A separation theorem for nonlinear systems

Kilicaslan, Sinan; Banks, Stephen P.

doi:10.1016/j.automatica.2008.11.019

Cited by 13 publications

(16 citation statements)

References 16 publications

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“…(2.29). This theorem, however, has also been shown to hold in some nonlinear feedback systems 13,21) and quantum systems. 22) Whether our result can be generalized in these systems remains to be studied.…”

Section: Discussionmentioning

confidence: 97%

Jarzynski Equality Modified in the Linear Feedback System

Fujitani

Suzuki

2010

J. Phys. Soc. Jpn.

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It is known that one can derive the Jarzynski equality in a stochastic process of a classical system by assuming the local detailed balance. We study how the equality is modified in the linear feedback system. There, measurement is performed on the state following a linear Langevin equation, the measured variable is linear to the state variable with white sensor noise, the Kalman filter estimates the state by utilizing measured values in the past, and a linear regulator controls the state dynamics. Although a stochastic process produced by this dynamics is non-Markovian because of the feedback loop, it is known in the control theory that a Markov process for the estimation can be separated from the whole process. To the exponent in the Jarzynski equality, we find an additional term, whose average gives the mutual information between state variables and measured variables in the Markov process for the estimation. The resultant equality holds whether the gain is optimal or not.

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Section: Discussionmentioning

confidence: 97%

Jarzynski Equality Modified in the Linear Feedback System

Fujitani

Suzuki

2010

J. Phys. Soc. Jpn.

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“…This needs not be the case for nonlinear systems or non-quadratic objective functions. Extensions of the separation theorem have been developed in Kilicaslan and Banks (2009) based on successive approximations of (1) in terms of a linear system with time-varying parameters. Particle filter methods can also be employed to average the future cost over the current distribution of the estimated state variables (Andrieu et al, 2003).…”

Section: Approximation Methods and Linear Systemsmentioning

confidence: 99%

Control Theory and Experimental Design in Diffusion Processes

Hooker

Lin²,

Rogers

2015

SIAM/ASA J. Uncertainty Quantification

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This paper considers the problem of designing time-dependent, real-time control policies for controllable nonlinear diffusion processes, with the goal of obtaining maximally-informative observations about parameters of interest. More precisely, we maximize the expected Fisher information for the parameter obtained over the duration of the experiment, conditional on observations made up to that time. We propose to accomplish this with a twostep strategy: when the full state vector of the diffusion process is observable continuously, we formulate this as an optimal control problem and apply numerical techniques from stochastic optimal control to solve it. When observations are incomplete, infrequent, or noisy, we propose using standard filtering techniques to first estimate the state of the system, then apply the optimal control policy using the posterior expectation of the state. We assess the effectiveness of these methods in 3 situations: a paradigmatic bistable model from statistical physics, a model of action potential generation in neurons, and a model of a simple ecological system.

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“…It should be noted that Gauthier and Kupka [53] and Bounit and Hammouri [54] prove the stability for the aim that consists of stabilizing system with the aid of estimated state (also known as separation principle) given by the Kalman‐like observers [59]. The separation principle for bilinear systems in Gauthier and Kupka [53] and Bounit and Hammouri [54] is extended in Kilicaslan and Banks [46] to the stochastic nonlinear systems by the Kalman–Bucy filters. As mentioned above, in a recent work [2], the authors introduce the methods in Kilicaslan and Banks [46] to derive the Nash equilibria of the bilinear stochastic differential game over the finite‐time horizon.…”

Section: Introductionmentioning

confidence: 99%

Infinite‐time partially observed nonzero‐sum bilinear affine‐quadratic stochastic differential game and its application to competitive advertising

Tao

Zhang

Yang

2023

Asian Journal of Control

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Due to the conjunction of huge expenditure and latent inefficiencies, competitive advertising has always been occupying a front‐and‐center place in marketing, among which a bilinear system is one of the most important. Because much real business is two giants playing among other emerging challengers and the players can only inspect the market through a related noisy process by the random sampling survey, the sales‐advertising dynamics is a partially observed nonzero‐sum stochastic differential game. However, by the intrinsic difficulties of mathematics, there is no extant research to augment the analysis on the infinite‐time horizon, which is of particular significance to implementing the online sales campaign and evaluating the consequence of endless competition. Motivated by this, controllability and stability analysis are explored first for the iterative approximation to bilinearity. Then, the simulation provides a rapid prototyping and automatic approach for the most efficient strategies. Moreover, four implications have been discovered by comparison with eight examples as the application of four encounters on finite‐ and infinite‐time horizons. Several new insights have been revealed for those struggling to develop successful advertising strategies.

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A separation theorem for nonlinear systems

Cited by 13 publications

References 16 publications

Jarzynski Equality Modified in the Linear Feedback System

Jarzynski Equality Modified in the Linear Feedback System

Control Theory and Experimental Design in Diffusion Processes

Infinite‐time partially observed nonzero‐sum bilinear affine‐quadratic stochastic differential game and its application to competitive advertising

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