Engineering Applications of Artificial Intelligence
Online set-point optimisation cooperating with predictive control of a yeast fermentation process: A neural network approach
Introduction
Model predictive control (MPC) refers to a computer control strategy in which an explicit dynamic model of the process is used to predict its future behaviour over some time horizon (Camacho and Bordons, 1999, Maciejowski, 2002, Rossiter, 2003, Tatjewski, 2007). Various MPC algorithms have been successfully used for years in numerous advanced industrial applications (Qin and Badgwell, 2003), in particular in chemical engineering. It is because MPC algorithms have a unique ability to take into account constraints imposed on both process inputs (manipulated variables) and outputs (controlled variables). Constraints are very important in practice, they usually determine quality, economic efficiency and safety. Moreover, MPC techniques are very efficient when applied to process with many inputs and many outputs.
To maximise economic profits MPC algorithms cooperate with set-point optimisation, the purpose of which is to calculate online optimal set-points for MPC (Blevins et al., 2005, Brdys and Tatjewski, 2005, Engell, 2007, Tatjewski, 2007, Tatjewski, 2008). Usually, the multilayer (hierarchical) structure is used in which the control layer keeps process at given operating points and the optimisation layer calculates these set-points (Findeisen et al., 1980). It is also possible to integrate set-point optimisation and MPC optimisation into one optimisation problem (Tvrzska de Gouvea and Odloak, 1998, Tatjewski, 2007, Zanin et al., 2000, Zanin et al., 2002). Alternatively, an integrated predictive optimiser and constraint supervisor can be used (Tatjewski et al., 2009). It provides the control layer with set-points calculated for both optimality and constraint handling.
When the classical multilayer control structure is used, it is usually assumed that disturbances are slowly varying when compared to the dynamics of the process (Tatjewski, 2007). In such a case, the steady-state nonlinear set-point optimisation problem can be solved reasonably less frequently than the MPC controller executes. Provided that the dynamics of disturbances is much slower than the dynamics of the plant, such an approach gives good or satisfactory results. In many practical cases, however, dynamics of disturbances is comparable with the process dynamics. Very often disturbances, for example flow rates, properties of feed and energy streams, etc., vary significantly and not much slower than the dynamics of the controlled process. In such cases operation in the classical structure with frequency of set-point optimisation much lower than that of MPC may result in significant loss of economic effectiveness (Tatjewski, 2007, Tatjewski, 2008). Ideally, nonlinear set-point optimisation should be repeated online as often as MPC is activated. Because of high computational complexity, it is usually not possible. Moreover, nonlinear optimisation may terminate in local minima. Hence, nonlinear set-point optimisation is rarely used online.
In order to reduce the computational complexity, in the simplest case for set-point optimisation a constant linear steady-state model derived from the dynamic model used in MPC can be used (Kassmann et al., 2000, Qin and Badgwell, 2003, Tatjewski, 2007, Tatjewski, 2008). As a result, one obtains the steady-state target optimisation (SSTO) structure in which set-points are calculated from an easy to solve linear programming problem. It can be solved online as frequently as MPC is activated. Naturally, for nonlinear processes such an approach may give economically wrong operating points. There are two solutions to this problem. It is possible to estimate and take into account the uncertainty in steady-state gain in the framework of a robust steady-state target calculation (Kassmann et al., 2000). Alternatively, in the adaptive steady-state target optimisation (ASSTO) structure the comprehensive nonlinear steady-state model is linearised online and set-point optimisation becomes a linear programming task (ławryńczuk et al., 2008a, Qin and Badgwell, 2003, Tatjewski, 2007, Tatjewski, 2008). A yet another approach is to use piecewise-linear steady-state target optimisation (ławryńczuk et al., 2008b, Tatjewski, 2008, Tatjewski et al., 2006). The set-point optimisation layer can be also replaced by a neural network which approximates the solution to the set-point optimisation problem (ławryńczuk and Tatjewski, 2010).
This paper details implementation of a computationally efficient neural ASSTO structure which cooperates with a suboptimal MPC algorithm. Two neural models of the process are used online. For set-point optimisation, a steady-state neural model is linearised online and the set-point is calculated from a linear programming problem. For MPC, a dynamic neural model is linearised online and the control policy is calculated from a quadratic programming problem. In consequence of linearisation of neural models, the necessity of online nonlinear optimisation is eliminated. The proposed structure is applied to a simulated yeast fermentation process. Fermentation is one of the most important biochemical processes. Because properties of the process are nonlinear, the classical PID controller and the MPC algorithm based on a linear model are unable to control the process efficiently as demonstrated in ławryńczuk (2008) and Nagy (2007). Results obtained in the ASSTO structure are comparable with those achieved in a computationally demanding structure with nonlinear optimisation used for set-point optimisation and MPC.
Neural networks (Haykin, 1999, Ripley, 1996), due to their advantages, can be successfully used for modelling and control of nonlinear processes, e.g. Hussain (1999) and Nørgaard et al. (2000). Neural networks are universal approximators (Hornik et al., 1989), hence neural models are usually very precise. Moreover, unlike fundamental models (Luyben, 1990, Marlin, 1995) (which are composed of algebraic and differential equations), neural models have a simple structure and relatively a limited number of parameters. Neural models directly describe input–output relations of process variables, complicated systems of equations do not have to be solved online in set-point optimisation and MPC. The literature concerned with MPC based on various types of neural models is very rich, e.g. Alexandridis and Sarimveis (2005), Hussain (1999), ławryńczuk (2009), da Cruz Meleiro et al. (2009), Nørgaard et al. (2000), Peng et al. (2007), Tatjewski (2007), Yu and Gomm (2003) and references therein.
This paper is organised as follows. Section 2 shortly reminds the classical multilayer control structure, both set-point and MPC optimisation tasks are defined. Next, in Section 3, the multilayer structure with the ASSTO layer which cooperates with the suboptimal MPC algorithm is presented, linearisation of dynamic and steady-state neural models is discussed. In Section 4 development of neural models of the yeast fermentation biochemical reactor is thoroughly described. The proposed structure is compared in terms of accuracy and computational burden with the structure in which nonlinear optimisation is used online. Finally, Section 5 concludes the paper.
Section snippets
The classical multilayer control structure
The standard multilayer system structure is depicted in Fig. 1 (Findeisen et al., 1980, Tatjewski, 2007, Tatjewski, 2008). The basic control layer is responsible for safe operation of the process. It has direct access to input (manipulated) variables of the process. The supervisory control layer (also named the advanced control layer) calculates online set-points for the basic control layer. The local steady-state optimisation (LSSO) layer calculates online economically optimal set-points for
The structure
The multilayer neural adaptive steady-state target optimisation (ASSTO) structure (Tatjewski, 2007, Tatjewski, 2008) which cooperates with MPC is depicted in Fig. 2. It consists of the nonlinear set-point optimisation (LSSO) layer, the adaptive steady-state target optimisation layer and the MPC layer. Two neural models are used online. A steady-state neural model is used in LSSO and in ASSTO layers, whereas a dynamic model is used in MPC. Nonlinear LSSO set-point optimisation, in which the
Yeast fermentation reactor
The considered yeast fermentation reactor is shown in Fig. 6. The reactor is modelled as a continuous stirred tank with the constant substrate feed flow, the outlet flow from the reactor containing the product, the substrate and the biomass is also constant. The reactor contains three distinct components: the biomass, which is a suspension of yeast fed into the system and evacuated continuously, the substrate, which is the solution of glucose feeding the micro-organism (Saccharomyces cerevisiae
Conclusions
The described structure with adaptive steady-state target optimisation (ASSTO) and the MPC-NPL algorithm is computationally efficient and gives trajectories very similar to those obtained when for set-point optimisation and MPC nonlinear optimisation is used online. For set-point optimisation and MPC steady-state and dynamic neural models are linearised online. In consequence, set-point optimisation is an easy to solve linear programming task, MPC optimisation is a quadratic programming
Acknowledgement
The work presented in this paper was supported by Polish national budget funds for science.
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