes. Obtaining a closed-loop
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2020, Computers and Chemical EngineeringCitation Excerpt :This can be carried out by changing the parameters of the model and observing the change in the evolution of the set points. This is achieved by modifying the activation energies of the model to match the two case studies in François et al. (2016), outlined in Table 2, whilst keeping the other parameters the same as the nominal model. As can be seen from Fig. 14, the change in the parameters of the model greatly changes the model optimum.
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2019, Computers and Chemical EngineeringCitation Excerpt :The choice of evaluating the gradients at c does not pose any conceptual difficulty since the set points are applied to the plant. However, this would be problematic if the gradients were evaluated at uk, since the computed uk is not applied to the plant and typically differs from the plant inputs achieved by the controlled plant at steady state (François et al., 2016; Papasavvas et al., 2017). In this example, using the Lagrangian formulation, the number of modifiers is reduced from 8 to 4, namely, two first-order modifiers corresponding to the Lagrangian gradient with respect to the two inputs and two zeroth-order modifiers for the two constraints.
Real-Time Optimisation of Closed-Loop Processes Using Transient Measurements
2019, Computer Aided Chemical EngineeringCitation Excerpt :The main advantage of MA lies in the mathematically proven capacity to converge to a KKT point of the process, even in the presence of plant-model mismatch, a case for which the standard two-step approach (Jang et al., 1987) typically fails (Forbes et al., 1994). Recently, an extension to MA has been proposed allowing for the optimisation of a controlled plant using an open-loop model (François et al., 2016), which is mathematically proven to reach a plant KKT point upon convergence. Such cases occur for complex plants using built-in control systems while the model has been developed in the lab.