Optimal Control Tools Research Articles

Improving operation in an industrial MDF flash dryer through physics-based NMPC

Nonlinear model predictive control (NMPC) has increased popularity thanks to the availability of black-box models, systematically obtained from plant data via machine-learning procedures. Although this may be a good approach in the medium term, pure data-driven models quickly become outdated with changes in the process operation, so re-identification (training) routines are periodically required. Moreover, if the NMPC includes any economic objective to drive the process to a more efficient operation, these models often go beyond their validity range, which translates in a degraded control performance (high plant-model mismatch). In this work, the above issues are considered to model the drying section of an industrial medium density fibreboard (MDF) plant. The main contribution of this work is proposing a lumped-parameter grey-box model built upon first principles and completed with experimental equations, obtained from constrained regression with plant data. Upon this model, a moving-horizon estimator (MHE) is designed to estimate unmeasured inputs and states. Then, a mixed economic-tracking NMPC, which explicitly considers actuator-stiction compensation, is proposed to drive the dryer to the humidity setpoint by the most efficient path. Both optimisations are solved within the system sampling interval thanks to an efficient discretisation of the system dynamics by orthogonal collocation, and to the high computational performance achieved by modern software tools for numerical optimal control. The controller was firstly tested in simulation with a rigorous model of a generic MDF dryer, and it is currently implemented in the actual plant achieving significant performance improvements.

Managing a hybrid energy smart grid with a renewable energy source

AbstractThe aim of this work is to develop tools for optimal power flow management control in a micro grid (MG). The latter consists of a wind turbine, energy storage system, two gas turbines (GTs), and the main grid. Unlike the traditional approach, which is limited to the distribution of active power, this paper models an electrical system to coordinate and optimize the flow of both active and reactive power using discrete controls. The proposed optimal power distribution strategy has two objectives. First, it aims at forecasting over a time horizon of 24 hours the optimal distribution of the active and reactive power required for each power source connected to the MG. The proposed management incorporates the forecasts of consumption, weather, and tariffs. Second, it aims at reducing the CO2 emissions rate by optimizing both the operating point of the two GTs and the usage of the storage unit. The proposed optimization problem for the energy management system is solved using the Bellman algorithm through dynamic programming. Its effectiveness is illustrated with various simulations carried out in the Matlab environment. The supremacy of the proposed management algorithm is highlighted by comparing its performance with conventional (restricted) management.

Energy Modeling with Nonlinear-Autoregressive Exogenous Neural Network

The model-based predictive control (MPC) is considered to be an effective tool for optimal control of building heating, ventilation, and air-conditioning (HVAC) systems. MPC need to update the operating set points of the local control loops that have a significant influence on the energy performance of the system. Performance of MPC relies on the accuracy of the system performance model. There are two commonly used modeling approach – conventional or analytical approach that is the way of process modeling for some time, but it tends to increase the online computational load as it requires a full mathematical description of the real system. Furthermore, such techniques rely on different simplifying assumptions that limit the accuracy of the performance model. A second commonly used technique is the data-driven approach. The neural network (NN) is the most potent data-driven approach. NN can accurately model complex nonlinear systems without even knowing the structure of the system and it also addresses the problem of the online computational load since the computational load moves to the offline training step. In order to set up neural network model-based predictive control (NNMPC), it is important to build a reliable energy model of HVAC system that can be used to perform multi-step-ahead prediction of system energy performance. In this paper, the energy modeling of the chiller plant is conducted. Data for the training of chiller plant energy model is generated from HVAC testbed build in TRNSYS simulation environment. The nonlinear-autoregressive neural network with exogenous input (NARX) is used to model the energy performance of the chiller plant. The NARX is a powerful method for forecasting of time series data and dynamic control problems. NARX model is first trained in the open-loop form with the actual output instead of feedback, using back-propagation with the Levenberg-Marquardt method; this model can be used to perform only one-step-ahead prediction. Open-loop NARX model is transformed into a closed-loop form, by connecting the internal feedback, i.e. actual output is replaced by predicted output, to perform multi-step-ahead prediction (for predictive control). Comparative analysis of developed NARX-based chiller model is carried out with respect to process data from testbed, which demonstrated the good accuracy of the NARX-based chiller model.

Optimization of Microalgae Selection: Highlighting turnpike features

We consider a simplified dynamical system to represent the competition between two species of microalgae in a chemostat. The model is derived from the classical Droop model, assuming the substrate level can be directly controlled. An optimal control problem (OCP) is formulated, with the objective of finding the substrate concentration which would maximize the fraction of the species of interest over a fixed finite time-window. We characterize the substrate-based control strategy that steers the model trajectories and achieves species separation. Our objective is to highlight through a numerical optimal-synthesis - based on direct optimal control tools - the existence of a turnpike feature that appears in the optimal control law as well as in the optimal model trajectories and their co-states.

An Optimal Control Strategy Separating Two Species of Microalgae in Photobioreactors

We investigate a minimal-time control problem in a chemostat continuous photo-bioreactor model that describes the dynamics of two distinct microalgae populations. More precisely, our objective is to optimize the time of selection - or separation - between two species of microalgae. We focus in this work on Droop’s model which takes into account an internal quota storage for each microalgae species. Using Pontryagin’s principle, we develop a dilution-based control strategy that steers the model trajectories to a suitable target in minimal time. Our study reveals that singular arcs play a key role in the optimization problem. A numerical optimal-synthesis, based on direct optimal control tools, is performed throughout the paper, thereby confirming the optimality of the provided feedback-control law, which is of type bang-singular.

Penerapan Metode Economic Order Quantity Pada Rancangan Aplikasi Inventory Control System

Tujuan dari penelitian ini adalah penerapan metode Economic Order Quantity (EOQ) pada perancangan aplikasi inventory control system untuk mengendalikan persediaan barang pada studi kasus di Bagian Gudang Sudin Damkar wilayah Jakarta Selatan. Penelitian ini dilakukan adanya berbagai permasalahan pengendalian persediaan, seperti: pengendalian persediaan yang tidak ekonomis, stok yang tidak aman, serta pemesanan kembali yang sulit. Kemudian, untuk menyelesaikan berbagai masalah dalam pengendalian persediaan ini, perlu menerapkan metode EOQ dalam aplikasi sistem kontrol persediaan. Perancangan aplikasi sistem dalam penelitian ini dengan menggunakan pendekatan Unified Modeling Language (UML).Hasil penelitian produk terapan ini adalah suatu aplikasi sistem yang mempermudah para pengguna sistem untuk dapat mengendalikan persediaan yang dimiliki dengan cepat, mudah dan akurat dan menghasilkan suatu laporan secara grafis yang sangat membantu dalam pengambilan keputusan. Aplikasi sistem ini diharapkan akan menjadi alat yang sangat berguna untuk pengendalian persediaan secara optimal.

On an optimal control strategy in a kinetic social dynamics model

Abstract Kinetic models have so far been used to model wealth distribution in a society. In particular, within the framework of the kinetic theory for active particles, some important models have been developed and proposed. They involve nonlinear interactions among individuals that are modeled according to game theoretical tools by introducing parameters governing the temporal dynamics of the system. In this present paper we propose an approach based on optimal control tools that aims to optimize this evolving dynamics from a social point of view. Namely, we look for time dependent control variables concerning the distribution of wealth that can be managed, for instance, by the government, in order to obtain a given social profile.

Hybrid Electric Vehicles: Energy Management Strategies [Bookshelf

Hybrid electric vehicles have a long history, dating as far back as the development of the Lohner-Porsche Mixte in 1899. A hybrid vehicle, by definition, is one with multiple propulsion power sources, one example being the combination of an internal combustion engine plus an electric drive. The tools of dynamic programming and optimal control lend themselves quite elegantly to the hybrid vehicle energy management problem. This has motivated a very substantial body of peer-reviewed conference and journal papers on the application of optimal control theory to hybrid vehicle energy management. This textbook addresses a critical gap in that it introduces students, scholars, and practitioners to the main ideas in the literature and the interconnections between these ideas.

Geometric optimal control and applications to aerospace

This article deals with applications of optimal control to aerospace problems with a focus on modern geometric optimal control tools and numerical continuation techniques. Geometric optimal control is a theory combining optimal control with various concepts of differential geometry. The ultimate objective is to derive optimal synthesis results for general classes of control systems. Continuation or homotopy methods consist in solving a series of parameterized problems, starting from a simple one to end up by continuous deformation with the initial problem. They help overcoming the difficult initialization issues of the shooting method. The combination of geometric control and homotopy methods improves the traditional techniques of optimal control theory.A nonacademic example of optimal attitude-trajectory control of (classical and airborne) launch vehicles, treated in details, illustrates how geometric optimal control can be used to analyze finely the structure of the extremals. This theoretical analysis helps building an efficient numerical solution procedure combining shooting methods and numerical continuation. Chattering is also analyzed and it is shown how to deal with this issue in practice.

Preface.

This volume was inspired by the topics presented at the international conference "Micro and Macro Systems in Life Sciences" which was held on Jun 8-12, 2015 in Będlewo, Poland. System biology is an approach which tries to understand how micro systems, at the molecular and cellular levels, affect macro systems such as organs, tissue and populations. Thus it is not surprising that a major theme of this volume evolves around cancer and its treatment. Articles on this topic include models for tumor induced angiogenesis, without and with delays, metastatic niche of the bone marrow, drug resistance and metronomic chemotherapy, and virotherapy of glioma. Methods range from dynamical systems to optimal control. Another well represented topic of this volume is mathematical modeling in epidemiology. Mathematical approaches to modeling and control of more specific diseases like malaria, Ebola or human papillomavirus are discussed as well as a more general approaches to the SEIR, and even more general class of models in epidemiology, by using the tools of optimal control and optimization. The volume also brings up challenges in mathematical modeling of other diseases such as tuberculosis. Partial differential equations combined with numerical approaches are becoming important tools in modeling not only tumor growth and treatment, but also other diseases, such as fibrosis of the liver, and atherosclerosis and its associated blood flow dynamics, and our volume presents a state of the art approach on these topics. Understanding mathematics behind the cell motion, appearance of the special patterns in various cell populations, and age structured mutations are among topics addressed inour volume. A spatio-temporal models of synthetic genetic oscillators brings the analysis to the gene level which is the focus of much of current biological research. Mathematics can help biologists to explain the collective behavior of bacterial, a topic that is also presented here. Finally some more across the discipline topics are being addresses, which can appear as a challenge in studying problems in systems biology on all, macro, meso and micro levels. They include numerical approaches to stochastic wave equation arising in modeling Brownian motion, discrete velocity models, many particle approximations as well as very important aspect on the connection between discrete measurement and the construction of the models for various phenomena, particularly the one involving delays. With the variety of biological topics and their mathematical approaches we very much hope that the reader of the Mathematical Biosciences and Engineering will find this volume interesting and inspirational for their own research.

Optimal stationary control of diesel engines using periodic control

Measurements and optimal control are used to study whether the fuel economy of a diesel engine can be improved through periodic control of the wastegate, illustrating how modern optimal control tools can be used to identify non-trivial solutions that can improve performance. The measurements show that the pumping torque of the engine is changed when the wastegate is controlled in a periodic manner versus stationary even if the mean position is the same. If this decreases the fuel consumption or not is seen to be frequency and operating point dependent. The measurements indicate that the phenomenon occurs in the time scales capturable by mean value engine models (MVEM). The operating points are further analyzed using a MVEM and optimal control. It is shown that whether the optimal solution exhibits periodic oscillations or not is operating point dependent, but is not due to the instantaneous nature of the controls. Even if an actuator model is added the oscillations persist for reasonable time constants, the frequency of the oscillations is however affected. Further it is shown that the periodic control can be predicted by optimal periodic control theory and that the frequency of the control affects the resulting efficiency.

Initial three-dimensional low-thrust trajectory design

This paper presents a method for rapid generation of three-dimensional low-thrust trajectories that utilizes Fourier series for shaping the position vector. The generated trajectories are feasible with respect to the given thrust acceleration constraints. An objective function is defined representing the overall mission cost, i.e. minimum ΔV. Four missions from Earth to Mars, the near Earth asteroid 1989ML, comet Tempel 1 and asteroid Dionysus are considered for assessing the performance of the algorithm. The selected missions present a range of various difficulties with different levels of thrust acceleration constraints. The Fourier series technique is flexible in generating various shapes rather than using one global shape. The proposed method is capable of rapid generation of sub-optimal feasible trajectories that are totally different from and comparable to the solutions of the state-of-the-art three-dimensional shape-based methods. This feature is quite favorable at the preliminary stages of low-thrust mission designs where various trajectory alternatives are required. The results also show that the obtained trajectories can be used as initial guesses for high fidelity optimal control tools.

Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes

This contribution concerns the development of generic methods and tools for robust optimal control of high-pressure liquid chromatographic separation processes. The proposed methodology exploits a deterministic robust formulation, that employs a linearization of the uncertainty set, based on Lyapunov differential equations to generate optimal elution trajectories in the presence of uncertainty. Computational tractability is obtained by casting the robust counterpart problem in the framework of bilevel optimal control where the upper level concerns forward simulation of the Lyapunov differential equation, and the nominal open-loop optimal control problem augmented with the robustified target component purity inequality constraint margin is considered in the lower level. The lower-level open-loop optimal control problem, constrained by spatially discretized partial differential equations, is transcribed into a finite dimensional nonlinear program using direct collocation, which is then solved by a primal-dual interior point method. The advantages of the robustification strategy are highlighted through the solution of a challenging ternary complex mixture separation problem for a hydrophobic interaction chromatography system. The study shows that penalizing the changes in the zero-order hold control gives optimal solutions with low sensitivity to uncertainty. A key result is that the robustified general elution trajectories outperformed the conventional linear trajectories both in terms of recovery yield and robustness.

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete Linear Quadratic Regulator (LQR) problem to find a locally stabilizing controller for a 40 DOF system with six constraints.

Analyzing human-swarm interactions using control Lyapunov functions and optimal control

A number of different interaction modalities have been proposed for human engagement with networked systems. In this paper, we establish formal guarantees for whether or not a given human-swarm interaction (HSI) strategy is appropriate for achieving particular multi-robot tasks, such as guiding a swarm of robots into a particular geometric configuration. In doing so, we define what it means to impose an HSI control structure on a multi-robot system. Control Lyapunov functions are used to establish feasibility for a user to achieve a particular geometric configuration with a multi-robot system under some selected HSI control structure. Several examples of multi-robot systems with unique HSI control structures are provided to illustrated the use of CLFs to establish feasibility. Additionally, we also uses these examples to illustrate how to use optimal control tools to compute three metrics for evaluating an HSI control structure: attention, effort, and scalability.

Exploiting Differential Flatness and Pseudospectral Optimization to Improve the Computational Efficiency of Health-Conscious Lithium-Ion Battery Control

This paper examines the health-conscious optimal control of lithium-ion batteries. The paper focuses specifically on the optimizing battery control policies to minimize solid electrolyte interphase (SEI) layer growth. However, its approach is applicable to other degradation mechanisms. The paper exploits the fact, established by the mathematical control literature1–4, that Fick’s second law of diffusion is differential flat for both linear and nonlinear diffusion problems. Differential flatness only applies to each diffusion medium in the battery separately. In the single particle model (SPM), for instance, each electrode’s diffusion sub-model alone is differential flat. Ensuring differential flatness for the entire battery is a two-step process. First, one must truncate redundant integrators from the two electrodes’ diffusion sub-models. Second, one must then relate the truncated integral state variable to the remaining state variables through an affine transformation. The differential flatness of the resulting battery model makes it possible to solve the health-conscious battery management problem efficiently using pseudospectral methods. This exploitation of the structure (i.e., differential flatness) of lithium-ion battery dynamics for more computationally efficient trajectory optimization is novel and significant. The existing literature already examines the problem of health-conscious optimal battery charging and discharging5,6. However, the optimal control tools used in this literature are not tailored to exploit the structure (e.g., differential flatness) of lithium-ion battery models. This can lead to higher computational burdens compared to the novel and more efficient approach we present here. The approach has the added advantage that it can be used for both offline trajectory optimization and online model-predictive control.The paper demonstrates the above ideas using the SPM. The optimization problem is formulated by equation [1-6] and the SPM with film growth model. The cost function in Equation [1] aims to track the reference state of charge (SOC) and at the same time minimize the film growth rate, which is defined by equation [7]-[14] in Ramadass’s paper7. The weight β represents a tradeoff/balance between aggressive charging and battery degradation. Constraints [2-6] place limits on battery charge/discharge current, reflecting battery management hardware capabilities. Furthermore, these constraints limit battery SOC to prevent over-charging and over-discharging. Finally, these constraints also place bounds on battery state variables contributing to damage phenomena such as lithium plating and mechanical degradation. For instance, we bound the concentration gradients in the battery cell and the overpotentials driving the lithium plating side reaction8–10. The optimization approach presented in this paper transforms these equations and constraints from a dynamic programming problem to a nonlinear programming (NLP) problem. Solving this NLP problem using traditional optimization methods leads to a health-conscious battery management policy. The value of this paper lies not in this policy, but rather in the computational efficiency with which it is obtained. Specifically, to the best of the authors’ knowledge, this paper represents the first attempt to exploit lithium-ion battery model structure for more efficient solution of the health-conscious optimal management problem.ACKNOWLEDGMENTSThe research was funded by ARPA-E AMPED program grant # 0675-1565. The authors gratefully acknowledge this support.REFERENCES1. M. FLIESS, J. LÉVINE, P. MARTIN, and P. ROUCHON, Int. J. Control, 61, 1327–1361 (1995).2. B. Laroche, P. Martin, and P. Rouchon, Int. J. Robust Nonlinear Control, 10, 629–643 (2000).3. M. Fliess and R. Marquez, Int. J. Control, 73, 606–623 (2000).4. T. Meurer, Automatica, 47, 935–949 (2011).5. R. Klein and N. Chaturvedi, Am. Control … (2011).6. V. Boovaragavan and V. R. Subramanian, J. Power Sources, 173, 1006–1011 (2007).7. P. Ramadass, B. Haran, P. M. Gomadam, R. White, and B. N. Popov, J. Electrochem. Soc., 151, A196 (2004).8. J. Christensen and J. Newman, J. Solid State Electrochem., 10, 293–319 (2006).9. X. Zhang, A. M. Sastry, and W. Shyy, J. Electrochem. Soc., 155, A542 (2008).10. N. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and A. Kojic, IEEE Control Syst. Mag., 30, 49–68 (2010).

Optimal Control Approach for Turning Process Planning Optimization

This paper considers the optimal process planning of turning operations where the machining time is minimized subject to a variety of equipment and process constraints. A new approach is proposed, in which virtual dynamics are incorporated, the process planning problem is formulated as a time-optimal dynamic system control problem, and optimal control tools are applied. The independent variable is changed from time to the instantaneous machined length to convert the time-optimal control problem to a finite-horizon control problem. Process and equipment constraints are incorporated using penalty terms in the cost function based on an approximation of the constrained variables. A recently developed method, called the Finite-horizon State Dependent Riccati Equation (Finite-SDRE), is used to solve the problem for the single tool turning process. Then, the developed method is extended and applied to parallel turning operations. Simulation studies are conducted to analyze the performance of the method for single-tool and parallel-tool turning operations, and the proposed method is shown to be very effective for solving such process planning problems.

Description of the Attainable Sets of One-Dimensional Differential Inclusions

The role played by the attainable set of a differential inclusion, in the study of dynamic control systems and fuzzy differential equations, is widely acknowledged. A procedure for estimating the attainable set is rather complicated compared to the numerical methods for differential equations. This article addresses an alternative approach, based on an optimal control tool, to obtain a description of the attainable sets of differential inclusions. In particular, we obtain an exact delineation of the attainable set for a large class of nonlinear differential inclusions.

Calibration of building thermal models using an optimal control approach

The prediction of a building's thermal behaviour within a short time horizon is necessary in many energy management applications. A numerical model can serve this purpose provided a good accuracy is obtained through a suitable calibration procedure. The paper deals with a model calibration procedure based on short-time on-site and weather measurements. It builds upon optimal control theory: an adjoint model is introduced to derive the gradient of a least squares cost function at a low computational cost. Two problems are solved. The first one is a non-linear model training problem. It consists in identifying the main influencing parameters of the system of partial differential equations that form the tendency model. The second problem is a linear identification problem that consists in identifying the unknown internal gains. This second problem can be solved in real-time in a continuous monitoring process. Both problems are solved within the same framework and same tools, illustrating the efficiency of the optimal control tools in this context. We give simulation results that show the performance of the calibration procedure under uncertainties on input parameters.

An atmosphere protection subsystem in the thermal power station automated process control system

Matters concerned with development of methodical and mathematical support for an atmosphere protection subsystem in the thermal power station automated process control system are considered taking as an example the problem of controlling nitrogen oxide emissions at a gas-and-oil-fired thermal power station. The combined environmental-and-economic characteristics of boilers, which correlate the costs for suppressing emissions with the boiler steam load and mass discharge of nitrogen oxides in analytic form, are used as the main tool for optimal control. A procedure for constructing and applying environmental-and-economic characteristics on the basis of technical facilities available in modern instrumentation and control systems is presented.