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Multi-objective Optimal Control of Chemical Processes using ACADO Toolkit

Abstract Many practical chemical engineering problems involve the determination of optimal trajectories given multiple and conflicting objectives. These conflicting objectives typically give rise to a set of Pareto optimal solutions. To enhance real-time decision making efficient approaches are required for determining the Pareto set in a fast and accurate way. Hereto, the current paper illustrates the use of the freely available toolkit ACADO Multi-Objective ( www.acadotoolkit.org ) on several chemical examples. The rationale behind ACADO Multi-Objective is the integration of direct optimal control methods with scalarisation-based multi-objective methods enabling the exploitation of fast deterministic gradient-based optimisation routines.

Towards Online Model Predictive Control on a Programmable Logic Controller: Practical Considerations

Given the growing computational power of embedded controllers, the use of model predictive control (MPC) strategies on this type of devices becomes more and more attractive. This paper investigates the use of online MPC, in which at each step, an optimization problem is solved, on both a programmable automation controller (PAC) and a programmable logic controller (PLC). Three different optimization routines to solve the quadratic program were investigated with respect to their applicability on these devices. To this end, an air heating setup was built and selected as a small-scale multi-input single-output system. It turns out that the code generator (CVXGEN) is not suited for the PLC as the required programming language is not available and the programming concept with preallocated memory consumes too much memory. The Hildreth and qpOASES algorithms successfully controlled the setup running on the PLC hardware. Both algorithms perform similarly, although it takes more time to calculate a solution for qpOASES. However, if the problem size increases, it is expected that the high number of required iterations when the constraints are hit will cause the Hildreth algorithm to exceed the necessary time to present a solution. For this small heating problem under test, the Hildreth algorithm is selected as most useful on a PLC.

Tuning of NMPC controllers via multi-objective optimisation

Abstract Nonlinear Model Predictive Control (NMPC) is a powerful technique that can be used to control many industrial processes. Different and often conflicting control objectives, e.g., reference tracking, disturbance rejection and minimum control effort, are typically present. Most often these objectives are translated into a single weighted sum (WS) objective function. This approach is widespread because it is easy to use and understand. However, selecting an appropriate set of weights for the objective function is often non-trivial and is mainly done by trial and error. The current study proposes a systematic procedure for tuning Nonlinear MPC based on multi-objective optimisation methods. Advanced methods allow an efficient solution of the multi-objective problem providing a systematic overview of the controller behaviour. Moreover, through analytic relations it is possible to link a solution obtained with these novel methods to a set of weights for a weighted sum objective function. Applying this set of weights causes the WS to generate the same solution as obtained with the advanced method. Hence, an appropriate controller can be selected based on the alternatives generated by the advanced method, while the corresponding weights for a WS can be derived for implementing the controller in practice. The procedure is successfully tested on two benchmark applications: the Van de Vusse reactor and the Tennessee Eastman plant.

Tuning of predictive controllers for drinking water networked systems

Abstract In this paper, two tuning strategies for a multi-objective predictive controller applied to a drinking water network (DWN) are proposed. A control-oriented DWN model is briefly reviewed, together with its management objectives. A comparison of methods to explore the Pareto front of the multi-objective optimisation (MOO) problem behind the predictive controller is presented with an effective normalisation method for the model predictive control (MPC) objectives. The proposed tuning strategies, applied to a real-life case study, are compared. Finally, simulation results show that the proposed MPC tuning strategies outperform the baseline results.

An interactive decision-support system for multi-objective optimization of nonlinear dynamic processes with uncertainty

We develop an interactive decision support system for dynamic process optimization.The system considers multiple and conflicting objectives to increase sustainability.Operational risk due to parametric uncertainty is included as additional objective.Optimal trade-off solutions and operating policies are interactively visualized.The system is illustrated for the optimal operation of a chemical reactor. The manufacturing industry is faced with the challenge to constantly improve its processes, e.g., due to lower profit margins, more strict environmental policies and increased societal awareness. These three aspects are considered as the pillars of sustainable development and typically give rise to multiple and conflicting objectives. Hence, any decision made will require trade-offs to be evaluated and compromises to be made. To support decision making an interactive multi-objective framework is presented to optimize dynamic processes based on mathematical models. The framework includes a numerically efficient strategy to account for parametric uncertainty in the models and it allows to directly minimize the operational risks arising from this uncertainty. Hence, for the first time expert knowledge on the trade-offs between traditional objective functions and operational risks is readily and interactively available for the practitioners in the field of dynamic systems. The introduced interactive framework for multi-objective dynamic optimization under uncertainty is successfully tested for a three and five-objective fed-batch reactor case study with uncertain feed temperature and heat transfer parameters.

Robustifying optimal experiment design for nonlinear, dynamic (bio)chemical systems

Abstract Dynamic experiments that yield as much information as possible are highly valuable for estimating parameters in nonlinear dynamic processes. Techniques for model-based optimal experiment design enable to systematically design such experiments. However, these experiments depend on the current best estimate of the parameters, which are not necessarily the true values. Consequently, in real experiments (i) the information content can be lower than predicted and (ii) state constraints can be violated. This paper presents a novel, computationally tractable formulation that enables the robustification of optimally designed experiments with respect to (i) information content and (ii) constraint satisfaction. To this end, the objective function is the expected value of a scalar function of the Fisher information matrix, which is efficiently computed using the sigma point method. This approach already has a robustifying effect. The sigma point method also enables the efficient computation of constraints’ variance–covariance matrix, this can be exploited for further robustification.

Multi-objective optimal control of dynamic bioprocesses using ACADO Toolkit

The optimal design and operation of dynamic bioprocesses gives in practice often rise to optimisation problems with multiple and conflicting objectives. As a result typically not a single optimal solution but a set of Pareto optimal solutions exist. From this set of Pareto optimal solutions, one has to be chosen by the decision maker. Hence, efficient approaches are required for a fast and accurate generation of the Pareto set such that the decision maker can easily and systematically evaluate optimal alternatives. In the current paper the multi-objective optimisation of several dynamic bioprocess examples is performed using the freely available ACADO Multi-Objective Toolkit (http://www.acadotoolkit.org). This toolkit integrates efficient multiple objective scalarisation strategies (e.g., Normal Boundary Intersection and (Enhanced) Normalised Normal Constraint) with fast deterministic approaches for dynamic optimisation (e.g., single and multiple shooting). It has been found that the toolkit is able to efficiently and accurately produce the Pareto sets for all bioprocess examples. The resulting Pareto sets are added as supplementary material to this paper.

Dynamic optimization of biological networks under parametric uncertainty

BackgroundMicro-organisms play an important role in various industrial sectors (including biochemical, food and pharmaceutical industries). A profound insight in the biochemical reactions inside micro-organisms enables an improved biochemical process control. Biological networks are an important tool in systems biology for incorporating microscopic level knowledge. Biochemical processes are typically dynamic and the cells have often more than one objective which are typically conflicting, e.g., minimizing the energy consumption while maximizing the production of a specific metabolite. Therefore multi-objective optimization is needed to compute trade-offs between those conflicting objectives. In model-based optimization, one of the inherent problems is the presence of uncertainty. In biological processes, this uncertainty can be present due to, e.g., inherent biological variability. Not taking this uncertainty into account, possibly leads to the violation of constraints and erroneous estimates of the actual objective function(s). To account for the variance in model predictions and compute a prediction interval, this uncertainty should be taken into account during process optimization. This leads to a challenging optimization problem under uncertainty, which requires a robustified solution.ResultsThree techniques for uncertainty propagation: linearization, sigma points and polynomial chaos expansion, are compared for the dynamic optimization of biological networks under parametric uncertainty. These approaches are compared in two case studies: (i) a three-step linear pathway model in which the accumulation of intermediate metabolites has to be minimized and (ii) a glycolysis inspired network model in which a multi-objective optimization problem is considered, being the minimization of the enzymatic cost and the minimization of the end time before reaching a minimum extracellular metabolite concentration. A Monte Carlo simulation procedure has been applied for the assessment of the constraint violations. For the multi-objective case study one Pareto point has been considered for the assessment of the constraint violations. However, this analysis can be performed for any Pareto point.ConclusionsThe different uncertainty propagation strategies each offer a robustified solution under parametric uncertainty. When making the trade-off between computation time and the robustness of the obtained profiles, the sigma points and polynomial chaos expansion strategies score better in reducing the percentage of constraint violations. This has been investigated for a normal and a uniform parametric uncertainty distribution. The polynomial chaos expansion approach allows to directly take prior knowledge of the parametric uncertainty distribution into account.

Symmetric algorithmic differentiation based exact Hessian SQP method and software for Economic MPC

Economic Model Predictive Control (EMPC) is an advanced receding horizon based control technique which optimizes an economic objective subject to potentially nonlinear dynamic equations as well as control and state constraints. The main contribution of this paper is an algorithmic differentiation (AD) based real-time EMPC algorithm including a software implementation in ACADO Code Generation. The scheme is based on a novel memory efficient, symmetric AD approach for real-time propagation of second order derivatives. This is used inside a tailored multiple-shooting based SQP method, which employs a mirrored version of the exact Hessian. The performance of the proposed auto-generated EMPC algorithm is demonstrated for the optimal control of a nonlinear biochemical reactor benchmark case-study. A speedup of a factor more than 2 can be shown in the CPU time for integration and Hessian computation of this example.

An economic objective for the optimal experiment design of nonlinear dynamic processes

State-of-the-art formulations of optimal experiment design problems are typically based on a design criterion which allows us to optimize a scalar map of the predicted variance-covariance matrix of the parameter estimate. Famous examples for such scalar objectives are the A-criterion, the E-criterion, or the D-criterion, which aim at minimizing the trace, maximum eigenvalue, or determinant of the variance-covariance matrix. In this paper, we propose a different way of deriving an economic design criterion for the optimal experiment design. Here, the corresponding analysis is based on the assumption that our ultimate goal is to solve an optimization problem with a given economic objective that depends on uncertain parameters, which have to be estimated by the experiment. We illustrate the approach by studying a fedbatch bioreactor.