Methods And Algorithms For Control Input Placement In Complex Networks
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Controllability of Complex Networks at Minimum Cost
Author: Gustav Lindmark
language: en
Publisher: Linköping University Electronic Press
Release Date: 2020-04-30
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.
Methods and algorithms for control input placement in complex networks
Author: Gustav Lindmark
language: en
Publisher: Linköping University Electronic Press
Release Date: 2018-09-05
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”.
Decentralized Estimation Using Conservative Information Extraction
Author: Robin Forsling
language: en
Publisher: Linköping University Electronic Press
Release Date: 2020-12-17
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.