Approximation of box dimension of attractors using the subdivision algorithm

In this paper, the design of a data-rate constrained observer for a dynamical system is presented. This observer is designed to function both in discrete time and continuous time. The system is connected to a remote location via a communication channel which can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust toward losses in the communication channel. Upper bounds on the required communication rate to implement the observer are provided in terms of the upper box dimension of the state space and an upper bound on the largest singular value of the system’s Jacobian. Results that provide an analytical bound on the required minimum communication rate are then presented. These bounds are obtained by using the Lyapunov dimension of the dynamical system rather than the upper box dimension in the rate. The observer is tested through simulations for the Lozi map and the Lorenz system. For the Lozi map, the Lyapunov dimension is computed. For both systems, the theoretical bounds on the communication rate are compared to the simulated rates.

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“…Let Assumption 5 hold. Then, for any compact invariant set S 0 of the map (27), the following inequality is valid…”

Section: The Smoothened Lozi Mapmentioning

confidence: 99%

“…Unfortunately, there are still no general analytical techniques for computing the two aforementioned dimensions for chaotic attractors. Numerical methods remain the main tool used by scientists and engineers to estimate these dimensions [27]. In this paper, an alternative to this numerical approach is developed.…”

Section: Introductionmentioning

confidence: 99%

Data-Rate Constrained Observers of Nonlinear Systems

Voortman

Pogromsky

Matveev

et al. 2019

Entropy

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“…Similarly in this paper, we optimize the solution with respect to LQR performance. Unlike in [16], we divide state space into simple boxes as it is performed in [17]. Partitioning of state space is used also in [18], where authors define time-varying piecewise affine (PWA) feedback control law defined over nonconvex regions for hybrid systems.…”

Section: Introductionmentioning

confidence: 99%

“…Partitioning of state space is used also in [18], where authors define time-varying piecewise affine (PWA) feedback control law defined over nonconvex regions for hybrid systems. 933 We follow the state space discretization approach but apply a more direct subdivision algorithm, proposed by the author and collaborators in [17], that bypasses the need to apply a graph algorithm. Although the algorithm introduced in [10] provides solutions if the state space is sufficiently partitioned, with our approach, we show that it is feasible to find optimal control even for coarse divisions of the state space, and we show how overall performance improves when the state space is further subdivided, which is not possible with [10] approach due to already mentioned need for sufficient partitioning.…”

Section: Introductionmentioning

confidence: 99%

Subdivision algorithm for optimal control

Taraba

2012

Intl J Robust & Nonlinear

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SUMMARYWe present a novel algorithm for optimal control of nonlinear systems based on a subdivision algorithm. The algorithm presented in this paper is an alternative to a set‐oriented approach for optimal feedback stabilization. We compare the proposed algorithm to the set‐oriented approach, contrast these two approaches, and use examples to show that the new algorithm produces comparable results. Also, we demonstrate by example that we receive a precomputed optimal solution. The main contribution of the paper is understanding how cost function improves with further subdivision of state space and smaller memory footprint of the final solution in comparison with set‐oriented approach. Copyright © 2012 John Wiley & Sons, Ltd.

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“…The difficulty of using these dimensions is that it can often not be computed analytically and one has to employ numerical methods to obtain estimates of the dimension (see e.g. [27]). Another solution is to use the Lyapunov dimension which upper bounds the two previous dimensions [13].…”

mentioning

confidence: 99%

A Data Rate Constrained Observer for Discrete Nonlinear Systems

Voortman

Pogromsky

Matveev

et al. 2018

2018 IEEE Conference on Decision and Control (CDC)

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In this paper, we develop a communication protocol for the observation of discrete time, possibly unstable, dynamical systems over communication channels with limited communication capacity. We develop an observer based on the upper box dimension for one-way communication channels that leads to a certain type of observability. This communication scheme preserves observability under communication losses which makes the communication scheme robust towards communication losses without feedback in the communication channel. Using Lyapunov-like techniques, we provide bounds on the minimum communication rate required to implement this observer. We also use the Lyapunov dimension to provide analytical upper bounds on the communication rate. We compute an analytical upper bound and an exact expression for the Lyapunov dimension of the smoothened Lozi map. This bound is then tested in simulations of the communication protocol for the observation problem of the smoothened Lozi map.

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Approximation of box dimension of attractors using the subdivision algorithm

Cited by 7 publications

References 16 publications

Data-Rate Constrained Observers of Nonlinear Systems

Data-Rate Constrained Observers of Nonlinear Systems

Subdivision algorithm for optimal control

A Data Rate Constrained Observer for Discrete Nonlinear Systems

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