# Poster Session on Thursday, September 12, 15:10 - 16:30

## Poster

last update of the PDF files of the posters: September 26, 9:40 am

- Andrés Botero-Halblaub: MPC as an Advanced Process Control Tool in an Industrial Environment [PDF]
- Max Demenkov: Embedded optimization using zonotopes [PDF]
- Tommy Etling: Optimal Experimental Design to Determine Parameters in Heat Transfer Problems [PDF]
- Gustavo Goretkin: Asymptotically Optimal Sample-Based Motion Planing for Linear Dynamical Systems [PDF]
- Teresa Daniela Grilo: Optimal control of passive particles advected by two-dimensional point vortices [PDF]
- Esteban A. Hernandez-Vargas: Optimality Conditions to mitigate HIV escape [PDF]
- Johannes Huber: Utilizing Nonlinear Model Predictive Control for Online Trajectory Optimization [PDF]
- Péter Koltai: Ignorant policies for centralized distributed controllers [PDF]
- Manuel Kudruss: Real-Time Experimental Design [PDF]
- Jan Kuratko: Optimization Based Search for Error Trajectories of Hybrid Dynamical Systems [PDF]
- Sofia Oliveira Lopes: Preditive Control in Irrigation Planning Problems [PDF]
- Mário Lopes: Optimal Control Methods for Infinite-Horizon Economic Growth Models
- Adeleh Mohammadi: Robust Model Predictive Control Formulation for Systems with Polytopic Uncertainty [PDF]
- Tatiana Odzijewicz: Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives [PDF]
- Rien Quirynen: Code generated integrators for fast optimal control [PDF]
- Ana Filipa Ribeiro: Optimal control and numerical approaches in a problem of management of hydroelectric resources [PDF]
- Ivan Samylovskiy: On the numerical investigation of the singular arcs in the trolley-like problem in the presence of a nonlinear friction and a bounded fuel expenditure [PDF]
- Tim Schwickart: Eco-Driving Assistance System for Electric Vehicles Using Model Predictive Control [PDF]
- Florin Stoican: Some remarks on the combinatorial properties of the explicit MPC [PDF, updated at Sep 23]
- Chrysovalantou Ziogou: The effect of a search space reduction algorithm to an NMPC problem — Online deployment to a fuel cell system

## Abstracts

### Andrés Botero-Halblaub (Linde AG, Engineering Division, Pullach, Germany): MPC as an Advanced Process Control Tool in an Industrial Environment

In Linde's field of applications, the process industry, the main part of supervisory applications can be classified as MIMO systems. To match the dynamic requirements for a more and more volative and fluctuating energy market, Linde has developed its proprietary advanced process control applications all grouped in a compendium of tools called APCS. The most usual applications for these controls software are air separation units (ASU with aprox. 30 inputs, and aprox. 10 outputs). Further applications are hydrogen plants, and liquid natural gas (LNG) plants. Among the advanced controls, binary operations for plant start-up processes, feed-forward and feedback calculation of controller set points subject to constraints. This poster will show applications for APC and LMPC in the process industry.

### Max Demenkov (Institute of Control Sciences, Russian Academy of Sciences): Embedded optimization using zonotopes

Zonotope is the image of a unit cube under an affine projection. Recently, their use in control gained some popularity. They can e.g. describe null-controllable regions of linear discrete input-constrained systems or evolution of a state set in hybrid systems. We consider linear inverse problem, in which one have to find a solution "u" of y=Au from the given "y" and A. Matrix A has more columns than rows and unknown vector of variables "u" is constrained to a box.

This formulation arises e.g. from control allocation and model predictive control approaches. Our goal is to restore "u" in real-time from the given "y". This problem can have infinite number of possible solutions. That is why some convex function of "u" needs to be optimized as well. The problem is a typical example of linear or quadratic programming application. Nevertheless, LP or QP can be prohibited due to its large algorithmic complexity or safety requirements (especially in aerospace applications).

We propose a method, which is analogous to interval global optimization but can be applied to our linear constrained problem. We represent the set of "Au" as an affine projection of a box, i.e. a zonotope, with the possibility of its real-time construction in the form of linear inequalities. Using this form, we effectively check and discard boxes that cannot contain a solution. The algorithm is guaranteed to find a solution in predefined and finite number of steps if nonzero accuracy is given. The drawback here is the zonotope complexity (i.e. number of facets) depending on the number of rows in A. In comparizon with the so called explicit or multi-parametric programming (i.e. closed-form solution of mathematical progrqamming problem), our algorithm allows easy reconfiguration in case of changes in matrix A. It can be superior to other approaches for embedded realization in low-dimensional cases.

### Tommy Etling (TU Chemnitz): Optimal Experimental Design to Determine Parameters in Heat Transfer Problems

The general problem is to determine parameters by means of experiments in which the parameters cannot be measured directly. By contrast, they are calculated from a model with state variables which can be measured. This leads to an inverse (parameter estimation) problems. It is well known that measurements always contain errors, so this leads us to the question about the transmission of the measurement error to the error of the estimated parameters. Supposing that the measurement errors are normally distributed and independent we can find (approximated) confidence regions of the estimated parameters as a criterion for the evaluation of their quality. In a sequence of experiments we wish to use control variables and weights to improve this quality of the parameter estimation in an optimal way. Subsequent experiments are designed to complement previous ones. This is the basic setting of the classical Optimal Experimental Design (OED), see, e.g., [Körkel(2002)].

In a simple heat transfer example between two solid samples we wish to determine the heat transfer coefficient and the thermal conductivities by temperature measurements from a thermographic camera. We will use a sequential OED approach in which the parameter identification and the design of the subsequent experiment alternate. The underlying model is the linear heat equation with a nonlinear parameter-to-state coupling. We show, as in [Etling(2012)], the improvement of the parameter estimation quality by OED, compared to simple repetitions of the same experiment. Finally we give some results how to incorporate state-dependent variances in the measurement error.

### Gustavo Goretkin (Massachusetts Institute of Technology): Asymptotically Optimal Sample-Based Motion Planing for Linear Dynamical Systems

We propose a new method for applying RRT* to kinodynamic motion planning problems by using finite-horizon LQR to measure cost and to extend the tree of feasible trajectories. First, we introduce the method in the context of arbitrary affine dynamical systems with quadratic costs. For these systems, the algorithm is shown to converge to optimal solutions almost surely. Second, we extend the algorithm to non-linear systems with non-quadratic costs, and demonstrate its performance experimentally.

### Teresa Daniela Grilo (FCUP/FEUP, Portugal): Optimal control of passive particles advected by two-dimensional point vortices

The objective of this work is to develop a mathematical framework for modeling, control and optimization of dynamic control sy tems whose state variable is driven by interacting ODE's and PDE's. This framework should provide a sound basis for the design and control of new advanced engineering systems arising in many important classes of applications, some of which encompass gliders and mechanical fishes.

The research effort has been focused in applying necessary conditions of optimality for some classes of flows driven dynamic control systems, in particular, using the vortex methods. The control problem of moving a particle between two given points driven by this classes of flows have been solved by using the maximum principle.

(joint work with Fernando Lobo Pereira and Sílvio M.A. Gama)

### Esteban A. Hernandez-Vargas (Systems Immunology Department, Helmholtz Centre for Infection Research, Braunschweig): Optimality Conditions to mitigate HIV escape

Highly active antiretroviral therapies (HAARTs) provide a rapid drop in plasma viral load with a large reduction of infected cells in patients with HIV infection. Even though long periods of HAART are provided, latently infected cells are still detectable. Therefore, cellular reservoirs may contribute to HIV persistence promoting the emergence of resistant mutants.

Using simplified switched linear system models of HIV mutation and treatment with certain class of symmetry and finite horizon cost functions, we demonstrate that the optimal state and co-state trajectories lie on a sliding surface where infinitely fast switching may occur. Results suggest that in the absence of other practical constraints, switching rapidly between therapies is relevant. Simulations show the potential benefits of a proactive switching strategy to minimize viral load and delay the emergence of resistant mutant viruses.

(joint work with Patrizio Colaneri and Richard H. Middleton)

### Johannes Huber (Institute of Automation and Control Engineering, UMIT, Hall i.T., Austria): Utilizing Nonlinear Model Predictive Control for Online Trajectory Optimization

The application of nonlinear model predictive control (NMPC) to systems with fast dynamics and/or complex systems with many states is challenging. The time needed to derive the solution of the optimization problem online introduces a delay between the measurement and the control action. This can lead to destabilization or cause performance drawbacks. As an alternative, we present a two degree of freedom control scheme that consists of an NMPC based online trajectory optimization unit as feed-forward component and a nonlinear trajectory tracking controller for disturbance rejection. In this approach, the online trajectory optimization and feed-forward control signal deduction is realized by applying a nonlinear model predictive control to a model of the plant and simulate the resulting closed loop (model-control-loop) in real-time. The states of the simulated model are the reference trajectories for the nonlinear trajectory tracking controller, while control signals of the NMPC in the model-control-loop can be used as feed-forward component to control the real plant. In the simulated NMPC closed loop system, the delay caused by solving the optimization problem, as mentioned above, can be compensated. Furthermore it is easier to realize as we do have full (simulated) state knowledge and the NMPC algorithm uses exactly the same model as the one it is controlling within the model-control-loop. We will present experimental and simulated examples to demonstrate the functionality of our proposed concept.

(joint work with Michael Hofbaur)

### Péter Koltai (Technische Universität München): Ignorant policies for centralized distributed controllers

We investigate an optimization-based global feedback construction for a centralized distributed control problem. As an intermediate step, we consider regularity properties of the value function encoding the accumulated costs along a trajectory of the system. Seeking for regularity for weakly coupled systems, we find that we can not expect to obtain it.

To enforce regularity (and hence computational tractability), the global optimal control problem is split into many coupled problems, each having a desired regularity. The corresponding controllers are "ignorant": they care less about the subsystems they are not affecting directly.

### Manuel Kudruss (University of Heidelberg): Real-Time Experimental Design

In optimal control problems one finds uncertainties in parameters and states. It is important to quickly and well estimate them in order to improve the performance. We present a real-time capable algorithm for optimal experimental design that is based on a moving horizon estimation scheme. A quadcopter model in two dimensions serves as example and is used to compare the method to an offline experimental design.

### Jan Kuratko (Institute of Computer Science Academy of Sciences of the Czech Republic): Optimization Based Search for Error Trajectories of Hybrid Dynamical Systems

In our research we are concerned with hybrid dynamical systems. These are systems which feature both continuous and discrete state and evolution. Such systems are used for example as models for biological processes and embedded systems. We study the problem of finding any trajectory originating in a set of initial states and reaching a set of unsafe states of a hybrid dynamical system. Such a trajectory is called an error trajectory of the system. Note that we do not assume any control input. This task is a current problem in the field of safety verification of hybrid systems.

Obtaining such an error trajectory, if it exists, means to solve an optimal control problem with free end-points. This optimal control problem, however, is of a very specific nature (partially discrete behaviour, no control inputs, free endpoints) which motivates the need for a specific method for this problem.

We base our approach on simulation and minimizing the distance of a trajectory from the initial and unsafe sets. However, local optimization may get trapped in a local minimum, failing to find an error trajectory. In order to avoid this, and to take into account global information, we use techniques for reachability analysis of hybrid systems. Those techniques allow us to exclude states through which no error trajectory passes and to start local optimization close to a global optimum/error trajectory.

(joint work with Stefan Ratschan)

### Sofia Oliveira Lopes (University of Porto): Preditive Control in Irrigation Planning Problems

Most irrigation systems on sale in the market are based on on-off control with no prediction techniques; when the system detects that the soil is dry, it triggers the irrigation cycle and suspends it when the soil is saturated. The excess of water in the soil, frequently a result of these type of techniques, is responsible for a considerable water waste.

We address the problem of optimizing the water use in the irrigation of farm fields by means of the optimal control. Our aim is to develop an irrigation system in which soil moisture does not present abrupt variations in order to achieve an effectively minor water expenditure. To correct any discrepancies and to get the best estimate of the volume of irrigation water consumed with current weather data we have used preditive control techniques. In particulary, we replan our problem using the latest precipitation data available.

(joint work with F. A. C. C. Fontes*, R. Pereira, A. Manuela Gonçalves and G. Machado)

### Mário Lopes (University of Porto): Optimal Control Methods for Infinite-Horizon Economic Growth Models

We propose a framework to solve dynamic nonlinear infinite-horizon models like those found in the standard economic growth literature. We employ a direct method to solve the underlying optimal control problem, something novel in the economic literature. Instead of deriving the necessary optimality conditions and solving the originated ordinary differential equations, this method first discretizes and then optimizes, in effect transforming the prob- lem into a nonlinear programming problem to be optimized at each sampling instant. We incorporate the work of Fontes (2001) in order to transform the infinite-horizon problem into an equivalent finite-horizon representation of the model. This framework presents several advantages in comparison to the available alternatives that use indirect methods. First, no linearization is required, which sometimes can be erroneous. The problem can be studied in its nonlinear form. Secondly, it enables the simulation of a shock when the economy is not at its steady state, a broad assumption required by all available numerical methods. Thirdly, it allows for the easy study of anticipated shocks. It also allows for the analysis of multiple, sequential shocks. Finally, it is extremely robust and easy to use. We illustrate the application of the framework by solving the standard Ramsey-Cassman-Koops exogenous growth model and the Uzawa-Lucas endogenous two-sector growth model.

(joint work with F. Fontes and D. Fontes)

### Adeleh Mohammadi (KU Leuven): Robust Model Predictive Control Formulation for Systems with Polytopic Uncertainty

In this poster, we consider a min-max model predictive Control (MPC) problem with convex cost and constraints for a linear system with polytopic uncertainty. The controller is designed to be able to control all the possible systems with parameters that can vary inside a polytope. The problem is formulated as Quadratically Constrained Quadratic Program (QCQP). Since this approach is based on a scenario tree formulation [Diehl 2007], the number of variables grows exponentially with the horizon length. The QCQP is then solved using the interior-point methods. Simulation result is then compared to a nominal MPC formulation. It is observed that the nominal MPC results in infeasibility of the optimization problem while the robust MPC controller can deal with uncertainties and the system has stable close loop response.

(joint work with Joris Gillis, Mario Zanon, and Moritz Diehl)

### Tatiana Odzijewicz (University of Aveiro, Portugal): Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives

We study dynamic minimization problems of the calculus of variations with Lagrangian functionals containing Riemann–Liouville fractional integrals, classical and Caputo fractional derivatives. Under assumptions of regularity, coercivity and convexity, we prove existence of solutions.

### Rien Quirynen (KU Leuven): Code generated integrators for fast optimal control

Algorithms for real-time, embedded optimization need to run within tight compu- tational times, and preferably on embedded control hardware for which only lim- ited computational power and memory is available. A computationally demanding step of shooting methods is often the model simulation with sensitivity generation. An implementation of code generation is presented for Implicit Runge-Kutta (IRK) methods with efficient sensitivity generation, which outperforms other solvers for the targeted applications. The tool supports index-1 Differential Algebraic Equa- tions (DAE) and continuous output functions, which are crucial for e.g. performing sensor fusion with measurements provided at very high sampling rates. In addi- tion, the importance of exploiting specific structures in a typical nonlinear model is shown.

(joint work with Moritz Diehl)

### Ana Filipa Ribeiro (FEUP - DEEC, Portugal): Optimal control and numerical approaches in a problem of management of hydroelectric resources

This work focuses on the study of a system of hydroelectric power stations for which the energy production must be optimized. In the systems considered it is possible to reverse the turbines and pump water up from a downstream reservoir to an upstream one. A simplified model for these systems is analysed in the context of optimal control theory with the fluxes of water to turbine or pump on each power station as control variables and the maximization of the profit of energy sale being the objective function. The presence of state constraints and the nonconvexity of the cost function contribute to an increase of complexity of the problem.

We obtain a global solution to this problem using a free software of Jieqiu Chen and Samuel Burer (for details, see article “Globally solving nonconvex quadratic programming problems via completely positive programming”). Two different approaches are adopted. In the first, after constructing a discretization of the problem, it is applied the Chen-Burer software (CB). In a second approach a numerical method of constructing projections of convex polyhedral sets (Bushenkov’s software) is used to reduce the dimension of the problem. After that the CB is applied and we restore all the variables with help of Simplex method. Results and execution time of the two procedures are compared.

Some theoretical analysis of the problem involving the Maximum Principle of Pontryagin is also undertaken.

### Ivan Samylovskiy (Lomonosov Moscow State University): On the numerical investigation of the singular arcs in the trolley-like problem in the presence of a nonlinear friction and a bounded fuel expenditure

Originally we deal with different simplifications of a well-kmown Goddard problem on a maximization of a result altitude of rocket flight. We have considered a number of possible simplifications of the original problem to find the simplest variant in which, however, the optimal trajectory is obtained in a non-trivial form and still may contains a singular arc. For such a problem we obtain a numerical algorithm to find the following:

- "treshold value" of a final time such that for all greater final times the optimal trajectory contains singular arc while for all smaller final times the optimal control is of bang-bang form.
- the start and the end times of a singular subarc in the case of bang-singular optimal control

For the computations and visualization we use C# / F# class libary for the numerical solution of ODE systems developed in collaboration with Microsoft Research, Cambridge specialists.

### Tim Schwickart (University of Luxembourg): Eco-Driving Assistance System for Electric Vehicles Using Model Predictive Control

The poster shows our planned realization of an energy efficient model-predictive cruise controller for electric cars. The basic principle of the controller is shown as well as some unconventional approaches how we want to bring the optimal control problem into a real time capable form. The energy consumption of the vehicle is modeled by several linear constraints on the problem that are always followed by a control variable representing the consumption. A control input taking discrete values is realized by an additional penalty term to the cost function. Finally, the planned experimental control setup is presented.

### Florin Stoican (Dept. of Automatic Control and Systems Engineering, Faculty of Automatic Control and Computers, "Politehnica" University of Bucharest): Some remarks on the combinatorial properties of the explicit MPC

Explicit MPC remains an interesting topic but is marred by numerical difficulties and an exponential increase in complexity in terms of constraints and space dimensions. These issues are due to the combinatorial nature of the problem, i.e., the critical regions and afferent affine laws are obtained by enumerating combinations of active constraints (which verify, e.g., the LICQ condition). While these issues have been extensively studied in the last decade, improvements can still be made by exploiting the particular structure of the problem. To this end, we consider a fairly typical (i.e., LTI dynamics, quadratic cost function and linear constraints) MPC formulation. In such a formulation, causality constraints ("at time instant 'k' there cannot be any influence from future control actions") imply a block-triangular structure for the input-associated matrix. Using this property, several directions of interest can be followed:

- Using results from combinatorial theory (i.e., Ehrhart polynomials) tighter bounds for the number of combinations of constraints which can be LICQ are obtained. These results offer a guarantee for memory allocation and assure selection of LICQ combinations (useful, e.g., in bi-level problems).
- A partial recurrence relation between explicit MPC laws over different prediction horizons can be deduced (and hence serve to minimize the computational effort by using previously known information).
- Critical regions with a common partial sequence of inputs can be merged (in fact only the first value of the sequence is of interest). Existing resultsprovide such a merging procedure but avoiding to compute the critical regions in the first place would be a significant improvement.
- Geometrical interpretations of the explicit MPC exist, but they can be improved by analyzing the facet lattice of the constraint polytope resulting from the MPC constraints (the overall constraint polytope is the result of successive intersections).

### Chrysovalantou Ziogou (Chemical Process and Energy Resources Institute, Center for Research and Technology Hellas, CPERI/CERTH, Greece): The effect of a search space reduction algorithm to an NMPC problem — Online deployment to a fuel cell system

The scope of this work is to present the implementation of a nonlinear model predictive control (NMPC) strategy for the control of a Polymer Electrolyte Membrane (PEM) fuel cell system. Overall during the operation of fuel cell systems various phenomena appear and their behavior is characterized by fast dynamics and nonlinearities. Based on such behavior a number of challenges appear that should be managed appropriately. Thus, control is vital for improving the response and the performance of the fuel cell system, and also for ensuring its durability and, most importantly, its safe operation.

In this work the solution of a multivariable NMPC problem is assisted by an algorithm that modifies the search space of the decision variables based on a piecewise affine function. This combination aims at the reduction of the computational requirements that arise by the nonlinear optimal problem which is solved online. The nonlinear dynamic model of the fuel cell is discretized based on orthogonal collocation on finite elements method. Furthermore, a solver that exploits the structure of the discetized system is selected which can handle the large, but sparse, nonlinear programming problem. The proposed framework is deployed to the automation system of the fuel cell unit and a set of indicative results are presented that demonstrate the effectiveness of the controller and reveals its capabilities with respect to the fulfilment of multiple desired objectives under constraints.

(joint work with S. Voutetakis, S. Papadopoulou)