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Regularization for Design
, 2014
"... An algorithmic bridge is starting to be established between sparse reconstruction theory and distributed control theory. For example, `1regularization has been suggested as an appropriate means for codesigning sparse feedback gains and consensus topologies subject to performance bounds. In recent ..."
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An algorithmic bridge is starting to be established between sparse reconstruction theory and distributed control theory. For example, `1regularization has been suggested as an appropriate means for codesigning sparse feedback gains and consensus topologies subject to performance bounds. In recent work, we showed that ideas from atomic norm minimization could be used to simultaneously codesign a distributed optimal controller and the communication delay structure on which it is to be implemented. While promising and successful, these results lack the same theoretical support that their sparse reconstruction counterparts enjoy – as things stand, these methods are at best viewed as principled heuristics. In this paper, we describe theoretical connections between sparse reconstruction and systems design by developing approximation bounds for control codesign problems via convex optimization. We also give a concrete example of a design problem for which our approach provides approximation guarantees.
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"... Control of coupled oscillator networks with application to microgrid technologies ..."
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Control of coupled oscillator networks with application to microgrid technologies
f n i, r
"... specific correlation with note that our method tor networks, provided rm. Moreover, because s have served as a paranchronization in various hed light more generally d could potentially give he termination of cardiac We consider the famous Kuramoto m ..."
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specific correlation with note that our method tor networks, provided rm. Moreover, because s have served as a paranchronization in various hed light more generally d could potentially give he termination of cardiac We consider the famous Kuramoto m
RESEARCH STATEMENT
"... My primary research interests in algebraic geometry lie in the Minimal Model Program and its applications, moduli spaces of stable maps (curves), and moduli spaces of branchvarieties. Recently I have also paid close attention to the new advances in computational algebraic geometry. 1 Minimal Model P ..."
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My primary research interests in algebraic geometry lie in the Minimal Model Program and its applications, moduli spaces of stable maps (curves), and moduli spaces of branchvarieties. Recently I have also paid close attention to the new advances in computational algebraic geometry. 1 Minimal Model Program and its applications In many branches of mathematics, classifications among objects up to certain relations are central themes. For example, topologists can classify topological spaces up to homeomorphism or up to the weaker relation of homotopy. Similarly, in algebraic geometry, we classify algebraic varieties up to either isomorphism or a weaker relation, birational equivalence (two varieties X and Y are birationally equivalent if there exist rational maps f: X − → Y and g: Y − → X such that g ◦f and f ◦ g are identity maps on some open subsets U ⊂ X and V ⊂ Y). Among algebraic varieties in the same birational equivalence class, we want to single out some “good ” representatives. Such good representatives are called minimal models. It is well known that every surface has a minimal model. Is there a minimal model for every higher dimensional algebraic variety? The answer was unknown for a long period of time even for threefolds. At first people tried to find a minimal model in the smooth category, but this turned out to be impossible. Gradually people realized that one can only
f n i, r
"... specific correlation with note that our method tor networks, provided rm. Moreover, because s have served as a paranchronization in various hed light more generally d could potentially give he termination of cardiac We consider the famous Kuramoto m ..."
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specific correlation with note that our method tor networks, provided rm. Moreover, because s have served as a paranchronization in various hed light more generally d could potentially give he termination of cardiac We consider the famous Kuramoto m
The Price of Synchrony: Evaluating the Resistive Losses in Synchronizing Power Networks
"... Abstract—In a network of synchronous generators, we investigate resistive power losses due to the power flow fluctuations required to keep the network in a synchronous state. Such fluctuations occur after a transient event or in the face of persistent stochastic disturbances. We term these losses t ..."
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Abstract—In a network of synchronous generators, we investigate resistive power losses due to the power flow fluctuations required to keep the network in a synchronous state. Such fluctuations occur after a transient event or in the face of persistent stochastic disturbances. We term these losses the “price of synchrony”, as they reflect real power flow costs incurred in synchronizing the system. In the case of small fluctuations at each node, we show how the total network’s resistive losses can be quantified using an H2 norm of a system of coupled swing equations subject to distributed disturbances. This norm is shown to be a function of transmission line and generator properties, to scale unboundedly with network size, and to be largely independent of network topology. This conclusion differentiates resistive losses from typical power systems stability notions, which show highly connected networks to be more coherent and thus easier to synchronize. In particular, the price of synchrony is more dependent on a network’s size than its topology. We discuss possible implications of these results to the design of future power grids, which are expected to have highlydistributed generation resources leading to larger networks with the potential for greater transient losses. I.
On Optimal Link Creation for Facilitation of Consensus in Social Networks
"... Abstract — We consider the problem of reaching consensus in a social network of agents described by the DeGroot model. We develop a measure for the efficiency with which consensus is reached, where the measure quantifies the transient behavior of public opinion around the consensus value. We then pr ..."
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Abstract — We consider the problem of reaching consensus in a social network of agents described by the DeGroot model. We develop a measure for the efficiency with which consensus is reached, where the measure quantifies the transient behavior of public opinion around the consensus value. We then propose an optimization problem that maximizes consensusreaching efficiency via the creation of new social links, subject to a total linkcreation budget. We employ the alternating direction method of multipliers, an algorithm wellsuited to large optimization problems, to find the optimal location and weights of the new links. We demonstrate the utility of our results through an example, where we observe that for a social network described by a regular graph the addition of new links leads to an augmented graph that resembles a smallworld network characterized by sparse longrange links. Index Terms — Alternating direction method of multipliers, consensus, DeGroot model, opinion dynamics, optimization, smallworld networks, social networks, sparsity, stochastic matrices. I.
1Dynamic mode decomposition with control
, 2014
"... Abstract—We develop a new method which extends Dynamic Mode Decomposition (DMD) to incorporate the effect of control to extract loworder models from highdimensional, complex systems. DMD finds spatialtemporal coherent modes, connects locallinear analysis to nonlinear operator theory, and provide ..."
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Abstract—We develop a new method which extends Dynamic Mode Decomposition (DMD) to incorporate the effect of control to extract loworder models from highdimensional, complex systems. DMD finds spatialtemporal coherent modes, connects locallinear analysis to nonlinear operator theory, and provides an equationfree architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an inputoutput model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, Dynamic Mode Decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate inputoutput models. The method is datadriven in that it does not require knowledge of the underlying governing equations, only snapshots of state and actuation data from historical, experimental, or blackbox simulations. We demonstrate the method on highdimensional dynamical systems, including a model with relevance to the analysis of infectious disease data with mass vaccination (actuation). I.
Control of coupled oscillator netw gies m r f n
"... specific correlation with note that our method tor networks, provided rm. Moreover, because s have served as a paranchronization in various hed light more generally d could potentially give he termination of cardiac We consider the famous Kuramoto m ..."
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specific correlation with note that our method tor networks, provided rm. Moreover, because s have served as a paranchronization in various hed light more generally d could potentially give he termination of cardiac We consider the famous Kuramoto m