Author: Gustav Lindmark
Publisher: Linköping University Electronic Press
ISBN: 9176852431
Category :
Languages : en
Pages : 36
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The control-theoretic notion of controllability captures the ability to guide a systems behavior toward a desired state with a suitable choice of inputs. Controllability of complex networks such as traffic networks, gene regulatory networks, power grids etc. brings many opportunities. It could for instance enable improved efficiency in the functioning of a network or lead to that entirely new applicative possibilities emerge. However, when control theory is applied to complex networks like these, several challenges arise. This thesis consider some of these challenges, in particular we investigate how control inputs should be placed in order to render a given network controllable at a minimum cost, taking as cost function either the number of control inputs or the energy that they must exert. We assume that each control input targets only one node (called a driver node) and is either unconstrained or unilateral. A unilateral control input is one that can assume either positive or negative values but not both. Motivated by the many applications where unilateral controls are common, we reformulate classical controllability results for this particular case into a more computationally-efficient form that enables a large scale analysis. We show that the unilateral controllability problem is to a high degree structural and derive theoretical lower bounds on the minimal number of unilateral control inputs from topological properties of the network, similar to the bounds that exists for the minimal number of unconstrained control inputs. Moreover, an algorithm is developed that constructs a near minimal number of control inputs for a given network. When evaluated on various categories of random networks as well as a number of real-world networks, the algorithm often achieves the theoretical lower bounds. A network can be controllable in theory but not in practice when completely unreasonable amounts of control energy are required to steer it in some direction. For unconstrained control inputs we show that the control energy depends on the time constants of the modes of the network, and that the closer the eigenvalues are to the imaginary axis of the complex plane, the less energy is required for control. We also investigate the problem of placing driver nodes such that the control energy requirements are minimized (assuming that theoretical controllability is not an issue). For the special case with networks having all purely imaginary eigenvalues, several constructive algorithms for driver node placement are developed. In order to understand what determines the control energy in the general case with arbitrary eigenvalues, we define two centrality measures for the nodes based on energy flow considerations: the first centrality reflects the network impact of a node and the second the ability to control it indirectly. It turns out that whether a node is suitable as driver node or not largely depends on these two qualities. By combining the centralities into node rankings we obtain driver node placements that significantly reduce the control energy requirements and thereby improve the “practical degree of controllability”.
Author: Gustav Lindmark
Publisher: Linköping University Electronic Press
ISBN: 9179298478
Category : Electronic books
Languages : en
Pages : 38
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Book Description
The control-theoretic notion of controllability captures the ability to guide a system toward a desired state with a suitable choice of inputs. Controllability of complex networks such as traffic networks, gene regulatory networks, power grids etc. can for instance enable efficient operation or entirely new applicative possibilities. However, when control theory is applied to complex networks like these, several challenges arise. This thesis considers some of them, in particular we investigate how a given network can be rendered controllable at a minimum cost by placement of control inputs or by growing the network with additional edges between its nodes. As cost function we take either the number of control inputs that are needed or the energy that they must exert. A control input is called unilateral if it can assume either positive or negative values, but not both. Motivated by the many applications where unilateral controls are common, we reformulate classical controllability results for this particular case into a more computationally-efficient form that enables a large scale analysis. Assuming that each control input targets only one node (called a driver node), we show that the unilateral controllability problem is to a high degree structural: from topological properties of the network we derive theoretical lower bounds for the minimal number of unilateral control inputs, bounds similar to those that have already been established for the minimal number of unconstrained control inputs (e.g. can assume both positive and negative values). With a constructive algorithm for unilateral control input placement we also show that the theoretical bounds can often be achieved. A network may be controllable in theory but not in practice if for instance unreasonable amounts of control energy are required to steer it in some direction. For the case with unconstrained control inputs, we show that the control energy depends on the time constants of the modes of the network, the longer they are, the less energy is required for control. We also present different strategies for the problem of placing driver nodes such that the control energy requirements are reduced (assuming that theoretical controllability is not an issue). For the most general class of networks we consider, directed networks with arbitrary eigenvalues (and thereby arbitrary time constants), we suggest strategies based on a novel characterization of network non-normality as imbalance in the distribution of energy over the network. Our formulation allows to quantify network non-normality at a node level as combination of two different centrality metrics. The first measure quantifies the influence that each node has on the rest of the network, while the second measure instead describes the ability to control a node indirectly from the other nodes. Selecting the nodes that maximize the network non-normality as driver nodes significantly reduces the energy needed for control. Growing a network, i.e. adding more edges to it, is a promising alternative to reduce the energy needed to control it. We approach this by deriving a sensitivity function that enables to quantify the impact of an edge modification with the H2 and H? norms, which in turn can be used to design edge additions that improve commonly used control energy metrics.
Author: Daniel Arnström
Publisher: Linköping University Electronic Press
ISBN: 9179296920
Category :
Languages : en
Pages : 45
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In model predictive control (MPC) an optimization problem has to be solved at each time step, which in real-time applications makes it important to solve these efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations is solved. The primary contribution of this thesis is a method which determines which sequence of subproblems a popular class of such active-set algorithms need to solve, for every possible QP instance that might arise from a given linear MPC problem (i.e, for every possible state and reference signal). By knowing these sequences, worst-case bounds on how many iterations, floating-point operations and, ultimately, the maximum solution time, these active-set algorithms require to compute a solution can be determined, which is of importance when, e.g, linear MPC is used in safety-critical applications. After establishing this complexity certification method, its applicability is extended by showing how it can be used indirectly to certify the complexity of another, efficient, type of active-set QP algorithm which reformulates the QP as a nonnegative least-squares method. Finally, the proposed complexity certification method is extended further to situations when enhancements to the active-set algorithms are used, namely, when they are terminated early (to save computations) and when outer proximal-point iterations are performed (to improve numerical stability).
Author: Magnus Malmström
Publisher: Linköping University Electronic Press
ISBN: 9179296807
Category :
Languages : en
Pages : 103
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In science, technology, and engineering, creating models of the environment to predict future events has always been a key component. The models could be everything from how the friction of a tire depends on the wheels slip to how a pathogen is spread throughout society. As more data becomes available, the use of data-driven black-box models becomes more attractive. In many areas they have shown promising results, but for them to be used widespread in safety-critical applications such as autonomous driving some notion of uncertainty in the prediction is required. An example of such a black-box model is neural networks (NNs). This thesis aims to increase the usefulness of NNs by presenting an method where uncertainty in the prediction is obtained by linearization of the model. In system identification and sensor fusion, under the condition that the model structure is identifiable, this is a commonly used approach to get uncertainty in the prediction from a nonlinear model. If the model structure is not identifiable, such as for NNs, the ambiguities that cause this have to be taken care of in order to make the approach applicable. This is handled in the first part of the thesis where NNs are analyzed from a system identification perspective, and sources of uncertainty are discussed. Another problem with data-driven black-box models is that it is difficult to know how flexible the model needs to be in order to correctly model the true system. One solution to this problem is to use a model that is more flexible than necessary to make sure that the model is flexible enough. But how would that extra flexibility affect the uncertainty in the prediction? This is handled in the later part of the thesis where it is shown that the uncertainty in the prediction is bounded from below by the uncertainty in the prediction of the model with lowest flexibility required for representing true system accurately. In the literature, many other approaches to handle the uncertainty in predictions by NNs have been suggested, of which some are summarized in this work. Furthermore, a simulation and an experimental studies inspired by autonomous driving are conducted. In the simulation study, different sources of uncertainty are investigated, as well as how large the uncertainty in the predictions by NNs are in areas without training data. In the experimental study, the uncertainty in predictions done by different models are investigated. The results show that, compared to existing methods, the linearization method produces similar results for the uncertainty in predictions by NNs. An introduction video is available at https://youtu.be/O4ZcUTGXFN0 Inom forskning och utveckling har det har alltid varit centralt att skapa modeller av verkligheten. Dessa modeller har bland annat använts till att förutspå framtida händelser eller för att styra ett system till att bete sig som man önskar. Modellerna kan beskriva allt från hur friktionen hos ett bildäck påverkas av hur mycket hjulen glider till hur ett virus kan sprida sig i ett samhälle. I takt med att mer och mer data blir tillgänglig ökar potentialen för datadrivna black-box modeller. Dessa modeller är universella approximationer vilka ska kunna representera vilken godtycklig funktion som helst. Användningen av dessa modeller har haft stor framgång inom många områden men för att verkligen kunna etablera sig inom säkerhetskritiska områden såsom självkörande farkoster behövs en förståelse för osäkerhet i prediktionen från modellen. Neuronnät är ett exempel på en sådan black-box modell. I denna avhandling kommer olika sätt att tillförskaffa sig kunskap om osäkerhet i prediktionen av neuronnät undersökas. En metod som bygger på linjärisering av modellen för att tillförskaffa sig osäkerhet i prediktionen av neuronnätet kommer att presenteras. Denna metod är välbeprövad inom systemidentifiering och sensorfusion under antagandet att modellen är identifierbar. För modeller såsom neuronnät, vilka inte är identifierbara behövs det att det tas hänsyn till tvetydigheterna i modellen. En annan utmaning med datadrivna black-box modeller, är att veta om den valda modellmängden är tillräckligt generell för att kunna modellera det sanna systemet. En lösning på detta problem är att använda modeller som har mer flexibilitet än vad som behövs, det vill säga en överparameteriserad modell. Men hur påverkas osäkerheten i prediktionen av detta? Detta är något som undersöks i denna avhandling, vilken visar att osäkerheten i den överparameteriserad modellen kommer att vara begränsad underifrån av modellen med minst flexibilitet som ändå är tillräckligt generell för att modellera det sanna systemet. Som avslutning kommer dessa resultat att demonstreras i både en simuleringsstudie och en experimentstudie inspirerad av självkörande farkoster. Fokuset i simuleringsstudien är hur osäkerheten hos modellen är i områden med och utan tillgång till träningsdata medan experimentstudien fokuserar på jämförelsen mellan osäkerheten i olika typer av modeller.Resultaten från dessa studier visar att metoden som bygger på linjärisering ger liknande resultat för skattningen av osäkerheten i prediktionen av neuronnät, jämfört med existerande metoder.
Author: Erik Hedberg
Publisher: Linköping University Electronic Press
ISBN: 9179297404
Category :
Languages : en
Pages : 64
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Book Description
The two topics at the heart of this thesis are how to improve control of industrial manipulators and how to reason about the role of models in automatic control. On industrial manipulators, two case studies are presented. The first investigates estimation with inertial sensors, and the second compares control by feedback linearization to control based on gain-scheduling. The contributions on the second topic illustrate the close connection between control and estimation in different ways. A conceptual model of control is introduced, which can be used to emphasize the role of models as well as the human aspect of control engineering. Some observations are made regarding block-diagram reformulations that illustrate the relation between models, control and inversion. Finally, a suggestion for how the internal model principle, internal model control, disturbance observers and Youla-Kucera parametrization can be introduced in a unified way is presented.
Author: Kristoffer Bergman
Publisher: Linköping University Electronic Press
ISBN: 9176850579
Category :
Languages : en
Pages : 91
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Book Description
During the last decades, motion planning for autonomous systems has become an important area of research. The high interest is not the least due to the development of systems such as self-driving cars, unmanned aerial vehicles and robotic manipulators. In this thesis, the objective is not only to find feasible solutions to a motion planning problem, but solutions that also optimize some kind of performance measure. From a control perspective, the resulting problem is an instance of an optimal control problem. In this thesis, the focus is to further develop optimal control algorithms such that they be can used to obtain improved solutions to motion planning problems. This is achieved by combining ideas from automatic control, numerical optimization and robotics. First, a systematic approach for computing local solutions to motion planning problems in challenging environments is presented. The solutions are computed by combining homotopy methods and numerical optimal control techniques. The general principle is to define a homotopy that transforms, or preferably relaxes, the original problem to an easily solved problem. The approach is demonstrated in motion planning problems in 2D and 3D environments, where the presented method outperforms both a state-of-the-art numerical optimal control method based on standard initialization strategies and a state-of-the-art optimizing sampling-based planner based on random sampling. Second, a framework for automatically generating motion primitives for lattice-based motion planners is proposed. Given a family of systems, the user only needs to specify which principle types of motions that are relevant for the considered system family. Based on the selected principle motions and a selected system instance, the algorithm not only automatically optimizes the motions connecting pre-defined boundary conditions, but also simultaneously optimizes the terminal state constraints as well. In addition to handling static a priori known system parameters such as platform dimensions, the framework also allows for fast automatic re-optimization of motion primitives if the system parameters change while the system is in use. Furthermore, the proposed framework is extended to also allow for an optimization of discretization parameters, that are are used by the lattice-based motion planner to define a state-space discretization. This enables an optimized selection of these parameters for a specific system instance. Finally, a unified optimization-based path planning approach to efficiently compute locally optimal solutions to advanced path planning problems is presented. The main idea is to combine the strengths of sampling-based path planners and numerical optimal control. The lattice-based path planner is applied to the problem in a first step using a discretized search space, where system dynamics and objective function are chosen to coincide with those used in a second numerical optimal control step. This novel tight combination of a sampling-based path planner and numerical optimal control makes, in a structured way, benefit of the former method’s ability to solve combinatorial parts of the problem and the latter method’s ability to obtain locally optimal solutions not constrained to a discretized search space. The proposed approach is shown in several practically relevant path planning problems to provide improvements in terms of computation time, numerical reliability, and objective function value.
Author: Robin Forsling
Publisher: Linköping University Electronic Press
ISBN: 9179297242
Category :
Languages : en
Pages : 110
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Book Description
Sensor networks consist of sensors (e.g., radar and cameras) and processing units (e.g., estimators), where in the former information extraction occurs and in the latter estimates are formed. In decentralized estimation information extracted by sensors has been pre-processed at an intermediate processing unit prior to arriving at an estimator. Pre-processing of information allows for the complexity of large systems and systems-of-systems to be significantly reduced, and also makes the sensor network robust and flexible. One of the main disadvantages of pre-processing information is that information becomes correlated. These correlations, if not handled carefully, potentially lead to underestimated uncertainties about the calculated estimates. In conservative estimation the unknown correlations are handled by ensuring that the uncertainty about an estimate is not underestimated. If this is ensured the estimate is said to be conservative. Neglecting correlations means information is double counted which in worst case implies diverging estimates with fatal consequences. While ensuring conservative estimates is the main goal, it is desirable for a conservative estimator, as for any estimator, to provide an error covariance which is as small as possible. Application areas where conservative estimation is relevant are setups where multiple agents cooperate to accomplish a common objective, e.g., target tracking, surveillance and air policing. The first part of this thesis deals with theoretical matters where the conservative linear unbiased estimation problem is formalized. This part proposes an extension of classical linear estimation theory to the conservative estimation problem. The conservative linear unbiased estimator (CLUE) is suggested as a robust and practical alternative for estimation problems where the correlations are unknown. Optimality criteria for the CLUE are provided and further investigated. It is shown that finding an optimal CLUE is more complicated than finding an optimal linear unbiased estimator in the classical version of the problem. To simplify the problem, a CLUE that is optimal under certain restrictions will also be investigated. The latter is named restricted best CLUE. An important result is a theorem that gives a closed form solution to a restricted best CLUE. Furthermore, several conservative estimation methods are described followed by an analysis of their properties. The methods are shown to be conservative and optimal under different assumptions about the underlying correlations. The second part of the thesis focuses on practical aspects of the conservative approach to decentralized estimation in configurations where the communication channel is constrained. The diagonal covariance approximation is proposed as a data reduction technique that complies with the communication constraints and if handled correctly can be shown to preserve conservative estimates. Several information selection methods are derived that can reduce the amount of data being transmitted in the communication channel. Using the information selection methods it is possible to decide what information other actors of the sensor network find useful.
Author: Fredrik Ljungberg
Publisher: Linköping University Electronic Press
ISBN: 9179298400
Category :
Languages : en
Pages : 124
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As marine vessels are becoming increasingly autonomous, having accurate simulation models available is turning into an absolute necessity. This holds both for facilitation of development and for achieving satisfactory model-based control. When accurate ship models are sought, it is necessary to account for nonlinear hydrodynamic effects and to deal with environmental disturbances in a correct way. In this thesis, parameter estimators for nonlinear regression models where the regressors are second-order modulus functions are analyzed. This model class is referred to as second-order modulus models and is often used for greybox identification of marine vessels. The primary focus in the thesis is to find consistent estimators and for this an instrumental variable (IV) method is used. First, it is demonstrated that the accuracy of an IV estimator can be improved by conducting experiments where the input signal has a static offset of sufficient amplitude and the instruments are forced to have zero mean. This two-step procedure is shown to give consistent estimators for second-order modulus models in cases where an off-the-shelf applied IV method does not, in particular when measurement uncertainty is taken into account. Moreover, it is shown that the possibility of obtaining consistent parameter estimators for models of this type depends on how process disturbances enter the system and on the amount of prior knowledge about the disturbances’ probability distributions that is available. In cases where the first-order moments are known, the aforementioned approach gives consistent estimators even when disturbances enter the system before the nonlinearity. In order to obtain consistent estimators in cases where the first-order moments are unknown, a framework for estimating the first and second-order moments alongside the model parameters is suggested. The idea is to describe the environmental disturbances as stationary stochastic processes in an inertial frame and to utilize the fact that their effect on a vessel depends on the vessel’s attitude. It is consequently possible to infer information about the environmental disturbances by over time measuring the orientation of a vessel they are affecting. Furthermore, in cases where the process disturbances are of more general character it is shown that supplementary disturbance measurements can be used for achieving consistency. Different scenarios where consistency can be achieved for instrumental variable estimators of second-order modulus models are demonstrated, both in theory and by simulation examples. Finally, estimation results obtained using data from a full-scale marine vessel are presented. I takt med att marina farkoster blir mer autonoma ökar behovet av noggranna matematiska farkostmodeller. Modellerna behövs både för att förenkla utvecklingen av nya farkoster och för att kunna styra farkosterna autonomt med önskad precision. För att erhålla allmängiltiga modeller behöver olinjära hydrodynamiska effekter samt systemstörningar, främst orsakade av vind- och vattenströmmar, tas i beaktning. I det här arbetet undersöks metoder för att skatta okända storheter i modeller för marina farkoster givet observerad data. Undersökningen gäller en speciell typ av olinjära modeller som ofta används för att beskriva marina farkoster. Huvudfokus i arbetet är att erhålla konsistens, vilket betyder att de skattade storheterna ska anta rätt värden när mängden observerad data ökar. För det används en redan etablerad statistisk metod som baseras på instrumentvariabler. Det visas först att noggrannheten i modellskattningsmetoden kan förbättras om datainsamlingsexperimenten utförs på ett sätt så att farkosten har signifikant nollskild hastighet och instrumentvariablernas medelvärde dras bort. Den här tvåstegslösningen påvisas vara fördelaktig vid skattning av parametrar i den ovan nämnda modelltypen, framför allt då mätosäkerhet tas i beaktning. Vidare så visas det att möjligheten att erhålla konsistenta skattningsmetoder beror på hur mycket kännedom om systemstörningarna som finns tillgänglig på förhand. I fallet då de huvudsakliga hastigheterna på vind- och vattenströmmar är kända, räcker den tidigare nämnda tvåstegsmetoden bra. För att även kunna hantera det mer generella fallet föreslås en metod för att skatta de huvudsakliga hastigheterna och de okända modellparametrarna parallellt. Denna idé baserar sig på att beskriva störningarna som stationära i ett globalt koordinatsystem och att anta att deras effekt på en farkost beror på hur farkosten är orienterad. Genom att över tid mäta och samla in data som beskriver en farkosts kurs, kan man således dra slutsatser om de störningar som farkosten påverkas av. Utöver detta visas det att utnyttjande av vindmätningar kan ge konsistens i fallet med störningar av mer generell karaktär. Olika scenarion där konsistens kan uppnås visas både i teori och med simuleringsexempel. Slutligen visas också modellskattningsresultat som erhållits med data insamlad från ett fullskaligt fartyg.
Author: Per Boström-Rost
Publisher: Linköping University Electronic Press
ISBN: 9176850757
Category :
Languages : en
Pages : 86
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This thesis studies a class of sensor management problems called informative path planning (IPP). Sensor management refers to the problem of optimizing control inputs for sensor systems in dynamic environments in order to achieve operational objectives. The problems are commonly formulated as stochastic optimal control problems, where to objective is to maximize the information gained from future measurements. In IPP, the control inputs affect the movement of the sensor platforms, and the goal is to compute trajectories from where the sensors can obtain measurements that maximize the estimation performance. The core challenge lies in making decisions based on the predicted utility of future measurements. In linear Gaussian settings, the estimation performance is independent of the actual measurements. This means that IPP becomes a deterministic optimal control problem, for which standard numerical optimization techniques can be applied. This is exploited in the first part of this thesis. A surveillance application is considered, where a mobile sensor is gathering information about features of interest while avoiding being tracked by an adversarial observer. The problem is formulated as an optimization problem that allows for a trade-off between informativeness and stealth. We formulate a theorem that makes it possible to reformulate a class of nonconvex optimization problems with matrix-valued variables as convex optimization problems. This theorem is then used to prove that the seemingly intractable IPP problem can be solved to global optimality using off-the-shelf optimization tools. The second part of this thesis considers tracking of a maneuvering target using a mobile sensor with limited field of view. The problem is formulated as an IPP problem, where the goal is to generate a sensor trajectory that maximizes the expected tracking performance, captured by a measure of the covariance matrix of the target state estimate. When the measurements are nonlinear functions of the target state, the tracking performance depends on the actual measurements, which depend on the target’s trajectory. Since these are unavailable in the planning stage, the problem becomes a stochastic optimal control problem. An approximation of the problem based on deterministic sampling of the distribution of the predicted target trajectory is proposed. It is demonstrated in a simulation study that the proposed method significantly increases the tracking performance compared to a conventional approach that neglects the uncertainty in the future target trajectory.
Author: Andreas Bergström
Publisher: Linköping University Electronic Press
ISBN: 9179299172
Category :
Languages : en
Pages : 119
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Book Description
The measurements of radio signals are commonly used for localization purposes where the goal is to determine the spatial position of one or multiple objects. In realistic scenarios, any transmitted radio signal will be affected by the environment through reflections, diffraction at edges and corners etc. This causes a phenomenon known as multipath propagation, by which multiple instances of the transmitted signal having traversed different paths are heard by the receiver. These are known as Multi-Path Components (MPCs). The direct path (DP) between transmitter and receiver may also be occluded, causing what is referred to as non-Line-of-Sight (non-LOS) conditions. As a consequence of these effects, the estimated position of the object(s) may often be erroneous. This thesis focuses on how to achieve better localization accuracy by accounting for the above-mentioned multipath propagation and non-LOS effects. It is proposed how to mitigate these in the context of positioning based on estimation of the DP between transmitter and receiver. It is also proposed how to constructively utilize the additional information about the environment which they implicitly provide. This is all done in a framework wherein a given signal model and a map of the surroundings are used to build a mathematical model of the radio environment, from which the resulting MPCs are estimated. First, methods to mitigate the adverse effects of multipath propagation and non-LOS conditions for positioning based on estimation of the DP between transmitter and receiver are presented. This is initially done by using robust statistical measurement error models based on aggregated error statistics, where significant improvements are obtained without the need to provide detailed received signal information. The gains are seen to be even larger with up-to-date real-time information based on the estimated MPCs. Second, the association of the estimated MPCs with the signal paths predicted by the environmental model is addressed. This leads to a combinatorial problem which is approached with tools from multi-target tracking theory. A rich radio environment in terms of many MPCs gives better localization accuracy but causes the problem size to grow large—something which can be remedied by excluding less probable paths. Simulations indicate that in such environments, the single best association hypothesis may be a reasonable approximation which avoids the calculation of a vast number of possible hypotheses. Accounting for erroneous measurements is crucial but may have drawbacks if no such are occurring. Finally, theoretical localization performance bounds when utilizing all or a subset of the available MPCs are derived. A rich radio environment allows for good positioning accuracy using only a few transmitters/receivers, assuming that these are used in the localization process. In contrast, in a less rich environment where basically only the DP/LOS components are measurable, more transmitters/receivers and/or the combination of downlink and uplink measurements are required to achieve the same accuracy. The receiver’s capability of distinguishing between multiple MPCs arriving approximately at the same time also affects the localization accuracy.
