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Stochastic differential dynamic programming

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Stochastic differential dynamic programming

作者： Theodorou, Evangelos Tassa, Yuval Todorov, Emo Departments of Computer Science and Neuroscience Computational Learning and Motor Control Lab. University of Southern California United States Interdisciplinary Center for Neural Computation Hebrew University Jerusalem Israel Department of Computer Science and Engineering University of Washington Seattle WA United States

ISBN: (纸本)9781424474264

Although there has been a significant amount of work in the area of stochastic optimal control theory towards the development of new algorithms, the problem of how to control a stochastic nonlinear system remains an open research topic. Recent iterative linear quadratic optimal control methods iLQG [1], [2] handle control and state multiplicative noise while they are derived based on first order approximation of dynamics. On the other hand, methods such as Differential Dynamic Programming expand the dynamics up to the second order but so far they can handle nonlinear systems with additive noise. In this work we present a generalization of the classic Differential Dynamic Programming algorithm. We assume the existence of state and control multiplicative process noise, and proceed to derive the second-order expansion of the cost-to-go. We find the correction terms that arise from the stochastic assumption. Despite having quartic and cubic terms in the initial expression, we show that these vanish, leaving us with the same quadratic structure as standard DDP. © 2010 AACC.

关键词： Additive noise

Non-Euclidean Contraction Theory for Robust Nonlinear Stability

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arXiv 2021年

作者： Davydov, Alexander Jafarpour, Saber Bullo, Francesco The Department of Mechanical Engineering The Center for Control Dynamical Systems and Computation University of California Santa Barbara93106-5070 United States

We study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the `1 and `∞ norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide (i) incremental input-to-state stability and finite input-state gain properties for contracting systems, and (ii) a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system. Copyright © 2021, The Authors. All rights reserved.

关键词： Dynamical systems

Separating real and virtual objects from their overlapping images 4th

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4th European Conference on Computer Vision, ECCV 1996

作者： Ohnishi, Noboru Kumaki, Kenji Yamamura, Tsuyoshi Tanaka, Toshimitsu Department of Information Engineering Nagoya University Furo-cho Chikusa-ku464-01 Japan Bio-Mimetic Control Research Center RIKEN 8-31 Rokuban 3-chome Atsuta-ku456 Japan Computation Center Nagoya University Furo-cho Chikusa-ku464-01 Japan

ISBN: (纸本)3540611231

We often see scenes where an object's virtual image is reflected on window glass and overlaps with an image of another object behind the glass. This paper proposes a method for separating real and virtual objects from the overlapping images. Our method is based on the optical property that light reflected on glass is polarized, while light transmitted through glass is less polarized. It is possible to eliminate reflected light with a polarizing filter. The polarization direction, however, changes even for planar glass and is not easily determined without information about the position and orientation of the glass and objects relative to the camera. Our method uses a series of images obtained by rotating a polarizing filter. Real objects are separated by selecting the minimum image intensity among a series of images for each pixel. The virtual image of objects is obtained by subtracting the image of the real objects from the image having the maximum image intensity among a series of images for each pixel. We present experiments with actual scenes to demonstrate the effectiveness of the proposed method. © Springer-Verlag Berlin Heidelberg 1996.

关键词： Glass

Stochastic strategies for robotic surveillance as stackelberg games

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arXiv 2020年

作者： Duan, Xiaoming Paccagnan, Dario Bullo, Francesco Mechanical Engineering Department the Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States

This paper studies a stochastic robotic surveillance problem where a mobile robot moves randomly on a graph to capture a potential intruder that strategically attacks a location on the graph. The intruder is assumed to be omniscient: it knows the current location of the mobile agent and can learn the surveillance strategy. The goal for the mobile robot is to design a stochastic strategy so as to maximize the probability of capturing the intruder. We model the strategic interactions between the surveillance robot and the intruder as a Stackelberg game, and optimal and suboptimal Markov chain based surveillance strategies in star, complete and line graphs are studied. We first derive a universal upper bound on the capture probability, i.e., the performance limit for the surveillance agent. We show that this upper bound is tight in the complete graph and further provide suboptimality guarantees for a natural design. For the star and line graphs, we first characterize dominant strategies for the surveillance agent and the intruder. Then, we rigorously prove the optimal strategy for the surveillance agent. Copyright © 2020, The Authors. All rights reserved.

关键词： Stochastic systems

Markov chain-based stochastic strategies for robotic surveillance

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arXiv 2020年

作者： Duan, Xiaoming Bullo, Francesco Department of Mechanical Engineering Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States

This article surveys recent advancements of strategy designs for persistent robotic surveillance tasks with the focus on stochastic approaches. The problem describes how mobile robots stochastically patrol a graph in an efficient way where the efficiency is defined with respect to relevant underlying performance metrics. We first start by reviewing the basics of Markov chains, which is the primary motion model for stochastic robotic surveillance. Then two main criteria regarding the speed and unpredictability of surveillance strategies are discussed. The central objects that appear throughout the treatment is the hitting times of Markov chains, their distributions and expectations. We formulate various optimization problems based on the concerned metrics in different scenarios and establish their respective properties. Copyright © 2020, The Authors. All rights reserved.

关键词： Stochastic systems

Graph-theoretic small-gain theorems for metzler matrices and monotone systems

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arXiv 2019年

作者： Duan, Xiaoming Jafarpour, Saber Bullo, Francesco Mechanical Engineering Department Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from the input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we define and compute two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Then, based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and sufficient for the stability of Metzler systems;based on the sum-interconnection gains, we obtain novel graph-theoretic conditions for the stability of Metzler systems. All these conditions highlight the role of cycles in the interconnection graph and unveil how the structural properties of the graph affect stability. Finally, we extend our results to the nonlinear monotone system and obtain similar sufficient conditions for global asymptotic stability. Copyright © 2019, The Authors. All rights reserved.

关键词： Graph theory

Markov chains with maximum return time entropy for robotic surveillance

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arXiv 2018年

作者： Duan, Xiaoming George, Mishel Bullo, Francesco Mechanical Engineering Department Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States

Motivated by robotic surveillance applications, this paper studies the novel problem of maximizing the return time entropy of a Markov chain, subject to a graph topology with travel times and stationary distribution. The return time entropy is the weighted average, over all graph nodes, of the entropy of the first return times of the Markov chain;this objective function is a function series that does not admit in general a closed form. The paper features theoretical and computational contributions. First, we obtain a discrete-time delayed linear system for the return time probability distribution and establish its convergence properties. We show that the objective function is continuous over a compact set and therefore admits a global maximum;a unique globally-optimal solution is known only for complete graphs with unitary travel times. We then establish upper and lower bounds between the return time entropy and the well-known entropy rate of the Markov chain. To compute the optimal Markov chain numerically, we establish the asymptotic equality between entropy, conditional entropy and truncated entropy, and propose an iteration to compute the gradient of the truncated entropy. Finally, we apply these results to the robotic surveillance problem. Our numerical results show that, for a model of rational intruder over prototypical graph topologies and test cases, the maximum return time entropy chain performs better than several existing Markov chains. Copyright © 2018, The Authors. All rights reserved.

关键词： Linear systems

Voltage Stabilization in Microgrids via Quadratic Droop control

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Voltage Stabilization in Microgrids via Quadratic Droop Cont...

IEEE Annual Conference on Decision and control

作者： John W. Simpson-Porco Florian Dorfler Francesco Bullo Center for Control Dynamical Systems and Computation University of California Santa Barbara Department of Electrical Engineering University of California Los Angeles

ISBN: (纸本)9781467357159

Motivated by the growing interest in energy technology and smart grid architectures, we consider the problem of voltage stability and reactive power balancing in low-voltage electrical networks equipped with DC/AC inverters ("microgrids"). It is generally believed that high-voltage equilibria of such networks are stable, but the locations of these equilibria are unknown, as is the critical network load where stability is lost. Inspired by the "control by interconnection" paradigm developed for port-Hamiltonian systems, we propose a novel droop-like inverter controller which is quadratic in the local voltage magnitude. Remarkably, under this controller the closed-loop network is again a well-posed electrical circuit. We find that the equilibria of the quadratic droop-controlled network are in exact correspondence with the solutions of a reduced power flow equation. For general network topologies, we study some simple yet insightful solutions of this equation, and for the frequently-encountered case of a parallel microgrid, we present a concise and closed-form condition for the existence of an exponentially stable high-voltage network equilibrium. Our condition establishes the existence of a critical inductive load for the network, which depends only on the network topology, admittances, and controller gains. We compare and contrast our design with the conventional droop controller, investigate the relationship between the two, and validate the robustness of our design through simulation.

关键词： Inverters Microgrids Equations Voltage control Mathematical model Vectors Reactive power

Synchronization of coupled oscillators: The taylor expansion of the inverse kuramoto map

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arXiv 2018年

作者： Huang, Elizabeth Y. Jafarpour, Saber Bullo, Francesco Department of Mechanical Engineering Center of Control Dynamical Systems and Computation University of California Santa Barbara93106-5070 United States

Synchronization in the networks of coupled oscillators is a widely studied topic in different areas. It is well-known that synchronization occurs if the connectivity of the network dominates heterogeneity of the oscillators. Despite extensive study on this topic, the quest for sharp closed-form synchronization tests is still in vain. In this paper, we present an algorithm for finding the Taylor expansion of the inverse Kuramoto map. We show that this Taylor series can be used to obtain a hierarchy of increasingly accurate approximate tests with low computational complexity. These approximate tests are then used to estimate the threshold of synchronization as well as the position of the synchronization manifold of the network. Copyright © 2018, The Authors. All rights reserved.

关键词： Synchronization