# Observability and Synchronizability

Christophe LETELLIERLuis A. AGUIRRE
24/11/2012

According Henri Poincaré’s works, a dynamical system must be investigated in the state space spanned by the variables required for a full description of any of its states. When a dynamical system - or any system evolving in time - is investigated from measurements, all its variables are commonly not recorded and we are not ensured to have all the required information to fully distinguish (nor observe) all the state of the original state space. This is the so-called observability problem.

The observability helps to address the question whether some measurements are sufficient to provide the required information for distinguishing all the state of the original state space. This is determined by using the observability matrix as introduced by Hermann & Kerner [1]. Such observability matrix is nothing else than the jacobian matrix of the change of coordinates between the original state space and the reconstructed space spanned by the Lie derivatives of the measurements [2]. The system is fully observable when the determinant of this matrix never vanishes. Otherwise, there is a singular observability manifold in the original state space which cannot be observed through the measurements [3]. Depending on the influence of this singular observability manifold, the observability of the dynamical system through the measured variable(s) is more or less good ; we introduced observability coefficients to quantify the quality of the measurements with respect to observability [4] [5]. There is a graphical interpretation of these observability coefficients [6]. We also developed a symbolic computations of these observability coefficients [7]. The question of assessing the observability coefficients from measurements ; a first attempt was nevertheless performed [8].

The relationship between observability and synchronizability was numerically investigated in [9]. Obsevability has a strong influence on our ability to obtain a global model from given measurements [10] [11] [12] [13]

 System Variable x Variable y Variable z Chua (1984) 0.78 0.84 1.00 Lorenz (1963) 0.78 0.36 0.36 Rossler (1976) 0.88 1.00 0.44

[1] R. Hermann & A. J. Krener, Nonlinear controllability and observability, IEEE Transactions in Automatic Control, 22 (5), 728-740 ; 1997.

[2] C. Letellier, L. A. Aguirre & J. Maquet, Relation between observability and differential embeddings for nonlinear dynamics, Physical Review E, 71, 066213, 2005.

[3] M. Frunzete, J.-P. Barbot & C. Letellier, Influence of the singular manifold of non-observable states in reconstructing chaotic attractors, Physical Review E, 86, 026205, 2012.

[4] C. Letellier, J. Maquet, L. Le Sceller, G. Gouesbet & L. A. Aguirre, On the non-equivalence of observables in phase space reconstructions from recorded time series, Journal of Physics A, 31, 7913-7927, 1998.

[5] C. Letellier & L. Aguirre, Investigating nonlinear dynamics from time series : the influence of symmetries and the choice of observables, Chaos, 12, 549-558, 2002.

[6] C. Letellier & L. A. Aguirre, A graphical interpretation of observability in terms of feedback circuits, Physical Review E, 72, 056202, 2005.

[7] C. Letellier & L. A. Aguirre, Symbolic observability coefficients for univariate and multivariate analysis, Physical Review E, 79, 066210, 2009.

[8] L. A. Aguirre & C. Letellier, Investigating observability properties from data in nonlinear dynamics, Physical Review E, 83, 066209, 2011.

[9] C. Letellier & L. A. Aguirre, Interplay between synchronization, observability, and dynamics, Physical Review E, 82, 016204, 2010.

[10] C. Letellier, L. A. Aguirre, J. Maquet & Aziz-Alaoui, Should all the species of a food chain be counted to investigate the global dynamics ?, Chaos, Solitons & Fractals, 13, 1099-1113, 2002.

[11] C. Lainscsek, C. Letellier & I. Gorodnitsky, Global modeling of the Rössler system from the z-variable, Physics Letters A, 314 (5-6), 409-427, 2003

[12] J. Maquet, C. Letellier & L. A. Aguirre, Scalar modeling and analysis of a 3D biochemical reaction model Journal of Theoretical Biology, 228 (3), 421-430, 2004.

[13] J. Maquet, C. Letellier & L. A. Aguirre, Global models from the Canadian Lynx cycles as a first evidence for chaos in real ecosystems, Journal of Mathematical Biology, 55 (1), 21-39, 2007.

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