In this paper, first the overall modeling approach for an optimized control of a hot-gas cycle for solar thermal power plants in the Modelica language is pointed out. The emphasis of the modeling work lies on the development of dynamic component models to be used in control systems. Depending on the control task, the discretization of the models has to be adapted. Main components of the hot-gas cycle are the solar thermal receiver and the storage system. The steam cycle is preliminarily only included as heat sink. Second, for control purposes a linear model-based controller (MPC) was implemented in Modelica based on an external state-of-the-art QP solver [2] linked to the Modelica model. The performance of the MPC is compared with a basic automation scheme based on classical PID controllers.

We present CasADi, an open-source software framework for numerical optimization. CasADi is a general-purpose tool that can be used to model and solve optimization problems with a large degree of flexibility, larger than what is associated with popular algebraic modeling languages such as AMPL, GAMS, JuMP or Pyomo. Of special interest are problems constrained by differential equations, i.e. optimal control problems. CasADi is written in self-contained C++, but is most conveniently used via full-featured interfaces to Python, MATLAB or Octave. Since its inception in late 2009, it has been used successfully for academic teaching as well as in applications from multiple fields, including process control, robotics and aerospace. This article gives an up-to-date and accessible introduction to the CasADi framework, which has undergone numerous design improvements over the last 7 years.

We present an extension of the CasADi numerical optimization framework that allows arbitrary order NLP sensitivities to be calculated automatically and efficiently. The approach, which can be used together with any NLP solver available in CasADi, is based on a sparse QR factorization and an implementation of a primal-dual active set method. The whole toolchain is freely available as open-source software and allows generation of thread-safe, self-contained C code with small memory footprint. We illustrate the toolchain using three examples; a sparse QP, an optimal control problem and a parameter estimation problem.

In this paper we present a systematic and efficient approach to deal with uncertainty in Nonlinear Model Predictive Control (NMPC). The main idea of the approach is to represent the NMPC setting as a real-time decision problem under uncertainty that is formulated as a multi-stage stochastic problem with recourse, based on a description of the uncertainty by a scenario tree. This formulation explicitly takes into account the fact that new information will be available in the future and thus reduces the conservativeness compared to open-loop worst-case approaches. We show that the proposed multistage NMPC formulation can deal with significant plant-model mismatch as it is usually encountered in the process industry and still satisfies tight constraints for the different values of the uncertain parameters, in contrast to standard NMPC. The use of an economic cost function leads to a superior performance compared to the standard tracking formulation. The potential of the approach is demonstrated for an industrial case study provided by BASF SE in the context of the European Project EMBOCON. The numerical solution of the resulting large optimization problems is implemented using the optimization framework CasADi.

In the last years, the use of an economic cost function for model predictive control (MPC) has been widely discussed in the literature. The main motivation for this choice is that often the real goal of control is to maximize the profit or the efficiency of a certain system, rather than tracking a predefined set-point as done in the typical MPC approaches, which can be even counter-productive. Since the economic optimal operation of a system resulting from the application of an economic model predictive control approach drives the system to the constraints, the explicit consideration of the uncertainties becomes crucial in order to avoid constraint violations. Although robust MPC has been studied during the past years, little attention has yet been devoted to this topic in the context of economic nonlinear model predictive control, especially when analyzing the performance of the different MPC approaches. In this work, we present the use of multi-stage scenario-based nonlinear model predictive control as a promising strategy to deal with uncertainties in the context of economic NMPC. We make a comparison based on simulations of the advantages of the proposed approach with an open-loop NMPC controller in which no feedback is introduced in the prediction and with an NMPC controller which optimizes over affine control policies. The approach is efficiently implemented using CasADi, which makes it possible to achieve real-time computations for an industrial batch polymerization reactor model provided by BASF SE. Finally, a novel algorithm inspired by tube-based MPC is proposed in order to achieve a trade-off between the variability of the controlled system and the economic performance under uncertainty. Simulations results show that a closed-loop approach for robust NMPC increases the performance and that enforcing low variability under uncertainty of the controlled system might result in a big performance loss. (C) 2014 Elsevier Ltd. All rights reserved.

We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m ≳ s(ln s)<;sup>3<;/sup>lnN. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m × N-matrices, by considering what is called effective sparsity. We also present a condition on the so-called restricted isometry constants, δ<;sub>s<;/sub>, ensuring sparse recovery via ℓ<;sup>1<;/sup>-minimization. We show that is sufficient and that this can be improved further to almost allow for a sufficient condition of the type .

In t his paper, nonlinear model predictive control (NMPC) is applied to the start-up of a combined-cycle power plant. An object-oriented first-principle model library expressed in the high-level language Modelica has been written for the plant and used to set up the simulation and optimization models. The NMPC optimization problems are both encoded, using a high-level notation, and solved in the open-source framework JModelica.org. The results demonstrate the effectiveness of the framework and its high-level description. It bridges the gap between an intuitive physical modeling format and state of the art numerical optimization algorithms. Promising closed-loop control results are shown for plant start-up when the NMPC model contains parametric errors and the simulation model, corresponding to the real plant, is subject to disturbances.

This paper demonstrates model-based dynamic optimization through the coupling of two open source tools: OpenModelica, which is a Modelica-based modeling and simulation platform, and CasADi, a framework for numerical optimization. The coupling uses a standardized XML format for exchange of differential-algebraic equations (DAE) models. OpenModelica supports export of models written in Modelica and the optimization language extension using this XML format, while CasADi supports import of models represented in this format. This allows users to define optimal control problems (OCP) using Modelica and optimization language specification, and solve the underlying model formulation using a range of optimization methods, including direct collocation and direct multiple shooting. The proposed solution has been tested on several industrially relevant optimal control problems, including a diesel-electric power train.

The airborne wind energy (AWE) paradigm proposes to generate energy by flying a tethered airfoil across the wind flow at a high velocity. Although AWE enables flight in higher altitude and stronger wind layers, the extra drag generated by the tether motion imposes a significant limit to the overall system efficiency. To address this issue, two airfoils with a shared tether can reduce overall system drag. Although this technique may improve the efficiency of AWE systems, such improvement can only be achieved through properly balancing the system trajectories and parameters. This brief tackles that problem using optimal control. A generic procedure for modeling multiple-airfoil systems with equations of minimal complexity is proposed. A parametric study shows that at small and medium scales, dual-airfoil systems are significantly more efficient than single-airfoil systems, but they are less advantageous at very large scales.

This paper demonstrates model-based dynamic optimization through the coupling of two open source tools: OpenModelica, which is a Modelica-based modeling and simulation platform, and CasADi, a framework for numerical optimization. The coupling uses a standardized XML format for exchange of differentialalgebraic equations (DAE) models. OpenModelica supports export of models written in Modelica and the Optimica language extension using this XML format, while CasADi supports import of models represented in this format. This allows users to define optimal control problems (OCP) using Modelica and Optimica specifications, and solve the underlying model formulation using a range of optimization methods, including direct collocation and direct multiple shooting. The proposed solution has been tested on several industrially relevant optimal control problems, including a diesel-electric power train, a free-floating robot, and a stirred-tank.

A large number of practical algorithms for Optimal Control Problems (OCP) relies on a so-called condensing procedure to exploit the given structure in the quadratic programming (QP) subproblems. While the established structure-exploiting condensing algorithm is of cubic complexity in the horizon length, in this technical note we propose a novel algorithm that is only of quadratic complexity in horizon length. We present numerical results confirming that the proposed algorithm is faster in practice and discuss implications for related QP solution algorithms that rely on the elimination of the state sequence.

The thermodynamical efficiency of a solar power tower power plant with an open volumetric air receiver depends among others on the operational strategy of the receiver. This strategy includes, on the one hand, controlling the distribution of irradiated power on the receiver surface via aim point optimization, and on the other hand, controlling the air mass flow rate and its distribution by choosing suitable dimensions of fixed orifices and controlling air flaps. The maximum mass flow rate of the receiver as an indication of the thermal power is commonly used as a quality function when assessing new component designs, comparing different operational strategies, or evaluating the role of aim point optimization for the open volumetric air receiver technology. In this paper, a method is presented to maximize the mass flow rate of the receiver using given technical capabilities of the receiver technology like orifices and air flaps for a desired air outlet temperature of the receiver. The method is based on dynamic programming, a general technique for solving decision making problems where a complex problem can be split up into a sequence of simpler ones. The potential of the method is demonstrated for a prototype solar thermal power tower with open volumetric air receiver technology.

We demonstrate how CasADi, a recently developed, free, open-source, general purpose software tool for nonlinear optimization, can be used for dynamic optimization in a flexible, interactive and numerically efficient way. CasADi is best described as a minimalistic computer algebra system (CAS) implementing automatic differentiation (AD) in eight different flavors. Similar to algebraic modeling languages like AMPL or GAMS, it includes high-level interfaces to state-of-the-art numerical codes for nonlinear programming, quadratic programming and integration of differentialalgebraic equations. CasADi is implemented in self-contained C++ code and contains full-featured front-ends to Python and Octave for rapid prototyping. In this paper, we discuss CasADi from the perspective of the developer or advanced user of algorithms for dynamic optimization for the first time, leaving out details on the implementation of the tool. We demonstrate how the tool can be used to model highly complex dynamical systems directly or import existing models formulated in the algebraic modeling language AMPL or the physical modeling language Modelica. Given this symbolic representation of the process models, the resulting optimal control problem can be solved using a variety of methods, including transcription methods (collocation), methods with embedded integrators (multiple shooting) as well as indirect methods.

We present CasADi, a free, open-source software tool for fast, yet efficient solution of nonlinear optimization problems in general and dynamic optimization problems in particular. To the developer of algorithms for numerical optimization and to the advanced user of such algorithms, it offers a level of abstraction which is notably lower, and hence more flexible, than that of algebraic modeling languages such as AMPL or GAMS, but higher than working with a conventional automatic differentiation (AD) tool.CasADi is best described as a minimalistic computer algebra system (CAS) implementing automatic differentiation in eight different flavors. Similar to algebraic modeling languages, it includes high-level interfaces to state-of-the-art numerical codes for nonlinear programming, quadratic programming and integration of differential-algebraic equations. CasADi is implemented in self-contained C++ code and contains full-featured front-ends to Python and Octave for rapid prototyping. In this paper, we present the AD framework of CasADi and benchmark the tool against AMPL for a set of nonlinear programming problems from the CUTEr test suite.

We consider an elementary identity for double singular integrals in the plane and show that one can apply this to deduce inversion and product formulae for the Hilbert transform and inversion formulae for the affine and weighted Radon transforms. We will be able to allow many of the previously known weights for which there is an inversion formula for the weighted Radon transform and also pose some new conditions on which weights that can be used.

Large interconnected systems consist of a multitude of subsystems with their own dynamics, but coupled with each other via input-output connections. Each subsystem is typically modelled by ordinary differential equations or differential-algebraic equations. Simulation and optimal control of such systems pose a challenge both with respect to CPU time and memory requirements. We address optimal control of such systems by applying “distributed multiple shooting”, a generalization of the direct multiple shooting method, which uses the decomposable structure of the system in order to obtain a highly parallel algorithm. The interconnections are allowed to be infeasible during the iterations but are driven to feasibility by a Newtontype optimization algorithm. We evaluate the performance of the distributed multiple shooting method on a large scale estimation problem.

Distributed multiple shooting is a modification of the standard multiple shooting method which takes into account the structure of certain large-scale systems in order to obtain a better controller design flexibility and high parallelizability. The aim of this paper is to extend the framework where distributed multiple shooting can be deployed and to propose a new solution method based on adjoint-based sequential quadratic programming. A numerical experiment shows that this can lead to considerable savings in computational time for the sensitivity generation.