The maneuverability and freedom provided by unmanned aerial vehicles (UAVs) make these an interesting choice for transporting objects in settings such as search and rescue operations, construction, and smart factories. A commonly proposed method of transport is by using cables attached between each UAV and the payload. However, the geometrical constraints posed by these attachments typically result in a system with highly complex dynamics. Although not an issue for conventional PID control schemes, these complex dynamics make the direct application of model predictive controllers (MPCs) infeasible for real-time usage. For this reason, much of the previous work has focused on treating the payload as a disturbance, thereby losing the ability to predict its effect on the UAVs. Contrary to this, this thesis presents an MPC that both captures the dynamics of the payload, and is capable of real-time usage. This is made possible by a parametrized linearization of the original system, and results in greatly improved performance compared to the disturbance model approach. The controller is derived for a system with two UAVs that transport a bar-like payload and verified both in simulations and physical experiments. The resulting control system is able track a multitude of setpoints, including rotations of both payload and UAVs, as well as lateral translations. Furthermore, it is able to attenuate external disturbances well, and dampens and prevents oscillations more efficiently when compared to the disturbance based approach. The resulting MPC solving time is on the order of milliseconds. Additionally, an initial attempt to decentralize the system is made, and the resulting controller experimentally tested on the UAV–bar system, resulting in a lower MPC solving time (2:5 times faster on average), but worsened performance in terms of position tracking of the bar. / Den manövrerbarhet och frihet som möjliggörs av användandet utav obemannade luftfarkoster (drönare) gör dessa till tämligen intressanta kandidater för lasttransport inom områden såsom sök- och räddningsuppdrag, byggnadskonstruktion och s.k. smarta fabriker. En vanligen förespråkad transportmetod består utav att förse systemet med kablar som fästs mellan last och drönare. De geometriska restriktioner som denna lastkoppling innebär resulterar emellertid ofta i system med väldigt komplicerad dynamik och interaktionskrafter. Även om detta inte innebär något problem för konventionella PID reglersystem så omöjliggör detta det direkta applicerandet utav modellprediktiv reglering (MPC) för realtidsbruk. Av denna anledning har tidigare verk fokuserat på att behandla lasten och dess inverkan på drönarna som en störning, men med detta därmed förlorat möjligheten att förutspå dess effekt på drönarna. I kontrast till detta, kommer det i detta verk att presenteras en MPC som både fångar lastens dynamik och är snabb nog för realtidsanvändning. Detta görs möjligt utav en parametriserad linjärisering utav originalsystemet och ger märkbart bättre resultat än den störningbaserade modellen. Reglersystemet appliceras på ett system bestående utav två drönare och en stång-liknande last och resultatet verifieras både i form av numeriska simuleringar och fysiska experiment. Det resulterande systemet klarar av både rotationer utav last och drönare samt translationer i alla riktningar. Dessutom är systemet kapabelt att hantera externa störningar och både dämpar och förhindrar oscillationer bättre i jämförelse med reglersystem baserat på störningsmodeller. Lösningstiden för MPC-regulatorn är i storleksordningen millisekunder. Utöver detta görs ett initialt försök i att decentralisera tidigare nämnda MPC och det resulterande reglersystemet utvärderas experimentellt på samma drönarsystem som tidigare. Detta resulterar i en lägre lösningstid (2.5 ggr snabbare i genomsnitt), men även i försämrad prestanda med avseende på reglering av stångens position.

<p>This thesis considers the problems of modelling and control of batch processes, a class of finite duration chemical processes characterized by their absence of equilibrium conditions and nonlinear, time-varying dynamics over a wide range of operating conditions. In contrast to continuous processes, the control objective in batch processes is to achieve a non-equilibrium desired end-point or product quality by the batch termination time. However, the distinguishing features of batch processes complicate their control problem and call for dedicated modelling and control tools. In the initial phase of this research, a predictive controller based on the novel concept of reverse-time reachability regions (RTRRs) is developed. Defined as the set of states from where the process can be steered inside a desired end-point neighbourhood by batch termination subject to input constraints and model uncertainties, an algorithm is developed to characterize these sets at each sampling instance offline; these characterizations subsequently play an integral role in the control design. A key feature of the resultant controller is that it requires the online computation of only the immediate control action while guaranteeing reachability to the desired end-point neighbourhood, rendering the control problem efficiently solvable even when using the nonlinear process model. Moreover, the use of RTRRs and one-step ahead type control policy embeds important fault-tolerant characteristics into the controller. Next, we address the problem of the unavailability of reliable and computationally manageable first-principles-based process models by developing a new data-based modelling approach. In this approach, local linear models (identified via latent variable regression techniques) are combined with weights (arising from fuzzy c-means clustering) to describe global nonlinear process dynamics. Nonlinearities are captured through the appropriate combination of the different models while the linearity of the individual models prevents against a computationally expensive predictive controller. This modelling approach is also generalized to account for time-varying dynamics by incorporating online learning ability into the model, making it adaptive. This is accomplished by developing a probabilistic recursive least squares (PRLS) algorithm for updating a subset of the model parameters. The data-based modelling approach is first used to generate data-based reverse-time reachability regions (RTRRs), which are subsequently incorporated in a new predictive controller. Next, the modelling approach is applied on a complex nylon-6,6 batch polymerization process in order to design a trajectory tracking predictive controller for the key process outputs. Through simulations, the modelling approach is shown to capture the major process nonlinearities and closed-loop results demonstrate the advantages of the proposed controller over existing options. Through further simulation studies, model adaptation (via the PRLS algorithm) is shown to be crucial for achieving acceptable control performance when encountering large disturbances in the initial conditions. Finally, we consider the problem of direct quality control even when there are limited quality-related measurements available from the process; this situation typically calls for indirectly pursuing the control objective through trajectory tracking control. To address the problem of unavailability of online quality measurements, an inferential quality model, which relates the process conditions over the entire batch duration to the final quality, is required. The accuracy of this type of quality model, however, is sensitive to the prediction of the future batch behaviour until batch termination. This "missing data" problem is handled by integrating the previously developed data-based modelling approach with the inferential model in a predictive control framework. The key feature of this approach is that the causality and nonlinear relationships between the future inputs and outputs are accounted for in predicting the final quality and computing the manipulated input trajectory. The efficacy of the proposed predictive control design is illustrated via simulations of the nylon-6,6 batch polymerization process with a different control objective than considered previously.</p> / Doctor of Philosophy (PhD)

<p>Process operability represents the ability of a process plant to operate satisfactorily away from the nominal operating or design condition, where flexibility and dynamic operability are two important attributes of operability considered in this thesis. Today's companies are facing numerous challenges, many as a result of volatile market conditions. Key to sustainable profitable operation is a robust process supply chain. Within a wider business context, flexibility and responsiveness, i.e. dynamic operability, are regarded as key qualifications of a robust process supply chain.</p> <p>The first part of this thesis develops methodologies to rigorously evaluate the dynamic operability and flexibility of a process supply chain. A model is developed which describes the response dynamics of a multi-product, multi-echelon supply chain system. Its incorporation within a dynamic operability analysis framework is shown, where a bi-criterion, two-stage stochastic programming approach is applied for the treatment of demand uncertainty, and for estimating the Pareto frontier between an economic and responsiveness criterion. Two case studies are presented to demonstrate the effect of supply chain design features on responsiveness. This thesis has also extended current paradigms for process flexibility analysis to supply chains. The flexibility analysis framework, where a steady-state supply chain model is considered, evaluates the ability to sustain feasible steady-state operation for a range of demand uncertainty.</p> <p>The second part of this thesis develops a decision-support tool for supply chain management (SCM), by means of a robust model predictive control (MPC) strategy. An effective decision-support tool can fully leverage the qualifications from the operability analysis. The MPC formulation proposed in this thesis: (i) captures uncertainty in model parameters and demand by stochastic programming, (ii) accommodates hybrid process systems with decisions governed by logical conditions/rulesets, (iii) addresses multiple supply chain performance metrics including customer service and economics, and (iv) considers both open-loop and closed-loop prediction of uncertainty propagation. The developed robust framework is applied for the control of a multi-echelon, multi-product supply chain, and provides a substantial reduction in the occurrence of back orders when compared with a nominal MPC framework.</p> / Master of Applied Science (MASc)

The freight transport plays an important role in the development of the economy. However, this comes with an important contribution to greenhouse gas emission. Recently a shift toward heavy-duty electric vehicles has been made, but some issues still need to be tackled. One of them is to develop ways to quickly recharge the vehicle’s batteries without damaging them. In this thesis, we highlight that not only the current but also the battery temperature need to be carefully managed in order to prevent damages during a charging session. To show that, an electrical, thermal and aging model of Li-ion battery is developed. A charging strategy based on a Model Predictive Control algorithm is proposed. The algorithm controls both the battery current and cooling system in order to achieve the optimal balance between the charging speed and the preservation of the battery. The resulting algorithm is tested, in simulation, against a conventional constant current charging in different charging scenarios. The results show an important increase in performance and highlight the role of the battery cooling system in the preservation of the battery. / Godstransporterna spelar en viktig roll för ekonomins utveckling. Detta innebär dock ett betydande bidrag till utsläppen av växthusgaser. På senare tid har en övergång till tunga elfordon skett, men vissa frågor måste fortfarande lösas. En av dem är att utveckla metoder för att snabbt ladda fordonsbatterierna utan att skada dem. I den här avhandlingen lyfter vi fram att inte bara strömmen utan även batteritemperaturen måste hanteras noggrant för att förhindra skador under en laddning. För att visa detta utvecklas en elektrisk, termisk och åldrande modell för Li-ion-batterier. En laddningsstrategi baserad på en algoritm för modellförutsägbar styrning föreslås. Algoritmen styr både batteriströmmen och kylsystemet för att uppnå en optimal balans mellan laddningshastighet och bevarande av batteriet. Den resulterande algoritmen testas i simulering mot en konventionell konstantström laddning i olika laddningsscenarier. Resultaten visar en betydande ökning av prestanda och belyser batterikylsystemets betydelse för bevarandet av batteriet.

The aim of this project was to compare two control algorithms designed to steer an autonomous vehicle. The comparison was made using a simulated environment to evaluate the performance of both controllers. The simulation used in this project was designed in Python and used an algorithm which randomly constructed roads from predefined road segments to create paths for the vehicle to follow. In this environment the Linear Time Varying (LTV)-Model Predictive Controller (MPC) and Pure Pursuit Controller (PPC) algorithms were evaluated. The thesis compared how well they follow paths, the average control cost of completing tasks, how well they handle input constraints, and the computational time for each algorithm. The data was collected by driving along three sets of randomly generated roads with both control algorithms. One set mostly straight, one with some turns, and one with mostly turns. An Analysis of Variance (ANOVA) test was used to make the comparison between the performance of the two algorithms. The results showed that both algorithms performed well. The PPC had low computation time and used less control, but it also had larger position errors. The LTV-MPC had higher computation time, but smaller position errors at the cost of larger control values. The conclusion is that the MPC is preferable if computational capabilities are available. Room for future work exists in the form of comparing additional controller types for autonomous vehicles and exploring different tuning parameters for the MPC controller. The simulation could also be expanded to more accurately reflect real world conditions. / Målet med detta projekt var att jämföra två kontrollalgoritmer avsedda för att styra en självkörande bil. Jämförelsen gjordes med hjälp av en simulering som utformades i Python. Den använde sig av en algoritm som slumpmässigt satte ihop vägar från förkonstruerade delar för att skapa banor för den självkörande bilen att följa. I denna miljö har vi testat två algoritmer, en LTV-MPC och en PPC. Vi jämförde hur pass väl de följer banor som skall likna riktiga vägar, hur mycket styrning de använder sig av för att bedöma energianvändning, hur väl de förhåller sig till begränsningar på acceleration och styrning, och den beräkningstiden som krävdes för att köra vår algoritm. Datan samlades genom att köra längs med tre grupper av slumpmässigt genererade vägar med båda kontrollalgoritmerna. En grupp innehöll huvudsakligen raka sträckor, en innehöll en del svängar, och en innehöll mycket svängar. ANOVA-testet användes för att göra jämförelsen mellan resultatet av dessa två algoritmer. Resultatet visade att båda algoritmer presterar väl. PPCn hade låg beräkningstid och mindre styrvärden, men större positionsfel. MPCn hade högre beräkningstid och större styrvärden, men mindre positionsfel. Slutsatsen är att MPCn är att föredra om beräkningsmöjligheterna finns tillgängliga. Det finns utrymme för framtida arbete i form av att jämföra fler kontrollalgoritmer och att utforska fler parameter justeringar för MPCn. Utöver det finns det även utrymme för en simulation som reflekterar verkligheten noggrannare.

La Commande prédictive ou Commande Optimale à Horizon Glissant (COHG) devient de plus en plus populaire dans de nombreuses applications pratiques en raison de ses avantages importants tels que la stabilisation et la prise en compte des contraintes. Elle a été bien étudiée pour des systèmes en dimension finie même dans le cas non linéaire. Cependant, son extension aux systèmes en dimension infinie n'a pas retenu beaucoup d'attention de la part des chercheurs. Ce travail de thèse apporte des contributions à l'application de cette approche aux systèmes de lois de conservation. Nous présentons tout d'abord une preuve de stabilité complète de la COHG pour certaines classes de systèmes en dimension infinie. Ce résultat est ensuite utilisé pour les systèmes hyperboliques 2x2 commandés aux frontières et appliqué à un problème de contrôle de canal d'irrigation. Nous proposons aussi l'extension de cette stratégie au cas de réseaux de systèmes hyperboliques 2x2 en cascade avec une application à un ensemble de canaux d'irrigation connectés. Nous étudions également les avantages de la COHG dans le contexte des systèmes non linéaires et semi-linéaires notamment vis-à-vis des chocs. Toutes les analyses théoriques sont validées par simulation afin d'illustrer l'efficacité de l'approche proposée. / The predictive control or Receding Horizon Optimal Control (RHOC) is becoming increasingly popular in many practical applications due to its significant advantages such as the stabilization and constraints handling. It has been well studied for finite dimensional systems even in the nonlinear case. However, its extension to infinite dimensional systems has not received much attention from researchers. This thesis proposes contributions on the application of this approach to systems of conservation laws. We present a complete proof of stability of RHOC for some classes of infinite dimensional systems. This result is then used for 2x2 hyperbolic systems with boundary control, and applied to an irrigation canal. We also propose the extension of this strategy to networks of cascaded 2x2 hyperbolic systems with an application to a set of connected irrigation canals. Furthermore, we study the benefits of RHOC in the context of nonlinear and semi-linear systems in particular with respect to the problem of shocks. All theoretical analyzes are validated by simulation in order to illustrate the effectiveness of the proposed approach.

In advanced process control, the Model Predictive Control (MPC) may be considered the most important innovation in recent years and the standard tool for industrial applications due to the fact that it keeps the plant operating in the constraints more profitable. However, like every control algorithm, the MPC after some time in operation rarely works as originally designed. Thus, to preserve the benefits of MPC systems for a long period of time, their performance needs to be monitored and evaluated during the operation. This task require the presence of reliable and effective tools to detect when the controller performance is below of the desirable, to define the need, or not, of recommissioning the system. Thus, the objective of this work is development of techniques for monitoring and evaluating the performance of multivariable predictive controllers, being developed two new tools: LQG benchmark Modified and IHMC benchmark. The results obtained from numerical simulations were satisfactory and consistent with the technical literature applied in the developments of the evaluators, which were used in the monitoring of the control system MPC of the oil-water-gas three-phase separation process, offering an appropriate solution and providing subsidies for implementations in real industrial systems. / Em controle avançado de processos, o controlador preditivo ou MPC (Model Predictive Control) pode ser considerado como a mais importante inovação dos últimos anos e a ferramenta padrão para aplicações industriais, devido ao fato do MPC manter a planta operando dentro das suas restrições de forma mais lucrativa. Entretanto, como todo algoritmo de controle, o MPC depois de algum tempo em operação dificilmente funciona como quando fora inicialmente projetado. Desta forma, com o objetivo de manter os benefícios dos sistemas MPC por um longo período de tempo, seu desempenho precisa ser monitorado e avaliado durante a operação. Esta tarefa requer a presença de ferramentas efetivas e confiáveis para detectar quando o desempenho do controlador estiver abaixo do desejável, para definir a necessidade, ou não, de um recomissionamento do sistema. Destarte, aborda-se neste trabalho o desenvolvimento de técnicas para monitoramento e avaliação de desempenho de controladores preditivos multivariáveis, sendo desenvolvidas duas novas ferramentas: LQG benchmark Modificado e IHMC benchmark. Os resultados obtidos a partir de simulações numéricas foram satisfatórios e coerentes com a literatura técnica aplicada no desenvolvimento dos avaliadores, os quais foram utilizados no monitoramento do sistema de controle MPC do processo de separação trifásica água-óleo-gás, oferecendo assim uma solução apropriada e fornecendo subsídios para implementações em sistemas industrias reais.

Les actionneurs électromécaniques supplantent les solutions d’actionnement hydrauliques concurrentes dans un nombre croissant d’applications industrielles, particulièrement en aéronautique. Ils le doivent à leurs rendements élevés et à une relative simplicité de commande. Fort de ces avantages, cette technologie a été mise en place dans le cadre du projet ELEXC, en vue de remplacer les vérins hydrauliques sur une mini-excavatrice tout-électrique et s’attaquant pour la première fois au secteur du bâtiment et des travaux publics. Cependant, cette intégration s’accompagne de doutes sur la performance et la durabilité de ces actionneurs dans ce cas précis d’utilisation. Pour répondre à ces interrogations, ce manuscrit se focalisera dans un premier volet sur la caractérisation des performances d’une structure spécifique, permise par la mise en place d’un banc de test multi-énergies instrumenté. Cette étude aboutira à la mise en place d’un modèle par lois physiques dont les paramètres auront été identifiés expérimentalement.Deux solutions innovantes vont ensuite être proposées, visant respectivement à réduire les répercussions, notamment énergétiques, de la caractéristique discrète du convertisseur à découpage, et à parer au manque de compliance de l’actionneur tout en conservant ses capacités de creusage. Elles seront toutes deux associées au concept de commande prédictive, mais offriront chacune une spécificité par rapport à l’implémentation standard de ce type de commande tout en conservant ses avantages. La première prendra la forme d’un modulateur fondé sur le principe d’une modulation de largeur d’impulsion vectorielle, dont la séquence sera sélectionnée par un contrôleur prédictif en rapport avec une fonction de coût alliant les ondulations de courant, les pertes par commutation et la tension de mode commun spécifiques aux séquences. La seconde consistera en un contrôleur d’impédance à boucle de position interne, dont les paramètres d’impédance seront variables et définis par un contrôleur prédictif afin de permettre une bonne pénétration dans l’environnement tout en limitant la dynamique et la valeur maximale de l’effort de compression sur la transmission. / Nowadays, the electro-mechanical actuators (EMA) are seen as an alternative to the conventional hydraulic cylinders in a growing number of industrial applications, including the aeronautic sector, thanks to their high efficiency and relatively simple commandability. According to this features, this actuator technology was selected to replace the hydraulic cylinders of the actuation system of a full-electric compact excavator in the ELEXC project. However, some concerns arose from the lack of existing examples with EMA functioning in relatively severe working conditions implying collisions of the actuators with stiff environments. Therefore, this manuscript will first focus on the characterization of the electro-mechanical actuator and of its performances in this specific case of application, enabled by the development of a multi-energy test-rig. Beside the validation of the EMA capacities, a physic law model will be developed, and its parameters tuned according to the experimental results. In a second part of this manuscript, two novel control laws will be proposed, aiming respectively to reduce the negative repercussions of the discrete characteristic of the electrical converter, and to counterpart the lack of natural compliance of the mechanical actuator. They will be both based on the concept of predictive control, but will differ from the conventional implementation of this type of controller. The first proposed topology is a modulator based on a vector Pulse Width Modulation, in which the PWM sequence to be used is defined online by a predictive controller according to a cost function that takes into account the current ripples, the switching losses and the common mode voltages specific to the sequences. Next, a position-based impedance controller, whose impedance parameters are defined online by a predictive controller, is introduced, in order to enable the EMA to penetrate a stiff environment with a relatively low dynamic increase of compression force on the mechanical transmission.

Indiana University-Purdue University Indianapolis (IUPUI) / Heating, ventilation, and air conditioning (HVAC) is the technology responsible to maintain temperature levels and air quality in buildings to certain standards. In a commercial setting, HVAC systems accounted for more than 50% of the total energy cost of the building in 2013 [13]. New control methods are always being worked on to improve the effectiveness and efficiency of the system. These control systems include model predictive control (MPC), evolutionary algorithm (EA), evolutionary programming (EP), and proportional-integral-derivative (PID) controllers. Such control tools are used on new HVAC system to ensure the ultimate efficiency and ensure the comfort of occupants. However, there is a need for a system that can monitor the energy performance of the HVAC system and ensure that it is operating in its optimal operation and controlled as expected. In this thesis, an air handling unit (AHU) of an HVAC system was modeled to analyze its performance using real data collected from an operating AHU using a wireless monitoring system. The purpose was to monitor the AHU's performance, analyze its key parameters to identify flaws, and evaluate the energy waste. This system will provide the maintenance personnel to key information to them to act for increasing energy efficiency. The mechanical model was experimentally validated first. Them a baseline operating condition was established. Finally, the system under extreme weather conditions was evaluated. The AHU's subsystem performance, the energy consumption and the potential wastes were monitored and quantified. The developed system was able to constantly monitor the system and report to the maintenance personnel the information they need. I can be used to identify energy savings opportunities due to controls malfunction. Implementation of this system will provide the system's key performance indicators, offer feedback for adjustment of control strategies, and identify the potential savings. To further verify the capabilities of the model, a case study was performed on an air handling unit on campus for a three month monitoring period. According to the mechanical model, a total of 63,455 kWh can be potentially saved on the unit by adjusting controls. In addition the mechanical model was able to identify other energy savings opportunities due to set point changes that may result in a total of 77,141 kWh.

In recent years, mobile robots have gained tremendous attention from the entire society: the industry is aiming at selling more intelligent products while the academia is improving their performance from all perspectives. Real world examples include autnomous driving vehicles, multirotors, legged robots, etc. One of the challenging tasks commonly faced by all game players, and all robotics platforms, is to plan motion or locomotion of the robot, calculate an optimal trajectory according to certain criterion and control it accordingly. Difficulty of solving such task usually arises from high-dimensionality and complexity of the system dynamics, fast changing conditions imposed as constraints and necessity for real-time deployment. This work proposes a method over the aforementioned mission by solving an optimal control problem in a receding horizon fashion. Unlike the existing Sequential Linear Quadratic [1] algorithm which is a continuous-time variant of Differential Dynamic Programming [2], we tackle the problem in a discretized multiple shooting fashion. Sequential Quadratic Programming is employed as optimization technique to solve the constrained Nonlinear Programming iteratively. Moreover, we apply trust region method in the sub Quadratic Programming to handle potential indefiniteness of Hessian matrix as well as to improve robustness of the solver. Simulation and benchmark with previous method have been conducted on robotics platforms to show the effectiveness of our solution and superiority under certain circumstances. Experiments have demonstrated that our method is capable of generating trajectories under complicated scenarios where the Hessian matrix contains negative eigenvalues (e.g. obstacle avoidance). / De senaste åren har mobila robotar fått enorm uppmärksamhet från hela samhället: branschen siktar på att sälja mer intelligenta produkter samtidigt som akademin förbättrar sina prestationer ur alla perspektiv. Exempel på verkligheten inkluderar autonoma körande fordon, multirotorer, robotar med ben, etc. En av de utmanande uppgifterna som vanligtvis alla spelare och alla robotplattformar står inför är att planera robotens rörelse eller rörelse, beräkna en optimal bana enligt vissa kriterier och kontrollera det därefter. Svårigheter att lösa en sådan uppgift beror vanligtvis på hög dimensionalitet och komplexitet hos systemdynamiken, snabbt föränderliga villkor som åläggs som begränsningar och nödvändighet för realtidsdistribution. Detta arbete föreslår en metod över det tidigare nämnda uppdraget genom att lösa ett optimalt kontrollproblem på ett vikande horisont. Till skillnad från den befintliga Sequential Linear Quadratic [1] algoritmen som är en kontinuerlig tidsvariant av Differential Dynamic Programming [2], tar vi oss an problemet på ett diskretiserat multipelfotograferingssätt. Sekventiell kvadratisk programmering används som optimeringsteknik för att lösa den begränsade olinjära programmeringen iterativt. Dessutom tillämpar vi trust region-metoden i den sub-kvadratiska programmeringen för att hantera potentiell obestämdhet av hessisk matris samt för att förbättra lösarens robusthet. Simulering och benchmark med tidigare metod har utförts på robotplattformar för att visa effektiviteten hos vår lösning och överlägsenhet under vissa omständigheter. Experiment har visat att vår metod är kapabel att generera banor under komplicerade scenarier där den hessiska matrisen innehåller negativa egenvärden (t.ex. undvikande av hinder).